Equations of State for Fluids and Fluids Properties

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Transcript of Equations of State for Fluids and Fluids Properties
EQUATIONS of STATE for FLUIDS and FLUID MIXTURES
E X P E R I M E N T A L T H E R M O D Y N A M I C S V O L U M E V
EXPERIMENTAL THERMODYNAMICS SERIES:
Calorimetry of Nonreacting Systems EXPERIMENTAL THERMODYNAMICS VOLUME I Edited by J.P. McCullough and D.W. Scott Butterworths, London, 1968
Experimental Thermodynamics of Nonreacting Fluids EXPERIMENTAL THERMODYNAMICS VOLUME II Edited by B. Le Neindre and B. Vodar Butterworths, London, 1975
Measurement of the Transport Properties of Fluids EXPERIMENTAL THERMODYNAMICS VOLUME III Edited by W.A. Wakeham, A. Nagashima and J.V. Sengers Blackwell Scientific Publications, Oxford, 1991
Solution Calorimetry EXPERIMENTAL THERMODYNAMICS VOLUME IV Edited by K.N. Marsh and P.A.G. O'Hare Blackwell Scientific Publications, Oxford, 1994
Equations of State for Fluids and Fluid Mixtures EXPERIMENTAL THERMODYNAMICS VOLUME V Edited by J.V. Sengers, R.F. Kayser, C.J. Peters, and H.J. White, Jr. Elsevier, Amsterdam, 2000
Measurement of Thermodynamic Properties of Single Phases EXPERIMENTAL THERMODYNAMICS VOLUME VI Edited by A.R.H. Goodwin, K.N. Marsh, and W.A. Wakeham Elsevier, Amsterdam, in press
Measurement of Thermodynamic Properties of Multiple Phases EXPERIMENTAL THERMODYNAMICS VOLUME VII Edited by R.D. Weir and T.W. de Loos Elsevier, Amsterdam, in press
International Union of Pure and Applied Chemistry Physical Chemistry Division Commission on Thermodynamics
EQUATIONS of STATE for FLUIDS and FLUID MIXTURES
P a r t I
E X P E R I M E N T A L T H E R M O D Y N A M I C S
V O L U M E V
E D I T E D B Y
J.V. S E N G E R S
Institute for Physical Science and Technology and Department of Chemical Engineering, University of Maryland, College Park, MD 20742, U.S.A.
and
Physical and Chemical Properties Division National Institute of Standards and Technology, Gaithersburg, AID 20899, U.S.A.
R.F. K A Y S E R Technology Services National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A.
C.J. P E T E R S Laboratory of Applied Thermodynamics and Phase Equilibria, Faculty of Applied Sciences, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
H.J. W H I T E , Jr. Institute for Physical Science and Technology and Department of Chemical Engineering, University of Maryland, College Park, MD 20742, U.S.A.
2 0 0 0
E L S E V I E R
Amsterdam  Lausanne  New Y o r k  Oxford  Shannon  S i n g a p o r e  Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
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CONTENTS
List of Contributors xvii Preface xix
PART I
Introduction J. V. Sengers, R.F. Kayser, C.J. Peters, and H.J. White, Jr.
Fundamental Considerations
M.B. Ewing and C.J. Peters
2.1 2.2
2.3
2.4
2.5 2.6 2.7 2.8 2.9 2.10
2.11
Introduction 6 Basic Thermodynamics 6 2.2.1 Homogeneous Functions 9 2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations 10 Deviation Functions 11 2.3.1 Residual Functions 14 2.3.2 Evaluation of Residual Functions 15 Mixing and Departure Functions 15 2.4.1 Departure Functions with Temperature, Molar Volume and Composition 15
as the Independent Variables 2.4.2 Departure Functions with Temperature, Pressure and Composition as the
Independent Variables 19 Mixing and Excess Functions 20 Partial Molar Properties 22 Fugacity and Fugacity Coefficients 23 Activity Coefficients 25 The Phase Rule 26 Equilibrium Conditions 27 2.10.1 Phase Equilibria 27 2.10.2 Chemical Equilibria 29 Stability and the Critical State 30 2.11.1 Densities and Fields 30 2.11.2 Stability 30 2.11.3 Critical State 31
The Virial Equation of State 3 5 J.P.M. Trusler
3.1 Introduction 3.1.1 The Volumetric Behaviour of Real Fluids 3.1.2 The Virial Equation of State 3.1.3 Temperature Dependence of the Virial Coefficients 3.1.4 Composition Dependence of the Virial Coefficients 3.1.5 The Pressure Series 3.1.6 Convergence of the Virial Series
36 36 37 38 38 40 40
vi
3.2
3.3
3.4
3.5
3.2.1 3.2.2 3.2.3 3.2.4 Theory 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5
Thermodynamic Properties of Gases Perfectgas and Residual Properties Helmholtz Energy and Gibbs Energy PerfectGas Properties Residual Properties
Properties of the Grand Canonical Partition Function The Virial Equation of State from the Grand Partition Function The Virial Coefficients in Classical Mechanics Quantum Corrections The Virial Coefficients in Quantum Mechanics: Helium and Hydrogen
Calculation and Estimation of Virial Coefficients 3.4.1 The Virial Coefficients of Model Systems 3.4.2 Estimation of Virial Coefficients Summary
41 41 42 43 47 48 48 49 51 55 60 67 67 70 72
Cubic and Generalized van der Waals Equations A. Anderko
4.1
4.2 4.3 4.4
4.5 4.6
4.7
Cubic Equations of State for Pure Components 4.1.1 Historical Perspective 4.1.2 Temperature Dependence of Parameters 4.1.3 Functional Form of the Pressure Volume Relationship Equations Based on the Generalized van der Waals Theory Simple Equations Inspired by the Generalized van der Waals Theory Methods for Extending Equations of State to Mixtures 4.4.1 Classical Quadratic Mixing Rules 4.4.2 CompositionDependent Combining Rules 4.4.3 DensityDependent Mixing Rules 4.4.4 Combining Equations of State with ExcessGibbsEnergy Models Explicit Treatment of Association in Empirical Equations of State Equations of State as Fully Predictive Models 4.6.1 Correlations for Binary Parameters 4.6.2 GroupContribution Equations of State 4.6.3 Utilization of Predictive ExcessGibbsEnergy Models Closing Remarks
75
76 76 79 82 86 92 94 94 98
100 102 107 116 116 117 119 119
Perturbation Theory T. Boublik
5.1 5.2 5.3
Introduction Basic Concepts of Perturbation Theory Perturbation Theories of Pure Simple Fluids 5.3.1 Van der Waals Equation of State 5.3.2 Equations of State and Radial Distribution Functions of Hard Spheres 5.3.3 Perturbation Expansion for Fluids with Infinitely Steep Repulsions 5.3.4 SecondOrder Perturbation Term 5.3.5 Perturbation Expansion for Fluids with Soft Repulsions 5.3.6 Quantum Effects
127
128 131 133 133 136 140 142 144 148
5.4
5.5
5.6
5.7
vii
Mixtures of Simple Fluids 148 5.4.1 Mixtures of Hard Spheres 148 5.4.2 Perturbation Expansion for Mixtures with Infinitely Steep Repulsions 150 5.4.3 Perturbation Expansion for Mixtures with Soft Repulsions 150 Perturbation Theories of Molecular Fluids 151 5.5.1 Pair Potential of Molecular Fluids 151 5.5.2 Equations of State and Distribution Function of Hard Nonspherical 154
Molecules 5.5.3 Perturbation Expansion for Pure Molecular Fluids 157
5.5.3.1 Fluids of KiharaMolecules 157 5.5.3.2 Fluids of Molecules with Multicenter Pair Potential 159 5.5.3.3 Fluids of Molecules with Electrostatic Interactions 160
5.5.4 Water 163 Mixtures of Molecular Fluids 163 5.6.1 Mixtures of Nonelectrolytes 163 5.6.2 Mixtures of Electrolytes and Aqueous Solutions 165 Conclusions 166
Equations of State from Analytically Solvable IntegralEquation Approximations Yu. V. Kalyuzhnyi and P. T. Cummings
6.1
6.2
6.3
169
6.3.2
6.3.3
Introduction 170 6.1.1 OrnsteinZernike Equation and Closure Relations 174 6.1.2 Short Review of Recent Progress in IntegralEquation Theory 176 Baxter/Wertheim Factorization of the OrnsteinZernike Equation 179 6.2.1 OneComponent Monatomic Fluids 180 6.2.2 MultiComponent Monatomic Fluids 182 6.2.3 Remarks 183 Equation of State for Analytically Solvable Models of Simple Fluids 183 6.3.1 HardSphere Fluid 183
6.3.1.1 Solution of the PYA for the HardSphere Fluid and Fluid Mixtures 184 6.3.1.2 Equation of State for HardSphere Fluid and Fluid Mixtures 186 6.3.1.3 Semiempirical Correction to the PYA: The Generalized MSA 188 Adhesive HardSphere Fluid 189 6.3.2.1 Solution of the PYA for the Adhesive HardSphere Fluid and
Fluid Mixtures 190
6.3.2.2 Equation of State for the Adhesive HardSphere Fluid and Fluid Mixtures 192
6.3.2.3 PercusYevick Theory for the HardSphere Chain Fluid 195 HardCore Yukawa Fluids 196 6.3.3.1 Solution of the MSA for the HardCore Yukawa Fluid and Fluid
Mixtures 196
6.3.3.2 Equation of State for the HardCore Yukawa Fluid and Fluid Mixtures 201
6.3.4 Charged HardSphere Fluids 204 6.3.4.1 Solution of the MSA for the Primitive Model of Electrolyte 204
Solutions 6.3.4.2 Equation of State for the Primitive Model of Ionic Fluids and
Fluid Mixtures 207
6.3.4.3 Corrections to the MSA: the Generalized MSA and the Associative MSA 209
viii
6.4
6.5
6.6
Equation of State for Analytically Solvable Models of Molecular Fluids 212 6.4.1 InteractionSite Model Fluids 213
6.4.1.1 SiteSite OrnsteinZernike Equation and Closure Conditions 214 6.4.1.2 SSOZPYA for the Fluid of Fused HardSphere Diatomic
216 Molecules
6.4.1.3 SSOZMSA for the Fluid of Fused Charged HardSphere Diatomic Molecules 218
6.4.1.4 Corrections to the SSOZ Formalism: ChandlerSilbeyLadanyi 219 Equation
6.4.2 Hard Spheres with Dipolar Interactions 222 6.4.2.1 Analytic Solution of MSA for Dipolar HardSphere Fluid 223
Equation of State for Analytically Solvable Models of Associating Fluids 228 6.5.1 TwoDensity OZ Equation and Closure Conditions: Relation to the CLS
229 Equation
6.5.2 TwoDensity PYA for Dimerizing HardSphere Models 233 6.5.3 TwoDensity PYA for the SmithNezbeda Primitive Model of Associating 237
Fluids 6.5.4 TwoDensity MSA: RPM of Electrolytes and Dimerizing HCY Fluid 239 Conclusion 241
Quasilattice Equations of State for Molecular Fluids N.A. Smirnova and A. V. Victorov
7.1 7.2 7.3 7.4 7.5 7.6
255
Introduction 256 Basic Features of Lattice Theories; Structure of a Quasilattice EOS 257 Influence of Molecular Size and Shape 259 ContactSite Models for Fluids with Strong Directional Attractive Interactions 264 Results of Thermodynamic Modeling by the Quasilattice EOS 270 Conclusions 282
The CorrespondingStates Principle J.F. Ely and I.M.F. Marrucho
8.1 8.2 8.3
8.4
8.5 8.6
Introduction Theoretical Considerations Determination of Shape Factors 8.3.1 Other Reference Fluids 8.3.2 Exact Shape Factors 8.3.3 Shape Factors from Generalized Equations of State Mixtures 8.4.1 Van der Waals OneFluid Theory 8.4.2 Mixture CorrespondingStates Equations Applications of the Extended CorrespondingStates Theory Conclusions
289
290 292 295 296 298 302 306 308 309 310 316
Mixing and Combining Rules S.L Sandler and H. Orbey
9.1
9.2
9.3 9.4
Mixing Rules for Cubic and Related Equations of State 9.1.1 The Van der Waals OneFluid Model 9.1.2 NonQuadratic Combining Rules for Van der Waals OneFluid Model 9.1.3 Mixing Rules that Combine an Equation of State with an
ActivityCoefficient Model 9.1.3.1 HuronVidal Model 9.1.3.2 WongSandler Model 9.1.3.3 Combination of ExcessEnergy Models and Equations of State
at Low or Zero Pressure 9.1.4 Commentary on Mixing Rules for Cubic Equations of State Mixing Rules for the Virial Family of Equations of State 9.2.1 The Van der Waals OneFluid Model 9.2.2 Generalized Extended Virial Equations of State and Their Mixing Rules Mixing Rules Based Upon Theory and Computer Simulation Conclusions
ix
321
323 323 327
330
330 332
340
346 346 346 348 352 354
10 Mixtures of Dissimilar Molecules E. Matteoli, E.Z Hamad, and G.A. Mansoori
10.1 10.2 10.3 10.4
10.5 10.6
10.7 10.8 10.9
Introduction Background Grand Canonical Ensemble Analytic Theory of Dissimilar Mixtures 10.4.1 Analytic Solutions for Low to Moderate Pressures Binary Mixtures Test of C 0 Closure 10.6.1 HardSphere Mixtures 10.6.2 LennardJones Mixtures 10.6.3 Real Mixtures VaporLiquid Equilibria LiquidLiquid Equilibria Summary
359
360 360 361 363 364 366 367 367 368 368 372 376 378
11 Critical Region M.A. Anisimov and J. V. Sengers
ll.1 11.2 11.3 11.4 11.5 ll.6
11.7 11.8 11.9 11.10
Introduction Critical Region in the Van der Waals Theory Asymptotic Scaled Equation of State Corrections to the Asymptotic Scaled Equation of State Crossover between Scaling and Classical Asymptotic Critical Behavior Crossover between Scaling and Classical Nonasymptotic Critical Behavior in the Extended Critical Region Global Crossover Equation of State for OneComponent Fluids Isomorphic Scaled Equation of State for NearCritical Binary Fluids Renormalization of Critical Exponents Isomorphic Description of FluidFluid Phase Equilibria
381
382 383 386 396 401
406
412 414 419 421
11.11 Crossover between VaporLiquid and LiquidLiquid Critical Phenomena 11.12 Summary and Outlook
424 427
12
PART II
Associating Fluids and Fluid Mixtures E.A. Mfiller and K.E. Gubbins
12.1
12.2
12.3 12.4
12.5 12.6
Introduction 12.1.1 Associating Fluids 12.1.2 Experimental Evidence of Association Chemical Theories 12.2.1 Chemical Theory 12.2.2 Coupling of Equations of State with Chemical Approaches 12.2.3 QuasiChemical Theory and GroupContribution Methods Statistical Mechanical Theories Wertheim's Theory 12.4.1 Brief Account of the Theory
12.4.1.1 Molecules with One Site 12.4.1.2 Mixtures of Molecules with Multiple Associating Sites
12.4.2 Tests of the Theory against Molecular Simulations 12.4.2.1 Hard Spheres with Two Association Sites 12.4.2.2 Mixtures of Associating LennardJones Spheres
12.4.3 Limitations ofthe TPT1 Theory 12.4.4 Extensions and Applications to Complex Fluids
12.4.4.1 TPT2 Chains and Polymers 12.4.4.2 Inter and Intramolecular Bonding 12.4.4.3 Water and Electrolytes 12.4.4.4 Closedloop Liquidliquid Immiscibility 12.4.4.5 Amphiphilic Systems 12.4.4.6 Associating Liquid Crystals 12.4.4.7 Inhomogeneous Fluids
The Saft EOS and Applications Conclusions
435
436 436 438 441 441 441 443 444 445 445 445 447 447 448 449 450 451 451 453 455 456 457 459 460 460 466
13 Polydisperse Fluids D. Browarzik and H. Kehlen
13.1
13.2
Continuous Thermodynamics 13.1.1 Influence of Polydispersity on Phase Equilibrium 13.1.2 Approaches to Polydispersity 13.1.3 Fundamentals of Continuous Thermodynamics 13.1.4 Generalized Formalism of Continuous Thermodynamics 13.1.5 Analytical Distributions 13.1.6 Advantages of Using Continuous Thermodynamics Application of Continuous Thermodynamics to Equations of State 13.2.1 Fugacity Coefficients 13.2.2 Phase Equilibrium 13.2.3 Flash Problem
479
480 480 482 485 490 493 494 497 497 499 502
13.3 Critical Review of Past Work and Future Challenges 13.3.1 Description of Polydispersity and Numerical Procedures 13.3.2 Phase Equilibria of Real Systems 13.3.3 Stability of Polydisperse Fluids 13.3.4 Conclusions
xi
504 505 508 515 518
14 Equations of State for Polymer Systems S.M. Lambert, Y. Song, and J.M. Prausnitz
14.1
14.2
14.3
14.4
Introduction 14.1.1 Overview 14.1.2 Organization Equations of State for Pure Polymer Liquids 14.2.1 Cell Models
14.2.1.1 Prigogine's Cell Model (PCM) 14.2.1.2 FloryOrwollVrij (FOV) 14.2.1.3 PerturbedHardChain (PHC) 14.2.1.4 Further Modifications and Discussion
14.2.2 LatticeFluid (LF) Models 14.2.2.1 SanchezLacombe (SL) 14.2.2.2 Costas and Sanctuary (CS) 14.2.2.3 PanayiotouVera (PV) 14.2.2.4 MeanField LatticeGas (MFLG) 14.2.2.5 Further Modifications and Discussion
14.2.3 Hole Models 14.2.3.1 SimhaSomcynsky (SS) 14.2.3.2 Further Modifications and Discussion
14.2.4 TangentSphere Models 14.2.4.1 Generalized Flory Theories 14.2.4.2 Statistical AssociatedFluid Theory (SAFT) 14.2.4.3 Perturbed HardSphereChain (PHSC)
Extension to Mixtures 14.3.1 FloryOrwollVrij (FOV) 14.3.2 PerturbedHardChain (PHC) 14.3.3 LatticeFluid (LF) 14.3.4 MeanField LatticeGas (MFLG) 14.3.5 SimhaSomcynsky Hole Model 14.3.6 Statistical AssociatedFluid Theory (SAFT) 14.3.7 Perturbed HardSphereChain (PHSC) 14.3.8 Further Developments of SphereChain Models 14.3.9 Cubic Equations of State for Polymer Mixtures 14.3.10 Specific Interactions Conclusion
523
524 524 524 524 525 525 527 528 529 530 530 531 533 534 535 536 536 538 539 54O 543 544 547 547 551 553 559 56O 563 567 574 575 577 582
15 SelfAssembled Systems R. Nagarajan and E. Ruckenstein
15.1 15.2
Introduction Thermodynamic Principles of SelfAssembly 15.2.1 Multicomponent Solution Model
589
592 595 595
xii
15.3 15.4
15.5
15.6
15.2.2 15.2.3 15.2.4 15.2.5 15.2.6
15.2.7 15.2.8 15.2.9
Modeling SelfAssembly from a Kinetic Perspective Pseudophase Model of Aggregation Estimation of Critical Micelle Concentration and Micelle Size Bounds on CMC and Micelle Size Size Dispersion of Micelles and Concentration Dependence of Micelle Size Micelle Charge Concentration of Singly Dispersed Surfactant Beyond the CMC SpheretoRod Transition
Molecular Packing in SelfAssembled Structures SelfAssembly in Aqueous Solutions 15.4.1 Model of the Standard GibbsEnergy Difference for Aggregates
15.4.1.1 Transfer Gibbs Energy of the Surfactant Tail 15.4.1.2 Deformation Gibbs Energy of the Surfactant Tail 15.4.1.3 Aggregate CoreWater Interfacial Gibbs Energy 15.4.1.4 HeadGroup Steric Interactions 15.4.1.5 HeadGroup Dipole Interactions 15.4.1.6 HeadGroup Ionic Interactions
15.4.2 Computational Approach 15.4.2.1 MaximumTerm Method 15.4.2.2 Calculations for Rodlike Micelles
15.4.3 Molecular Constants for Surfactants 15.4.4 Influence of GibbsEnergy Contributions on Aggregation Behavior 15.4.5 Influence of Tail and Head Groups on Aggregation Behavior 15.4.6 Influence of Ionic Strength on Aggregation Behavior 15.4.7 Influence of Temperature on Aggregation Behavior 15.4.8 Transition from Spherical to Rodlike Micelles 15.4.9 Formation of Bilayer Vesicles Micelles with Poly(ethylene oxide) Head Groups 15.5.1 Approach to Modeling HeadGroup Interactions 15.5.2 Uniform Concentration Model
15.5.2.1 HeadGroup Mixing Gibbs Energy 15.5.2.2 HeadGroup Deformation Gibbs Energy 15.5.2.3 Aggregate CoreWater Interfacial Gibbs Energy
15.5.3 Nonuniform Concentration Model 15.5.3.1 HeadGroup Mixing Gibbs Energy 15.5.3.2 HeadGroup Deformation Gibbs Energy
15.5.4 Predicted Aggregation Behavior SelfAssembly of Surfactant Mixtures 15.6.1 Size and Composition Distribution of Aggregates 15.6.2 Gibbs Energy of Formation of Mixed Micelles
15.6.2.1 Transfer Gibbs Energy of Surfactant Tail 15.6.2.2 Deformation Gibbs Energy of the Surfactant Tail 15.6.2.3 Aggregate CoreWater Interfacial Gibbs Energy 15.6.2.4 HeadGroup Steric Interactions 15.6.2.5 HeadGroup Dipole Interactions 15.6.2.6 HeadGroup Ionic Interactions 15.6.2.7 Gibbs Energy of Mixing of Surfactant Tails
15.6.3 Predictions of Molecular Theory 15.6.3.1 Nonionic HydrocarbonNonionic Hydrocarbon Mixtures 15.6.3.2 Ionic HydrocarbonIonic Hydrocarbon Mixtures 15.6.3.3 Ionic HydrocarbonNonionic Hydrocarbon Mixtures
596 597 598 598
600
603 604 604 606 610 610 610 611 611 612 613 614 615 615 616 617 618 620 625 626 628 630 631 632 632 633 634 635 636 636 637 637 641 642 644 644 645 645 646 646 647 648 648 649 650 652
xiii
15.6.3.4 Anionic HydrocarbonCationic Hydrocarbon Mixtures 659 15.6.3.5 Anionic FluorocarbonNonionic Hydrocarbon Mixtures 661 15.6.3.6 Anionic HydrocarbonAnionic Fluorocarbon Mixtures 666
15.7 Surfactant SelfAssembly in NonPolar Media 667 15.7.1 Gibbs Energy of Aggregation 668
15.7.1.1 Interaction between Head Groups 670 15.7.1.2 QuasiChemical Bonding between Head Groups 670 15.7.1.3 Steric Interactions between Surfactant Tails 671 15.7.1.4 TailSolvent Interactions 672
15.7.2 Predictions from the Model 672 15.7.2.1 GibbsEnergy Contributions 673 15.7.2.2 Size Distribution of Aggregates 673 15.7.2.3 Influence of Surfactant Molecular Structure 677 15.7.2.4 Influence of Temperature and Solvent Polarity 681
15.8 SelfAssembly of Surfactants in Polar NonAqueous Solvents 683 15.8.1 Gibbs Energy of Aggregation 683 15.8.2 Prediction of Solution Behavior 684
15.8.2.1 Aggregate Size Distribution 685 15.8.2.2 Critical Micelle Concentration 686 15.8.2.3 Aggregate Polydispersity and ConcentrationDependent 686
Aggregate Size 15.8.2.4 GibbsEnergy Contributions 688 15.8.2.5 Aggregation Behavior ofAlkyl Pyridinium Bromides 691 15.8.2.6 Aggregation Behavior ofAlkyl Trimethyl Ammonium Bromides 693 15.8.2.7 Comparison with Experimental Observations in the Literature 694
15.8.3 Micellization in Mixed Solvent System of WaterEthylene Glycol 695 15.8.3.1 Aggregation Behavior of Cetyl Pyridinium Bromide 697
15.9 Solubilization in Surfactant Aggregates 698 15.9.1 Size Distribution of Micelles Containing Solubilizates 701 15.9.2 Gibbs Energy of Solubilization 702
15.9.2.1 Micelle CoreWater Interfacial Gibbs Energy 702 15.9.2.2 HeadGroup Interactions for Poly(ethylene oxide) Surfactants 704 15.9.2.3 Surfactant TailSolubilizate Mixing Gibbs Energy 705
15.9.3 Predictions of Solubilization Behavior 705 15.9.3.1 Computational Approach 706 15.9.3.2 Solubilization in Ionic Surfactant Solutions 706 15.9.3.3 Solubilization in Micelles of Poly(ethylene oxide) Surfactants 708 15.9.3.4 SolubilizationInduced RodtoSphere Transition 710
15.9.4 Solubilization of Binary Hydrocarbon Mixtures 716 15.9.4.1 Gibbs Energy of Solubilization of Binary Mixtures 718 15.9.4.2 AggregateWater Interfacial Energy 718 15.9.4.3 Surfactant TailSolubilizate Mixing Gibbs Energy 718 15.9.4.4 Predictions of the Binary Solubilization Model 719
15.10 Microemulsions 719 15.10.1 Geometrical Characteristics of Microemulsions 721 15.10.2 Size Distribution of Droplet Microemulsions 723 15.10.3 Size and Composition Dispersion of Droplets 726 15.10.4 Calculation of Interfacial Tension 727 15.10.5 Bicontinuous Microemulsions 727 15.10.6 Gibbs Energy of Formation of the Film Region 730
15.10.6.1 Transfer Gibbs Energy of Surfactant and Alcohol Tails 730 15.10.6.2 Deformation Gibbs Energy of Surfactant and Alcohol Tails 731
xiv
15.10.6.3 Interfacial Gibbs Energy of FilmWater Contact 15.10.6.4 HeadGroup Steric Interactions 15.10.6.5 HeadGroup Dipole Interactions 15.10.6.6 HeadGroup Ionic Interactions 15.10.6.7 Gibbs Energy of Mixing inside the Film Region
15.10.7 Predictions for Droplet Microemulsions 15.10.8 Prediction of a Bicontinuous Microemulsion
731 732 732 732 733 733 737
16 Ionic Fluids H. Krienke and J. Barthel
16.1 16.2 16.3 16.4
16.5 16.6 16.7 16.8
16.9
16.10 16.11 16.12 16.13 16.14
Introduction Low Concentration Chemical Model IntegralEquation Methods at the MM Level Classical Systems with LongRange Forces 16.4.1 Correlation Functions and Screening 16.4.2 Thermodynamic Functions Effective Ionic Interactions Solvation of Ions in Apolar Solvents Molecular Correlation Functions Invariant Expansions MSA for Charged and Dipolar Hard Shperes 16.8.1 MSA of Charged Hard Spheres 16.8.2 MSA of Mixtures of Ions and Dipoles Molecular Pair Correlations 16.9.1 Interaction Site Model (ISM) 16.9.2 Molecular OrnsteinZernike (MOZ) Theory 16.9.3 SiteSiteOrnsteinZernike (SSOZ) Theory 16.9.4 Monte Carlo Simulation Solvent Properties Ion Solvation in Molecular Solvents Ion  Ion Potential of Mean Force Ion Association Concluding Remarks
751
752 755 758 763 763 766 767 768 771 772 773 776 781 781 782 784 785 785 787 792 792 798
17 Ionic Fluids Near Critical Points and at High Temperatures J.M.H. Levelt Sengers, A.H. Harvey, and S. Wiegand
17.1 17.2
17.3
Introduction Criticality and Ionic Fluids 17.2.1 Criticality and Critical Exponents 17.2.2 Criticality and Range of Forces 17.2.3 Phase Separation due to Coulombic Interactions 17.2.4 The Restricted Primitive Model, RPM 17.2.5 Criticality of the RPM 17.2.6 Tricriticality and ChargeDensity Waves in Ionic Fluids Critical Behavior in OneComponent Charged Systems 17.3.1 Water 17.3.2 Liquid Metals near the VaporLiquid Critical Point 17.3.3 Molten Salts near the VaporLiquid Critical Point
805
806 806 806 807 809 810 811 812 812 812 812 813
17.4
17.5
17.6 17.7
17.8
Experiments in Partially Miscible Ionic Liquids 17.4.1 The RPM as a Guide 17.4.2 Character of Criticality in Nonaqueous Ionic Solutions 17.4.3 Character of Criticality in Aqueous Ionic Solutions 17.4.4 Viscosity 17.4.5 Summary 17.4.6 Crossover: a Perspective Solution Thermodynamics l~ar Critical Points 17.5.1 Principal Issues 17.5.2 Thermodynamic Properties of Interest 17.5.3 Criticality of Dilute Solutions 17.5.4 Ionic Effects on Critical Behavior Gibbs Energy Models for HighTemperature Aqueous Electrolyte Systems Helmholtz Energy Models; Equations of State 17.7.1 Models Assuming Complete Ionic Dissociation 17.7.2 Models Assuming no Ionic Dissociation 17.7.3 Models with Partial Ionic Dissociation 17.7.4 Solid Solubility Calculations Conclusions
XV
814 814 817 823 824 824 825 827 827 828 828 832 833 836 836 838 84O 840 841
18 Multiparameter Equations of State R. T Jacobsen, S.G. Penoncello, E.W. Lemmon and R. Span
18.1 18.2 18.3
18.4
18.5 18.6
18.7 18.8 18.9 18.10
Introduction The Development of a Thermodynamic Property Formulation The Use of LeastSquares Fitting in Developing an Equation of State 18.3.1 The LeastSquares Method 18.3.2 Multiproperty Fitting 18.3.3 Constraints 18.3.4 Optimizing the Functional Form of Equations of State PressureExplicit Equations of State 18.4.1 The BenedictWebbRubin Equation of State 18.4.2 The MartinHou Equation of State 18.4.3 The Bender Equation of State 18.4.4 The JacobsenStewart Equation of State Thermodynamic Properties from PressureExplicit Equations of State Fundamental Equations 18.6.1 The Equation of Keenan, Keyes, Hill, and Moore 18.6.2 The Equation of Pollak 18.6.3 The Equations of Haar and Gallagher and Haar, Gallagher, and Kell 18.6.4 The Equation of Schmidt and Wagner 18.6.5 The Equation of Jacobsen, Stewart, Jahangiri, and Penoncello 18.6.6 Recent Equations of State 18.6.7 Transition (or Unified) Equations Thermodynamic Properties from HelmholtzEnergy Equations of State Comparisons of Property Formulations Recommended Multiparameter Equations of State Equations of State for Mixtures 18.10.1 The Virial Equation of State for Mixtures 18.10.2 Extended CorrespondingStates Methods 18.10.3 Mixture Properties Using HelmholtzEnergy Equations of State
849
85O 851 852 852 854 854 854 856 856 856 857 857 858 859 860 860 861 862 863 864 867 867 868 873 874 876 876 877
Subject Index 883
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LIST OF CONTRIBUTORS
xvii
• M.A. Anisimov (U.S.A.) • A. Anderko (U.S.A.)
• J. Barthel (Germany) • T. Boublik (Czech Republic)
• D. Browarzik (Germany)
• P.T. Cummings (U.S.A.)
• J.F. Ely (U.S.A.)
• M.B. Ewing (U.K.)
• K.E. Gubbins (U.S.A.) • E.Z. Hamad (Saudi Arabia)
• A.H. Harvey (U.S.A.)
• R.T Jacobsen (U.S.A.) • Yu.V. Kalyuzhnyi (U.S.A.) • R.F. Kayser (U.S.A.) • H. Kehlen (Germany)
• H. Krienke (Germany)
• S.M. Lambert (U.S.A.)
• E.W. Lemmon (U.S.A.) • J.M.H. Levelt Sengers (U.S.A.)
• G.A. Mansoori (U.S.A.) • I.M.F. Marrucho (U.S.A.)
• E. Matteoli (Italy)
• E.A. Mtiller (Venezuela)
• R. Nagarajan (U.S.A.)
• H. Orbey (Turkey)
• S.G. Penoncello (U.S.A.)
• C.J. Peters (The Netherlands)
• J.M. Prausnitz (U.S.A.) • E. Ruckenstein (U.S.A.)
• S.I. Sandler (U.S.A.)
• J.V. Sengers (U.S.A.)
• N.A. Smirnova (Russia) • Y. Song (U.S.A.) • R. Span (Germany)
• J.P.M. Trulser (U.K.)
• A.I. Victorov (Russia)
• S. Wiegand (Germany) • H.J. White, Jr. (U.S.A.)
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xix
P R E F A C E
The present volume on Equations of State for Fluids and Fluid Mixtures has been prepared under the auspices of Commission 1.2 on Thermodynamics of the International Union of Pure and Applied Chemistry (IUPAC). When the Commission requested me to pursue first a feasibility study and subsequently the production of a volume that would assess the range of approaches to develop equations for calculating thermodynamic properties of fluids and fluid mixtures, I was fortunate in finding Richard F. Kayser and Cor J. Peters and, subsequently, also Howard J. White Jr. as coeditors willing to assist me with this arduous task. The resulting volume deals with a broad range of equations of state for fluids and fluid mixtures including complex fluids like polydisperse fluids, polymer systems, self assembled systems and ionic fluids.
This volume was originally scheduled to appear in 1998. However, after the volume had been in the production state for several months, the publisher assigned by IUPAC requested permission not to proceed with publication. Fortunately, after a second favorable independent review of the material in the book, Elsevier agreed to publish this volume. Because of the resulting delay of the publication schedule, the authors were provided with an opportunity to make (minor) revisions in their manuscripts which were submitted in the fall of 1999. It is hoped that, in spite of this delay, the reader will find this volume to be a valuable reference concerning the status of equations of state for fluids and fluid mixtures.
On behalf of all the editors, I thank the authors for their patience in dealing with the unexpected delay of the publication of their chapters. The Physical and Chemical Properties Division of the National Institute of Standards and Technology (NIST) provided the necessary office support for the editorial processing of this volume. We are especially indebted to Laurell R. Phillips, Jr. and Carol A. Thomas for their outstanding and dedicated service in preparing the volume for the publisher. The pleasant interaction with Huub Manten of Elsevier is also gratefully acknowledged.
Jan V. Sengers
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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 Intemational Union of Pure and Applied Chemistry. All rights reserved
1 INTRODUCTION
J. V. Sengers a'b, R. F. Kayser c, C. J. Peters d, and H. J. White, Jr. a
alnstitute for Physical Science and Technology and Department of Chemical Engineering University of Maryland College Park, MD 20742, U.S.A.
bphysical and Chemical Properties Division National Institute of Standards and Technology Gaithersburg, MD 20899, U.S.A.
C Technology Services National Institute of Standards and Technology Gaithersburg, MD 20899, U.S.A.
dLaboratory of Applied Thermodynamics and Phase Equilibria Faculty of Applied Sciences Delft University of Technology Julianalaan 136, 2628 BL Delft, The Netherlands
The voluminous literature on equations of state for fluids and their mixtures and the closely related equations for the Helmholtz energy or Gibbs energy over a range of variables, and their wide use in science, engineering and industry attest to their importance.
Commission 1.2 on Thermodynamics of the International Union of Pure and Applied Chemistry (IUPAC) considered the desirability of preparing a volume on equations of state for fluids and fluid mixtures under IUPAC sponsorship. The volume in question would cover the entire range of approaches to developing equations of state, including their theoretical bases and practical uses and their strengths and limitations. Furthermore, the volume would not be limited to equations of state for simple fluids and fluid mixtures, but would also include the more important classes of complex fluids and mixtures.
The Commission requested assistance in obtaining a broader view of the desirability and feasibility of such a volume. As a result, a committee, consisting of three of the editors of this volume, sent out a questionnaire to a wide range of experts and potentially interested parties. In addition to desirability, the questionnaire asked for suggestions for topics to be included and
the names of appropriate potential authors. The results of the questionnaire were strongly positive with respect to desirability of the
volume. The suggestions with regard to content, specific topics and potential authors were many and varied, as would be expected. Summarizing the suggestions made and integrating them with the original IIYPAC concept resulted in the following features: 1. All principal approaches to developing equations of state should be covered. 2. The theoretical basis and practical use of the various types of equations of state would be
covered. 3. The strengths and limitations of each type would be addressed. 4. In each case, what is available and what additional work is needed should be pointed out. 5. In addition to simple fluids and fluid mixtures, more complex systems would be covered.
These would include associating fluids, ionic fluids, polydisperse systems, polymers, and micelle forming and other self organizing systems. The editors have selected authors who are knowledgeable in the fields covered and asked
them to pay particular attention to the desired information listed in items 2, 3 and 4 above. There are some other areas, not mentioned specifically above, which deserve attention: The critical region is unique and has received a great deal of theoretical attention in recent
years. In addition, mixtures in the critical region have proved to have interesting properties with practical engineering potential.
Another area, namely that of chemically reacting fluids and mixtures, has not been covered in a general way, although it is treated in some of the chapters on more complex systems insofar as it pertains to these specific topics. Some limited aspects of this area have been studied for some time. For example, fluids that readily form dimers and mixtures containing weakly ionizing molecules, but the more general subject appears not to have received the study that the importance of the topic deserves. The editors hope that this area will receive additional theoretical and experimental study in the near future.
An increasingly more important area of research is the use of computer simulations for the calculation of thermodynamic properties of simple and complex fluids. This subject would deserve an indepth coverage of its own and, because of size limitations, it was decided not to include the topic in the present volume. The structure of the volume closely follows the requirements suggested by the IUPAC commission and amplified and reinforced by the resulting questionnaire.
Chapter 2 provides the basic thermodynamic foundation that underlies all of the remaining chapters.
The next six chapters deal with the various approaches that have produced enlightening and useful equations of state with the emphasis on single fluids. They are: Chapter 3, the virial equation; Chapter 4, the van der Waals and related cubic equations; Chapter 5, perturbation theory; Chapter 6, the use of analytically solvable integral equations; Chapter 7, quasilattice equations; and Chapter 8, the correspondingstates principle.
Mixing and combining rules are dealt with in Chapter 9 with emphasis primarily on mixtures of simple fluids. Mixtures of dissimilar fluids are considered in Chapter 10.
The basic approach to the critical region with emphasis on the simpler fluids and their mixtures is given in Chapter 11.
The remaining chapters deal with more complex fluids and their mixtures. Associating fluids and their mixtures are the subject of Chapter 12. Polydisperse fluids are
the topic of Chapter 13. These are fluids of so many components that the composition of the fluid has to be considered in terms of a continuous distribution function, crude oil being an example. Polymer systems are considered in Chapter 14, and selfassembling systems are covered in Chapter 15. Selfassembling mixtures contain components that spontaneously form micelles and related structures.
The next two chapters deal with ionic fluid mixtures. Ionic fluids in general form the subject of Chapter 16 and ionic fluids and chemically reacting fluids in the critical region are the subject of Chapter 17.
Finally, multiparameter equations of state are covered in Chapter 18. Since these provide equations of state for individual fluids, they might have been grouped with the earlier chapters involved primarily with individual fluids. However, these equations are used for substances, such as water or ethylene, for which accurate data for a number of thermodynamic properties are available over wide ranges of temperature and pressure. Since the thermodynamic properties are interrelated, thermodynamic consistency is needed over the entire range. The resulting complexities put these equations in a special category.
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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved
2 F U N D A M E N T A L C O N S I D E R A T I O N S
M.B. Ewing a and C.J. Peters b
a University College London, Chemistry Department, Christopher Ingold Laboratories 20 Gordon street, London WC1H OAJ, United Kingdom bDelft University of Technology, Faculty of Chemical Technology and Materials Science Laboratory of Applied Thermodynamics and Phase Equilibria Julianalaan 136, 2628 BL Delft, The Netherlands
2.1 Introduction 2.2 Basic Thermodynamics
2.2.1 Homogeneous Functions 2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations
2.3 Deviation Functions 2.3.1 Residual Functions 2.3.2 Evaluation of Residual Functions
2.4 Mixing and Departure Functions 2.4.1 Departure Functions with Temperature, Molar Volume and Composition as the
Independent Variables 2.4.2 Departure Functions with Temperature, Pressure and Composition as the
Independent Variables 2.5 Mixing and Excess Functions 2.6 Partial Molar Properties 2.7 Fugacity and Fugacity Coefficients 2.8 Activity Coefficients 2.9 The Phase Rule 2.10 Equilibrium Conditions
2.10.1 Phase Equilibria 2.10.2 Chemical Equilibria
2.11 Stability and the Critical State 2.11.1 Densities and Fields
2.11.2 Stability 2.11.3 Critical State
References
2.1 INTRODUCTION
This chapter provides a thermodynamic toolbox and contains most of the important basic relations that are used in other chapters. The scope is restricted almost exclusively to the second law of thermodynamics and its consequence, but the treatment is still intended to be exemplary rather than definitive. New results are not presented as befits a discussion of fundamentals which are necessarily invariant with time.
2.2 BASIC THERMODYNAMICS
The state of a system may be described in terms of a small number of variables. For a phase in the absence of any external field, the second law of thermodynamics may be written as
d U = T d S  p d V + ~ ~tidn i (2.1) i
This equation shows that the change dU in the energy U may be described in terms of simultaneous changes dS in the entropy S, dV in the volume V, and dn~ in the amounts n~ of the C components. It is often convenient to eliminate the size of the phase by writing Equation (2.1) in terms of intensive variables. For example, division by the total amount
N = ~ n i i
gives
(2.2)
alUm = T d g m  pdVm + 2 ~idxi (2.3) i
where the subscript m denotes a molar quantity and the mole fraction of component i is defined by
X i = ?l /N (2.4)
Equation (2.1), or Equation (2.3) for intensive variables, is the most fundamental expression of the second law of thermodynamics. However, entropy, in particular, is not a very convenient experimental variable and, consequently, altemative forms have been derived from the fundamental equation (2.1). Introduction of the following characteristic functions:
Enthalpy H = U + p V (2.5) Hemholtz energy A = U  TS (2.6)
Gibbs energy G = U + p V  TS = H  TS = A + p V (2.7)
and use of Legendre transformations (1,2)with the fundamental equation (2.1) gives the following alternative forms of the second law
d H = TdS + Vdp + ~ ~idni i
dA =  Sd T  p d V + ~ g idn i i
dG =  S d T + Vdp + ~ gidni (2.10) i
Another modification of Equation (2.1), frequently used in statistical mechanics, is
(2.8)
(2.9)
d ( p V ) = S d V + p d V + E n i d ~ i (2.11) i
or, as the GibbsDuhem equation, in the study of phase equilibria
0 = S d T  Vdp + ~ rtid~t i (2.12) i
From Equation (2.1) and Equations (2.8)  (2.10) it can be seen that
T~p,I~ i
~t i =
S, V,I~ i S,p,fi i T~ V,I~ i
(2.13)
where the subscripted fig means that the amounts nj of all the components are constant except for component i. The quantity g~ is the chemical potential of species i. In terms of intensive variables these equations are
dHm = TdS m + VmdP + ~ ~t idx i (2.14) i
d A m   S m d Z  pdVm + ~ ~l'idxi (2.15) i
dGm =  SmdT + VmdP + ~ ~idx i (2.16) i
d(o Vm) = SmdT + p d V m + Z nid~ t i (2.17) i
0 = S m d T  VmdP + E xid~ti (2.18) i
For a system of constant composition, Equation (2.1) and Equations (2.8)  (2.10) reduce to
d U = TdS  p d V
d H = TdS + Vdp
dA =  S d T  p d V
d G =  Sd T + Vdp
(2.19) (2.20) (2.21) (2.22)
A large number of thermodynamic relations may be derived from the above equations by conventional manipulations. Table 2.1 summarizes the most frequently used equations.
Table 2.1 Frequently used thermodynamic relationships with general validity.
d U = TdS  p d V
dA =  S d T  p d V
d H = TdS + Vdp
dG :  S d T + Vdp
v T p T
av) = Op)  ~ T= ~)
Maxwell:
U/,,
Helmholtz:
OU
 ~ , r
GibbsHelmholtz:
OT v T2 [ OT p T 2
2.2.1 Homogeneous Functions
A homogeneous function F of the first order in any number of the variables x, y, z . . . . is defined by:
F(X~, Zy, ~ , ...)  ZY(x, y, z, ...) (2.21)
where ;~ is an arbitrary number. If each independent variable is made X times larger, the function F increases ~. times. For large enough systems, all extensive thermodynamic properties are homogeneous and of the first order in amounts n, at fixed temperature and pressure. For homogeneous functions of the first order, Euler's theorem applies:
y,z,. , x,z,...
+ z + "'" (2.22) x,y,...
10
Equation (2.22) relates the value of the function to the values of its derivatives. For ~ = 1, Equation (2.22) reduces to:
• = + z + "'" (2.23) y,z,.., x,z .... x,y,...
By applying Euler's theorem to the various characteristic functions U = U(S, V, n l, n2,... ), H = H(S, p, n l, n2,... ), A = A(V, T, nl, n 2...) and G = G(p, T, nl, nz,...), respectively, the following expressions result
U = TS  p V + Z ni~i ( 2 . 2 4 ) i
n : TS + E n i~ti ( 2 . 2 5 ) i
A :  p V + ~ Rig i (2.26) i
G = ~_~ ni~i (2.27) i
For further details on Euler's theorem see references 13.
2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations
The quantities U(S, V, n 1, n 2, ...), H(S, p, n 1, n 2, ...), A(T, V, hi, n z . . . . ), and G(T, p , nl, n z . . . . ) are examples of thermodynamic potentials from which all properties of a system can be obtained without the need for integration. For example, Equation (2.10) gives directly the heat capacity at constant pressure
p,n D77),,,n
and the isothermal compressibility
KT =  ~P T,n Op2) T,n "~P T,n
The avoidance of integration is often an advantage in theoretical applications of thermodynamics because there are no troublesome constants of integration. On the other hand, very often the derivatives needed involve variables that are difficult to measure experimentally. Even with the Gibbs energy surface, which is closely linked to the convenient experimental variables of temperature, pressure and composition, differences in the Gibbs energy can be studied only at equilibrium. At constant composition the characteristic functions U = U(S, V, n l, n 2 .... ),
11
H = H(S, p, n 1, n2,. . . ), A = A(V, T, n 1 ,n2,. . . ) and G = G(p, T, nl, t/z,. . . ) reduce to U = U(S, V), H = H(S, p), A = A(V, T) and G = G(p, T). By differentiation of these functions and use of the fundamental equations (2.5) and (2.18)  (2.20), the value of any thermodynamic property can be expressed in terms of the derivatives of each characteristic function. Table 2.2 summarizes the most relevant results. Further details can be found elsewhere (3).
2.3 Deviation Functions
Almost all definitions of molar properties for mixtures lack an unambiguous definition in the sense that they can be related directly to measurable properties. Therefore, it is common practice to compare an actual mixture property with its corresponding value obtained from an arbitrary model, for instance, an equation of state. This approach leads to the introduction of deviation functions. For a general mixture molar property M m the deviation function is defined by (4):
M S = M'a_£ tual _ M m°del (2.28)
An important aspect in this definition is the choice of the independent variables. Almost all analytical equations of state are expressions explicit in pressure: that is, temperature, molar volume (or density) and composition x = x~, x 2, ... are the natural independent variables. Therefore, Equation (2.28) can be rewritten into:
MD(T, V, n) = Mactual(r, V, n) Mm°del(T, V, n) (2.29)
where n denotes the amounts n 1, t/z, """ The value of M obtained from the model is evaluated at the same values of the independent variables as used for the actual mixture property. Alternatively, although not of direct interest for equationofstate models, temperature, pressure and composition may be a suitable choice as independent variables for deviation functions, i.e.:
MD(T, p, n) = Mactual(r~ p, n) Mm°a~l(T, p, n) (2.30)
In this case the value of M obtained from the model is evaluated at the same values of T, p, and n as used for the actual mixture property. Both approaches are interrelated as follows (4):
~ 0 M model MD(T'V'n) = MD(T'p'n) + ( Op )V,n dp (2.31)
Pr
In this equation Pr is the reference pressure at which the molar volume of the mixture obtained from the model is equal to molar volume of the actual mixture at the same temperature and composition as the mixture. A necessary feature of the model is that such a Pr exists.
12
= 0 o~..~
o~,~
c~
0
o~._~
r~
o~,~
o c~
0
E
~1 ~ I
cb
I
E~
~L
J
rb c~
I
f ~
~1 ~
I
E~
~ J
I
f ~
I
I
I
Cb
I I
~1~ ~1 ~
~L
J ~L
J
I
~1 ~ I
E~
~ J
I I
fir ~
I
~1 ~ c~
I
As pointed out in the previous section, the calculation of deviation functions requires a choice
13
~1 ~ cO
I I
r ~
~1 ~ cO
I
fr _
~
~1 ~
I
f ~
I I
fr ~
cO
I
f ~
c~b
I
ro
f ~
ro c~
C
As pointed out in the previous section, the calculation of deviation functions requires a choice
14
2.3.1 Residual Functions
As pointed out in the previous section, the calculation of deviation functions requires a choice of an appropriate model. If the model system is chosen to be an ideal gas mixture, which is an obvious choice for fluid mixtures, then the deviation functions are called residual functions. With temperature, volume, and composition as independent variables Equation (2.29) becomes
MD(T, V, n) = MR(T, V, n) = Mactual(T~ V, n)/k/ag(T, V, n) (2.32)
or with T and p as independent variables
MD(T, p, n) = MR(T, p, n) = Mactual(T~ p , n) Mig(T, p, n) (2.33)
which is a particular form of Equation (2.30). The two sets of residual functions are related by Equation (2.31) in the form:
MR(T,V,n)=MR(T,p,n)+P_f(OMig) dp (2.34)
Pr Op T,n
where the reference pressure Pr = RT/Vm is sufficiently low for the thermodynamic property M of the real fluid to have the idealgas value.
Some thermodynamic properties (U, H, Cv and Cp) of an ideal gas are independent of pressure, while, others like S, A and G are not. Consequently, from Equation (2.33), it can be seen easily that the following equations hold:
UR(T, V,n) = UR(T,p,n) (2.35)
I1R( T, V,n) = H~( T,p,n) (2.36)
CRv( T, V, n ) = CRv( T, V, n )
CRp(T,V,n)=CR(T,p,n)
S" (T,V,n) = SR(T,p,n) = nlnZ
(2.37)
(2.38)
(2.39)
AR(T, V,n) = Ag(T,p,n) + RTInZ (2.40)
GR(T, V,n) = Gg(T,p,n) + RTInZ (2.41)
with Z =p V/nRT as the compression factor.
15
2.3.2 Evaluation of Residual Functions
With temperature, molar volume and composition as the independent variables, general expressions for the residual functions of thermodynamic properties are readily obtained from Equation (2.32). Abbott and Nass (4) have given a definitive list of expressions for various properties as residual functions and these are summarised in Table 2.3.
For temperature, pressure and compositions as the independent variables, Abbott and Nass (4) also evaluated general expressions for the residual functions. Table 2.4 summarizes the results. For further details see references 2 and 4.
2.4 Mixing and Departure Functions
In terms of the independent variables, temperature, pressure and composition, departure functions compare the value of a general thermodynamic property M(T,p,n) with the corresponding property in the idealgas state and at a reference pressure Pr, that is/h0g(T, Pr, n). According to the ideal gas law, the reference pressure Pr is related to the reference volume V r = nRT/pr. Similarly, as was the case for the residual functions, the independent variables, temperature, molar volume and composition can also be used to define departure functions. Based on these independent variables, the general thermodynamic property M(T, V, n) is compared with the corresponding idealgas property _aa0g(T, V, n).
2.4.1 Departure Functions with Temperature, Molar Volume and Composition as the Independent Variables
The following equality can be derived for the departure function of a general thermodynamic property M(T, V, n) "
T,n
 o M i g d V + dV O V r,, O V r,n
r
(2.42)
Applying conventional thermodynamic manipulations on Equation (2.42), the following relations can be obtained:
U  u i g = ! { / (  ~ T )v,n p}dV (2.43)
V,n  p} dV + nRT(Z 1) (2.44)
16
Table 2.3 Residual functions with volume or density as an independent variable. (p = n/V and Z= pV/nRT). M R (T, V, n) Residual Function
R =
H R  
 ~"~o~ ~~/~,~ do
nRT 2 OZ d 9  ~ + n R T ( Z  1) O " p, n [3
R _ _
A R .
a R ._~
C n m
v
 nR   ~ p,n P
n R T f ( Z  1 ) dp o P
P
n R T f ( Z  1) do + nRT(Z  1) o P
(oz I }o R  ~ ) o , n + 2 ~ p,n P
C R _.~ P R t ~ O~TI l!t I ~~1 t 1 C v  R + R Z + Z + p
p ,n T,n
pR = pRT(Z  1)
17
Tab le 2.4 Residual functions with pressure as the independent variable. (9 = n/Vand Z =pV/nRT). M ~ (T, p, n) Residual Function
R
R .__
A R  
G R  
R
v
C P
_ nRT 2 OZ dp _ nRT(Z  1)
0 " p ,n P
_ nRT2 OZ dp
0 " p ,n P
 nR  ~ p,n
P
nRT f (Z  1) dp  nRT(Z  1) P
0
9
n R T f ( Z  1) dp o P
+ 2 I } ~)p , , , p,n P
C v  R + R Z + Z + p p ,n T,n
p R __ R T ~ ( Z  1) P
18
(2.45)
Note, that if the reference volume V r is replaced by the actual volume V of the system, then the corresponding residual functions are recovered (Table 2.3).
A  Aig = p  d V  nRTI V (2.46)
G  G ig =  t)  d V  nRT1 V + n R T ( Z  1) (2.47)
The thermodynamic properties for the reference state are defined by the following relations:
g i g = E nig] g= Z nig] g  n R T ~ n i (2.48) i i i
sig = E niS] g  R ~ nilnn i (2.50) i i
n i g = E niH] g (2.49) i
A ig = ~ n c 4 / g + R T ~ n i l n n i = ~ n , ' t l ~ g  T.~n,S/g  R T ~ n i + R T ~ n i l n n i (2.51) i i i i i i
G i g = E niG] g+ R T ~ nihlni = E ntH~ g  T ~ niS]g + R T ~ nihln i (2.52) i i i i i
From Equations (2.48) through (2.52), the following mixture properties for ideal gases are obtained:
mmixU = uig  E niU] g = 0 (2.53) i
Amixn = n i g  Z n f l ] g  0 (2.54) i
AmixS = sig  E niS] g =  R E nilnni (2.55) i i
AmixA = A ig _ Z n~] g = R T ~ nilnn i (2.56) i i
Amixa = Gig  Z nia] g = R T E nilnni (2.57) i i
Departure functions are conveniently evaluated from Equation (2.46) as the generating
19
function. The following calculation procedure is the appropriate route to follow:
1. Equation (2.46) gives (A  A ig)
2. ( S  sig) = _ ( c 3 ( A  Aig)/ OT V,n
3. (U U ig) : (A A ig) + T(S S ig)
(2.58)
(2.59)
4. (H /T g) : (A A ig) + I"(8 S ig) + nRT(Z 1) (2.60)
5. (G Gig) = (A A ig) + nRT(Z 1) (2.61)
2.4.2 Departure Functions with temperature, Pressure and Composition as the Independent Variables
In this case, the general thermodynamic property is M(T, p, x) and the following equality for the departure function can be derived:
m(T,p,n)mig(T,Pr,n) = ~ T,n Op T,n Op T,n Pr
The following relations can be obtained from Equation (2.62):
g  gig _ V  ~ p,n d p  n R T ( Z  1) (2.63)
n  nig = i { V  ~( ~TIp,n}dP (2.64)
7 ;r (2.65)
A _Aig _ V _ n T dp  (2.66)
G  G ig = V  n T dp + nRTI P
Note, if the reference pressure Pr is replaced by the actual pressure p of the system, then the corresponding residual functions are recovered (Table 2.4).
20
Again, the thermodynamic properties for the reference state can be obtained from the Equations (2.48)  (2.52). In order to obtain the departure functions for this set of independent variables, the following procedure is recommended:
1. Equation (2.67) gives (G  G ig )
2. ( S  sig) :  { O(GGig),} OT ,p,n
3. (U ~g) = (G G 0g) + T(S s~g)  nRT(Z 1)
(2.68)
(2.69)
4. (H /T g) = (G G ig) + T(S S ~g) (2.70)
5. (A A ig) = (G  G'g)  n R T ( Z  1) (2.71)
Further details on departure functions can be found elsewhere (5).
2.5 Mixing and Excess Functions
Deviation functions for mixtures are concemed mainly with variation in composition rather than pressure or density. Consequently, it is convenient to use molar quantities. Molar excess functions are defined by
M E = M g = Mm actual  Mim d (2.72)
where M~ is the molar value for the ideal mixture at the same temperature and pressure. If, for
example,/W d = pd then the molar volume of the ideal mixture (id) is given by
Vim d = E i x i V i * (2.73)
where Vi* is the molar volume of pure component i.
Other thermodynamic properties for ideal mixtures are defined by:
gim d = E x i g i * i
= , g* H~ Ex, i i
S~ : E x,Si*  RE xilnxi i i
id .4* + R T ~ xi lnx i Am = E Xt i i i
Gim d = E x i G i * + R T E x i l n x i i i
From Equations (2.73) through (2.78), it readily can be seen that for the mixing properties
(2.74)
(2.75)
(2.76)
(2.77)
(2.78)
21
the following expressions hold:
Amixgm = Vim d  £ x i V i * = 0
Amixgm = g i d  ~ x i g i * : 0
Amix//m : g ~  Z x i g i * : 0
AmixSm : Sim d _ Z x i S i * : RZxi lnx i i
id • AmixAm = Am  Z x t A i = R T ~ xilnx i
i
Amixam = Gim d  ~ x i G i * = R T Z x i l n x i i
(2.79)
(2.80)
(2.81)
(2.82)
(2.83)
(2.84)
Again an important aspect in this definition is the choice of the independent variables. From Equation (2.72) two different definitions of excess functions can be obtained:
ME(T,V,n) = M(T,V,n)  Mid(T,V,n) (2.85)
ME(T,p,n)  M(T,p,n)  Mid(T, pV, n) (2.86)
The two approaches are related by
MZm(T,V,n) E ~ OMimd) dp (2.87) = Mm(T'p'n) + Op r,~
Pr
Abbott and Nass (4) pointed out that in this case p is the pressure for which the molar volume of the ideal solution is the same as that of the real solution at given temperature and composition. This pressure is obtained by solving
Vm(T,p,x ) = ~ xiVi*(T~Pr) (2.88) i
A required feature of the model is that Equation (2.88) can be solved for the pressure. For Equation (2.87), Abbott and Nass (4) proposed an approximate relation suitable for practical purposes:
= Mm(T'p'n) + (2.89) Op r,~ Z x i K i V i *
i
In Equation (2.89), K i X is the isothermal compressibility of pure i:
22
, 1 ~:i :  , (2.90)
V i O P ) T
Based on Equation (2.89), Abbott and Nass (4) summarize expressions for the various thermodynamic properties.
Excess functions and residual functions are related. From Equations (2.33) and (2.86), it can be seen that the following equality holds:
ME(T, p, n) = M~(T, p, n) {)14¢a(T, p, n) Mag(T, p, n)} (2.91)
Considering Equations (2.48) to (2.52) and (2.74) to (2.78), one can see readily that for an arbitrary extensive thermodynamic property the following relation holds:
ig (2.92)
2.6 Partial Molar Properties
A partial molar property M~ of an arbitrary extensive thermodynamic property M = M(T, p,
n 1, n2.... ) is defined by the equation: Partial molar properties give information about the change of the total property due to addition of
(2.93) T,P,~i
an infinitely small amount of species i to the mixture. From Equation (2.17) it becomes apparent that, by definition, the chemical potential is the partial molar Gibbs energy, i.e., gi = Gr For this arbitrary thermodynamic property M the following expressions can be derived (2):
d M = d p + d Z + E Mi dn i T,n p ,n i
where M = ~2 n tM i, or equivalently, i
(2.94)
M m  ~ x M i (2.95) i
From Equations (2.94) and (2.95) the generalized GibbsDuhem equation is readily obtained:
dp + d T  ~ n = T,n p ,n i
(2.96)
23
In the case, when M = G, Equation (2.12) is recovered.
2.7 Fugacity and Fugacity Coefficients
The fugacity of a real fluid mixture with constant composition and at constant temperature is defined by the equation:
d G = R T dlnf (2.97)
Combination of this equation with its ideal gas equivalent leads to:
d ( G  6 ¢g) = dG R = R T d l n ( f ) = RTdln~0 P
(2.98)
In this equation G R is the residual Gibbs energy and q~ is the fugacity coefficient. Integration of Equation (2.98) yields:
G R = R T lntp (2.99)
Comparison of Equation (2.97) and its equivalent in Table 2.4 leads to: P
ln~0 : f(z 1)d f (2.100) 0 P
This expression shows that an equation of state can be used to evaluate the fugacity coefficient. Similar expressions hold for an arbitrary pure component i:
dG~ = R T dlnf
and: R
G i : R T ln(,0i
In a real solution for species i the defining equation reads:
d G i : R T dlnj~.
From the definition of the residual Gibbs energy G R = G G ~g, it follows that: R _ G~g
G i = G i
and for an ideal gas at constant temperature it holds that:
~i  G~ g + R T lny i,
From Equations (2.103) through (2.105) the following expression results:
(2.101)
(2.102)
(2.103)
(2.104)
(2.105)
t ~ ) dln~. d ( a i  a ] g) = d a ? = R r dl ~i ~ = R r (2.106)
24
Note that from Equation (2.106) the relationship between the fugacity f. and the
fugacitycoefficient ~i in the mixture for component i follows:
~ .  f" (2.107) YiP
Integration of Equation (2.106) leads to a similar expression as represented by Equation (2.102):
Gi ~ = RT lndi (2.108)
For M= G, Equation (2.93) can be rewritten into:
(2.109)
Substitution of Equations (2.102) and (2.108) into Equation (2.109) gives:
ln~i = ( O(ln~)] On i r ,p ,n , i
(2.110)
Since ln~i is related to ln(p i as a partial molar property, from Equation (2.95) the following relation holds:
ln~ = ~ x iln(i (2.111) i
At constant temperature and pressure, the following GibbsDuhem equation can be obtained from Equation (2.96):
Sxid(]n~i) = 0 (2.112) i
Fugacities and/or fugacitycoefficients in mixtures are easily evaluated from all kinds of equationofstate models. In case the equation of state has pressure and temperature as the independent variables, the following, generally valid, relationship can be applied:
RT ln~/  RT 1 t ~ / i = (Vi  pRT)dp (2.113)
In this equation the partial :molar volume V~ is to be evaluated from the equation of state and Equation (2.93). For a pure substance the partial molar volume V~ becomes equivalent to the molar volume Vi* and Equation (2.113) simplifies to Equation (2.100). Most equationofstate models have temperature and volume as the independent variables. In that case the following, generally valid, expression can be used for evaluation of the fugacity and/or fugacitycoefficient in mixtures at constant temperature and composition:
25
RT. ~ ] d V  R T lnZ (2.114)
V
In this equation Z = p V / n R T i s the compressibility factor of the mixture; the partial derivative in Equation (2.114) can be obtained from the equation of state under consideration. For a pure substance Equation (2.114) reduces to:
oo
_~~ RT_ RTln~o i = R T l n f = f [  ~ ] d V  R T l n Z + R T ( Z  1) (2.115)
P v "
For an extensive treatment of the fugacity concept, one is referred elsewhere (1,2,6).
2.8 Activity Coefficients
Although activity coefficients, in general, are most conveniently evaluated from models which are specifically designed for the condensed phase only, this section demonstrates how the concept of activity is related to a similar formalism introduced for the fugacity concept. Equation (2.102) in Section 2.7 relates the residual Gibbs energy and the fugacity. In a similar way the excess Gibbs energy (Section 2.5) is related to the activitycoefficient formalism, which may be useful in describing the nonideality of a condensed phase.
From Equation (2.86) the following equation is readily obtained by applying Equation (2.93):
E i d G i = G i  G i (2.116)
Integration of Equation (2.102) at constant temperature and pressure from the pure state of ,
component i, where G i  G i and f. = f. , to an arbitrary composition in the solution
yields:
Gi  Gi * = R 7 1 ~ ~ i ) (2.117)
Application of Equation (2.93) to Equation (2.78) gives:
i d • G i = G i + R T lnx i (2.118)
Consequently, from the latter two equations the following result is obtained:
G i  G i = R T 1 = R T In ~i (2.119)
where, by definition, 3',.  f , . / (x~) is called the activity coefficient of component i in the solution.
In a similar way, as fugacity coefficients are related to the residual Gibbs energy, identical expressions for the activity coefficient can be related to the excess Gibbs energy. For instance, the partial molar excess Gibbs energy of species i is related to the activity coefficient by:
26
G~ = RT lnYi (2.120)
Since G ~ is the partial molar property of G E , consequently, lnyi is also a partial molar property of G, the following two relations of general validity can be derived:
G E = R T ~_, xi ln~ ~ (2.121) i
RTlny, =(O(nGE)) O n i T,P,~ i
(2.122)
Additionally, at constant temperature and pressure, from Equation (2.96) it follows that:
E xtd(ln'Yi) = 0 ( 2 . 1 2 3 ) i
For further details one is referred elsewhere (13,5,6).
2.9 The Phase Rule
For a system consisting of C components and P phases in equilibrium the number of intensive variables required to specify the state of the system (that is the number of degrees of freedom F) is given by the Gibbs phase rule:
F  C  P + 2  R (2.124)
In this equation R represents the number of restrictions imposed on the system. While the value for isothermal, isobaric or isochoric changes is obvious, the restrictions imposed by chemical reactions are often more subtle. For example, liquid water will exist as a mixture of H20, H +, H3 O+, and OH but, if C is taken as 4, then the requirements of electroneutrality and the ionic equilibria lead to R = 3 and the system still behaves, quite correctly, as a pure component. Two restrictions are particularly important when studying phases in equilibrium. IfPa~ phases have the same composition, then there are (Paz  1) phase boundaries across which the (C1) compositions must be the same and the additional restriction is:
R = (Paz  1)(C  1) (2.125)
and the number of degrees of freedom is: F  C(2  Paz) +1 (2.126)
This makes it clear that 3 phases can have the same composition only in the special case of a pure component, i.e., at its triple point. If Paz = 2, then F =1, and an azeotropic line always results irrespective of the number of components.
The second special case applies to the critical state. Here the Pc phases that become identical at the critical state are considered separately from the Pnc phases that behave normally. In this case, the additional restriction is R = 2P c  1, and the number of degrees of freedom becomes:
27
F = C  Pnc + 2  (2P c  1) (2.127)
Consequently, a critical point (F = 0) is unique for a pure component while, for a binary mixture, a critical line (F = 1) is the simplest case and critical endpoints (P,~ = 1, Pn~ = 2) are unique points. We note also that since F cannot be negative:
C ~ 2P c  3 (2.128)
As a consequence, a tricritical point cannot exist in a binary mixture. The combination of the restrictions for an azeotrope and a critical state shows that it is not possible for two azeotropic phases to become identical in a critical state, i.e., the critical and the azeotropic composition must be different. The restrictions that follow from the phase rule are simple consequences of geometry and are useful because they reduce the number of variables that must be used to describe the state of a system: small values o f f should result in simple equations of state. However, the phase rule gives no guidance as to which variables should be chosen.
2.10 Equilibrium Conditions
For a closed system with an arbitrary number of components and phases in which the temperature and pressure are uniform, the following combined statement of the first and second law is applicable (2,6):
dU t + p d V t  T d S t <_ 0 (2.129)
In this equation t refers to the total value of the various properties. The inequality symbol applies for infinite small changes between nonequilibrium states and the equality symbol holds for infinite small changes between equilibrium states, i .e., reversible processes.
For practical purposes, from Equation (2.129), the following alternative equation in terms of the Gibbs energy can be obtained:
dG t + S t d T  Vtdp <_ 0 (2.130)
Since in this equation temperature and pressure, which are easily controlled in experiments, are the independent variables, Equation (2.130) is the most useful representation of Equation (2.129). At constant temperature and pressure, Equation (2.130) reduces to:
(dat)p,T < 0 (2.131)
This equation states that at constant temperature and pressure any irreversible process proceeds in such a direction that the total Gibbs energy of a closed system will decrease. At equilibrium the Gibbs energy has reached a minimum value for the given temperature and pressure.
2.10.1 Phase Equilibria
For a closed multicomponent system, Equation (2.131) can be used to derive the equilibrium conditions between two or more phases in a system at constant temperature and pressure. If we indicate the various phases by ~, [3, T ..... n, and the various species by 1,2,3,....,C, the following
28
equilibrium conditions in terms of the chemical potential result:
a ~ 7 _ n ~t i = ~t = ~t i  . . . = ~t i, i = 1,2,3,. .... ,C (2.132)
Alternatively, it easily can be shown (2,6) that phase equilibrium can also be defined in terms of the fugacity:
f~/ = f f = f7 = = ~ 1,2,3 ..,C (2.133) • ° ° / ' , o , ,
For a simple vaporliquid equilibrium Equation (2.133) reads:
fv =~!, i = 1,2,3,. .... ,C (2.134)
Substitution of Equations (2.107) and (2.119) into this equilibrium condition gives:
" v 1 Yi~i P  xiT;f, i = 1,2,3,. .... ,C (2.135)
This formalism is the gammaphi approach for calculating vaporliquid equilibria. The fugacity coefficient (/9 i of each component that accotmts for the nonideality of the vapor phase can be evaluated from an equation of state model, while the activity coefficient Yi to describethe nonideal behavior of the liquid phase can be obtained from an excess Gibbs energy model.
The fugacity~ of pure species i can be obtained from the relation (2,6,7):
RT 1 = f V.dp (2. 1 36) f/)
In case the temperature is appreciably below critical and for not too high pressures, Equation (2.136) can be approximated by:
inf. _ V](p  Pi ) (2.137) f" Rr
Substitution of f,.  ~iPi gives:
ft . l = ~oi*P;exp[ V~(p  Pi )] (2.138) RT
In the latter expression the exponential is the Poynting factor. The contribution of this term
becomes significant only at higher pressures. For an ideal vapor phase (q~i= 1) , the liquid phase
is an ideal solution (Yi = 1) and if the Poynting factor does not contribute (low pressures), Equation (2.135) reduces to Raoult's law.
29
Equations of state, in principle, are able to describe the vapor and liquid phase ~V ~1
simultaneously, i.e., both the (j9 i and (/9 i c a n be evaluated from an equation of state model. Substitution of Equation (2.107) for both the vapor and liquid phase into Equation (2.134) leads to the phase equilibrium conditions:
"v " l yiqgi = xiq~ i , i = 1,2,3,. .... ,C (2.139)
Details of the various approaches to model vaporliquid equilibria from equations of state can be found elsewhere (2,3,59).
2.10.2 Chemical Equilibria
Application of Equation (2.131), the general criterion for chemical equilibrium, is conveniently expressed in terms of the chemical potential of each species present in the equilibrium mixture:
E Vi~i  0 , i = 1,2,3 ...... ,C (2.140) i
In this equation v~ are the stoichiometric coefficients, which for products are taken with a positive sign and for reactants are taken with a negative sign. At constant temperature and composition, Equation (2.103) reads:
d ~ i   d a i  R T d l R j ~ i (2.141)
Integration of this equation from a standard state of pure species i to its actual state in solution gives:
0 gi g~ + R T In ~ = g i + R T In d i (2.142)
In this equation d i is the activity of component i in the mixture. Substitution of this expression
into the condition for chemical equilibrium gives the important relation:
/•. 0  Vi~ i
Il(ai) v'  exp (2.143) i R T
Since the righthand side of Equation (2.143) is a function of temperature only, this term can be written as:
0 Ar GO  E vi~ti   R T In K(T) (2.144)
i
Equation (2.144) defines the thermodynamic equilibrium constant, which is a function of temperature only. From Equations (2.143) and (2.144) it can be seen how the equilibrium constant is related to the activities of the various species in the mixture. Since the activities of the reacting
30
species are related to their fugacities, equations of state can be used for their evaluation. Further details on chemical equilibria can be found elsewhere (1,3,7).
2.11 STABILITY AND THE CRITICAL STATE
2.11.1 Densities and Fields
Griffiths and Wheeler (11) divided thermodynamic properties into two classes: fields f(for example T, p, and ~t) that must be uniform throughout a system at equilibrium; and densities 9 (for example S, V, and n) which, in general, are discontinuous across a phase boundary although they are uniform throughout each phase. With this nomenclature, the fundamental equation (2.1) for the second law of thermodynamics can be written, very compactly, in the form:
dU : ~ f j d p j (2.145) J
where the density U(S, V, n 1, .., nc) is the thermodynamic surface and it should be noted that the hydrostatic field is p rather than p. Equation (2.145) shows that conjugate densities and fields are related by
fj = (OU/Ogj)Oj o (2.146)
where the subscript bj indicates that all the densities except 9j are held constant. Griffiths and
Wheeler used an equivalent definition
9j :  (0fo/0fj)fj (2.147)
where f0 is a thermodynamic potential.
2.11.2 Stability
Just as the fields, which are the first derivatives (OU/09), characterise equilibrium, the curvature of the thermodynamics surface, which depends on the second derivatives (0 zU/09Jogk), determines the stability of the system. The stability determinant for a system with C components may be written as
D(S,V,n 1, ..., riG_l) =
" "
Uvs Uvv Urn, ""
U.1S U.1V Unlll 1 ""
i i i "..
(2.148)
31
where the elements of the determinant are given by
/ Pj, Ok n l , "", nc
'x oZu with S, V, (2.149)
In a stable system with C components, the thermodynamic surface U(S, V, nl,..., nc_l) lies above its tangent plane and has positive curvature and, consequently, all the (C + 1) determinants D(S), D(S, V), D(S, V, nO,..., D(S, V, n 1, ..., nc_ 0 are positive. Furthermore, since the variables may be chosen in any order, many more determinants may be formed and they are all positive. However, with C components there are only (C + 1) independent variables and a set of (C + 1) determinants is sufficient to establish the conditions for stability. For example, the system is stable provided
 _> 0 V, n CV (2.150)
D(S,V) D(S'IOD(S): D(S)(Op) D(S) 1   = > 0
D(S) ~ T, n VKT (2.151)
D(S,V,n,)DS (O~t) = >_ 0 (2.152) D(S,V, nl) = ~(~~ ( , V ) : D(S,V) ~ r,p, nl
and so on. The ratios of the determinants are obtained from
D(pl,",pj ) _ [ Ofj.] (2.153)
D(p 1 , ..., lgj_ 1) Opj ) fi<j, Ok>g
which was derived by Gibbs (12).
2.11.3 Critical State
The critical state is the limit of stability at which all the determinants that were positive in Section 2.11.2 become zero. However, in the usual case where a transition between two phases is terminated, the critical state imposes only 3 additional restrictions, irrespective of the number of components. Similarly, although all the discontinuities in the densities vanish because the phases become identical, it is sufficient to consider the behaviour of the system with respect to a single density and to formulate the restrictions in terms of higherorder derivatives
2 2 3 4 (0 g/ooj )O j = 0"~ (03 g/opj )~j = O; (o4 g/opj )Oj > 0; (2.154)
The conditions are often defined in other thermodynamic surfaces where the variables more closely match an equation of state or the experimental conditions. For example, a gasliquid critical point in a pure fluid is usually defined by
32
((~p/O V)T,n = O; ((]2p/(~ V2)T,n : 0"~ (03p/O V3)T,n < 0 (2.155)
which may be written in terms of the A(T, V, n) surface as where the notation introduced in Equation (2.149) has been extended in an obvious way.
/ Azv = ~  ~ ) = ~  ~ ) = 0; A4v =  ~ ] V , n > 0 (2.156)
Temperature is assumed to be uniform and constant for both Equations (2.155) and (2.156). However, if the H(S, p, n) surface is used, then an entirely equivalent set of conditions is obtained
H2S :   ~ )  ' ~ ) : O; n4s =  ' ~ ) p , n > 0 (2.157)
but now pressure is assumed to be uniform and constant. Equation (2.153) shows the relation between these surfaces and U(S, V,n), since A2v = D(S, V)/D(S) and Hzs = D( V,S)/D( V). Conditions equivalent to Equations (2.156) or (2.157) are obtained from the U(S,V,n) surface and Equation (2.154) with pj = V or S. While Equation (2.155) is more familiar through its use of (p,V,T) variables and association with gasliquid critical points in pure fluids, each set of conditions can be used to describe the same critical state. For example Equation (2.157) or pj = S in Equation (2.154) might be very appropriate for a calorimetric study of a gasliquid critical point.
The experimental conditions of a critical state in a binary mixture are closely matched by the Gibbs energy G(T, p, n~, n2) and the relation with the U(S, V, n 1, n2) surface is established with Equation (2.153) in the form
D(S,V, nl)/D(S,V) = (O~]l/Onl)T,p,n2 (2.158)
which leads to the following conditions for the critical state: 2 2 3
(O~tl/Onl)T,p, n2 = O; (0 ~tl/Onl)T,p, n2 = O; (03~l/Onl)T,p, n2 >_ 0 (2.159)
The GibbsDuhem Equation (2.18) allows these conditions to be expressed (1,10) in terms of the molar Gibbs energy G m and a mole fraction x
Gzx = (c~2G/(]xz)T,p = 0"~ G3x = (03G/Ox3)v,p = 0; G4x = (04G/Ox4)T,p ~ 0 (2.160)
Since most equations of state have temperature, molar volume, and composition as independent variables, while the Gibbs energy is explicit in temperature, pressure, and composition; a formulation of the critical conditions in terms of the Helmholtz energy is required. The following equations allow a transformation between G(T, p, x) and A(T, V, x) (1,10):
Gzx = Azx  (Avx)Z/Azv (2.161)
G3x = A3x  3Avzx(Avx/Azv) + 3Azvx(Avx/Azv) 2  A3v(Avx/A2v ) (2.162)
The Helmholtz energy A(T, V, x) and the derivatives required for Equations (2.161) and (2.162) may be obtained from any Equation of state that gives the pressure through
33
, o A(V,T,x) : (1  x)A((V°,T) + xA 2 (V ,T) +
V RT{(1  x)ln(1  x) + xlnx}  fpdV
V °
(2.163)
In this equation AI*(V°,T) and Az(V°,T) are the molar Helmholtz energies of the pure
components and V ° is a reference volume. Multicomponent systems are handled in a similar way. For example, a ternary mixture can
be described in terms of four variables and Equation (2.153) gives
D(S, V, nl,n2)/D(S, V, nl) = (O].t2/On2)T,p,gl,n 3 (2.164)
and the conditions for a critical state are therefore 2 2
(C~tZ/(~nZ)T,p,~,,n 3 = O; (0 ~l,2/OnZ)T,p,p,,n 3 = O; (03].t2/On23)T,p,~tl,n3 > 0 (2.165)
The defining equations for higherorder critical points are straightforward in terms of the Gibbs energy and the composition variables. For instance, for a tricritical point in a (pseudo) binary mixture the following 2P c  1 = 5 conditions have to be satisfied (1,10):
G2x : O; G3x : O; Gax : O; Gsx : O; G6x > 0 (2.166)
For higherorder critical points, the transformation equations from the Gibbs into the Helmholtz energy become extremely complex.
Acknowledgements
The authors are indebted to J.M.H. Levelt Sengers for numerous suggestions to improve the manuscript. The support of M.M. Abbott is also gratefully acknowledged.
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6. J.M. Prausnitz, R.N. Liehtenthaler, and E. Gomes de Azevedo, Molecular Thermodynamics of FluidPhase Equilibria, 2nd ed., Prentice Hall, Englewood Cliffs (1986).
7. J.M. Smith, H.C. Van Ness and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 5th ed., McGrawHill, New York (1996).
8. S. Malanowski and A. Anderko, Modelling Phase Equilibria, Wiley, New York (1992). 9. A. Anderko, Fluid Phase Equilib. 61, 145 (1990).
10. J.S. Rowlinson and F.L. Swinton, Liquids and Liquid Mixtures, 3rd ed., Butterworth, London (1982).
11. R.B. Griffiths and J.C. Wheeler, Phys. Rev. A 2, 1047 (1970). 12. J.W. Gibbs, Collected Works, Vol. 1, Yale University Press, New Haven (1928,1948)
Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 35
3 THE VIRIAL EQUATION OF STATE
J. P. M. Trusler
Department of Chemical Engineering and Chemical Technology Imperial College of Science, Technology and Medicine London SW7 2BY, United Kingdom
3.1 Introduction 3.1.1 The Volumetric Behaviour of Real Fluids 3.1.2 The Virial Equation of State 3.1.3 Temperature Dependence of the Virial Coefficients 3.1.4 Composition Dependence of the Virial Coefficients 3.1.5 The Pressure Series 3.1.6 Convergence of the Virial Series
3.2 Thermodynamic Properties of Gases 3.2.1 Perfectgas and Residual Properties 3.2.2 Helmholtz Energy and Gibbs Energy 3.2.3 PerfectGas Properties 3.2.4 Residual Properties
3.3 Theory 3.3.1 Properties of the Grand Canonical Partition Function. 3.3.2 The Virial Equation of State from the Grand Partition Function. 3.3.3 The Virial Coefficients in Classical Mechanics 3.3.4 Quantum Corrections 3.3.5 The Virial Coefficients in Quantum Mechanics: Helium and Hydrogen
3.4 Calculation and Estimation of Virial Coefficients 3.4.1 The Virial Coefficients of Model Systems 3.4.2 Estimation of Virial Coefficients
3.5 Summary References
36
3.1 I N T R O D U C T I O N
3.1.1 The Volumetric Behaviour of Real Fluids
The virial equation of state is a relation between pressure p, temperature T and amountof substance density Pn which may be applied to real gases. For a perfect gas, these quantities are related by the well known equation
p = p,,RT (3.1)
or simply
Z = 1 (3.2)
where R is the universal gas constant and Z =p/PnRT is called the compression factor. For real fluids, the compression factor is found to behave in a rather more complicated
way than Equation (3.1) predicts. Such behaviour is illustrated in Figure 3.1 where experimentally determined values of Z are plotted as a function of Pn along a number of isotherms for fluid methane (1,2). At temperatures above the critical, the isotherms are smooth continuous functions of density but, at subcritical temperatures, they are split into two branches which each terminate on the vapourliquid coexistence curve. At low densities, the isotherms converge in accordance with the experimentally proven result that
1.2
1.0
0.8
N 0 . 6
0.4
0.2
0.0
, , , / /
_
 ~ ~
[ I I E
0 5 10 15 20 25
Pn (m01"dm3)
Figure 3.1 Compression factors of fluid methane as a function of density (1,2).  . . . . . . , saturation curve; isotherms calculated from the equation of state of Setzmann and Wagner (3); c, critical point. O, 160 K; O, 190.555 K; II, 200 K; m, 220 K; ~, , 273.15 K; A,, 373.15 K; 'IF, 573.15 K.
37
Z ( T , p n ff 0)  1 (3 .3)
for all nonreacting gases. This behaviour will be referred to as the perfectgas limit. At the other extreme of high densities, the isotherms adopt positive slopes and Z starts to increase rapidly as the fluid becomes nearly incompressible. It is also instructive to examine the same experimental isotherms as a function of pressure as in Figure 3.2. Here, one can see that both branches of a subcritical isotherm terminate on the coexistence curve at the saturated vapour pressure. The high compressibility of the fluid in the nearcritical state is also apparent.
1.0
0.8
0.6
0.4
0.2
0.0
' ' ' ' ' ' ' ' I
1 2 5 10 20
p (MPa)
Figure 3.2 Compression factors of fluid methane as a function of pressure (13). Symbols and curves as for figure 1 e x c e p t  , tie line.
3.1.2 The Virial Equation of State
The accurate representation of p(T,Pn ) over wide ranges of temperature and density is a
problem of great practical importance to which there is no simple solution. The virial equation of state
p = p .RT {1 + Bp. + Cp2. + Dp3 +...} (3.4)
is one of the many hundreds of equations which have been proposed in an attempt to obtain a more accurate description of the behaviour of real fluids than is given by Equation (3.1). This equation may be viewed as a MacLaurin series in which (p/RT) is expanded in powers of Pn about the perfectgas limit with densityindependent coefficients B, C, D ... known as virial
38
coefficients. Conventionally, B is called the second viral coefficient, C the third virial coefficient, D the fourth and so on. Since it is an experimentally proven fact that Z is a function of both T and Pn, one must expect B, C, D, ... to be functions of the temperature and, as Z(T , Pn ) is not generally the same function for all gases, the virial coefficients for mixtures must also be functions of the composition.
If the virial equation of state had no better foundation than the empirical arguments introduced above, it is doubtful that it would have received the attention that it has. Indeed, some modem empirical equations of state (see, e.g., reference 3) are of much wider applicability and their greater complexity is not usually a real disadvantage. However, the importance of the virial equation is that it has a rigorous theoretical foundation in statistical thermodynamics which provides exact analytic relations between the virial coefficients and the interactions between molecules in isolated clusters. One finds that B depends upon interactions between pairs of molecules, C upon interactions in a cluster of three molecules, D upon interactions in a cluster of four molecules, and so on.
3.1.3 Temperature Dependence of the Virial Coefficients
One of the significant advantages of the virial equation of state is the existence of a very large database of experimentally determined virial coefficients (4). The second virial coefficient is by far the most well studied but C is known with some accuracy for a number of fluids and a few values of D have been reported. In view of this situation, the temperature dependence of the virial coefficients will be discussed first in the light of the experimental results. Figure 3.3 shows values of the second and third virial coefficients of methane over a wide range of temperature. The critical temperature T c and the Boyle temperature 7 9 (at which B = 0) are also indicated in this figure. At high temperatures, B is positive and varies only slowly with temperature while, at lower temperatures, it becomes large and negative. Like B, C is a slowly varying positive quantity at high temperatures and a rapidly varying negative one at low temperatures; in between these extremes it passes through a maximum. The curves shown in Figure 3.3 were calculated from intermolecular pair and triplet potential energy functions determined for methane from speedofsound measurements (5). The behaviour shown in Figure 3.3 is typical of almost all gases although, of course, the magnitude of the virial coefficients and the characteristic temperatures such as 7 9 vary from substance to substance.
The temperature dependence of the higher virial coefficients is not well established experimentally but theoretical evaluations for model systems (see Section 3.3.6) indicate that these virial coefficients too are positive at high temperatures with a rapid divergence towards large negative values at temperatures below the critical (6).
3.1.4 Composition Dependence of the Virial Coefficients
For a pure gas, the phase rule requires that the pressure be a function of the two variables T and Pn only. Similarly, in the case of a mixture composed of v components, with amounts of substance n 1, n 2, n 3 ... n v present in volume V, the pressure must be a function of the (v+ 1)
variables T, nl/V, n2/V, n3/V.., nv /V and an expansion ofp, at constant temperature, in powers o f n l / V , n2/V , n3/V.., nv /Vshould exist. Carrying out this expansion, one again arrives at the
39
50 O
E co
E 100 O 
150
T c T B
   
200 0 600
I I I I
200 300 400 500
T (K)
5000
4000
3000
2000
1000
O
E to
E O
v
O
Figure 3.3 Second and third virial coefficients of methane (1,2). O, II B ; O, r! c. Curves are calculated from intermolecular pair and triplet potentials determined from speed of sound measurements (5).
virial equation of state with Pn =(2iVlni)/V but with virial coefficients that depend upon
composition as follows:
i=1 j   1
Cmix =£££XiXjXkC~ik i=1 j = l k = l
Omix "~£~£XiXjXkXlOijkl i = l j = l k = l /=1
etc.
(3.5)
Here, x, = n,/(E[=lni) denotes the mole fraction of substance s in the mixture and Bg, Co.k,
D/jkl, .. denote a set of virial coefficients which depend only on T. Clearly, Bss, Csss, Dssss ... are the virial coefficients of pure s. The remaining coefficients are known as interaction virial coefficients and are related by the theory to interactions within clusters of dissimilar molecules• It is notable however that the composition dependence of the mixture virial coefficients (quadratic, cubic, quartic, etc. in the mole fractions) was arrived at without
4O
recourse to molecular theory. Interaction second virial coefficients B/j have been measured for many systems and the results collated by Dymond and Smith (4) but there are few experimental results for the higherorder terms.
3.1.5 The Pressure Series
In both experimental work and practical applications it is sometimes convenient to use (T, p) as the independent variables in place of (T, Pn)" For this purpose, an expansion of Z in p is used which may be written as:
Z = 1 + B ' p + C'p 2 + D ' p 3 +. . . (3.6)
The coefficients of this series are uniquely related to the virial coefficients as may be shown by eliminating p from the fight hand side of Equation (3.6) using Equation (3.4) and collecting terms. The first few such relations are given in Table 3.1. The composition dependence of the coefficients B', C', D' .. in a mixture may be determined by combining these relations with Equations (3.5).
Table 3.1 Relations between coefficients in the density and pressureexplicit expansions of Z.
Pressure Series Density Series
B ' = B / R T B = R T B '
c ' = ( c  , ~ ) / ( R r ) ~
D' = ( D  3BC + 2B 3) / (RT) 3
c = (RT)~(C' +/~ '~ )
D : ( R T ) 3 ( D ' +3B 'C ' + B '3)
3.1.6 Convergence of the Virial Series
It is important to recognize that the virial series might not converge for all experimentally realisable densities. The true radius of convergence is unknown in general and the experimental evidence is somewhat ambiguous. It is certainly possible to fit experimental compressionfactor isotherms that extend over a wide range of densities to a truncated form of Equation (3.4). At supercritical temperatures, such representations appear to be satisfactory at densities of up to two or three times the critical (see, e.g., reference 7) while, at subcritical temperatures, good results may be obtained up to the density of the saturated vapour. However, it is often argued that the coefficients obtained in this way are not the true virial coefficients (8); indeed, the value of B determined in a fit to highdensity data may differ noticeably from that obtained by analysis of precise lowdensity data (9). Whether or not such differences should be attributed solely to experimental uncertainties, or to the limitations of Equation (3.4) itself, is really not clear. On the other hand, it is known that the equation of
41
state is nonanalytic at the critical point and it is therefore certain that some fluid states exist in the neighbourhood of the critical point at which the virial series fails to converge. Some theoretical results pertaining to the issue of convergence will be mentioned later in connection with model systems (see Section 3.3.6).
Whatever the ultimate radius of convergence might be, the question of real practical importance is to establish the region in which the series is rapidly convergent. Often, the virial series is truncated after the term in C; we are then interested in the associated truncation error. Provided that sufficiently accurate values of B and C are available, compression factors may be obtained with an accuracy of order 10 4 at supercritical temperatures for densities up to roughly one sixth of the critical and the truncation error may be expected to rise to 10 2 at about three quarters of the critical density. These estimates of truncation error are based largely on theoretical expectations of the magnitude of D and higher virial coefficients (6). The predicted divergence of these coefficients at low temperatures suggests that there might be a range of temperatures around the critical in which somewhat larger truncation errors occur. At lower temperatures still, the rapid decline in the saturated vapour density is such that the series converges rapidly for all accessible states of the gas.
3.2 THERMODYNAMIC PROPERTIES OF GASES
In this section, a general prescription is presented by means of which the thermodynamic properties of a fluid may be expressed in terms of an equation of state. Explicit results are presented for the virial equation of state.
3.2.1 Perfectgas and Residual Properties
It is often convenient to express the thermodynamic properties of a fluid as the sum of a perfectgas term and a residual term (10). The utility of this separation lies first in the availability of widelyapplicable theory for the prediction, estimation and correlation of perfectgas properties (11); and second in the dearth of such theory to account exactly for the effects of molecular interactions with the consequent need for some measure of empiricism.
For generality, consider a mixture of v components characterised by amounts of substance nl, n2, n 3 ... n v (denoted by the vector nV), temperature T, pressure p and volume V. If (T,V,n") are taken as the independent variables, a thermodynamic property X(T,V,n") may be written in the form
X(T,V,n v)= xpg(T,V,n v)+ xres(T,V,n v) (3.7)
where xPg(T,V,n v) denotes the property of a hypothetical perfect gas with the specified values of (T,V,n v) and where xres(T,V,n v) is the residual term. A similar decomposition may be applied with (T,p,n v) as the independent variables
X(T,p,n")= xPg(T,p,n")+ xres(T,p,n") (3.8)
but one should note that, although X(T,V,n")= X(T,p,n") , the perfectgas terms and the residual terms in Equations (3.7) and (3.8) generally differ. This is a consequence of the fact that, while (T,V,p,n") characterise the state of the real fluid, the state of the hypothetical
42
perfect gas depends upon whether (T,V,n ~) or (T,p,n ~) are specified. The difference between xPg(T,p,n") and xPg(T,V,n ") may be obtained by noting that the volume of the
v hypothetical perfect gas is nRT/p in the one case, where n = ~i=l ni, but V in the other so that
XPg(T,V, nV)= xPg(T,p, nV)+ (¢~X pg/c:~V)T,nvdV RT/p
(3.9)
and
~n~" ( O X p g / 0 V)T,,,~ dV XreS(T'V'nV)= xres(T'p'nV) R~'/p (3.10)
In the case of properties for which the perfectgas term depends only on (T,n"), one then has XPg(T,V, nV)= xPg(T,p, nV).
3.2.2 Helmholtz Energy and Gibbs Energy
All of the thermodynamic properties of a homogeneous phase may be obtained from the Helmholtz energy or from the Gibbs energy, as discussed in Chapter 2. When (T,V,n v) are the independent variables, the Helmholtz energy A(T,V,n v) is the appropriate choice and the fundamental thermodynamic equation for the phase is
dA = SdT  pdV + £~Aidn i (3.11) i=1
where S is the entropy and Pi is the chemical potential of component i. The partial derivatives of A with respect to T, V or n i then give S, p or/A i respectively. Once these quantities are obtained, the other state functions such as enthalpy H, energy U and Gibbs energy G follow from the appropriate combinations of A, TS and pV. Quantities such as heat capacity and compressibility may be obtained from second derivatives of the Helmholtz energy combined, where necessary, with the Maxwell relations (12).
When (T,p,n v) are the independent variables, the properties of a homogeneous phase are best obtained from the Gibbs energy G(T, p, n v) and the fundamental equation in the form
dG =SdT + Vdp + £]Aidn i (3.12) i=1
S, V and ~i are then obtained from the firstorder partial derivatives, other state functions from combinations of G, TS and p V, and the remaining properties from secondorder partial derivatives of G.
Equations (3.11) and (3.12) are, of course, quite general and apply to any homogeneous phase. In the following sections, these relations are applied to obtain, first, expressions for the properties of the perfect gas and, second, expressions for the residual properties of a real gas in terms of either the virial coefficients or the coefficients of Equation (3.6).
3.2.3 PerfectGas Properties
43
The molar Helmholtz energy A~ g = APg/n of a pure perfect gas is a function only of (T, Pn ) and may conveniently be written
T Ipn(c~Apng/~;~Pn)TdPn A~g(T,p,) = A~ + ITO ( ~ A ~ g / g T ) o d r + o: (3.13)
where T ° and p O are reference values of the temperature and density at which A~ g takes the value Am °. Am ° is often specified in terms of reference values of molar energy and molar entropy: Am ° =Um °  T°S°m . The integration with respect to temperature is best carried out after noting that
o (~A~ g / ~ T)pn "=  S Pm g =  o ( CPv ,gm / T ) d T  S m (3.14)
where C ~ is the isochoric perfectgas heat capacity and is a function only of temperature. V,rn For the integration over density, it is also convenient to note that
(~Am pg / ~Pn ) T  P / P ~  R T / P n (3.15)
Then, carrying out the integrations in Equation (3.13), one obtains the molar Helmholtz energy of the perfect gas as
o o
A~ g '  C'~,, g d T  T C~,, g d ln T + R T l n ( p . / p. ) + Um  TS ~ o V,m o V,m (3.16)
Provided that C ~ (T) is known and that T ° o o V,m , pn, U m and S m a re defined, all of the thermodynamic properties of the pure perfect gas may then be obtained through Equations (3.11) and (3.16). Typically in the construction of an equation of state, the perfectgas heat capacity is correlated by means of a suitable function which may then be integrated to obtain an analytic approximation to A~ g ( r ,Pn) .
For a mixture of v components, the molar Helmholtz energy is given by (11)
v v Affr~. ( T , P n , X V ) = Z x i A P g + R T Z x i l n x i (3.17)
i=1 i=1
where A/°g is the molar Helmholtz energy of pure i and x v denotes the set of mole fractions Xl, X 2, X 3 ... X v.
The principal perfectgas thermodynamic properties of pure substances and multi component mixtures determined from Equations (3.11), (3.16)and (3.17)with (T, Pn,X") as the independent variables are summarised in Table 3.2.
An equivalent set of equations to describe the properties of the perfect gas may also be obtained in terms of the Gibbs energy. Following arguments similar to those used to derive Equation (3.16), one may show that
44
GPm g : o Cl~gp,m d T  T o C~pg.md lnT + R T l n ( p / P°) + Hm  (3.18)
Here, Cp,~m = C~V,m q R is the isobaric perfectgas heat capacity and T °, p °, H m° and S m° are reference values of temperature, pressure, molar enthalpy and molar entropy. In order to make Equations (3.16) and (3.18) consistent, one must choose the same reference states; this requires that p° = RT°Pn ° and H°m = U°m + RT. Mixture properties may be obtained by means of the equation
v
G ~ ( T , p , x ~ ) = xi Gpg ] e Z 2 x i l n x i (3.19) i=1 i=1
The perfectgas thermodynamic properties determined from Equations (3.12), (3.18) and (3.19) with ( T , p , x v) as the independent variables are summarised in Table 3.3.
It should be emphasised that the relations in Tables 3.2 and 3.3 give identical values of the thermodynamic properties when the reference states are defined as above and when Pn and p are related by p = pnRT. However, as indicated in Section 3.2.2, when these relations are applied to obtain the perfectgas component of the thermodynamic properties of a real fluid, the results generally depend upon the choice of the independent variables. This completes the discussion of perfectgas properties.
Table 3.2 Thermodynamic properties of pure gases and gaseous mixtures with (T ,p , ,xu) as the independent variables.
* Here B, = T ( d B / d T ) , B2 = T 2 ( d 2 ~ l d T 2 ) etc. for a pure gas, and B = B,,,, , B, = T(dB,, I d T ) , etc. for a mixture.
P V1
Residual Properties
Pure gas or mixture*
= R T { B P , + f c p : + : D ~ : + . . . )
S , ' =  R { ( B + B l ) p n + f ( ~ + ~ l ) p ~ + ~ ( ~ + ~ l ) p ~ + )
U,"" = RT{BIp,, + iC ,p i + f DIP:+...)
CTm = R{(2B,+B2)p, + i (2Cl+C2)pf + j( 2Dl+D2)pi+...}
H;; = R T { ( B  ~ , ) p , , + (c:c,)pf + ( D  ~ D , ) ~ ; + . . . J
G,"" = RT{2Bp, + t C p : ++DP;+...)
paS = R T { ~ ( ~ X ~ B ~ , > P , + + ( C ~ X ~ ~ ~ C ~ ~ ) P S , I I j
+ ~ ( ~ ~ C X ~ X , X ~ D ~ ) P ~ )+...I i j k
PerfectGas
Pure gas
Xg = IT: C&dT  T IT: C;gmd ln T
+RT l r ~ ( ~ , / ~ ~ ) +u: TS:
S ; g = ~ : + & a g m d l n ~  ~ l n ( p n / d )
U:g = U. + IT: G a d T
C,":m = CyPgm ( T )
H:g = U. + RT + IT: G t d T
GF = AEg + RT
&PP p 3 * ~ g = G;
Properties
Mixture " U
&fX = ~ x i ~ ~ % R T ~ x , ~ n x , i=l !=I
" U
S : f x = ~ ~ i ~ ~ g  ~ ~ ~ , l n x i ,=I i=l
U
U::x = x x i v g i=l
U
C,"f;nix = C xiqg i (T) i=l
" H:~ = 1 xi H ~ P
,=I
" U
Gzx = C X ~ G P ~ + R T C X , l n x , ,=I ,=I
,us = ps*pg + RT In X ,
Table 3.3 Thermodynamic properties of pure gases and gaseous mixtures with ( T , p , x u ) as the independent variables. P m
* Here B: = T ( d B r / d T ) , Bl = T2(d2B' /d T 2 ) etc. for a pure gas, and B' = Bk, , B; = T(dBA,/d T ) , etc. for a mixture. p: is given in terms of
Residual Properties
Pure gas or mixture*
PerfectGas Properties
U u
4, = z x i ~ y g + ~ ~ z x i l n x i i=l i=I
M = p3*pg + RT lnx,
the interaction virial coefficients defined in Equation (3.5) with B = B, etc.
Pure gas
=  R T { ~ C p 2 + 5 D f p 3 + . . . )
psres = R T [ { ~ ~ x ~ B , ,  B)(pIRT) i
+ { + z x x i x j c i i ,  ~ B ~ x , B , + + B 2  C ) ( P / R T ) ~ i j i
+  3 ~ ~ ~ x ~ x ~ ~ ~ ~ i j k i j
Mixture
47
3.2.4 Residual Properties
All residual thermodynamic properties may be expressed in terms of an equation of state of the form p = p(T, Pn) or On = [Dn(T~P); Equation (3.4) is an example of the former and Equation (3.6) of the latter. The residual part of the Helmholtz energy for a phase of constant composition may be obtained by combining the identity
A~e=(T,p,,) = Io °" {(OAm/OP,)r(OAff/OP=)rIdp= (3.20)
in which (OAm/OP,)r =p/p~, with Equation (3.4) for the pressure. All other residual properties may then be derived by manipulation of the result. Table 3.2 gives expressions for A~ es and for the other five common residual thermodynamic functions S res, Ur~ es, C~m, H re=
and G re~ in terms of the virial coefficients and with (T,p,,x ~) as the independent variables.
Also given in Table 3.2 is the expansion of the residual part of the chemical potential/a s of component s in a multicomponent mixture. This is useful in a vapourliquid equilibrium calculation where the virial equation of state is used for the vapour phase.* Such calculations are usually carried out for conditions of constant temperature and pressure and, to evaluate the chemical potential for specified values of (T,p,x ~), one must first solve Equation (3.4) for Pn" The partial fugacity fs of component s in a mixture is often used in place of the chemical potential to determine the equilibrium condition between phases. This quantity may be defined by the relation
ln(f~ / p) = (/a= /a~ *pg) / RT (3.21)
where /a~*Pg is the perfectgas chemical potential of pure s at the temperature and pressure in
question. In turn, f~ may be conveniently expressed in terms of the dimensionless partial fugacity coefficient d?= =(f=/x=p) which, in view of Equations (3.19) and (3.21), is most closely related to the residual chemical potential at specified (T,p,x v):
Rrln~=(T, Pn,XV)= /are=(T,p,x") = r e s " T , X v , X v /a= t ,P~ )  RTlnZ(T, Pn )
(3.22)
The second part of Equation (3.22) may be obtained by using Equation (3.10). The residual part of the Gibbs energy may be obtained in a manner analogous to that used to obtain A~ es.
Table 3.3 gives the expansions of G res, S re=, H res, C~m, U re=, A re= and /a ~e= in terms of the
coefficients of Equation (3.6) with (T,p,x") as the independent variables. Other thermodynamic properties may be obtained through standard manipulations of the relations given in Tables 3.2 and 3.3.
* The chemical potential of components in the coexisting liquid cannot be obtained from the Virial equation but may be deduced from, for example, an activity coefficient model.
48
3.3 THEORY
In this section, the theoretical basis of the virial equation of state will be outlined and the relationships between the virial coefficients B, C, D, ... and intermolecular potential energies will be derived. Although the virial equation derives its name from the virial theorem of Clausius, only the more powerful statisticalmechanical development based on the grand canonical ensemble will be considered here (8,13,14). The development is carried out in both classical and quantum mechanics and the results are compared with experiment.
3.3.1 Properties of the Grand Canonical Partition Function
The ensemble methods of statistical mechanics have been discussed by many authors (11,13,15) and the well known results which form the starting point in the derivation of the virial equation are therefore reviewed here only briefly. The grand canonical ensemble is a hypothetical isolated assembly of some large number of systems of volume V each separated from its neighbours by walls which are permeable to both matter and energy. Since the entire ensemble is isolated it must be isothermal and each of its systems is therefore a replica on a macroscopic scale of a single open isothermal system with volume V, temperature T and absolute activity '~i for molecules of type i (~t, i "~" e~/Rr).
The number N i of molecules of type i in each system of the ensemble is not fixed but may fluctuate about mean values N i which, according to the postulates of statistical mechanics, are taken to be identical with the observable value of this property in the real system. The value of N i and of all the other thermodynamic properties of the open isothermal system may be obtained from a single function ~ called the grand canonical partition function.
For the general case of a mixture containing v components, the grand canonical partition function is given by
oo
Nl=O N2=O Nv=O c = l
(3.23)
The quantity 6)U,N~...Uv which appears as a coefficient in Equation (3.23) is the canonical
ensemble partition function for a closed system containing the specified numbers of molecules in volume V at temperature T. This quantity is given by
ON, N~...N" = ~_~ exp(Es/kT ) (3.24) J
where Ej is the energy of quantum state J and the summation is performed over all possible states of the system with the given volume and numbers of molecules.
The grand partition function is related to thermodynamic properties of the system by the equation
3= exp(pV/kT) (3.25)
This may be combined with the fundamental thermodynamic equation for the system with (T, V,~ v) as the independent variables (11)
49
d ( p V ) = S d T + p d V + R T ~ n~d ln3, i (3.26) i
to obtain all of the other thermodynamic properties. In particular, the amount of substance n i
of component i in the system is given by
L n i = N i = (OlnE/c31n3, i)T,V,~.j, i (3.27)
where L = R / k is Avogadro's constant. In the following sections, these relations are applied to derive the virial equation of state.
3.3.2 The Virial Equation of State from the Grand Partition Function
In principle, the thermodynamic properties of the system may be obtained directly from the grand partition function through Equations (3.23) and (3.25). Unfortunately, this requires computation of [~NlU2...U v for all possible values of N 1, N 2, .; this is an impossible task. A
practical alternative which leads to the virial equation is to develop ~ as a power series in some suitably chosen set of variables which are then eliminated in favour of the number densities N i / V . Several authors have given derivations of the virial series for onecomponent (8,11,13,15), twocomponent (16) and multicomponent systems (17,18). Here, the general case of the multicomponent system is considered in outline.
It is first convenient to make two definitions. An 'active number density' z i is defined for each component in the mixture by the equation
z i = ( O } ° / V ) ~ , i (3.28)
This quantity has the property that z i ~ N g / V as X / ~ 0. A configurational partition function
Q~S...) is also defined for the system when it contains a total of N molecules of types (id', "") by means of the relation
(3.29)
In these equations, the partition functions O and Q are labelled in the same notation; thus
O~ ° )  O u , N2...Uv, N = Ec \ lNc and (ij, ...) lists the type (species) of each molecule in turn. In
terms of these new variables, the grand partition function is given by
3 = 1+ ~Q(i)zi+ ~'(Q~O)/2,)zizj + "~'~~(Q}iJk)/3,)zizjzk+... i=l i=1 j=l i=1 j=l k=l
(3.30)
and takes the form of a power series in the z i in which the Nth group of terms pertains to a system of N1 molecules. It is then a s s u m e d that an analogous expansion ofp in powers of the z i exists:
50
p / k r  £ b(i)zi .[ £ £ b~O')zizj [ £ £ £ b~ijk)zizjzk .[ . . . i=1 i=1 j=l i=1 j=l k=l
(3.31)
The coefficients b~ j ) of this series may be related to the Q~J) by means of a simultaneous expansion of e x p ( p V / k T ) in powers of the z i and termbyterm comparison with Equation (3.30). The results of that operation are:
b(i) __ Q(i) /(I!V)= 1
b~U) : (Q~U)_ V 2)/(2!V)
b~iJ~) (Q~ij~) vQ~ij) VQ~ik) VQ~jk)+2V3 . . . . )/(3!V)
etc.
(3.32)
Incidentally, the quantities bff '), known as cluster integrals, which appear here play an important role in some formulations of statistical thermodynamics (15). They have the property that b~ j)~ 0 when any one or more members of the group of N molecules is remote from the others.
The final step in the derivation is to eliminate the z i from Equation (3.31) in favour of the number densities N i / V . This may be done by noting that N i / V is given according to Equations (3.27) and (3.28)by
N~IV = (zi lV)(aln~lOZi)r .v,zm
= (zilkT)(c9 p 10zi)r.v.=j.,, (3.33)
from which it follows that,
( v / N i / V = z i  b(i) + 2~'~ b~°)zj + 3~"~ b}O'k)zjzk +..
j=l j=l k=l
(3.34)
Equation (3.34) may be inverted to obtain a power series for z i in the number densities N i / V ,
Zi _ (~i/V)(a~i) q_ £a~iJ)(~i/V)_t_ ££a~ijk) (gjgk/V  2 )k.. "1 j=l j=l k=l
(3.35)
with coefficients:
a~ i) =1
a~O') :  2b~0) ._ ( ik ) ( ik ) b~ iJ) a~ Uk,  3bJ Uk, + 2b~U'(b~ °, + b~ik')+ 2b~ (b~ + )
etc.
(3.36)
51
Finally, combining Equation (3.31) with Equations (3.36) and collecting coefficients of (Ni /V) a, one obtains the virial equation of state with p, =(Y~iNi/LV) and with virial coefficients which are related to the cluster integrals as follows:
B U =Lb~ °) (ik) (jk) Co~ =  L 2 {2b~ 'jk)  41"~'(u)h(i~)3 ku2 '12 + "2~(~i)]'l(Jk)'2 + b~ 6 2 )}
etc.
(3.37)
These equations apply to any combination of like and unlike molecules and, by combination with Equations (3.32), the virial coefficients may be calculated from configurational partition functions. In the case of a onecomponent gas, the results simplify to B =  L b 2 and
C   L 2 (2b34b 2). General relationships have been derived for the onecomponent gas by which any virial coefficient may be expressed in terms of a set of cluster integrals and by which each cluster integral may be expressed in terms of configurational partition functions (14,19).
In the following sections, the evaluation of Q~J) will be considered in both classical and
quantum mechanics for N = 1, 2, 3, .... However, one can see already that, in making the virial expansion, the intractable Nbody problem posed by Equations (3.23) and (3.25) has been converted into a series of onebody, twobody, threebody .,. problems the first few of which one may expect to solve with relative ease.
3.3.3 The Virial Coefficients in Classical Mechanics
For almost all substances at accessible temperatures a classical treatment, augmented at low temperatures by small quantum 'corrections', may be used to obtain the configurational partition function and thus the virial coefficients with great accuracy. Only at low temperatures for the isotopes of hydrogen and helium is a fully quantummechanical treatment required. A fully classical treatment of Q~J) will therefore be considered first.
We begin by returning to Equation (3.24) for the canonical partition function. The energy levels over which the summation in that equation extends are those of the entire system and it is usual to make the assumption that they may be separated into two entirely independent sets. The first is associated with the internal degrees of freedom of the molecules and the second with the position, orientation and momentum of the molecules. With this assumption, which is amply justified for small rigid molecules, the partition function factorises into the product of two terms, the one (FN ~'j)) for the internal degrees of freedom and the other (0~u ~°) ) for the motion of the centres of mass:
@u(~ ) =.r(,Y)as(u)N =N (3.38)
By essentially the same assumption, ~7...) for N > 1 factorises into the product of N one molecule partition functions with one term for each molecule. The configurational partition function is then entirely independent of the internal degrees of freedom and is given by
52
= (I ( V l )'<. ! c=l
(3.39)
The centreofmass term is given in quantum mechanics by the sum
q~") = ~ e x p (  H j / k T ) (3.40) J
over all allowed (and distinguishable) translational quantum states of the Nmolecule system. The quantity Hj which appears in this expression is the total (kinetic + potential) energy of the N centres of mass when the system is in the quantum state labeled J. The classical limit of q ~ ) is obtained by replacing the discrete spectrum of translational states by a continuum and the summation by integrals over the position, orientation and momentum of the molecules (20):
( )iS 3 N v • h FI .c = ~c Arc! e x p (  H / kT) dP Ndr Ndo c = l
(3.41)
Here, H = 2~1(P i • P/2mi)+ U~ j''') is the classical equivalent of Hj, Pi is the momentum and
m i the mass of molecule i, U~ s ) is the total potential energy stored in the intermolecular
forces, and r N and o N denote the set of position and orientation vectors of the N molecules. In
addition, f2 c is a normalisation constant equal to the integral J'dco over all possible orientations in space of one molecule of component c.* The integration in Equation (3.41) is carried out over all possible values of linear momentum, over all positions of the molecules within the volume V and over all possible orientations of the molecules. The integration over momentum is easily carried out, yielding a factor (2rtmikT) 3/2 for each molecule, so that
q)(j...) = ( U v c=lA c 3Nc ,C2jNc N~,)l lexp(_U~j...)/kT)drUdoU (3.42)
where
A i = h/(2xmikT) 1/2 (3.43)
is a de Broglie wavelength. Since, for a system containing only one molecule, there is no
intermolecular potential energy, q~(c) is given by
* The appearance of the remaining terms in the prefactor of Equation (3.41) is explained as follows. Since molecules of the same species are indistinguishable, configurations which differ only by the assignments of labels to the individual molecules are actually identical and should not be counted separately. It is therefore necessary to divide by the Arc! possible assignments of labels amongst the N molecules for each component c in the system, h 3u is a normalisation factor for the
c
integrations over position and momentum chosen such that the classical and quantum treatments of the partition function converge in the limit of high temperatures.
53
cI91(c) = ( V / A 3c) (3.44)
and, finally, the configurational partition function is
(n t ls Q~J) = ~¢'~c NC exp(U~J)/kT)dr N d o N c=l
(3.45)
Note that, while 05u~/J'") is dimensionless, Q~J) has the dimensions of V u.
In order to evaluate the virial coefficients, knowledge is required of the potential energy of clusters of N molecules with N = 2, 3, .... For N = 2, the situation is comparatively simple and U 2 = u(r12,o12 ) is just the intermolecular potentialenergy function for a pair of molecules with separation r12 =lr 2 r l l and relative orientation given by a set of (bodyfixed) coordinates
denoted by co12. The precise quantitative determination of u(r~2,o12 ) for a range of interesting molecules remains one of the most resilient problems in chemical physics and our present knowledge is quite restricted. The pair potential is however known with considerable accuracy for several monatomic substances, especially He, Ne, Ar, Kr, Xe and their mixtures, and for a number of simple systems containing molecular hydrogen (21,22). Considerable progress has been made for other diatomic gases, for mixtures of diatomic and monatomic components, and for a few other systems such as water (22). For other polyatomic systems, it has so far proved impossible to establish the multidimensional potential energy surface in any precise way. In these cases only crude models are available, many of which neglect completely the orientational dependence of the pair potential. The origin and determination of the intermolecular pair potential has been reviewed by several authors (2123).
For virial coefficients above the second, U N is required for N > 3 and the question therefore arises as to whether or not this may be expressed simply as a sum of pair interaction energies. It is now well established that the approximation of pairwise additivity is in fact fairly poor (24) and that one should write
N1 N
U N  Z Z Uij "+" AuN (3.46) i=1 j= i+ l
where u o. = u(r0.,o U) and Au N represents the deviations from pairwise additivity in a cluster
of N molecules (8). Some information is available on the behaviour of Au 3 and, in particular, the asymptotic longrange form of the potential for atoms and sphericaltop molecules is known from the theory of dispersion forces to be (25)
Au 3 = v3(1 + 3cosOicosOjcosOk) / ( r i j r i~ r j~ ) 3 (3.47)
Here, 0 3 is a dispersion coefficient and 0 i is the angle subtended at molecule i by moleculesj
and k. 03 is related approximately to the coefficient C 6 of r "6 in the asymptotic longrange expansion of the twobody potential by 03 = J a C 6, where a is the polarizability. The corresponding result for linear molecules is also known (26).
The cluster integrals b~ U) and b~ °k) may now be evaluated from Equations (3.32) and (3.45) and the results are
54
/52( U)= (2!V"Q2)I ffo dridrjdc°idoj
= ( 3 ! V " Q 3 ) 1 fL f i j f i k f # + fijf ik + f i j f jk + fikf jkldridrydrkdc°idc° yd¢°k
+ (3 ! Vff23) 1 f(l+fj)(l+f/k)(l+fjk)f: k dridrjdrkd¢o idfh jdo3 k.
(3.48)
Here use has been made of the Mayer function f/j, defined by
f0 = exp(uu/kT)  1 (3.49)
and of an analogous function of the threebody potential:
f/jk = exp(Au3 / k T )  1 (3.50)
For neutral molecules, bothf/j andfj k approach zero rapidly as the separation of the molecules is increased thus ensuring that the integrals converge. Notice that, if Au 3 is zero, fijk vanishes.
Since the intermolecular potential energy depends only on the relative position and orientation of the molecules, it is convenient to transform the integration variables from the spacefixed to the bodyfixed frame. In the case of the position vectors, the transformations are
= 4 nri j dr, j V dridry 2 1 8n r r r dr dr dr V dridrflrk= 2 i j i k i k ij ik jk
(3.51)
and the integrations for b~ ~/k) then extend over all values of (rij,r/k,rjk) which form a triangle. It
is also convenient to extend formally the range of integrations over molecular separations to infinity, rather than just to the walls of the container, and this is justified by the rapid convergence of the integrals. For the orientation vectors, one may adopt any convenient set of angular coordinates provided that the appropriate normalisation constant is used. Equations (3.37), (3.48) and (3.51) are now combined to determine B and C.
The second virial coefficient corresponding to the interaction of molecules 1 and 2 is given in bodyfixed coordinates as
B12 =(2nL/£212) ~ I,~,2)f2r~22dr~2d°~12 (3.52)
where ~r~12 is a new normalisation factor equal to the integral ~ dCOl2 over all possible relative
orientations. In the case of the third virial coefficient corresponding to the interaction of molecules 1, 2
and 3, the effects of additive and nonadditive intermolecular potential energies separate so that
C123 = C ~ + AC123 (3.53)
55
The additive and nonadditive contributions are given by
cla2d3 d = (8~;2L 2/3~~23) If12f3f23 r12r13r23dr12dr13dr23dO~ldCO2do~3 (3.54)
and
AC123 = (871:2L 2/3~"~23 ) I(l+f12)(l+f3)(l+fE3)f23 r~2r13r23dr~2dr13dr23&o ldf9 2df93 (3.55)
where £2 123 = I d~ldo,)2d~ 3 . Explicit expressions for the additive and nonadditive parts of D have been given by Mason and Spurling (8); for higher virial coefficients it is preferable to use graphical notation to simplify the otherwise complicated expressions (15).
As a practical consideration, evaluation of B is easy by quadrature even for nonlinear molecules (a sixdimensional integral is required). Similarly, evaluation of C, D or E for monatomic molecules is quite easy but, as soon as orientational dependence of the intermolecular potential is included, one is faced with a significant computational task. For example, to evaluate C for three linear molecules, one must integrate over nine coordinates and it is then probably necessary to resort to MonteCarlo methods at least for the angular variables.
In the few cases where the intermolecular pair potential is accurately known, and quantum effects are unimportant, values of B calculated from Equation (3.52) are in close agreement with experiment. This situation is illustrated in Figure 3.4 by the example of krypton. The experimental third virial coefficients of krypton (2830) are also shown in Figure 3.4 together with values calculated by means of Equations (3.54) and (3.55) on the assumption that Au 3 is given by Equation (3.49) with the theoretical value of o 3 (23). In view of the considerable scatter amongst the experimental results, the agreement is satisfactory and certainly much better than when AC is neglected.
3.3.4 Quantum Corrections
Except for hydrogen and helium at low temperatures, quantum effects may be accounted for rather accurately by the addition to the results of the previous section of small quantum corrections. It turns out that these terms may be obtained by expansion of the quantum mechanical partition function about the classical limit in powers of h 2 (31). At the low temperatures where these corrections are significant, it is safe to treat the molecules as rigid rotators. Only translational and rotational terms will then be present in the expansion.
The treatment is most complete for monatomic systems where B is known correct to 0(]/6) (32) and C is known, including the effects of nonadditive intermo_lecular forces, correct to O(1/4) (33,34). In the case ofpolyatomic molecules, the expansion of B is less well developed and there appears to have been no calculations for C. The translational and rotational quantum corrections to B were worked out for linear molecules to O(h 2) by
Kirkwood (35) and later to O(h 4) by Wang Chang (36). However, Pack (37) has pointed out that, in the derivation presented by Wang Chang, the transformation from spacefixed coordinates to bodyfixed coordinates was made incorrectly with the consequence that a
56
50
., 100 '7,
0 E 150
e0
E 200 0
m 250
300
350
4 0 0 0
3 0 0 0 o E
2 0 0 0 o,2.
o 1 0 0 0
o
t .0 t 200 300 400 500 600
T ( K ) I I I , I , I , I
100 200 300 400 500 600 700
T (K)
Figure 3.4 Second virial coefficients of krypton. O, recommended (experimental) values from reference 4; ~ , calculated from the interatomic potential of Aziz and Slaman (27). Insert: third
• ~ C add IA C ; C add . virial coefficients. O, (28); l , (29); O, (30); , . . . . . . ,
coriolistype rotationtranslation coupling term was neglected. It has also been observed that some terms of O(h 4) are missing from Wang Chang's formulae (38). A revised semiclassical
expression for B,
B = Bc~as s + ( h Z / 2 # ) B t l + (h2/24)B)(~ ) + (h2/212/~"rl]R(2) "3t ( ]~2 /21 ~ )Bc l (3.56)
correct to O(h 2) has been derived and may be applied to all rigid molecules (both like and unlike) except asymmetric tops. In Equation (3.56), Bclas s is the classical second virial coefficient, # is the reduced mass and 11, I 2 are the moments of inertia of the two molecules.
#(°and Be1 determine respectively the leading translational, rotational The coefficients Btl, rl and coriolis contributions to the quantum corrections. Pack (37) gives a general formula in spacefixed coordinates that relates the leading quantum corrections to the translationrotation Hamiltonian. Specific expressions were also given in bodyframe coordinates fo r linear molecules and for the interaction between an atom and either a linear molecule or a spherical top molecule. Here, only the results for a system of linear molecules are given:*
* Altemative expressions for B r and B e are given in reference 37 for the case in which the intermolecular potential is expanded in spherical harmonics.
2zrL ]" ~{exp ( u l kT) 1} ,2 _ _ r122dr12do3 Bclass = ~('~2 0 (~12)
57
(3.57)
oo
zcL [ Sexp(_u/kV)(t3u/c3r)2 r122dr12d~12 gt' = 6(kT)3L'~2 ;(~,2)
(3.58)
B r i)   ,
oo
nL I [.exp(ulkT){(c~ulOOi )2 +(~ula~i) ~ cs#O/} ~l~d~,~dco,~ (3.59)
rtL Be' = 6(kr)3D~2 S ~exp(u/kr)F(r,O,,O2,~2 ) dr~2dco,2
0 (co12)
(3.60)
where (39)
F(r,O~,02,~2 ) = (0u/00~) 2 + (0u/002) 2 + {csc20~ + csc202  2}(0u/0~2) 2
 2(c9u/0442) sin ~2 {(0u/00~ ) cot 02 + (0u/002) cot 01}
+ 2 cos~2 {(Ou/OO~)(Ou/OO2)cotO ~ cotO2(Ou/O~2) 2 }
(3.61)
Here, 0 i and ~b i are bodyfixed polar and azimuthal angles, $12 = ($2$1), and
1 1 2
=
(o211) 1 1 0
(3.62)
Equations (3.57)  (3.60) are easily specialised to atomlinear molecule systems, where u no longer depends upon 02 and ~b12. Btl further reduces to the corresponding monatomic result when u is a function only of r. These formulae may also be applied to the interaction second virial coefficient B12.
An important theoretical result is that Bclas s always gives a lower bound for B itself so that the quantum effects are always positive (37). It appears that the semiclassical expansion of B in h 2 converges satisfactorily for neon and for heavier monatomic systems at all temperatures above the normal boiling temperature. This is illustrated in Figure 3.5 where B and, on an expanded scale, the first two translational quantum corrections are plotted for neon. For helium, the series converges satisfactorily only at temperatures above about 40 K. Except for the isotopes of hydrogen, molecular gases are generally sufficiently heavy that translational quantum effects are given with sufficient accuracy at temperatures of interest by the correction of 0 ( ] ; / 2 ) . Whether or not this is also true of the rotational term depends upon the moment of inertia and the degree of anisotropy of the intermolecular potential. For nonpolar molecules, the rotational and coriolis terms together are generally smaller than the translational term but for polar molecules, especially polar hydrides, the reverse is true. It appears that B c is always significantly smaller than B r. These points are illustrated in Figures 3.6 and 3.7, where B, Btl
58
30
40
"7, _== 0
E
E 0 v
10 0 E O3 E 20 0 "
i !
~ i ~ ' ~ " " 3 Btl .... ' t
Y t / 1 40 60 80 100
I I I T (I K) I
50 75 100 125 150
T (K)
Figure 3.5. Second virial coefficients of neon. 0 , recommended (experimental) values from reference 4; , calculated correct to the second quantum correction from the interatomic potential of Aziz and Slaman (40). Insert: Btl and Bt2.
15
10
±
6
o t~~___. Brl o k    ~ 2 7 ""B,~
1 0 0 200 300 T (K)
5 I , I. I i 100 200 300 400
T (K)
Figure 3.6. Second virial coefficients of hydrogen. O, recommended (experimental) values from reference 4; ~ , calculated correct to the first quantum corrections from an anisotropic intermolecular potential model. Insert: Btl, Brl and Be.
59
100
"T o 200 E
03 E o 300 133
400
500
4¢ ~ 2o
"" 15 ,_.... o E
10
rn < 5
0
I I I
200 250 300
J Brl
200 250 300 350 400
T (K) I I I
3 5 0 4 0 0 4 5 0
T K
Figure 3.7 Second virial coefficients of hydrogen chloride. O, experimental results of Schramm and Leuchs (42); , calculated correct to the first quantum correction from the Stockmayer potential (43) with s/k = 413 K and cr  0.2591 nm, where s and cr are the scaling parameters for energy and length in the isotropic part of the potential. The dipole moment was taken as 3.53 × 10 30
C.m (22). Insert: Btl and Brl.
and Brl are p lo t t ed for H 2 and for HC1; in these cases, Bcl is negl ig ib le . Fo r h y d r o g e n *, the second t rans la t iona l q u a n t u m cor rec t ion is also shown.
* The potential model used here is given by:
U(r, 0,, 02, ~b~2 ) = A exp(Br)[1 + C{P2 (c~) + P2 (c2)} ]  F(r)IC6r6 f 3(02 / 4rcs0)rSf~ + 9(02a/4ZCSo)f3]
where
2 2 2 f~ = 1  ~¢  3~c(1  ~) (c2qc 2) ~1~.32(s1s2c  2CLC2) 2, f2 = 1  5(c~ + c2)  1 5c, c 2 + 2(s~s2c  4qc2) 2,
f 3 = S4"JrS2 "Jr4 4C4+4C4, ~C = (alia±)/3~, ~ '  2 ~ _ 1 _ + 1 ~ 1 ] , C i = COS0/ , S i = s i n 0 i , C = COS~)12 ,
and
F(r) = exp[{(ro/r )  1} 2 ] r < r o
=1 r > r o
A and B were fitted to the hightemperature second virial coefficients while C and C6 were taken from (41): A/k = 3.545x107 K, B = 38.27 nm 1, C = 0.14, C6/k = 0.0899 K.nm 6, r0 = 0.3 nm.
Polarisabilities a I and a2_ and the quadrupole moment ® where taken from reference 24.
60
3.3.5 The Virial Coefficients in Quantum Mechanics: Helium and Hydrogen
For the isotopes of helium and hydrogen, the semiclassical expansion of the virial coefficients does not converge at low temperatures and, to obtain reliable results, a fully quantummechanical evaluation is then required. Although the relations given in Section 3.3.2 between the virial coefficients and the configurational partition function remain formally valid, the evaluation and treatment of the energy levels from which QN may be calculated is different in the full quantum treatment. The origin of these differences lies primarily in quantisation of the angular momentum.
It is important to recognise that we should observe quantum statistics as well as quantum mechanics in a correct treatment. It turns out that the statistical restrictions on the allowed states of a system become important at very low temperatures, and this aspect of the problem will be examined first. The essential limitations imposed by quantum statistics are well known and may be stated as follows. When a system is composed of two or more indistinguishable molecules, the only allowed states of the system are those which are either symmetrical or antisymmetrical with respect to the interchange of two particles. Only symmetrical states are found for systems composed of indistinguishable atoms or molecules which contain an even number of elementary particles; such systems are said to observe Bose Einstein quantum statistics. For systems composed of indistinguishable atoms or molecules which contain an odd number of elementary particles, FermiDirac quantum statistics apply and only antisymmetrical states are allowed. The adoption of correct quantum statistics restricts the number of quantum states from which the canonical partition function of a system of indistinguishable molecules is constructed. In the case of distinguishable molecules, no statistical restrictions apply.
The symmetry of a quantum state is determined by the parity of the wavefunction h TM
which describes it.* For indistinguishable atoms, the parity of h TM is determined by the nuclear spin and translational states of the system and, in the case of nonzero nuclear spin, a number of degenerate nuclearspin states is available. For molecules, the parity of 7* is determined by nuclearspin, translational, rotational and vibrational states. However, the isotopes of hydrogen are the only molecular systems where statistical quantum restrictions have a significant influence on the virial coefficients and, even then, the effects are restricted to temperatures below about 10 K. Under those conditions, only the lowest accessible vibrational and rotational states will be populated in the gas and the treatment of internal modes is thereby greatly simplified. It is however necessary to recognise the existence of (distinguishable) ortho and para H 2 and D 2 molecules.
In view of the forgoing remarks, only a single (but possibly degenerate) internal energy level need be considered for an isolated atom or molecule at temperatures where statistical effects are of significance. The internal partition function F~ is therefore given by
F1 = go exp(~o / kT) (3.63)
* A symmetric (or evenparity) state is characterised by a wavefunction which is invariant under the operation of exhanging the coordinates of any two molecules. Conversely, an antisymmetric (or oddparity) state has a wavefunction which, under the same operation, changes sign.
61
where e0 and go are the energy and degeneracy. Since the zero of energy may be defined at will, one might as well measure all energies relative to that of the isolated molecule in the lowest allowed energy level. In that case, e0 = 0 and
F~ = go (3.64)
In the case of an atom with nuclearspin quantum number s, go = (2s + 1) while, for a diatomic molecule, go = ( 2 s + l ) ( 2 j + l ) , where j is the rotational quantum number of the lowest allowed energy level. Since, for a system containing just a single molecule, the question of parity does not arise, the overall partition function O 1 is simply the product F~ q~l, where q~l is given by Equation (3.40) with its unrestricted sum over translational states. For all cases of practical importance, the sum over the translational quantum states may be approximated by an integral giving q~l = ( V / A 3) and thence
6) 1 = g o ( V / A 3) (3.65)
When the system contains two or more molecules, the calculation of the partition function is more involved. In the following treatment, attention will be restricted to the second virial coefficient and at first to the case of a onecomponent gas. The discussion starts with sphericallysymmetric systems and proceeds to the case of linear molecules. The corresponding results for unlike atoms or molecules are also obtained rather easily by dropping the symmetry constraints.
In the case where the intermolecular potential is (or is assumed to be) spherically symmetric, it is still convenient to treat the translational and internal degrees of freedom of the molecules separately. However, in order to satisfy the symmetry constraints, both the internal and translational parts of the partition function should be separated into terms arising from quantum states of given parity. ~2 is therefore split into two terms, 052 and q~2, which are obtained by summing only over translational states of the specified parity (+ or ). Similarly, the (g0)2 degenerate internal states are separated into symmetric and antisymmetric terms and the overall partition function 0 2 is then formed from products of internal and translational terms which have the required parity:
+ (3.66)
In this equation, W2 + and W 2 are statistical weights which are determined as follows. For
indistinguishable atoms or molecules which obey BoseEinstein statistics, (go)2W2 + is the
number of symmetric internal states and (g0)2W2  is the number of antisymmetric internal states of the two molecules each in its lowest allowed internalenergy level. In the case of FermiDirac statistics, the situation is reversed while, for distinguishable molecules, Boltzmann statistics apply and W2 + = W 2 = 1/2!. The statistical weights W2 ÷ and W 2 for the isotopes of helium and molecular hydrogen are given in Table 3.4. Consistent with the assumption of a spherical intermolecular potential, hydrogen molecules are treated as spherical particles with spin equal to the vector sum of the spins of the constituent nuclei.
62
Table 3.4. Statistical weights for the isotopes of helium and hydrogen (44,45).
System j s go W2 + W2 System j s
1 2 1/4 3/4 oD 2 0 0,2 3He 2
4He 0 1 1 0 pD 2 1 1 1 3 pH 2 0 0 1 1 0 HD 0 2,2
oH 2 1 1 9 5/9 4/9
g0 W2 + W 2
6 7/12 5/12
9 5/9 4/9
6 5/12 7/12
In view of Equation (3.66), the configurational partition function of a pure gas is given by
Q2 = 2 ( V / ~ 1 ) 2 ~ 2 = 2A6(Wz+(I)~ + W2(I)2) (3.67)
and the second virial coefficient becomes
B = W:+B + + W2B (3.68)
Here, B + and B involve only translational partition functions and are given by
B + =  ( L A 6 / V ) ( ~  ½ ~ ) (3.69)
An important theoretical consequence of the adoption of quantum statistics is that B does not vanish in the absence of intermolecular forces. Indeed one can show that, for noninteracting indistinguishable atoms or molecules (14),
• ~  ½ ~ =+_2s/2(V/A 3) (3.70)
so that the second virial coefficient, n(°) is given by " " e x c h ,
Be °) = W2 + B(°) + + W 2 B(O)  xch
= _ 2~LA 3(W2+ _ W2 )
with
(3.71)
B (°)+ = T2 ~ L A 3 (3.72)
In view of the existence of an exchange term even for noninteracting molecules, it is convenient for real systems to deal with that part of B + which arises from the intermolecular forces:
63
B +  B {°)± =  ( L A 6 / V ) ( ~ 2  ~o~±) (3.73)
The partition functions 052 and 05~ °)± which appear here are given by
052 = ~ exp(H,~/kT) t ~{2°} ± = ~ exp( H( f f / kT)
(y
(3.74)
where a represents the full set of six translational quantum numbers for the twomolecule system (subject to the implied symmetry restrictions) and H a and H2 °} are the translational
(kinetic + potential) energy of the system with and without intermolecular forces. Both Ho
and H(~ °) may be separated into the sum of two terms each associated with three quantum
numbers, the one term for the translational kinetic energy of the centre of mass, and the other for the (kinetic + potential) energy of the relative motion. Integration over the former yields a common factor of 2 3/2 (V/A 3) so that
B±B(°)± =23/2LA 31~. e x p (  H , ~ . / k T )  ~ exp(H(~°)/kT)) (3.75)
where a' indicates that the summations now exclude the centreofmass terms. In the present case, with an isotropic intermolecular potential, the possible states of
relative motion are characterised by the quantum numbers (l, ml, n ). Here, l is the quantum number for the orbital angular momentum 1, m t gives the orientation of 1 in space, and n determines the energy. It is convenient to divide these states into bound levels, with discrete energy levels C.nl , and states with a continuum of positive energy levels H~I; each such state is (2l+1) fold degenerate corresponding to the (2l+ 1) allowed values of m l. The summation over n for positive energies may be replaced by an integral over the continuous variable ~c,
exp(Hn~/ kT) = ~ exp(J6rc2 )( dn/drc ) drc n
(3.76)
where rc is defined such that Hn~t = h2rcz/21a, ~ is the reduced mass, [3=h2/21akT, and
(dn/dtc) is the density of quantum states in the continuum. A similar equation holds for the
summation over the energy levels 14 (°) but with a different density of states: (dn/d~c) (°} The * " n l
quantity (dn/d~c)(dn/d~c) (°) is known as the excess density of states and may be related rather easily to the quantummechanical phase shifts rh(rc ) of atomatom scattering theory (46):
(1/ ~ )( drlt / d~ ) = ( dn/ d~c )  ( dn/ d~ ) (°) (3.77)
Equation (3.75) then becomes
64
B +  B ( ° ) +   2 3 / 2 LA3lZeven(2l + 1)B l t
B  B (°) = 23/2 LA 3y' (21 + 1)B 1 l odd
(3.78)
where*
B t = ~ . , [ e x p (  e . n , l k T )  1] + (2fl/n) ~r/~(~:) exp(/7~c2)~c d~: n
(3.79)
One can therefore write the complete second virial coefficient as
B = Bdirect + Bexch (3.80)
where
Bdirect  2 v2 LA 3 ~ (2l + 1)B t (3.81) l
and
(3.82)
In the case of unlike atoms of types 1 and 2, Bexch = 0 and it is easy to show that
B12 = (LA 3/2)~~(2l + 1)B~ (3.83) l
where A 12= ( h2/2n#kT) 1/2. In order to evaluate the second virial coefficient, one must calculate from Equation (3.79)
the quantity B t for 1 = 0, 1, 2, . . . . 6nl and r/t are required for this purpose and both may be obtained by numerical solution of the radial SchrOdinger equation for the two atom system. Although once considered to be a computationally demanding procedure, such calculations are today routine and may be performed for helium over a wide range of temperatures. Numerical methods for the precise evaluation of the phase shifts have been described by Boyd et aL (47), while Cooley (48) and Cashion (49) have described methods of calculating the boundstate energies. In practise, numerical calculation of the sums over l are usually truncated at some value l = lma x and the remaining contribution of terms with l > lma x are estimated using the Born approximation. Similarly, the integrals over ~¢ may be truncated at some upper limit and appropriate choices, together with some highly accurate calculations, have been reported by Aziz and Slaman for 3He and 4He (50). It appears that the intermolecular potential of helium
* The form of this equation is arrived at after integration by parts and noting that, according to Levinson's theorm, rll(O)/n is equal to the number of bound states in the well for the given l.
65
supports only one bound state for 4He (l = n = 0) with an energy given by Eoo/k = 0.0016 K (50). In the case of 3He, there are no bound states.
The results of full quantummechanical calculations of B for the isotopes of helium, based on an accurate intermolecular potential (50), are compared with experiment in Figure 3.8. Remarkably, the agreement is within a few tenths of a cm3/mol over the entire temperature range. For comparison, the semiclassical results for 4He are also shown. It is interesting to note that, while the contribution of #(°) appears to be significant up to temperatures of order *"exch 100 K, the total I Be~ch [ given by Equation (3.82) drops off very rapidly with increasing temperature and is negligible (< 0.01 cm3.mo11) for 4He above 5 K and for 3He above 6 K (47). At higher temperatures, quantum effects are confined to the difference between Bdirect
and the classical second virial coefficient; the origin of such differences lies is the quantisation of angular momentum. At temperatures where the semiclassical treatment of B is valid, these differences are accounted for correctly by the quantum corrections and it would therefore be incorrect to then add n (°) to the result. ~'exch
25
25
"S 50 E
75 E O
v 100 rn
125
150
175
. . . . . . . . i . . . . . . . . i . . . . .
3He ~ ~ 7 ........ ':" . . . . . . . . . . . . . . . . . . . .
/ / \ V ,.o / "4 e ,0: ; ; 4
. . . . . . . I . . . . . . . . ~ T (K ) . . . . .
10 100 700
T (K)
Figure 3.8 Second virial coefficients of 3He and 4He. ~, (51); O, (52); l , (53); El, (54); ,ik, (55); IF, (56); 0 , (57) . ~ , quantummechanical calculation of B (50); .............. , semiclassical B for 4He to O(h 2 ); , semiclassical B for 4He to O(h 4 ).
Insert: ~ , Bexch" , . . . . . . , ~'exch#(0) (47).
Apart from 3He and 4He, a fullyquantum mechanical treatment is required only for the isotopes of hydrogen. For H 2 and D 2, but not for HD, the molecules exist in distinguishable ortho (o) and para (p) forms and the gas is best considered as a mixture with second virial coefficient
66
B z 2 X o X p B o p at  2  Xo Boo b xpBpp (3.84)
Since calculations indicate that Bexch is completely negligible in all cases at temperatures above 10 K, only B~ectwill be considered further here.
Molecular systems are characterised by the presence of angulardependent intermolecular forces. These have the effect of coupling the angular momentum of the individual molecules to the intermolecular axis and thereby lifting the degeneracy of the states of relative motion. Strictly, only the total angular momentum J is a rigorous constant of motion but, in systems containing H 2, HD or D 2, the anisotropy is weak and the angular momenta J1 and J2 of the individual molecules are very nearly conserved. Each state with orbital angular momentum 1 is associated with substates characterised by the quantum numbers J and K. Here, J is the quantum number for J and K is the quantum number for the internal rotation angular momentum K = J l + J2.
In view of the coupling of rotational and translational states, one must return to Equation (3.67) and proceed with the full partition function. In the case where only the lowest allowed rotational energy level is populated, the development is rather simple and parallels closely that given above. The boundstate energies of the twomolecule system are now characterised by (n, l, K, J) and written c,~Ks. Considering for generality the case of unlike molecules with
rotational quantum numbers jl andj2, the result is
_LA 3 12 Bl'2 direct = 2(2jl + 1)(2j2 + 1)
0; ) x ~ )"~(2J + 1) exp(eSntz~/kT)+ [(dn/d~)(dn/drz)(°)]exp([3rcZ)d~: \K=Ij~j2I l=0 J=llKI
The excess density of states in the continuum is given by (41,58)
( dn/ drc)  ( dn/ d~¢) (°) = (i/2r0 Tr (S dSt/dr¢) (3.86)
where S = S(~c) is the Smatrix of moleculemolecule scattering theory, St is the adjoint of S, and Tr denotes the trace operation over the full set of quantum numbers (l, K, J). The result of
J this operation may be expressed in terms of a set of generalised phase shifts r/tK(~:) and, for
each l, (2 Jl + 1)(2 J2 + 1) such terms appear. These last two equations may be applied to obtain Baireet for any combination of like or
unlike molecules or atoms. In the case of an atommolecule interaction, the sum over K drops out while, for atomatom systems, J = l and the sums over both K and J drop out. In the latter case, only the diagonal elements of the S matrix are nonzero and they are related to the scattering phase shifts by S~/(r¢)= exp[2ir/t(~)]. Equation (3.86) then reduces to Equation (3.77) and Equation (3.85) reduces to Equation (3.81).
J and S(~:) again involves solution of SchrOdinger's equation for the Determination of ~'nlK system and such calculations are tractable for the low temperatures at which the quantum treatment is required. The Smatrix may be determined to high accuracy in the closecoupled approximation while the boundstate energies may be obtained by the same general method
67
(59) or by the more efficient secular equation method (60). It is also possible to rely on spectroscopically determined values of ~'nlK J and it is worth remarking that, for the systems of interest, only n = 0 states have been observed and there is little evidence of splitting of the l states due to anisotropy of the intermolecular potential (61). The sums over K and J in Equation (3.85) could therefore reasonably be collapsed leaving a boundstate contribution to Bdirect identical in form with that of a monatomic system.
Although Equation (3.85) is restricted to molecules in a single rotational level, it is clearly possible to derive a more general expression in which the full set of molecular rotational energy levels is retained. Alternatively, the gas may be treated as a mixture of species having different values of j. Unfortunately, closecoupling calculations become very time consuming once more than a few rotational levels are populated. It is notable however that, for pH 2, thej = 2 level only reaches 1 per cent occupancy at about 75 K while, for oH 2, the temperature must reach 145 K before the j = 3 level reaches the same occupancy. For purposes of comparison, the semiclassical formula, including the translational corrections to O(h4), should be of sufficient accuracy at temperatures above about 100 K (see Figure 3.6).
3.4 CALCULATION AND ESTIMATION OF VIRIAL COEFFICIENTS
Although an extensive database of experimental virial coefficients is available (4), methods for the calculation and estimation of virial coefficients are important for a number of reasons. In the first place, there are still many systems (especially mixtures) for which B and C are not known at the temperatures of interest and, in these cases, estimation methods can be very useful. Even when experimental data are available, in may be desirable to establish a soundlybased correlation and, for that purpose, a model intermolecular potential with appropriately chosen parameters can be very powerful. Finally, it is sometimes useful (for example, in thermodynamic perturbation theory) to have essentially exact virial coefficients for some simple model potential.
3.4.1 The Virial Coefficients of Model Systems
Virial coefficients have been evaluated for several isotropic model intermolecular potentialenergy functions including the squarewell, LennardJones (n,6), exponentialsix and MaitlandSmith functions.* A number of tabulations of B, C , D and E are available (6,21,62,63) and, as mentioned above, the evaluation of B and C by quadrature is quite routine. The parameters of such a model may be optimised fairly easily in a fit to virial coefficients or related quantities such as the JouleThomson coefficient or acoustic virial coefficients. The model may then permit extrapolation or the estimation of other properties, often with high accuracy. It appears that simple isotropic potential models can offer a very good account of the second and third virial coefficients of nonpolar molecules provided that nonadditive threebody forces are included in the calculation of C (5,64). Equation (3.47), with u3 treated as an empirical parameter, appears to be quite adequate (64).
Although the importance of analytical and tabulated results has been greatly diminished by the speed of modem computers, there is one class of model systems, namely the convex hard bodies, which are of special importance as reference systems in statistical
* Descriptions of these models may be found in, for example, reference 21.
68
thermodynamics and for which the existence of tabulated results is important. The virial coefficients of such systems are independent of temperature and some of the results are mentioned here.
The simplest of the convex hard bodies is the hard sphere. Analytical expressions for B, C and D, and precise numerical results up to the eighth virial coefficient, are available for a gas composed of such molecules (13,65). Estimates of the ninth and tenth virial coefficients have also been made (65) and the results are given in Table 3.5. A considerable amount of work has also been done on nonspherical convex hard cores for which the second virial coefficient is given by B = L v ( 1 + ?'), where v is the volume of one molecule and ?' is a shape factor which may be determined from the mean radius of curvature, the surface area and v (66). For spherical hard cores, ?' = 3 while, for other shapes, ?' > 3 and increases with the eccentricity of the molecule. Virial coefficients up to the fifth have been computed numerically for hard ellipsoids of revolution (67), spherocylinders (68) and chains of fused hard spheres (69). Except for highly eccentric molecules, the results are found to correlate rather well with the shape factor ), (6 7).
Table 3.5 Virial coefficients of hard spheres of diameter d (13,65).
B = 2toLd3~3 = b o
c = (5/8) bo 2
D = 0.2869495 b 3
E  (0.110252 + 0.000001) b 4
F = (0.038808 + 0.000055) bo 5
G = (0.013071 + 0.000070) b 6
H = (0.00432 + 0.00010) bo 7
I~, 0.00142 bo 8
J ~ 0.00047 bo 9
It appears that the virial series for hard spheres is convergent up to the density at which computer simulations indicate that solidification occurs (70). For nonspherical hard cores, Pad6 approximants to the compression factor formed from the available virial coefficients also agree closely with the results of computer simulations up to very high densities (71). These observations suggest that the virial equation is valid for all fluid states of this class of system.
Hard bodies do not exhibit a vapourliquid phase transition but the addition to the intermolecular potential of an attractive field outside the hard core does give rise to this phenomenon. A simple and very interesting model of that kind is the one considered by van der Waals (72) in which the molecules are treated as spherical hard cores of diameter d with a weak but longrange attractive interaction such that the intermolecular potential is:
=E / ~ 3 d < r < 6 d
=0 r > 6 d
69
(3.87)
The second virial coefficient for this system is b0{1(e/kT)} but the dimensionality of the
cluster integrals is such that all higher virial coefficients are identical with those of the corresponding hard sphere system. Consequently, for this model system the virial series is also convergent for all fluid states, both gas and liquid, and the equation of state is analytic everywhere including at the critical point. However, for systems with an intermolecular potential o f f i n i t e range, it is known that the equation of state is nonanalytic at the critical point and that the virial series must, therefore, have a finite radius of convergence on isotherms close to the critical temperature. The convergence of the virial series elsewhere for this class of system is not established theoretically owing, at least in part, to the absence of a complete set of virial coefficients. For LennardJones molecules, virial coefficients up to the fifth have been computed (6) and found to diverge towards oo as T ~ 0; the results are plotted in Figure 3.9. This behaviour is usually taken as evidence that the virial series does not converge for such molecules in the liquid state. However, a satisfactory virial expansion might exist for both gas and liquid branches below the critical temperature if the higher virial coefficients adopt finite positive values (73). The question therefore remains open.
.... ! . . . . . . . . 09 ~  (''(~ O 
~" 1
0
~._ 2 D* B* .,. >
0
"U
~ 4 n'
5 0.5 1 2 5 10
T Ot
Figure 3.9 Reduced virial coefficients for the LennardJones T*= k T / s , B*= C/bo , C* = C / b g, D* = D / b 3 and E* = E / b 4 .
(12,6) potential (6).
70
3.4.2 Estimation of Virial Coefficients
For cases in which experimental values of B and C are unavailable, some method of prediction is required and for this purpose the principle of corresponding states is usually applied. In its simplest form, the principle applies to systems whose intermolecular pair potentials may be written in the form u(r )= e F ( r / or), where e and cr are scaling
parameters which characterise a particular substance and F is a universal function. Systems which obey this relation are said to be conformal. In all conformal systems to which classical statistical mechanics applies, the reduced second virial coefficient, B*= B/(ffT~Zt73), is a
universal function of T*= kT/e. Thus, the second virial coefficient of one conformal substance (labeled i) may be estimated from that of another (labeled 0) if the ratios si/e o and cri/cr 0 are known. From a theoretical point of view, the most satisfactory way of relating the scaling parameters to measurable properties is by means of the Boyle temperature and the so called Boyle volume V B, equal to T(dB/dT) at T = T B. In terms of these quantities, B/V B is,
according to the principle, a universal function of T / T B and, if pairwise additivity of the intermolecular forces is assumed, C/(VB) 2 is another universal function of T / T B. This method of selecting the scaling parameters has the disadvantage of requiting some measurements of B(T) in the first place. Furthermore, T B is inconveniently high for most substances.
Practical correlations of virial coefficients employ as scaling parameters the critical temperature T ¢ and the characteristic molar volume RTC/p ~, where pC is the critical pressure, and seek to i'epresent B ( f f / R T ~) and C(pURT~) 2 as universal functions of the new reduced
temperature T r = T / T ~. Although the principle, as stated above, applies to only a small number of simple fluids, Pitzer (74) was able to show that many different kinds of molecular complexity may be accounted for by the inclusion of a third parameter co which he called the acentric factor. This parameter is defined in terms of the vapour pressure p°, by the equation
co = 1log,0 {pC (T r = 0.7)/pC} (3.88)
such that it is essentially zero for the simple fluids Ar, Kr and Xe. For other fluids, values between 0 and about 0.4 are usually found. The second virial coefficient is given in this extended principle of corresponding states by
B(pC/RT c) = B o + coB, (3.89)
where B 0 and B 1 are dimensionless functions of T r. Expressions for B 0 and B 1 w e r e given by Pitzer and Curl (75) in their original formulation but here the more recent correlations proposed by Tsonopoulos (76) for nonpolar gases are given:
B o =0.14450.3300/Tr0.1385/Tr20.0121/Tr30.000607/Tr 8 l
B, = 0.0637 + 0.331 / Tr 2  0.423 / Tr 3  0.008 / Tr s J (3.90)
As it stands, Equation (3.89) is useful only for essentially nonpolar gases, but Tsonopoulos
showed (76) that the addition of the t e r m a/Tr 6 to the righthand side of that equation
71
permitted the results for polar gases to be correlated also. A further term b/Tr 8 is necessary for
associating compounds. While there seems to be no simple universal correlation for the additional parameters a and b, correlations of the former in terms of the dipole moment may be established within certain classes of compounds such as ketones or ethers (76).
In order to apply these formulae to mixtures, some method is required for determining the
acentric factor coU and the pseudocritical constants T/f and p~ for the unlike interactions. In
the present case, extended van der Waals onefluid mixing rules are applied (76) in terms of which
co O'= I(coi "{" coJ)
Tiu ~  ( 1  ku)~/Ti~T~?
c c c c c c c c c Pu :4~. { ( P i V i / T i ) + ( p j V j / T j )} / {(V/C) 1/3  ~  ( V j )1/3}
(3.91)
where k u is a binary interaction parameter, which may be optimised against experimental data,
and V,. c is the critical molar volume of pure i. When an experimental value is unavailable,
Vic= (RTiCZ~/p~) may be estimated from the correlation of the critical compression factor
proposed by Lee and Kesler (77): Z/~ = 0.2905 0.085coi. For interactions between polar and
nonpolar molecules, the polar te rm a/Tr 6 is dropped while for polarpolar interactions an
arithmeticmean combining rule is recommended for a. The third virial coefficients of pure nonpolar gases have also been correlated using a
similar model by Orbey and Vera (78) and the results are:
C(pC/RTC) 2 = C o + co C 1 ]
C o = 0.01407 + 0.02432 / Tr 28 0.00313 / Tr I°5
C 1 = 0.02676 + 0.0177 / Tr 28 + 0.040 / Tr 3  0.003 / Tr 6  0.00228 / Tr ~°5
(3.92)
Several methods have been proposed for the estimation of the interaction third virial coefficients C,jk in mixtures. Orbey and Vera (78) follow Chueh and Prausnitz (79) in proposing the relation
Cu~ = ( CuC,.k Cjk ) v3 (3.93)
in which C, 7 is evaluated from Equations (3.92) with the acentric factor and pseudocritical constants pertaining to the binary pair i and j. In a recent test against accurate experimental results for binary mixtures (80), this method was found to give satisfactory estimates.
Although none of the formulae discussed in this section are very accurate, gas densities and partial fugacity coefficients estimated by such methods for nonpolar and slightlypolar pure gases and mixtures may be accurate enough for many engineering purposes up to about two thirds of the critical density (78,80). Furthermore, as the method may be applied knowing only the critical constants and the acentric factor for each component, it is very easy to apply.
72
3.5 S U M M A R Y
In one sense, being restricted to gases at low and moderate densities, the virial equation of state is of rather limited application. Nevertheless, it has the merits of a sound theoretical basis, a large database of experimentallydetermined virial coefficients, and the ability to describe mixtures exactly. The relations set out in Table 3.2 may be used to express a wide range of thermodynamic properties of pure gases and mixtures in terms of the virial coefficients, their temperature derivatives and the heat capacity of the ideal gas. The virial coefficients may be obtained from experiment, from knowledge of the intermolecular potentialenergy functions or from the principle of corresponding states.
The relations between the virial coefficients and intermolecular forces is of real importance and has therefore been explored in detail. Equations (3.52), (3.54), and (3.55) may be used to determine B and C from Ul2 and Au 3 at temperatures where quantum corrections are unimportant. Quantum effects have also been considered above in sufficient detail to permit their calculation for the atomic and molecular systems of interest. Because such calculations can now be performed routinely, even for nonspherical molecules, it is perfectly feasible to represent the equation of state correct at least to the third virial coefficient by means of prescribed pair and triplet potentials.
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25. Y. Midzuno and T. Kihara, J. Phys. Soc. Japan 11, 1045 (1956). 26. W. Ameling, K.P. Shukla, and K. Lucas, Mol. Phys. 58, 381 (1986). 27. R.A. Aziz and M.J. Slaman, Mol. Phys. 58, 679 (1986). 28. J.A. Beattie, J.S. Brierley, and R. J. Barriault, J. Chem. Phys. 20, 1615 (1952). 29. E. Whalley and W.G. Schneider, Trans. Am. Soc. Mech. Eng. 76, 1001 (1954). 30. N.J. Trappeniers, T. Wassenaar, and G.J. Wolkers, Physica 32, 1503 (1966). 31. Reference 13, Ch. 6. 32. Reference 8, p. 48. 33. T. Kihara, Adv. Chem. Phys. 1,267 (1958). 34. T. Yokota, J. Phys. Soc. Japan 15, 779 (1960). 35. J.G. Kirkwood, J. Chem. Phys. 1,597 (1933). 36. C.S. Wang Chang, PhD Thesis, University of Michigan (1944). 37. R.T. Pack, J. Chem. Phys. 78, 7217 (1983). 38. A. Pompe and T.H. Spurling, Aust. J. Chem. 26, 855 (1973). 39. B.H. Wells, Unpublished calculations. 40. R.A. Aziz and M. J. Slaman, Chem. Phys. 130, 187 (1989). 41. L. Monchick, Chem. Phys. Lett. 24, 91 (1974). 42. B. Schramm and U. Leuchs, Unpublished results (1979); Quoted in reference 4. 43. W.H. Stockmayer, J. Chem. Phys. 9, 398 (1941). 44. M.J. Offerhaus and J. de Boer, Proc. VII Int. Conf. Low Temp. Phys., Toronto (1960). 45. Reference 8, p. 46. 46. Reference 13, pp. 6972. 47. M.E. Boyd, S.Y. Larsen, and J.E. Kilpatrick, J. Chem. Phys. 50, 4034 (1969). 48. J.W. Cooley, J. Math. Comp. 15, 363 (1961). 49. J.K. Cashion, J. Chem. Phys. 48, 94 (1967). 50. R.A. Aziz and M. J. Slaman, Metrologia 27, 211 (1990). 51. K.H. Berry, Metrologia 15, 89 (1979). 52. D. Gugan and D. W. Michel, Metrologia 16, 149 (1980). 53. R.C. Kemp, W.R.G. Kemp, and L.M. Besley, Metrologia 23, 61 (1986). 54. B.E. Gammon, J. Chem. Phys. 64, 2556 (1976). 55. G.S. Kell, G.E. McLaurin, and E. Whalley, J. Chem. Phys. 68, 2199 (1978). 56. L.A. Guildner and R.E. Edsinger, J. Res. Natl. Bur. Stand. A:80, 7.03 (1976). 57. F.C. Matacotta, G.T. McConville, P.P.M. Steur, and M. Durieux, Metrologia 24, 61 (1987).
(Corrected values of B given in reference 44). 58. R. Dashen, SK. Ma, and H. J. Bemstein, Phys. Rev. 187, 345 (1969). 59. A.M. Dunker and R.G. Gordon, J. Chem. Phys. 64, 354 (1976). 60. R.J. Le Roy and S. Carley, Adv. Chem. Phys. 42, 353 (1980). 61. A.R.W. McKellar and H. L. Welsh, Can. J. Phys. 52, 1082 (1974). 62. A.E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41,413 (1964). 63. J.A. Barker and J.J. Monaghan, J. Chem. Phys. 36, 2564 (1962). 64. J.P.M. Trusler, W.A. Wakeham, and M.P. Zarari, Int. J. Thermophys. 17, 35 (1996). 65. E.J. Janse van Rensburg, J. Phys. A 26, 4805 (1993). 66. Reference 21, pp. 145146. 67. M. Rigby, Mol. Phys. 66, 1261 (1989). 68. W.R. Cooney, S.M. Thompson, and K.E. Gubbins, Mol. Phys. 66, 1269 (1989).
74
69. C. Vega, S. Lago, and B. Garzon, Jr. Chem. Phys. 100, 2182 (1994). 70. B.J. Alder and T.E. Wainwright, J. Chem. Phys. 33, 1439 (1960). 71. M.J. Maeso and J.R. Solana, Mol. Phys. 79, 1365 (1993). 72. J.D. van der Waals, On the continuity of the gaseous and liquid states (English translation, J. S.
Rowlinson, ed.). NorthHolland, Amsterdam (1980). 73. J.H. Gibbs, I. Pavlin, and A.J. Yang, J. Chem. Phys. 81, 1443 (1984). 74. K.S. Pitzer, J. Amer. Chem. Soc. 77, 3427 (1955). 75. K.S. Pitzer and R. F. Curl, Jr., J. Am. Chem. Soc. 79, 2369 (1957). 76. C. Tsonopoulos, AIChEJ. 20, 263 (1974); ibid. 21,827 (1975); ibid. 24, 1112 (1978). 77. B.I. Lee and M. G. Kesler, AIChEJ. 21,510 (1975). 78. H. Orbey and J. H. Vera, AIChEJ. 29, 107 (1983). 79. P.L. Chueh and J. M. Prausnitz, AIChE J. 13, 896 (1967). 80. H.B. Brugge, L. Yurtlas, J. C. Holste, and K. R. Hall, Fluid Phase Equilibria 51, 187 (1989).
Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 75
4 CUBIC AND GENERALIZED VAN DER WAALS EQUATIONS
Andrzej Anderko
OLI Systems Inc. 108 American Road Morris Plains, NJ 07950, U.S.A.
4.1 Cubic Equations of State for Pure Components 4.1.1 Historical Perspective 4.1.2 Temperature Dependence of Parameters 4.1.3 Functional Form of the Pressure  Volume Relationship
4.2 Equations Based on the Generalized van der Waals Theory 4.3 Simple Equations Inspired by the Generalized van der Waals Theory 4.4 Methods for Extending Equations of State to Mixtures
4.4.1 Classical Quadratic Mixing Rules 4.4.2 CompositionDependent Combining Rules 4.4.3 DensityDependent Mixing Rules 4.4.4 Combining Equations of State with ExcessGibbsEnergy Models
4.5 Explicit Treatment of Association in Empirical Equations of State 4.6 Equations of State as Fully Predictive Models
4.6.1 Correlations for Binary Parameters 4.6.2 GroupContribution Equations of State 4.6.3 Utilization of Predictive ExcessGibbsEnergy Models
4.7 Concluding Remarks References
76
Cubic and generalized van der Waals equations of state have been a subject of active research since van der Waals (1) proposed his famous equation in 1873. During the intervening years, they played a major role in the development of fluid state theories as well as in modeling fluid behavior for practical purposes. Currently, their development is motivated primarily by industrial needs for accurate processdesign calculations and supported by steadily advancing computational techniques.
The cubic and generalized van der Waals equations are not the most appropriate models for the accurate representation of purefluid properties, because they usually lack the necessary flexibility in some regions of the phase diagram. Also, they are not the most useful methods for understanding the properties of fluids from a microscopic perspective. For such purposes, their theoretical background is usually insufficiently rigorous. However, cubic and, to a lesser extent, generalized van der Waals equations are the most frequently used equations of state for practical applications. This is due to the fact that they offer the best balance between accuracy, reliability, simplicity and speed of computation. Additionally, their success stems from the importance of multicomponent mixtures for many industrial applications. Cubic equations of state are usually the models of choice for phaseequilibrium computations for multicomponent mixtures.
The purpose of this chapter is to review cubic and generalized van der Waals equations of state as convenient practical tools for modeling the properties of multicomponent fluids. The subject is extremely broad and it is virtually impossible to enumerate all significant approaches that appeared in the literature during the last 130 years. However, an attempt will be made to identify the most useful equations of state as well as those that appear to be most promising for future research.
4.1 C U B I C E Q U A T I O N S O F S T A T E F O R P U R E C O M P O N E N T S
4.1.1 Historical Perspective
The first equation of state that was capable of reasonably representing both gas and liquid phases was proposed by van der Waals in 1873 (1). Although van der Waals derived his equation in an intuitive manner, he was able to create a model of fluid behavior that was later proven to be qualitatively correct. According to van der Waals' assumptions, molecules have a finite diameter, thus making a part of the volume v not available to molecular motion. This increases the number of collisions with the walls of the vessel and, subsequently, increases the pressure. Therefore, the actual volume available to molecular motion is vb, where b is a characteristic constant for each fluid. On the other hand, intermolecular attraction decreases the pressure. It is reasonable to assume that the pressure decrease due to intermolecular attraction is proportional to the number of molecules in a volume unit and inversely proportional to volume. Therefore, the correction for intermolecular attraction becomes (a/v2). It can be further assumed that the available volume (vb) and the corrected pressure (P+a/v e) should obey the ideal gas law, so that
P + ( v  b) : RT (4.1)
This gives rise to a twoterm equation for pressure, namely
77
R T a P =   ( 4 . 2 )
v  b v 2
where the two terms correspond to repulsive and attractive contributions to pressure. Although the original van der Waals terms do not quantitatively represent the true repulsive and attractive forces, the concept of separating the repulsive and attractive terms in equations of state has proven to be extremely valuable for the representation of fluid properties. In fact, it is the cornerstone of the generalized van der Waals theory.
The original van der Waals equation has two features that make it very convenient for calculations. First, it can be rewritten as a cubic polynomial with respect to volume. Therefore, it can be solved analytically for volume or density. This feature has been retained by contemporary cubic equations of state. Second, the two van der Waals parameters a and b can be determined from criticalpoint coordinates by applying the criticalpoint conditions:
0/9 ~ 2 p ~= ~7 = 0 (4.3)
After simple algebraic manipulations, the values of the van der Waals parameters a and b at the critical point can be found as functions of the critical temperature Tc and pressure Pc:
2 2 27 R T c
a~ = ~ ~ (4.4) 64 p2
1 RT~ b~ =  ~ (4.5)
8P~
These values remain reasonable, although not accurate, in other regions of the PVT space. The van der Waals equation inspired a huge number of researchers to work on improved
equations of state. Partington (2) summarized the modifications proposed through the late 1940's. Here, we mention only two of the earlier researchers, whose equations can be regarded as precursors of modem cubic equations of state. In 1881, Clausius (3) replaced the volume in the van der Waals attractive term by (v+c), thus creating a threeparameter equation of state:
R T a P = (4.6)
v  b (v+c) ~
This approach resembles the volumetranslation technique, which became popular roughly a century later. In 1899, Berthelot (4) introduced an equation with a temperaturedependent attractive parameter, i.e., aT = a/T. Several decades later, the concept of temperaturedependent attractive parameters proved to be essential for the practical success of cubic equations of state for phaseequilibrium calculations.
In the first half of the 20th century, the van der Waals equation and its modifications became gradually superseded by the virial equation and its empirical modifications. From the theoretical viewpoint, the statisticalmechanical foundations of the virial equation seemed superior. At the same time, empirical virialtype equations like the BenedictWebbRubin EOS
78
(5) were shown to be empirically effective for pure fluids and promising for mixtures. However, the need for simple analytical tools for the calculation of fugacities for process design spurred a revival of interest in cubic equations of state. Redlich and Kwong (6) proposed the first cubic EOS that became widely accepted as a tool for routine engineering calculations of the fugacity. The RedlichKwong equation was constructed by postulating two reasonable boundary conditions in the low and highdensity limit. In the lowdensity limit, the equation was constrained to give a reasonable second virial coefficient:
constant B = b  ..1/5 (4.7)
I
At high densities, it was noted that the reduced volume at infinite pressure could be approximated by 0.26. This condition, together with the usual criticalpoint conditions, provided guidance for the functional form of the attractive term in which the v 2 term was replaced by v(v+b):
RT a P = ~  (4.8)
v  b T1/Ev(v + b)
In analogy to the van der Waals equation, the parameters a and b were calculated from critical point conditions"
R 2 2.5 a = ~a T~ (4.9)
RT~ (4.10) b=f~b pc
where f2a and ~"2 b a r e numerical constants equal to 0.42747 and 0.0867, respectively. The RedlichKwong equation proved to be very successful for the calculation of the properties of gas mixtures. However, it was still not adequate for the modeling of both gas and liquid phases. The simple temperature dependence of the attractive parameter was insufficient for the representation of vapor pressures. Also, liquid volumes were not predicted with acceptable accuracy.
The success of the RedlichKwong EOS stimulated many investigators to introduce a number of improvements. The early modifications of the RedlichKwong EOS were reviewed by Horvath (7), Tsonopoulos and Prausnitz (8), Wichterle (9) and Abbott (10). Here, we mention the modification due to Zudkevitch and Joffe (11) because of its general character. In 1970, Zudkevitch and Joffe introduced temperaturedependent parameters a and b by constraining the quantities Oa and f2b to match vapor pressures and liquid densities along the vaporliquid saturation line. In this way, the quantities f2a and f2b became temperature dependent at subcritical conditions, whereas they remained fixed at T>Tc. Although an EOS constructed in this way exactly reproduces the vapor pressure and liquid density, it is inconvenient because of the lack of analytical expressions for ~a and £'2b and the need for extensive purecomponent data.
79
Therefore, further research in this area focused on the development of fully analytical cubic equations of state. The developments concentrated in two areas: (1) Improving the temperature dependence of the attractive parameter to control the vapor pressure predictions and (2) Improving the P(v) functional form to optimize the prediction of volumetric properties.
4.1.1 Temperature Dependence of Parameters
For the prediction of vapor pressures, it is entirely sufficient to introduce a temperature dependent attractive parameter. Thus, it is perfectly possible (and, in fact, common) that an equation of state can correctly reproduce the vapor pressure even when it gives inaccurate PVT predictions. It is convenient to express the parameter a as a product of its value at the critical point (i.e., ac) and a dimensionless function of temperature a(T):
a=aca(T) (4.11)
In 1964, the first generalized a function was introduced by Wilson (12):
a = Tr[1 + (1.57 + 1.62~)T~' ] (4.12)
where Tr is the reduced temperature and co is the acentric factor. This equation was established by forcing the EOS to give reasonable values of the terminal slope of the vapor pressure curve. However, this did not guarantee good predictions far away from the critical point and, therefore, the Wilson form turned out to be insufficiently accurate.
The first a function that gained widespread popularity was introduced by Soave in 1972 (13):
a = [1 + m(1 Trl/2)] 2 (4.13)
where m is a function of the acentric factor:
m = 0.480 + 1.574co  0.175co 2 (4.14)
Soave developed his function by forcing the EOS to reproduce the vapor pressures at Tr=0.7. Therefore, the RedlichKwong EOS with Soave's a form (frequently referred to as RKS or SRK) accurately predicts the vapor pressures at reduced temperatures ranging from about 0.6 to 1.0. Since the a form is generalized in terms of the acentric factor, it is fully predictive for nonpolar compounds with acentric factors not exceeding about 0.6 (thus excluding heavy hydrocarbons).
The Soave function played a pivotal role in the development of cubic equations of state and greatly contributed to their acceptance as tools for vaporliquid equilibrium calculations. It was used in conjunction with other equations of state such as those developed by Peng and Robinson (14), Schmidt and Wenzel (15), Patel and Teja (16), Adachi et al. (17) and Watson et al. (18). Only the dependence of m on co had to be changed to accommodate other equations of state. Further improvements of the a form focused on three problems:
80
1. Introduction of adjustable parameters to correlate the vapor pressure of any fluids, without the restriction to nonpolar fluids; 2. Improvement of the behavior of the a form in the supercritical region; 3. Development of generalized forms that overcome the limitations of Soave's function at low reduced temperatures and for large acentric factors.
The introduction of substancespecific parameters is necessary to obtain a satisfactory correlation of vapor pressures of polar or associating compounds. Several empirical extensions of the Soave function have been proposed for this purpose. Among the most significant modifications, Mathias and Copeman (19) proposed a threeparameter form:
a = [1 + Cl(1  Trl/2)~c2(1  Trl/2) 2 tc 3 (1 _ Trl/2 ])3 2 (4.15)
Soave (20) proposed a twoparameter equation:
oc=l+m(l+Tr)+n(Tr 1) (4.16)
Stryjek and Vera (21) developed an equation with one generalized (k0) and one adjustable (kl) parameter:
a = {1 + [ko(cO) + k1(1 + Trl /2)(0.7  Tr)](1  Trl/2)} 2 (4.17)
Androulakis et al. (22) analyzed several two and threeparameter forms of the a function and recommended a threeparameter form:
a = 1+ d1(1 Tr 2/3) + d2(1 Tr2'3) 2 "~" d3(1 Tr2/3) 3 (4.18)
and its simplified version with d2=0. An optimum a function should be positive at all temperatures to prevent the attractive
parameter from switching from positive to negative values. It should decrease continuously with temperature and reach zero at infinite temperature. The original function of Soave does not satisfy these requirements because it becomes negative at some temperatures in the supercritical region and then increases. The above modifications of the Soave function may also share this deficiency for some parameter sets. The simplest way to obtain a qualitatively correct a function is to use an exponential form. Early exponential forms were proposed by Graboski and Daubert (23):
O~  C 1 exp(c 2 T r) (4.19)
and Heyen (24):
a  e x p [ c ( 1  420)
81
Melhem (25) analyzed several forms and recommended an exponential version of the Soave (20) equation:
a =exp[m(1T,)+n(1Tr ' ) ] 2 (4.21)
which has a similar accuracy, but extrapolates better above the critical temperature. Yu and Lu (26) developed a generalized exponential a form:
loglo a = M(oo)(A o + A IT r I. A2 T,2)(1 Tr) (4.22)
T w u et al. (27) derived a modified exponential a form and found that it is superior to other functions:
lna = N ( M  1 ) l n T r + L(1 Tr TM) (4.23)
Although the additional adjustable parameters provide the necessary flexibility, they do not guarantee the accurate simultaneous representation of vapor pressure and derivative properties. Therefore, it is advisable to obtain the a form parameters by multiproperty regression. It is especially convenient to use vapor pressures and heat capacity departures simultaneously because Cp can be related to the second derivative of the vapor pressure, c f Mathias and Klotz (28).
Another important area of research to improve the a functions is the development of their extensions to heavy hydrocarbons, which are characterized by low reduced temperatures and high acentric factors. Carrier et al. (29) and Rogalski et al. (30,31) developed such a method in conjunction with the PengRobinson EOS (14). They divided hydrocarbons into thirteen groups and calculated the parameter b from additive contributions. The attractive parameter of a heavy hydrocarbon was expressed as
a= a(Tu)[1 +ml (1  (T I Tb)l/2)i t m2(1 T I Tu) ] (4.24)
where
m 1 = A m + B
m 2 = Cm + D (4.25)
The constants A, B, C, and D have universal values and, therefore, the only characteristic parameters are a(Tb) and m. The value of a at the normal boiling point Tb is determined from one vaporpressure datum and m is calculated from group contributions. This fairly complicated technique gives very good results for vapor pressures of heavy hydrocarbons at low reduced temperatures.
A particularly simple and elegant approach has been proposed by Twu et al. (32), who observed that the a function should be linear with respect to the acentric factor, i.e.,
82
a = a (°) +co(a (1) o~ (°)) (4.26)
This is in sharp contrast with the original Soave form, which is equivalent to a fourthorder equation with respect to the acentric factor. The linearity of the a form is remarkably good and makes it possible to make extrapolations for compounds with high acentric factors. Both terms a (°) and a (1) are expressed by the same function of reduced temperature, see Equation (4.23). The method is applicable to any cubic equation of state.
A different approach has been proposed by Dohrn (33), who developed a correlation for twoparameter cubic equations by using the liquid density at 20°C and the normal boiling point as characteristic parameters instead of To, Pc and co. While this approach may be advantageous for large molecules, it does not address the issue of diminished accuracy at low reduced temperatures.
Besides the generalizations for hydrocarbons and other nonpolar fluids, several attempts have been made to generalize the a form for polar fluids. Such attempts are hampered by the lack of a truly effective characterization parameter for polar fluids that would be comparable to the acentric factor for nonpolar fluids. For example, Guo et al. (34) utilized the dipole moment and Valderrama et al. (35) used the product cozc where zc is the critical compressibility factor. Although such correlations are adequate for selected groups of polar compounds, they cannot be generally valid in the whole universe of polar, associating and nonpolar compounds. Therefore, the use of individual parameters for polar and associating fluids should be regarded as the only safe approach.
In contrast to the attractive parameter, the repulsive parameter b of cubic equations of state is usually kept independent of temperature. A temperaturedependent parameter b usually does not lead to any significant improvement of the equation's accuracy. As shown by Salim and Trebble (36), a temperaturedependent b is even dangerous because it results in the prediction of negative heat capacities at constant volume when the pressure exceeds a certain value.
4.1.2 Functional Form of the PressureVolume Relationship
While an appropriate temperature dependence of the attractive parameter is sufficient for the accurate representation of vapor pressures, the functional form of the pressurevolume relationship is important for the prediction of volumetric properties. To keep the cubic form of the equation of state, at most five parameters can be used on an isothermal basis. Since the twoparameter RedlichKwong EOS gives a reasonable representation of fluid properties, it might appear that a fiveparameter EOS would be vastly superior. Unfortunately, this intuitive reasoning is not true. In fact, the additional parameters can bring only a limited incremental improvement of the accuracy because of the fundamental limitations of the cubic form.
The most popular modification of the pressurevolume relationship, due to Peng and Robinson (14), does not involve any additional parameters beyond the original two. Peng and Robinson recognized that the critical compressibility factor of the RedlichKwong EOS, which is equal to 0.333, is greater than the compressibility factors of practically all fluids. Therefore, they postulated a function that reduces zc to 0.307:
RT a(T) P = ~ (4.27)
v  b v ( v + b ) + b ( v  b )
83
The PengRobinson EOS predicts better liquid volumes for mediumsize hydrocarbons and other compounds with intermediate values of the acentric factor. However, it is worse than the SoaveRedlichKwong EOS for compounds with small acentric factors.
When a third parameter is introduced into a cubic EOS, the critical compressibility factor becomes substance dependent. Although a threeparameter equation can be forced to predict the correct critical compressibility factor, the isotherms at low and h i ~ pressures are then distorted much more than can be tolerated for the purposes of modeling PVT relations. Better overall results are usually obtained when the apparent (calculated) compressibility factor ~ is greater than the real one Zc. Early threeparameter cubic equations of state were proposed by Fuller (37) and Heyen (24). These two equations involve temperaturedependent covolumes b, which adversely affects their accuracy for derivative properties.
A wellknown threeparameter cubic EOS was proposed by Schmidt and Wenzel (15), who designed their equation as a form of interpolation between the SoaveRedlichKwong and PengRobinson equations. This interpolation ensures a better representation of liquid volumes over a reasonable range of acentric factors. Schmidt and Wenzel analyzed a general form
RT a(T) P = (4.28)
v  b v 2 + ubv + wb2
and constrained the parameters u and w by
u = 1 w (4.29)
Additionally, they related w to the acentric factor by w=3co. The apparent critical compressibility factor was found to be a linear function of the acentric factor.
Other threeparameter cubic equations were proposed by Harmens ancl Knapp (38):
RT a(T) P =   (4.30)
v  b v 2 + v c b  ( c  1 ) b 2
and Patel and Teja (16)"
RT a(T) P = (4.31)
v  b v ( v + b ) + c ( v  b )
Patel and Teja treated the apparent compressibility factor as an adjustable parameter, which is temperature dependent for 0.9<Tr<l.0.
Yu and Lu (26) considered the fourparameter form proposed originally by Schmidt and Wenzel (Equation (4.28)) and arrived at a different constraint for the parameters u and w, i.e., uw = 3. Their final equation takes the form
RT a(T) P = ~  (4.32)
v  b v ( v + c ) + b ( 3 v + c )
where c = wb. Twu et al. (39) systematically analyzed the cubic equations of state on a uw diagram with respect to the representation of volumetric properties. They checked 21 possible
84 combinations of u and w for a number of alkanes, alcohols and glycols. The optimum constraint was found to be uw=4, which is closer to the result of Yu and Lu than to that of Schmidt and Wenzel. This gives the form
R T a(T) P = ~ (4.33)
v  b v(v + 4b) + c(v + b)
with the apparent critical compressibility factor optimized as a third parameter. In the development of these threeparameter equations, use was made of the fact that
acceptable representation of both low and highdensity regions requires that the apparent critical compressibility factor be treated as an empirical parameter. A simple, yet useful method to achieve this is the socalled volume translation. This concept, proposed by Martin (40) and refined by Peneloux et al. (41), is based on a translation along the volume axis, i.e.,
v~anslated _. Vioriginal  C i (4.34)
If the translation parameter c for a mixture is related to composition by a linear mixing rule, the partial molar volumes are also translated and the fugacities are multiplied by a compositionindependent coefficient so that the phaseequilibrium conditions (i.e., the isofugacity criteria) are not affected by the volume translation. Peneloux et al. (41) showed that this method markedly improves the prediction of liquid volumes except in the vicinity of the critical point. It should be noted that the original van der Waals EOS becomes, after volume translation, identical to the Clausius EOS (3). Martin (40) found the Clausius EOS to be the best cubic equation with respect to the representation of gasphase properties. Joffe (42) and Kubic (43) used the Clausius EOS in a modified form. Salerno et al. (44), Czerwienski et al. (45) and Androulakis et al. (22) presented a translated van der Waals EOS, called vdW711, that contains a temperaturedependent volume translation. The translating function was fitted to experimental liquid volumes including the critical volume and generalized in terms of the acentric factor.
A somewhat different volume translation was presented by Mathias et aL (46), namely
0.41 / V translated __ v°riginal + S b f c 0.41 + 6 (4.35)
where 6 is the bulk modulus,
v2( 6 =  (4.36)
R T r
and the correction functionfi is chosen so that the actual critical volume is obtained:
f e  Ve  ( r e °figinal k S) (4.37)
Although a volume translation, defined in such a way, is very accurate, it introduces a thermodynamic inconsistency into the calculations because the bulk modulus is a function of
85
the equation of state and, therefore, the fugacity and volume calculations are not decoupled as in Peneloux's method. It is interesting to note that the concept of volume translation has even been used to develop an equation of state that reproduces the solid phase (47).
It should be noted that the invariance of the isofugacity condition with respect to the (original) volume translation is only one example of the useful mathematical properties of the cubic equations. In particular, Leibovici (48) showed that the properties calculated from the genetic SchmidtWenzel EOS (see Equation (4.28)) can be divided into two classes, called variant and invariant properties. The variant properties depend on the individual parameters whereas the invariant properties depend only on the combined quantity
l + u + w K = ~ (4.38)
( u + 2 ) ~
The difference of fugacities is an example of an invariant property. Other invariant properties include entropy, heat capacity at constant P or V, enthalpy of vaporization, internal energy and Helmholtz energy. On the other hand, volume is an example of the variant properties, which also include enthalpy, sound velocity, JouleThomson coefficient, Gibbs energy, isothermal compressibility and adiabatic compressibility. Thus, the volume translation can be applied to improve the representation of any variant property while keeping the invariant properties constant.
In addition to the threeparameter cubic EOS discussed above, several authors tried to improve the flexibility of cubic equations by using four or five parameters. A fourparameter form was used by Trebble and Bishnoi (49) and Salim and Trebble (50):
RT a(T) P =   (4.39)
v  b v 2 + ( b + c ) v  ( b c + d 2)
Fiveparameter cubic equations were analyzed by Adachi et al. (51) in the traditional twoterm form, i.e.,
RT a(T)[v c (T)] P = (4.40)
v  b [ v  b][v d(T)][v + e(T)]
and by Kumar and Starling (52) in a genetic form, i.e.,
l + d, ( r )o + 4 ( r )o ~ z = ( 4 . 4 1 )
1 i d3 (T)p nt d4 (T)p2 t d5 (T)p 3
The KumarStarling EOS was more recently modified by Chu et aL (53), who added additional temperaturedependent functions in the nearcritical region and attempted to generalize the parameters for polar fluids.
The results obtained from the four and fiveparameter equations are only slightly better than those obtained from the threeparameter EOS. In most cases, the differences resulting from a more flexible PV dependence are masked by the temperature dependence of parameters. It should be noted that, in contrast to other cubic equations, the original Kumar
86
Starling EOS was not constrained at the critical point, which automatically improves the results in the farfromcritical regions.
The results of several authors indicate that cubic equations of state require at least three parameters for a reasonable representation of both vapor and liquid volumes. More than three parameters lead to only small incremental improvements. The attractive parameter has to be temperaturedependent. However, the temperature dependence of other parameters, such as the covolume, may lead to spurious results. It appears that the property that can be most easily correlated using cubic equations of state is the vapor pressure. It is customary that cubic equations can predict vapor pressure within 1  2 % when their parameters are generalized in terms of To, Pc and ¢o. With parameters fitted to an individual fluid's properties, the deviations can drop to a fraction of a percent. The liquid and vapor volumes can be predicted within 3  4 % by the more advanced equations with at least three parameters. All cubic equations give a very poor performance in a relatively wide "nearcritical" region' This is not only due to the fact that every cubic equation shows classical critical behavior. Another reason for this deficiency is the relative rigidity of the cubic form, which imposes limits on the quality of the representation of derivative properties such as residual enthalpy, entropy and heat capacity. Not surprisingly, the enthalpy of vaporization is usually the derivative property that is most accurately reproduced by cubic equations. This results from the fact that the enthalpy of vaporization is related to the slope of the vapor pressure curve by the ClausiusClapeyron equation. Therefore, the relative error of the predicted enthalpy of vaporization is usually very similar to the relative error of vapor pressure. However, the residual enthalpies of the liquid and the vapor phases are usually predicted with a significantly lower accuracy. Secondorder derivative properties such as the specific heat capacities Cp and Cv as well as the Joule Thomson coefficient are predicted with a still lower accuracy, but the results are usually reasonable (i.e., deviations of the order of 10  15 % are typical). The equations that are correctly constructed (i.e., with carefully introduced temperaturedependent parameters) produce consistent results and anomalies are usually not detected. One of the few anomalies that is commonly detected in cubic equations of state is their failure to qualitatively reproduce Cv in the critical region, c f Salim and Trebble (50). In fact, all cubic equations of state give physically incorrect Cv versus density relationships within ca. 2030 K from the critical point. Numerical values of deviations between experimentally determined properties (i.e., residual thermodynamic functions and P VT properties) and those predicted from selected cubic equations of state can be found in the papers of Trebble and Bishnoi (49), Salim and Trebble (50), Lielmezs and Mak (54), Mak and Lielmezs (55) and Solimando et al. (56) and in chapter 4 of Malanowski and Anderko (57).
For practical purposes, cubic equations can be easily tailormade to fit specific properties in a specified range, e.g., by adjusting the parameters of the temperaturedependent attractive term or by making a volume translation. However, it should be noted that such finetuning may result in inaccurate extrapolations, e.g., in the supercritical region.
4.2 E Q U A T I O N S B A S E D O N T H E G E N E R A L I Z E D V A N D E R W A A L S
T H E O R Y
Although the original van der Waals equation was developed on the basis of simple molecular concepts, cubic equations of state can be regarded as nothing more than comprehensive empirical correlations of fluid properties. Although a cubic EOS can work very well as a whole, neither its repulsive nor its attractive terms reflect the physical reality.
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Therefore, numerous researchers focused their efforts on the development of equations of state with clear physical foundations.
The basic premise of the generalized van der Waals theory is the separation of the repulsive and attractive contributions to pressure. Originally postulated by van der Waals in 1873, it was theoretically established several decades later by Zwanzig (58), Hemmer et al. (59) and Barker and Henderson (60) for some model intermolecular potentials. Subsequently, more practicallyoriented approaches were presented by Vera and Prausnitz (61), Sandier (62), Lee et al. (63), Lee and Sandier (64) and Abbott and Prausnitz (65). The development of equations of state based on the generalized van der Waals theory was strongly influenced by the progress in the theory of hardbody fluids, which are commonly used to model the effects of repulsive interactions. Since the structure of fluids has been shown to be determined primarily by repulsive forces, the equations of state for hardbody fluids can be conveniently used as reference terms for equations of state for real fluids.
Camahan and Starling (66) developed an accurate equation of state for hard spheres, i.e.,
1+~+~2 __~3 (4.42) Zrep = (1 _~)3
where ~ = zcpcr3/6 is the reduced density with a being the hardsphere diameter. This equation has been extended to hard convex bodies by Boublik and Nezbeda (67):
1 +(3a 2)~ +(3a 2  3 a + 1)~ 2  a 2 ~ 3 Zrep " (1  ~)3 (4.43)
where a is the nonsphericity parameter defined as a = RoSo/Vo, where R0, So and V0 are the mean curvature, mean surface and volume of the hard convex body, respectively. A different expression for the effect of hardcore geometry on molecular repulsions was formulated by Naumann et al. (68). Tildesley and Streett (69) obtained an accurate expression for dimers (or dumbbells). More recently, several authors (7072) obtained equations of state for hardsphere chain fluids.
Several authors combined the rigorous hardbody repulsive terms with empirical attractive terms that had been used earlier in cubic equations of state. In early attempts to utilize the hardcore repulsive terms to construct equations for real fluids, the simple van der Waals or RedlichKwong attractive terms were employed as perturbations by Carnahan and Starling (73). The CarnahanStarling  van der Waals EOS was further studied by Anderson and Prausnitz (74) and Dimitrelis and Prausnitz (75). More recently, the CarnahanStarling EOS was combined with the SchmidtWenzel attractive term (76) and the PatelTeja attractive term (77). The CarnahanStarling EOS was also used with attractive terms expressed by truncated virial expressions (7881). For example, Dohm and Prausnitz (81) formulated an attractive term that optimizes the representation of the critical isotherm for several fluids:
4a z = zrep (CS)  ~~ r/(1  1.41r/+ 5.077/2 ) (4.44)
where 7/= b / 4v.
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The combinations of the CarnahanStarling repulsive term and various empirical attractive terms are not more accurate than comparable cubic equations with respect to the representation of pure fluid properties. They are usually reported to give better predictions of volumes in the highdensity region, but this is mostly due to the fact that they are not constrained at the critical point, which facilitates the fitting of volumes. Thus, they are mostly useful for testing and developing mixing rules, for which a correct separation of repulsive and attractive terms may be advisable.
Further progress in the area of generalized van der Waals equations of state resulted from the use of molecularsimulation data, which made it possible to establish accurate attractive terms for selected intermolecular potentials. For example, Alder et al. (82) obtained an expression for the attractive term of a squarewell fluid:
zat ~ =~~mD,, , , , (u /kT)"(v° /v~ (4.45) n,m
where u is the characteristic energy, v ° is the hardcore volume and Dnm are numerical coefficients. An alternative expression for the attractive term of a squarewell fluid was proposed by Heilig and Franck (83), who constructed a Pad6 approximant from the second and third virial coefficients:
B1) Za~ (v 2  v C / B ) (4.46)
where both B and C are expressed in terms of the squarewell parameters. The theoretically based repulsive and attractive terms have been used by several authors to
construct generalized van der Waals  type equations of state. Chen and Kreglewski (84) and Kreglewski and Chen (85) combined Equations (4.43) and (4.45) into the BACK (Boublik AlderChenKreglewski) equations of state. They determined the parameters Onto from the residual energy and volumetric data of argon and treated them as universal constants. Both hardcore volume and characteristic energy were assumed to be temperaturedependent:
v ° = v°°[1_ Cexp(3u ° / kT)] 3 (4.47)
u / k  ( u ° / k)0 + / (4.48)
The BACK equation contains three parameters that have to be determined from experimental data, i.e., v °°, aand u°/k. The remaining parameters are assigned fixed values.
Beret and Prausnitz (86) and Donohue and Prausnitz (87) created the first generalized van der Waalstype equation of state that was designed for chains of segments. They utilized Prigogine's (88) arguments that the rotational and vibrational contributions to the canonical partition function can be factorized as
qr, v = (qr, v)ext (qr, v )int (4.49)
89
where (qr,v)ext and (qr,v)int are the external (densitydependent) and internal (density independent) contributions to the partition function, respectively. The essential assumption was that the external contributions could be calculated in essentially the same way as the contributions of translational motion. Since the contribution of each translational degree of freedom is (qrepqattr) and the total number of external degrees of freedom is assumed to be equal to a substancespecific parameter 3c, the canonical partition function is postulated to be
1 V N Nc Q=~.V(~3) (qrepqattr)(qr, v)int (4.50)
where A is the thermal de Broglie wave length. This partition function has two characteristic features. First, the idealgas term (V/A3) u is factorized so that the idealgas limit can be recovered (which is not the case in the original work of Prigogine). Second, the substance specific number of external degrees of freedom in a system of N molecules, equal to N¢, replaces the number of molecules in the part of the partition function that accounts for nonideality (i.e., (qrepqattr)UC). The partition function factorized in this way gives rise, according to elementary statistical thermodynamics, to a general expression for the compressibility factor:
+Za°)
where Zrep is calculated from the CamahanStarling equation corrected for the idealgas term and Zatt~ is obtained from the equation of Alder et al. (82). The complete equation was called PHCT (PerturbedHardChain Theory). Similarly to the BACK EOS, the PHCT equation contains three parameters for each substance: the characteristic temperature T* related to the depth of the potential well, characteristic volume v* related to the size of the segments and one third of the external degrees of freedom c. These parameters were found to correlate smoothly within homologous series of hydrocarbons.
The PHCT EOS was further refined by Morris et al. (89), who replaced the attractive term based on the squarewell potential by one based on the LennardJones potential. Jin et al. (90) recast this equation in terms of group contributions. Cotterman et al. (91) improved the performance of PHCT at low densities by separating the attractive Helmholtz energy into high and lowdensity contributions and applying a switching function to interpolate between them. Kim et al. (92) developed a simplified version of PHCT (SPHCT = Simplified Perturbed HardChain Theory) by replacing the complex attractive part of PHCT by a simpler expression that combines ingredients of the radial distribution function theory for squarewell molecules with the lattice statistics for chain molecules:
cv*Y Zattr = Z M ~ (4.52)
V+ v*Y
with
Y = exp(e.q / 2ckT)  1 (4.53)
90
where ZM is the maximum coordination number, e is the characteristic energy per unit surface area and q is the normalized surface area..
Chien et aL (93) developed their COR (Chain of Rotators) EOS by adopting a different approximation for the chainlike structure of molecules than that used in PHCT and related models. They suggested treating the first segment of a chainlike species like a free molecule. The subsequent segments are assumed to rotate about their neighbors. Therefore, the configurational (i.e., excluding the internal degrees of freedom) canonical partition function is given by
Oconf Nc = Qtqr Qa~ (4.54)
where Qt is the translational partition function calculated in analogy to that of a spherical molecule, qr is the partition function of an elementary rotator and c is the number of rotational degrees of freedom of the chain. Qt is calculated from the CamahanStarling expression whereas qr is evaluated by forcing Equation (4.54) to represent the partition ftmction of a hard dumbbell, i.e., Qdb 2N = Qtqr • The obtained expression is arbitrarily assumed to be valid for chains containing more than one segment. The Qattr contribution is calculated from the expression of Alder et al. given by Equation (4.45). The main difference between COR and PHCT is in the term that is not affected by the c parameter. In the case of PHCT, it is the ideal gas term. In COR, it is the singlesegment translation term.
Deiters (94) developed a different generalized van der Waals EOS by deriving a semi empirical approximation for the radial distribution function for squarewell fluids, involving corrections for nonspherical molecular shape, "soft" repulsive potential and threebody effects. The equation was shown to reproduce the P V T behavior in a large pressure range using only three characteristic parameters.
An important equation for fluids with chainlike molecules was proposed by Chapman et
al. (95) and Huang and Radosz (96). This equation, called SAFT (Statistical Associating Fluid Theory), is a combination of Wertheim's (97) perturbation theory for intermolecular association and chain formation with the usual expressions of Camahan and Starling and Alder et al. In the SAFT equation, the Helmholtz energy (and all thermodynamic functions that can be derived from it) is expressed as
A = A ig ..1_ A seg + A chain + A a~oc (4.55)
where the four terms on the right hand side represent contributions from the ideal gas, intermolecular interactions between segments, the formation of chains from segments and association. The segment Helmholtz energy is
A 'eg = r ( A hs + A disp) (4.56)
where r is the number of segments and A hs and A disp are the hardsphere and dispersive contributions for a single segment, respectively. The A hs and A disp terms are calculated from the CarnahanStarling and modified Alder et al. expressions, respectively. The A chain and A ass°c
terms are obtained from Wertheim's perturbation theory of association. For more information about this theory, the reader is referred to the chapter of this book that deals with associated fluids (Chapter 12).
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More recently, Song et aL (98) presented a different equation, called PHSC (Perturbed HardSphere Chain). It is based on Chiew's (72) PercusYevick integralequation theory for chain molecules. The methods for treating chain molecules in both SAFT and PHSC are firmly based on statistical mechanics and are superior to those used in PHCT or COR. Both SAFT and PHSC have been shown to be very useful primarily for systems containing polymers even though they can also be applied to other systems (e.g., SAFT was used for heavy hydrocarbons). A more complete discussion of PHSC and SAFT can be found in the chapter on equations of state for polymer systems (Chapter 14).
Several authors attempted to improve the performance of generalized van der Waalstype equations for polar fluids by adding an additive term that explicitly accounts for anisotropic interactions that result from the presence of dipoles and quadrupoles. A perturbation theory for mixtures of multipolar fluids was developed by Gubbins and Twu (99) following an earlier work of Stell et al. (100). Vimalchand and Donohue (101) incorporated the perturbation expansion of Gubbins and Twu into the PHCT EOS. The resulting equation, called PACT (PerturbedAnisotropicChain Theory) contains an additional Z ani term:
z = 1 + C(zreP + ziS° q_ zani) (4.57)
Similarly, Saager and Fischer (102) added dipolar and quadrupolar terms to the BACK equation. For the dipolar and quadrupolar terms, they used expressions fitted to molecular simulation data. Since the anisotropic term from the perturbation theory is quite complex, several simplified expressions have been developed. A particularly simple formula was proposed by Bryan and Prausnitz (103), who incorporated dipolar effects into a single term whose density dependence is identical to that of the CarnahanStarling term:
zrep+dipolar ._ 1 + ft ' lr/+ ft2Jr/2  ft31r/3 (4.58) (10) 3
where f i l l , f[21 and ft31 are functions of the reduced dipole moment. Brandani et al. (104) developed an equation of state on the basis of this term. Another simplification of the multipolar term was proposed by Sheng and Lu (105).
The main reason for introducing the multipolar terms into generalized van der Waals  type equations of state is to improve the predictions for mixtures. It has been found that smaller values of binary parameters are needed when the multipolar terms are taken into account and, in favorable cases, phase equilibria can be predicted without any binary parameters. The multipolar terms are, in general, not needed for improving the representation of pure fluid properties. It should be noted that the improvement for mixed fluids is usually limited to mixtures that contain multipolar and nonpolar, but not hydrogenbonding fluids. However, the usefulness of the dipolar term has also been demonstrated for systems such as NaC1 + H20 or KC1 + H20 at very high temperatures (106).
A different, more empirical approach to decoupling the effect of polar interactions from the rest of the equation of state has been proposed by Lee and Chao (107). They postulated that the properties of hydrogenbonding fluids could be treated using the dipolarfluid formalism and separated the nonpolar and polar contributions to the pressure of water:
P   Prep + Patt, np ~ Patt, p (4 .59)
92
where the subscripts np and p stand for the nonpolar and polar contributions, respectively. The nonpolar contribution was evaluated from the BACK EOS with parameters determined from the LennardJones constants that represent the nonpolar interactions in water. The polar contribution was obtained by forcing the complete equation of state to be identical with the accurate multiparameter EOS developed by Keenan et al. (108). Later, Lee and Chao (109) and Peng and Chao (110) developed closedform empirical expressions for Patt,p. The model was also extended to other polar fluids using a correspondingstates transformation. For mixtures, the model requires two binary parameters in the BACK EOS. The polar pressure is extended to mixtures using a correspondingstatesbased technique. In this approach, the effects of hydrogen bonding are not treated explicitly and the polar term effectively describes both dipoledipole and hydrogenbonding interactions. The equation was shown to give good accuracy for correlating phase equilibria in aqueous hydrocarbon systems. It should be noted, however, that equations of state that explicitly take into account hydrogen bonding (described here in Section 4.5) can give similar results for aqueous hydrocarbon systems with only one binary parameter and have a simpler form.
The main virtue of the equations based on the generalized van der Waals theory is their ability to represent the saturated properties and P V T data outside of the saturation region using only a few (typically three) substancedependent parameters. However, they contain a large number of universal parameters and their functional form is much more complex than that of cubic equations of state. Although the characteristic parameters of these equations are not correlated with the critical temperature, pressure and acentric factor, they usually follow well established patterns. For example, the parameters of the SAFT EOS follow regular linear correlations within some homologous series of hydrocarbons (96). This makes it possible to apply the equation to heavy hydrocarbons, for which very little experimental information is available. It is becoming increasingly evident, however, that this family of equations is most useful for systems containing polymers as well as small molecules. For such systems, cubic equations of state appear to be not very well suited whereas models like SAFT and PHSC are at their best.
4.3 S I M P L E E Q U A T I O N S I N S P I R E D BY T H E G E N E R A L I Z E D V A N
D E R W A A L S THEORY
Several researchers have presented methods that attempt to combine the simplicity of cubic equations of state with the theoretical background of equations based on the augmented van der Waals theory. This approach has resulted in the development of models that include reformulated cubic equations, quartic equations (which still can be solved analytically) and higherorder polynomials. In the majority of models of this type, the CarnahanStarling hard sphere term is replaced by a simple approximation, i.e.,
Zrep : V + C
v  b (4.60)
This term can be used to approximate the CamahanStarling expression by appropriately adjusting the parameter c or by expressing it as a function of b and other parameters.
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An early example of such an approximation is the expression of Scott (111), in which the parameter c is related to the covolume b, i.e.,
2 v + b = (4.61) ZreP 2V b
This equation was later combined with the RedlichKwong attractive term by Ishikawa et al.
(112) to yield a cubic EOS. A more complex cubic EOS was developed by Guo et al. (113) and Kim et al. (114), who simplified the COR EOS. The simplified equation, called CCOR (Cubic Chain of Rotators), contains analytical approximations for the hardcore and rotational pressure contributions in COR:
p = RT(1 + 0.77b / v) _ c R (O.055RTb / v) _ a ( T ) _ bd (4.62) v  0.42b v  0.42b v(v + c (T ) ) v(v + c ( T ) ) ( v  0.42b)
Unlike in most modem cubic equations of state, the critical compressibility factor in the CCOR EOS is constrained to match the experimental value. This appears to impair the equation's performance in some regions of P V T space.
Several authors developed quartic equations of state in the attempt to improve the prediction o f P V T properties. Among them, Kubic (115) developed a quartic equation of state by simplifying the BeretPrausnitz generalized van der Waals partition function:
R T 1 .19cbRT a P =  ~ (4.63)
v v ( v  0.42b) (v + d) 2
Soave (116) developed a quartic EOS by combining a modified form of Equation (4.60) with a genetic attractive term, i.e.,
b av t p = R T l + c  (4.64) v v  b ( v + d ) ( v + e )
Soave further reduced the number of independent adjustable parameters to three and found that the quartic equation was superior to cubic equations in the supercriticalfluid region. Another quartic EOS was developed by Shah et al. (117), who obtained a repulsive term by fitting molecularsimulation results for hard spheres:
R T f lk 1R T av + k o tic
P = (v  kob ~ + (v  k0fl) 2  v(v + e) (v  kofl ) (4.65)
where ko, kl, e and/3 are coefficients regressed from simulation data. In the above equation, the first two terms on the fighthand side form the repulsive contribution and the third term is an attractive contribution, obtained by fitting experimental data for argon.
An equation of a higher order was proposed by Anderko and Pitzer (118), i.e.,
94
+ a p r + ~p2 + ~p3 r (4.66) v bp~ r
where Pr is the reduced density and the parameters c, a, 13 and ~, are functions of reduced temperatures. Anderko and Pitzer found this equation to be very effective over large ranges of pressure and temperature.
From the practical point of view, one of the main reasons for using the equations based on the generalized van der Waals theory (whether simplified or not) is their superiority over cubic equations of state with respect to the representation of PVT properties, especially in the supercritical region and for dense liquids. This is usually achieved with a reasonable number of parameters. However, it should be kept in mind that most of the equations based on the generalized van der Waals theory are not constrained to reproduce the critical point (Deiters' equation (94), Soave's quartic equation (116) and the implementation of the SPHCT EOS by van Pelt et al. (231) are notable exceptions). This is due to the fact that their form is usually too complex to derive analytical relations for the critical point. As a result, the equations overshoot the critical temperature and pressure by considerable amounts. Such behavior is unavoidable when no nearcritical corrections are introduced and is very similar for practically all equations that are not constrained at the critical point. This is illustrated in Figure 4.1 for methane. It should be noted that, if the equations were constrained at the critical point, the overall numerical deviations between experimental data and predicted values would be worse. Thus, an equation that is not constrained at the critical point will usually perform better when compared with a large set of experimental data points. Therefore, the frequently published comparisons between generalized van der Waalstype equations that are not constrained at the critical point and cubic equations (which are usually constrained) should be taken with a grain of salt. Additionally, it should be noted that the overprediction of the critical point coordinates is a serious disadvantage for process simulation because it may result in qualitatively incorrect phase equilibrium calculations. Such a disadvantage is usually not compensated by a better prediction of PVT properties in most regions of the PVT phase diagram. This is one of several reasons why cubic equations are, by far, the most popular equations of state for industrial process design.
4.4 M E T H O D S F O R E X T E N D I N G E Q U A T I O N S O F S T A T E T O
M I X T U R E S
Representation of mixedfluid properties is the main, if not the only, purpose of using cubic and generalized van der Waals equations of state. Whenever only pure fluids are of interest, equations of this kind are not the method of choice. Here, we outline the various methods for extending the equations discussed above to mixtures.
4.4.1 Classical Quadratic Mixing Rules
The classical quadratic mixing rules can be easily deduced from the composition dependence of virial coefficients, which is known from basic statistical thermodynamics. If we
95
90
80
70
6O
n 50 o "It '"
40 13_
30
20
10
oC~ ~" • \
I o I I
cb I I
Q I
0 I !
O I 0 I ) I
I I I
0 5 10 15 20 25 30 P / mol d m 3
Figure 4.1 Vaporliquid coexistence curve for methane calculated from a typical van der Waalstype equation of state by fitting its parameters to vaporpressure and liquiddensity data without imposing criticalpoint constraints (solid line). Experimental data are shown as solid circles. Also, the calculated critical isotherm is shown as a dashed line and compared with experimental data (hollow circles).
expand the original van der Waals EOS in a power series around zero density, we get
Z a z = 1 + (4.67) i=1 RTv
This allows us to identify the second, third, and higher virial coefficients:
B = b  a / R T
(4.68) C b ~ D b 3 = , = , e t c .
To maintain consistency with the quadratic composition dependence of the second virial coefficient, the parameters a and b can be, at most, quadratic functions of composition. However, the cubic composition dependence of the third virial coefficient imposes a more stringent restriction that the parameter b should be only a linear function of composition. Very
96
similar conclusions can be reached for other equations of state. Thus, the simplest (although not the only possible) function for the attractive parameter is given by
a = ~ ~ XiXja O. (4.69)
where the crossterm a• is related to the purefluid terms aii and aj~ by the geometricmean rule with one adjustable binary parameter ku:
a u = (aiiab.) '/2 (1  ku) (4.70)
The covolume b is usually expressed by a linear mixing rule:
b = E x i b i (4.71)
However, many authors use a quadratic mixing rule for b in analogy with the mixing rule for a even though it violates the composition dependence of the third and higher virial coefficients. In addition to the results obtained from the virial equation, the classical mixing rules are supported by the results of Leland et al. (119,120), who derived them from the theory of radial distribution functions.
In the case of the generalized van der Waals equations, the mixing rules are either analogous to the quadratic mixing rules described above, or are derived from more sophisticated theories. For the ubiquitous CarnahanStarling term, an extension to mixtures has been derived by Boublik (121) and Mansoori et al. (122):
1 + (3DE / F  2)~ + (3E 3 / F 2  3DE / F + 1)~ 2  ( E 3 / F 2 )~3 z = (4 .72 )
( 1   ~ ) 3
with F = ~Y.i~i 3, E = ~_,xi~i 2 and D = Exicri. Dimitrelis and Prausnitz (75) found that this equation is superior to the onefluid approximation (essentially given by Equation (4.71)) when applied to the correlation of VLE in mixtures of nonpolar molecules that appreciably differ in size. The mixing rules for the attractive terms are equationspecific, but they generally follow the form of Equations (4.69) and (4.71).
The classical quadratic mixing rules are, in general, suitable for the representation of phase equilibria in multicomponent systems containing nonpolar and weakly polar components. Their performance has been extensively tested by Han et al. (123) for seven van der Waalstype equations including five cubic ones. It should be noted that very similar results are obtained from vaporliquid equilibrium calculations using various equations of state. This indicates that the form of the mixing rules is more important than the particular PKT relationship embodied in an EOS.
There are very few examples that would indicate the superiority of generalized van der Waalstype equations over cubic equations when it comes to the correlation of phase equilibria in typical nonpolar systems. Peters et al. (124) obtained better results from the SPHCT EOS (92) than from the SoaveRedlichKwong EOS for the correlation of the phase behavior of mixtm'es containing short and long nalkanes. Ashour and Aly (125) found that the correlation of VLE for mixtures containing CO2 and fattyacid esters could be somewhat improved when
97
the CamahanStarlingBoublikMansoori expression was used as a repulsive term. Such findings are corroborated, to a certain extent, by the study of the topology of phase diagrams obtained from the CamahanStarlingRedlichKwong EOS by Kraska and Deiters (126) and the SPHCT EOS by van Pelt et al. (232). Although the global phase diagrams obtained from these equations are similar to those obtained from the van der Waals EOS when the components have equal sizes, significant differences appear when the sizes are different (e.g., a new tricritical line).
In most practical applications, a linear mixing rule is used for the covolume b. However, several authors used a quadratic form for b even though it violates the cubic composition dependence of the third virial coefficient. In this way, a second binary parameter can be introduced. This parameter is usually found to be useful for correlating VLE in mixtures with components of very different sizes, such as hydrogen and hydrocarbons (127). Similarly, it usually improves results for solid  supercriticalfluid equilibria because of the large size differences in such systems. This was demonstrated by Caballero et al. (128) for a large number of solid + supercritical fluid systems.
An important application of cubic equations of state with classical quadratic mixing rules is the modeling of reservoirfluid phase behavior and volumetric properties. Research in this field was initiated by Katz and Firoozabadi (129), who showed that the PengRobinson EOS can be used to compute highpressure reservoir crude VLE by characterizing the heptaneplus fraction and using a methane  last fraction adjustable binary parameter. This subject was reviewed by Firoozabadi (130), who concluded that the prediction of VLE in reservoir fluids is influenced mainly by the binary parameters between C1, CO2, Nz and the last fraction. Cubic equations are, however, deficient in the critical region, where they have a tendency to fail to predict sharp changes of saturation pressures and liquiddropout curves. In general, the use of equations of state for modeling reservoir fluids requires judicious tuning of both pure component and binary parameters because both phase equilibria and volumetric properties are of interest. For example, AasbergPetersen and Stenby (131) obtained accurate volumes and liquiddropout curves by using a volumetranslated cubic EOS and proposing a characterization procedure for the C7+ fraction. The performance of various equations of state for modeling reservoir fluids was analyzed by Danesh et al. (132) and Anastasiades et al. (133).
Methods for improving the performance of cubic equations of state in the nearcritical region will not be discussed here because they are described in Chapter 11 of this book. However, it should be mentioned that some techniques have been proposed specifically for nearcritical hydrocarbon mixtures and may be applicable to reservoirfluid simulation (134 136).
In addition to phase equilibria, the van der Waalstype equations of state with classical mixing rules can be used to correlate thermodynamic functions such as excess enthalpy and excess volume. Several authors analyzed the prediction of excess enthalpies in mixtures containing nonpolar and weakly polar components (137139). As shown by Casielles et al.
(137) and Zebolsky and Renuncio (138), even complex shapes of HE vs. x curves can be reproduced (e.g. curves in which HE changes from positive to negative values depending on composition). Also, excess enthalpies in ternary systems can be reasonably predicted from binary data. A simultaneous representation of VLE and heats of mixing is possible, although it has not been investigated extensively. A particularly accurate simultaneous representation of VLE and HE has been obtained by Abdoul et al. (218) using their group contribution method.
98
Van der Waalstype equations of state are also capable of representing excess molar volumes of fluid mixtures. For example, Serbanovic et al. (140) examined the application of cubic equations of state for the calculation of excess volumes in hydrocarbon mixtures and obtained satisfactory results. However, there is very little chance that excess volumes can be calculated simultaneously with vaporliquid equilibria. The theoretical basis of the equations is too weak to predict consistently the different phenomena that manifest themselves in G E and V E. A successful simultaneous correlation can be obtained only in individual cases.
The classical quadratic mixing rules become inappropriate when applied to mixtures containing strongly polar and, in particular, associating components. A typical example of the failure of classical mixing rules for such systems is provided by mixtures of water and hydrocarbons. Fairly accurate results can be obtained only when different values of binary parameters are used for the waterrich and hydrocarbonrich phases. This is, however, unacceptable because it leads to a thermodynamic inconsistency (141). A very convincing example of the failure of classical mixing rules for alcohol + hydrocarbon systems was presented by Trebble (142). Trebble used the TrebbleBishnoi EOS, which makes it possible to use as many as four binary parameters by applying the quadratic mixing rules to its four constants. Even with four binary parameters, a satisfactory correlation of VLE for alcohol + hydrocarbon systems was not obtained. In the next three sections, we will discuss various techniques that have been used to create models that are applicable to strongly nonideal systems.
4.4.2 C o m p o s i t i o n  D e p e n d e n t C o m b i n i n g Rules
Since the classical quadratic mixing rules are not sufficiently flexible to correlate the phase behavior of mixtures containing strongly polar and associating components, several authors introduced empirical modifications to enhance their flexibility. These modifications effectively replace the compositionindependent combining rule in Equation (4.70) with compositiondependent expressions.
Panagiotopoulos and Reid (143), Stryjek and Vera (21) and Adachi and Sugie (144) proposed three mathematically equivalent combining rules with two binary parameters. In the notation of Panagiotopoulos and Reid (143), the final mixing rule takes the form:
a = E E x i x j ( a i i a j j ) ' / 2 0  k u + ( k i j  k j i ) x i ) (4.73)
Stryjek and Vera (21) also proposed another form, which they called a van Laartype mixing rule because of a similarity with the van Laar equation for the excess Gibbs energy:
a EEx ix j (a i i a j j ) I / 20 k i j k j i / ( x i k i j +xjkji)) (4.74)
Kabadi and Danner (145) proposed a twoparameter equation designed specifically for water + hydrocarbon mixtures:
2 [(a~,a22),/2 (1  k12 ) + x, G(1  Tr~ s )] a = x2all + x 2 a22 + 2xlx 2 (4.75)
99
where the subscript 1 denotes water. Correlations were established for the parameters k12 and G within homologous series of hydrocarbons. Michel et aL (146) and Daridon et al. (147) developed alternative correlations for mixtures containing water and hydrocarbons.
Schwartzentruber et al. (148) proposed a threeparameter expression to further enhance the flexibility of the Panagiotopoulos  Reid mixing rule:
a = Z Z x i x j ( a i i a j j ) ' / 2 I I  k u  l u m U x i  m j i x j ] i j mo.Xi + mjixj (4.76)
w h e r e kji = ko. , lji = lij, mji " 1  m U a n d kii = lii = 0. T h i s expression reduces to t h e
PanagiotopoulosReid mixing rule when m,j = 0. In general, the compositiondependent combining rules constitute the simplest method for
applying equations of state to complex systems. Their accuracy for binary systems is usually very satisfactory as shown by Sandoval et al. (149), Margerum and Lu (150) and others. However, the mixing rules described above suffer from a very serious deficiency, which was discovered for the first time by Michelsen and Kistenmacher (151). They are not invariant to dividing a component into a number of identical subcomponents. If a binary mixture with composition (Xl, x2) is treated as a temary system with composition (Xl,X2,X3), where the temary is formed by dividing component 2 into two pseudocomponents with identical properties, a different value for the parameter a will result. Therefore, the calculated properties of a mixture will depend on the number of pseudocomponents, which is in contrast to experimental evidence.
Several attempts have been made to eliminate this inconsistency while retaining the flexibility of the compositiondependent combining rules for mixtures. Mathias et al. (152) proposed a mixing rule that overcomes the problem of invariance:
/ 3 1/3 a = Z Z x i x j ( a i i a j j ) l / 2 ( 1  k u ) + Z x i xj[(aiiajj)l/2]l/3lji
i j i (4.77)
The first term is the classical quadratic mixing rule. The second term introduces a cubic composition dependence, when / j i   ltj or a quartic one, when /ji ~: ltj. It can be shown that Equation (4.77) reduces to the PanagiotopoulosReid mixing rule when lji = lij. Twu et al. (27) proposed a somewhat similar mixing rule:
E 13 Z HuUS Gu,/3(agiaii) '/6 xj
a = Z Z x i x j ( a i i a j j ) l / Z ( l _ k o . ) + Z x i J i j i ~'~/~GO.X j
J
(4.78)
where
H o =(kj/ k0)/:r (4.79)
100
G u = exp(/3~/H~/) (4.80)
This mixing rule can be used either in a twoparameter version (i.e., with only k O. and k j i
adjusted) or in a fourparameter version (i.e., with/3/j and/3ji adjusted in addition to k O. and kji). In both Equations (4.77) and (4.78), an improvement for multicomponent mixtures was obtained by introducing more symmetry into the nonquadratic part of the mixing rule, which bears resemblance to the composition dependence of the third virial coefficient. Another method for overcoming the MichelsenKistenmacher consistency problems was proposed by Schwartzentruber and Renon (153), who also introduced more symmetry into the cubic composition dependence of the mixing rule.
Figure 4.2 illustrates the performance of several compositiondependent combining rules for the strongly nonideal ternary system methanol + cyclohexane + hexane. It should be noted that all combining rules are capable of correlating vaporliquid equilibria for the binary subsystems with reasonable accuracy. However, a much more stringent test is provided by the prediction of VLE in the ternary system using the parameters obtained from binary data. In this case, the three mixing rules that suffer from the MichelsenKistenmacher inconsistency (i.e., PanagiotopoulosReid, StryjekVera and Schwartzentruber et al.) show a large loss of accuracy for the ternary system. Similar conclusions were also reached for other systems in the more comprehensive studies by Knudsen et al. (154) and Malanowski and Anderko (57). However, the mixing rule of Twu et al. (27) shows a more satisfactory behavior. Although it loses accuracy for the ternary system when only two binary parameters (i.e., k u and kji) are regressed from binary data, it becomes more accurate in its fourparameter form. The parameters of all the mixing rules discussed above usually show appreciable temperature dependence. The temperature dependence of these mixing rules cannot be established a prior i and has to be determined from experimental data at several temperatures.
4.4.3 DensityDependent Mixing Rules
It is evident from the functional form of the compositiondependent combining rules that they do not reproduce the quadratic composition dependence of the second virial coefficient and, thus, violate basic statistical thermodynamics. This is indicative of their weak theoretical background. Therefore, several authors came up with a theoretically appealing concept that better results could be obtained if the constants of a selected equation of state were allowed to depend not only on composition and, sometimes, temperature, but also on density. To illustrate this concept, we consider the mixing rule of Luedecke and Prausnitz (155), who postulated that the attractive parameter can be written as
a = Z Z x i x j a u +a "c (4.81)
where the first term is identical to the classical quadratic mixing rule, a/j is equal to (aiiajj)l/2(1 k,j) and anc is a contribution of noncentral forces which are due to differences in polarity, size and shape of molecules. An expression for this term is constructed by analyzing boundary conditions. First, since anc refers only to contributions from unlike pairs, then:
l i m a ne  0 (4.82) x i >0
101
Correlation of binary vapor  liquid equilibria Percent deviations
6.00 1 4.00 ]
cyc lohexane + hexane 2.oo
o.oo . . . . ~ . . . .
m e t h a n o l +
cyc lohexane
6.00 1 4.00 2.00 ~ 0.00 r][ ]DDr D
m e t h a n o l +
hexane
6.00 4.00 2.00 o.oo __1
PR SV SGR AEOS SRKS2 SRKS4
Prediction of ternary vapor liquid equilibria
m e t h a n o l +
cyc lohexane +
hexane
8.00
00o 4.00
2.00 D 0.00
PR SV SGR AEOS SRKS2 SRKS4
Figure 4.2 Application of several compositiondependent combining rules to the correlation and prediction of VLE in the ternary system methanol + cyclohexane + hexane. Vaporliquid equilibria in the three binary subsystems are correlated and the parameters are used to predict VLE in the ternary system. The following models are used in conjunction with the PengRobinson or SoaveRedlich Kwong EOS: PR  Panagiotopoulos and Reid's mixing rule; SV  Stryjek and Vera's van Laartype mixing rule; AEOS  Anderko's EOS incorporating association; SRKS2  Twu, Bluck, Coon and Cunningham's mixing rule with two binary parameters; SRKS4  the same model with four binary parameters.
Second, it can be assumed that, as temperature rises or as density falls, the importance of noncentral forces declines:
lim a ne = 0 (4.83) p/RT>O
The simplest nonquadratic approximation that is consistent with these boundary conditions introduces cubic composition dependence and linear density dependence:
102
a nc : ( p / R T ) Z Z x i x j ( x i c i o ) txjcj(i)) (4.84)
where ci(j) is a binary parameter that reflects noncentral forces when molecule j is infinitely dilute. The resulting mixing rule is
a = Z Z x i x j ( a i i a j j ) ' / 2 ( 1  k o . ) + ( p / R T ) E E X i X j ( X i C i ( ] ) I XjCj(i) ) (4.85)
and contains three binary parameters. In the lowdensity limit, it correctly reproduces the quadratic composition dependence of the second virial coefficient. While this mixing rule accurately correlates binary VLE and LLE, it was found to be inaccurate for ternary systems.
Following the early studies of Whiting and Prausnitz (156) and Mollerup (157), most densitydependent mixing rules were derived using the local composition concept to represent mixture nonideality. These methods have been reviewed by Danner and Gupte (158) and, later, by Anderko (159). We will not discuss them here because they have not been proven advantageous for practical calculations. The densitydependent mixing rules do not improve the results of phaseequilibrium computations over the densityindependent, composition dependent combining rules. The loss of accuracy for ternary and multicomponent systems, which has been discussed in Section 4.4.2, is also shared by the densitydependent mixing rules (25). At the same time, the densitydependent mixing rules eliminate the thirdorder density dependence of cubic equations of state and impose a higherorder dependence. Also, there are alternative techniques for satisfying the secondorder composition dependence of the second virial coefficient, which will be discussed in Section 4.4.4.
4.4.4 Combining Equations of State with ExcessGibbsEnergy Models
Excess Gibbs energy (gZ) models based on the local composition concept have proven very useful for the correlation of lowpressure phase equilibria. Therefore, much effort has been focused on combining them with equations of state. A detailed discussion of the combined gEEOS models is provided in Chapter 9 of this book. However, it is worthwhile to mention them here in order to provide an appropriate perspective for other methods discussed in this chapter.
Standard thermodynamics defines a simple relationship between the excess Gibbs energy and fugacity coefficients, which are calculated from an equation of state:
gE(T,P,x)=RTOn~= Zxiln~i) (4.86)
where ~ x and ~bi are the fugacity coefficients of the mixture and the ith pure component, respectively. This relationship was employed by several authors to combine excessGibbs energy models and equations of state so that the desirable features of both classes of models can be utilized.
Vidal (160) derived the infinitepressure limit of the excess Gibbs energy calculated from the RedlichKwong EOS with quadratic mixing rules. To arrive at an explicit expression, Vidal assumed that v = b and v E = 0 at infinite pressure.
gE (p = oo) = ln21 a / b + Z xiai / bi I
103
(4.87)
It is interesting to write this equation specifically for a binary system when the binary parameter k O. is equal to zero:
/[/ /2/ /2] 2 gE(p=oo)=ln2(Xlb11/( xzbz2 all/bll  a22/b22
t , ~ ) t , b bll b22 (4.88)
In this equation, xibii/b are the infinitepressure volume fractions and the terms [(aii/bii)ln2/bii] 1/2 can be identified with solubility parameters. Therefore, Equation (4.88) becomes essentially identical to the wellknown HildebrandScatchard regular solution theory:
gE(P: ~) = b*,02(6, 62) 2 (4.89)
where (I) i and ~i are the volume fractions and solubility parameters, respectively. This explains the inherent limitations of quadratic mixing rules because it becomes evident that they are applicable only to mixtures whose excess properties can be approximated by the regular solution theory. While this is a good approximation for hydrocarbon mixtures, it cannot be applied to systems containing strongly polar or associating components.
Vidal's (160) derivation can be generalized to other equations of state as shown by Huron and Vidal (161):
gZ(P: m) = A( a /b+ Zxiai /bil (4.90)
where A is a characteristic parameter for each cubic EOS used. This equation suggests a mixing rule for the attractive parameter in cubic equations of state:
a :b~_axiaii/biigE(P=c~)/A) (4.91)
Any excessGibbsenergy model can be selected to represent the infinitepressure excess Gibbs energy. Huron and Vidal (161) suggested using the NRTL equation. It should be noted, however, that the excess Gibbs energies at finite and infinite pressure are not identical and cannot be used interchangeably. Therefore, the parameters of the Huron  Vidal mixing rule have to be determined by fitting the complete equation of state to phaseequilibrium data. Parameters of the original NRTL model cannot be used for that purpose. The Huron  Vidal mixing rule was shown by several authors to give results similar to those from the NRTL model. In particular, it makes it possible to predict multicomponent phase equilibria from binary data with satisfactory accuracy (162,163). It was also shown to compare favorably with several compositiondependent combining rules by Knudsen et al. (154).
104
Kurihara et al. (164) proposed a modification of the HuronVidal mixing rule that separates the classical quadratic term from a "residual" term, which is calculated from the Wilson equation:
a = ~ ~ x i x j ( a i i a j j ) 1/2  (b/ln2)g E(resiaual) ( P = oo) (4.92)
Tochigi et al. (165) used this equation to predict temary vaporliquid equilibria from binary data and obtained satisfactory results.
More recently, the focus of research in this area shifted to developing mixing rules that not only share the numerical properties of equations for the excess Gibbs energy, but also can be used with exactly the same parameters as the original excessGibbsenergy models. For this purpose, the infinitepressure limit was abandoned in favor of a more realistic zeropressure limit. An early method for accomplishing this was proposed by Mollerup (166). It should be noted that the zeropressure limit creates problems whenever the equation of state does not predict a liquid root, which is needed to calculate the liquidphase fugacity coefficients and, subsequently, the excess Gibbs energy. For example, the zeropressure liquid root of the van der Waals EOS exists when a/RTb > 4. In other cases, it has to be determined by suitable extrapolation. For this purpose, two techniques have been proposed by Michelsen (167) and Heidemann and Kokal (168).
Michelsen (167) applied the HuronVidal technique of matching the excess Gibbs energy using the reference pressure of zero. In contrast to the HuronVidal mixing rule, the expression obtained is not explicit:
gE q ( a ) = ~_ziq(aii)+~f+ ~izi ln( b ] (4.93)
i • ~ b i i )
In this equation, a is a shortcut notation for a/bRT and the function q(a) is given, for the RedlichKwong EOS, by:
ln( u+ 1 t q(a) =  1  ln(u 1) a ,,~ (4.94)
with
u 1( ) = = a  1  ~/a 2  6a + 1 (4.95)
These equations cannot be used to evaluate the mixing rule under all conditions because the function q(a) is defined only for a>3+23/2. To maintain continuity for all values of a, Michelsen suggested two approximations. In the first approximation, a linear relation was used:
q(a) = qo + ql a (4.96)
This leads to an explicit mixing rule:
105
1EgE a ~ = ~ Z i a i i n t n t Z i In
i gl • (4.97)
where ql has a universal recommended value. Due to its similarity to the HuronVidal mixing rule, this equation was called the Modified HuronVidal FirstOrder mixing rule (MHV1). In the second approximation, Michelsen employed a quadratic approximation q(a)=qo +ql a +q2 a2, which yields the Modified HuronVidal SecondOrder (MHV2)
mixing rule:
( ~ i ) ( 2 ~ 23 gZ ~i b (4.98) ql a,~,,  . ziaii ] q2 a ~  . z i a i i = ~ + . z i ln~//
where ql and q2 are universal parameters. Dahl and Michelsen (169) demonstrated that the MHV2 mixing rule gives very good predictions of highpressure VLE using parameters obtained from lowpressure VLE regressions. In spite of its success, it should be noted that MHV2 requires a relatively arbitrary quadratic extension for the calculation of volumes. Also, similarly to the HuronVidal mixing rule, it fails to satisfy the quadratic composition dependence of the second virial coefficient.
Heidemann and Kokal's procedure (168) is, in spirit, similar to Michelsen's because it also involves the use of an auxiliary function q(a). Since this function is cubic, the amix parameter has to be found in an iterative way. However, it has similar properties. More recently, Boukouvalas et aL (170) proposed a linear combination of the algebraically similar MHV1 and HuronVidal mixing rules and obtained an improvement for mixtures containing components with appreciably different sizes. This combination is purely empirical because the MHV1 and HuronVidal mixing rules have been derived at different reference pressures and the reason for the improvement is unclear.
A different approach has been proposed by Wong and Sandler (171). Instead of trying to match the zeropressure excess Gibbs energy, Wong and Sandler observed that the excess Helmholtz energy is practically independent of pressure, which makes it almost the same at low and infinite pressure. At the same time, the zeropressure Helmholtz energy is equal to the zeropressure Gibbs energy. This makes it possible to identify the lowpressure excess Gibbs energy calculated from an activitycoefficient model with the infinitepressure excess Helmholtz energy calculated from an equation of state:
E (T,P = oo, x)= G(T,P(low),x) AEos (4.99)
Additionally, Wong and Sandler utilized the quadratic composition dependence of the second virial coefficient to relate the mixedfluid a and b parameters to their purefluid counterparts:
B(T,x) =b a = ~,~,xixjIbi j aij I (4.100) RT   ~)
106
where the crossterm for i¢j is related to the purefluid terms by a combining rule, i.e.,
b~i   ~ = 2 bii RT) (4.101)
with an additional binary parameter k, 7. The two conditions given by Equations (4.99) and (4.100) give the following mixing rules:
D amix = Q (4.102) RT 1  D
O b m = ~ (4.103) 1  D
where
/ a/ Q'22XiXj b   ~ o " (4.104)
a i Ge(x) = + ~ (4.105)
O Z X i b i R T crRT
and cr is an EOSspecific constant.
The WongSandler mixing rule contains one additional binary parameter k U (cf Equation (4.101)), which has to be regressed, preferably using lowpressure data. Alternatively, it can be obtained from the same excess Gibbs energy model that was used to formulate the mixing rule. Therefore, the application of this mixing rule is not automatic even though the parameters of the excess Gibbs energy model can be used unchanged. Wong et al. (172) showed that the WongSandler mixing rule remains accurate even when it is used at pressures and temperatures that are very remote from the conditions of the lowpressure VLE data that were used to regress the parameters. Huang and Sandier (173) compared the MHV2 and WongSandler mixing rules and found that the WongSandler formulation is more accurate, especially at high pressures and temperatures. Also, EscobedoAlvarado and Sandier (236) studied the application of the MHV1 and WongSandler mixing rules to liquidliquid equilibrium computations and concluded that the WongSandler rule is better for predicting highpressure LLE from lowpressure data although both mixing rules are appropriate for correlating the data.
An extension of the WongSandler mixing rule was proposed by Chou and Wong (174), who imposed the correct composition dependence on the third virial coefficient. Since the mixing rule for the third virial coefficient contains triplet interaction parameters, Chou and Wong utilized them to introduce ternary parameters for fitting ternary phase equilibrium data. It should be noted that this kind of additional flexibility limits the predictive character of the model.
107
In their original forms, the combined GEEOS models do not reduce to the classical quadratic mixing rules, which makes it difficult to use them in a unified computational environment for both hydrocarbons and polar and associated chemicals. This is important in the context of process and reservoir simulation, where users have a tendency to rely on the wellestablished quadratic mixing rules for hydrocarbon systems. Therefore, Orbey and Sandier (237) and Twu et al. (238,239) developed modifications of the WongSandler mixing rule so that the classical quadratic mixing rule can be recovered. The modification of Orbey and Sandier (237) is based on a reformulation of the excess Gibbs energy model so that it can reduce to the classical mixing rule. The approach of Twu et al. (238,239) relies on a different concept. It is based on equating the difference between the excess Helmholtz energies for the complex fluid and a classical van der Waals fluid with an analogous difference for the Helmholtz energy departure, i.e.,
A E E  Avd w = AA/~vdW (4105a)
where the subscript vdW denotes a van der Waals fluid for which the classical quadratic mixing rule is used. This expression makes it possible to define a mixing rule that combines the quadratic mixing rule with an excess Gibbs energy model.
The mixing rules based on excess Gibbs energy models have gained considerable popularity for practical calculations. This is due to the fact that they share their features with the activity coefficient models such as NRTL or UNIQUAC, which have been commonly used for engineering calculations for almost three decades, especially in the context of process simulation.
4.5 E X P L I C I T T R E A T M E N T O F A S S O C I A T I O N IN E M P I R I C A L
E Q U A T I O N S O F S T A T E
Molecular association and solvation strongly affect the thermophysical properties of pure and mixed fluids. In the case of purefluid PVT properties, association significantly increases the critical temperature, makes the second virial coefficient more negative and affects the shape of the vaporpressure curve. For a number of compounds, such as carboxylic acids or hydrogen fluoride, association drastically reduces the gasphase compressibility factor even at low densities. Derivative properties such as enthalpies are also strongly influenced. While the behavior of vaporpressure curves can be adequately modelled up to the critical temperature using the empirical techniques described in Section 4.1, the effect of strong association on the gasphase P V T properties makes it necessary to account for association in the equation of state. In addition to PVT properties, association profoundly influences the nonideality of mixtures, which affects phase equilibria. In principle, the multiparameter mixing rules discussed in Section 4.4 are sufficiently flexible to correlate the phase behavior of strongly nonideal, associated systems. However, these methods require a substantial number of binary parameters (typically 24 per binary) and are, therefore, only correlative. It is usually difficult to extrapolate their parameters beyond the range of experimental data and the prediction of phase equilibria in multicomponent systems from binary data is frequently unreliable. Therefore, several methods have been devised to incorporate association explicitly into the framework of equations of state.
108
The possibility of representing phase equilibria by postulating that molecules associate or solvate to form new species has been recognized for a long time (175). The chemical theory has been shown by many authors to be particularly useful when applied to mixtures containing one associating compound and some inert solvents. In this ease, the chemical approach made it possible to obtain an accurate representation of phase equilibria with a limited number of physically meaningful parameters. However, the application of the chemical theory to systems containing any number of associating species encountered substantial difficulties. The major drawback of the chemical theory has been its specific character, restricted to particular classes of mixtures, and the difficulty of extending the models that are valid for binaries to multicomponent mixtures. In this section we will show how these and other problems were treated by various investigators. Our discussion will be limited to the application of the chemical theory to the modeling of associated fluids. Statisticalmechanical methods for treating association will not be discussed because they are described in Chapter 12 of this book.
In general, there are two methods for incorporating association equilibria into an equation of state. The first one is conceptually simpler, but it is more difficult to use, as we will show later. It consists in simultaneously solving chemical equilibria between associated species and physical (phase) equilibria for all species in the mixture. A pure associated compound is assumed to be a mixture of a few associated species Ai, each of them containing i monomeric units in a multimer. Once the associates are selected, they are treated as pseudocomponents and the system of equations to be solved is as follows:
Chemical equilibria for the reaction iA1  Ai in each phase:
xi qg i (4.106) g i = i ni1 i
X l l " (D1
Phase equilibria:
fiPhasel (T~ P, x) =ft" phase2 (T~ P, x_) (4.107)
where the superscripts stand for any two phases in equilibrium. The association constant Ki depends only on temperature because the standard state for the Ai species in both the gas and liquid phases is pure Ai in the idealgas state. These equations can be solved numerically provided that the fugacity coefficients of each species are calculated from an equation of state that is suitable for multicomponent mixtures. For this purpose, Wenzel et al. (176) used the SchmidtWenzel (15) cubic EOS whereas Grenzheuser and Gmehling (177) employed the PHCT EOS (87). In general, the most difficult problem in this approach is the assigmnent of associated species, the number of which has to be minimized. For example, if we assumed the continuous linear association model, the number of equations would be infinite. Therefore, Grenzheuser and Gmehling (177) restricted their calculations to dimers, which makes their method applicable primarily to carboxylic acids. On the other hand, Wenzel et al. (176) allow for the existence of higher multimers. For example, methanol is assumed to be a mixture of monomers, tetramers and dodecamers. The selection of multimers is arbitrary and results from a compromise between the requirements of reproducing the data and minimizing the number of adjustable parameters. Nevertheless, the number of adjustable purecomponent parameters is very large (three parameters for each mono or multimer and two or three for each reaction)
109
and their determination is by no means simple. To determine purecomponent parameters, Wenzel et al. (176) utilize not only purecomponent data (i.e., vapor pressure, liquid and vapor volumes and critical properties), but also binary VLE data.
Extension of this scheme to mixtures is straightforward, provided that the mixture contains only one associating component. In this case the chemical equilibria are not changed and the number of phaseequilibrium equations only increases by the number of inert components that are present in the mixture. However, in the case of mixtures containing two or more associating components, a problem of crossassociates appears. Since it is impossible to deduce all parameters for crossassociates from binary data, only arbitrarily selected cross associates (usually crossdimers) are allowed for.
The computational scheme outlined above makes it possible to represent VLE and LLE with good accuracy using only one binary parameter as shown by Wenzel et al. (176) and Kolasinska et al., (178). Lang and Wenzel (179) were even able to simulate a third phase for a pure component by assuming the existence of a large oligomer (e.g., a 2540 mer). The parameters of that oligomer were adjusted to match the properties of the solid phase. This approach can be used as an alternative to introducing an additional physical term to simulate a solid phase in an equation of state as done by Schmidt and Wenzel (230).
However, the good features of this method for introducing association equilibria have been achieved at a considerable cost. First, the determination of purecomponent parameters is cumbersome and no clearcut algorithm exists for guessing which associates should be assumed to provide a good representation of phase equilibria. This can be done only by trial and error. Second, this approach cannot overcome the traditional deficiency of association models, i.e., the difficulty of formulating a consistent model for systems containing any number of associating and inert components.
A successful attempt to solve these problems has been made in the development of the second method for incorporating association into an equation of state. This method is based on solving the chemicalequilibrium expressions for an assumed association model analytically in order to derive an explicit equation of state with association builtin. The first model of this type was proposed by Heidemann and Prausnitz (180), who combined an EOS with the continuous linearassociation model. According to this model, the equilibrium constants for the consecutive association reactions Ai + A1 = Ai+l (i = 1 . . . . oo) are equal to each other (i.e., Ki,i+l = K). Heidemann and Prausnitz used a generalized van der Waals EOS:
p R T a = v Zrep(~) ~T1iat t (~) (4.108)
where a and b are the generalized van der Waals parameters andZrep and 1latt are functions of the reduced density ~. For associated species, the classical quadratic mixing rules have been used:
a = E Z x i x j ( a i a j ) '/2 (4.109)
b : Z xibi (4.110)
with the following combining rules for the multimers' parameters:
110
a i = i 2 a l (4.111)
b i = ib, (4.112)
Heidemann and Prausnitz obtained the remarkable result that, with these combining rules, the reduced density and, subsequently, the Zrep and Ilatt functions are independent of association. The effects of association enter the equation of state only through the factor nT/no, where nT is the total number of moles of the associated species and no is the number of moles of the compound in the absence of association. Then, the van der Waals parameters take the form:
a = (n o / n T)2 a, (4.113)
b = (n o / n r)bl (4.114)
and the equation of state becomes
p = n r RT~ a n o b Zr~P(~)~ 1Iatt(~) (4.115)
After evaluating the materialbalance and chemicalequilibrium equations, the factor nT/no becomes
n__ L = 2 (4.116) n o 1 + (1 + 4 R T K e g / v) 1/2
where
g = f[(Zr~ p  1) / ~]d~ (4.117) o
The expression for nT/no can be easily generalized for mixtures containing one associating component A and any number of inert ones:
n T 2xA ~ = + l  x A (4.118) n o 1 +(1 + 4 R T K e g x 4 / 1 ) ) 1/2
It should be noted that the final equation of state is strongly dependent on the combining rules for the parameters a and b of the multimers. These combining rules were tested by Wenzel and Krop (181), who analyzed the a and b parameters calculated for a series of hydrocarbons using the PengRobinson EOS (14). They found that the multipliers i 2 and i in Equations (4.111) and (4.112) are reasonable, but not very accurate. A closer inspection of the a and b values would suggest the multipliers i e where the exponent e lies between 1 and 2 (closer to 2 for a and closer to 1 for b). However, it is not worthwhile to establish a more accurate approximation for the parameters a and b of hydrocarbons for two reasons. First,
111
hydrocarbons provide only a very rough approximation of the properties of multimers and, second, any fractional exponent would preclude any further analytical derivations. It should be realized that the combining rules for the associates are purely hypothetical and serve merely as a tool for deriving a convenient closedform equation of state.
The original HeidemannPrausnitz (180) technique yields equations of state that can be very accurate for the calculation of PVT properties of pure fluids. This has been demonstrated by Twu et al. (182,183), who applied this technique in conjunction with the RedlichKwong EOS to carboxylic acids and hydrogen fluoride. They achieved a very good representation of gasphase PVT properties, which are extremely nonideal for these fluids. However, Twu et al. did not obtain any simplification for mixtures and continued to use a multiparameter mixing rule with compositiondependent combining rules for mixtures containing HF and acids. Thus, the original HeidemannPrausnitz method is fully successful for PVT properties, but still requires multiparameter mixing rules for mixtures. Therefore, several authors introduced modifications to develop a more convenient equation for mixtures.
Ikonomou and Donohue (184,185) used the concept of Heidemann and Prausnitz in conjunction with the PACT equation of state given by Equation (4.57). They assumed that the mixing rules for the associated species are analogous to those used for nonassociating components. As with the HeidemannPrausnitz EOS, the key step was to assign appropriate combining rules for the associated species:
3 3 O" 0. = 0"11
rj = jr~
qj = Jql
(4.119)
The form of the resulting equation of state, called APACT (Associated PerturbedAnisotropic Chain Theory) is significantly different from that obtained by Heidemann and Prausnitz in that it contains the ratio (nT/no) as a separate term:
n T PACT PACT (4.120) Z = ~ + Zre p + Zattr
n 0
This is a consequence of a different equation of state. In particular, it can be shown that the presence of the Prigogine parameter c, which multiplies the nonideal part of the equation, gives rise to the separation of the (nT/no) term from the rest of Equation (187). The factor (nT/no) is similar to that obtained by Heidemann and Prausnitz except for the term g(O, which is equal to zero in APACT. Thus, for a mixture containing one associating component and any number of inert ones, the factor (nT/no) becomes:
n T 2 x A
no I+(I+4RTKxA/v)l/2 +1 x A (4.121)
This equation was later extended to systems with multiple associating components (186). It should be noted that an analytical solution for (nT/no) cannot be found when more than one
112
associating component is present. Therefore, Ikonomou and Donohue (186) and Economou et al. (189) solved the chemical equilibria and material balances numerically and, subsequently, devised an analytical approximation. Economou and Peters (233) also developed a specialized version of APACT for systems containing hydrogen fluoride. The method of Ikonomou and Donohue was also used by Elliott et al. (188) in conjunction with a much simpler equation of state that approximates the PerturbedHardChain Theory. A somewhat similar model has also been proposed by Deiters (190).
Anderko (191195) developed an equation for associated mixtures that combines a cubic EOS with a chemical term that accounts for association. Initially, Anderko (191) added a chemical term z (~h) to the usual (or physical) compressibility factor, i.e.,
z = z ~h) + z (oh)  1 (4.122)
by generalizing the observation that the second and higher virial coefficients could be separated into physical and chemical contributions (196,197). Later, Anderko (195) rederived Equation (4.122) by applying the HeidemannPrausnitz technique with different combining rules. The term z (oh) is equivalent to the ratio n+/no, i.e.,
n T z (oh) = ~ (4.123) n o
whereas the term z (ph) is an equation of state for the hypothetical monomeric fluid and can be expressed by any cubic equation of state. Most calculations were performed, however, using the YuLu (26) EOS because of its accuracy for liquiddensity calculations. The combined model was called AEOS (Association + Equation of State).
For the continuous linear association model, the expression for z (oh) is identical to Equation (4.121) obtained by Ikonomou and Donohue. To extend the applicability of the equation to systems containing any number of associating components, Anderko (193) developed a multicomponent analog of the continuous linear association model. The model is based on the assumption that, 1. Only linear multimers containing any possible combinations of monomeric units occur in the mixture. There is no upper limit on the size of the multimer and 2. The multimers are formed in consecutive association reactions. For each of the elementary reactions the equilibrium constant depends on the chemical identity of the monomers that form the bond but not on the number of monomers that constitute the multimer. In this scheme a system containing n associating components is characterized by n selfassociation constants gii and n(n1)/2 crossassociation (solvation) constants Kij (i~j).
The derivation requires certain approximations because the system of chemical equilibrium and massbalance equations cannot be solved analytically. The final result is:
= nT (4.124)
no i=1 1 + [1 + 4 R T ( ~ KuxAo )) / v] 1/2 k=, j=l
where A (i) and B (k) denote the ith associating and kth inert components, respectively. This equation satisfies three important boundary conditions:
113
1. If the number of associating components is equal to one, Equation (4.124) reduces to Equation (4.121) for a mixture with one selfassociating component. 2. When all self and crossassociation constants are equal to each other, the associated species become indistinguishable. Then, the expression for z (oh) reduces to the purecomponent case. 3. In the lowdensity limit, the correct composition dependence of the chemical contribution to the second virial coefficient is recovered, i.e.,
B(Ch) = ££xAo)xAo~(RTKij ) (4.125) i j
Although the continuous linearassociation model is successful for a variety of associating compounds (e.g., alcohols, phenols, amines, etc.), some compounds require different association models. An important example is water because the structures of water clusters are threedimensional rather than linear. Another example is hydrogen fluoride, which preferentially forms associates of some intermediate sizes (such as 5, 6, 7,... mers) rather than long chains. At the same time, the analytical solution of the chemicalequilibrium and mass balance expressions is very difficult, if not impossible, for more complex association models. To overcome this difficulty, Anderko (195) noted that the nT/no term is always an algebraic function of the product RTKxA/v for any oneequilibriumconstant association model. In the simple case of a mixture containing one associating component, the term nT/no is
nT F( RTKx A ] (4.126) = x A + 1  x A
no x , V
where F is an algebraic (even though unknown a priori) function of the product RTKxA/v. This equation is valid irrespective of the characteristics of the association model. If, for brevity, we use the symbol q to denote the product RTKxA/v, then the function F(q)=2/[l+(l+4q) 1/2] will give us the expression for the continuous linearassociation model.
Lencka and Anderko (198) and Anderko and Prausnitz (199) utilized this result to arrive at a closedform expression for hydrogen fluoride. The preferential formation of intermediate size associated HF species can be expressed using a Poissontype distribution function (199):
K~J 1
Kjj+I = K ~ (4.127) j!
which can be compared with the simple relationship Kjj+a=K for the continuous linear model. For this model, the values of nv/no were found numerically and closely approximated using the function
8
1 + Z akq k k=l (4.128) F(q) = (1 + q)8
where ak are numerical coefficients and q denotes, again, the product RTKxA/v. For water, Anderko (195) developed a simple function:
114
1 F(q) = (4.129)
1 + q + a q 2
where a is a numerical constant. This function was later empirically improved by Shinta and Firoozabadi (200):
(4.130) F(q) = ~f~ + q + 3q 2
where ~ and 13 are, again, numerical constants. For any association model, the final AEOS equation of state contains five parameters for
each associating compound, i.e., the enthalpy and entropy of association to evaluate the temperature dependence of the association constant and the apparent critical temperature, critical pressure and acentric factor of a hypothetical monomer. The latter three parameters are used to calculate the physical contribution to the compressibility factor (zCPh)), which is expressed by a generalized cubic EOS. All of these parameters are fitted to match pure component vapor pressure, liquid density and (optionally) critical coordinates. Alternatively, the equationofstate parameters can be evaluated locally for each temperature of interest using vaporpressure and liquiddensity data generated from another equation of state as discussed by Anderko (201).
When applied to mixture calculations, the AEOS model is capable of reproducing VLE and LLE in strongly nonideal systems using only one binary parameter. This parameter is introduced in the most traditional way, i.e., by applying the classical quadratic mixing rule to the a parameter in the cubic equation of state that represents the physical contribution to the compressibility factor z (ph). For example, Shinta and Firoozabadi (200) demonstrated that AEOS is very effective for the simultaneous correlation of VLE and LLE in aqueous hydrocarbon systems. Similar results were obtained by Lencka and Anderko (198) and Anderko and Prausnitz (199) for mixtures containing hydrogen fluoride and various chlorinated and fluorinated hydrocarbons. Anderko and Malanowski (194) showed very good results for mixtures containing alcohols and phenols. Since only one binary parameter is needed, there are no problems with the intercorrelation of binary parameters. This enhances the accuracy of the model for multicomponent systems. As shown in Figure 4.2, AEOS predicts the VLE in the ternary system methanol + cyclohexane + hexane with a very good accuracy, which is similar to the accuracy of VLE correlation for the binary subsystems. Shinta and Firoozabadi (200,202) showed that AEOS accurately predicts the phase behavior of multicomponent systems encountered in reservoir engineering (i.e., water + H2S + CO2 + hydrocarbons). Also, they demonstrated that the behavior of water + reservoircrude systems (e.g., in highpressure steam distillation) can be modeled with the same degree of simplicity as for dry reservoircrude systems (202). In many cases, AEOS is purely predictive, i.e., good estimates can be obtained with the binary interaction parameters set equal to zero. This has been demonstrated for the industrially important systems containing HF and halocarbons (198, 199). For example, Figure 4.3 shows the representation of VLE and VLLE for the HF +
I : I I
20 / ~ . . . . . . . ~ . . . e . . ~ . . . . . . . . . .
' o
~ 1 5
m lO
o i i i i o.o 0.2 0.4 0.6 0.8 1.0
X H F
12 • . . . . . . . . .
lO
8
7:::, 6
n
4
2
0 0.0 0.2 0.4 0.6 0.8 1.0
X H F
115
Figure 4.3 Vaporliquidliquid equilibria calculated for the systems HF + CF3CC13 (upper graph) and HF + CHF2C1 (lower graph) using the AEOS equation (198) without any binary parameters (dotted lines) and with one regressed binary parameter (solid lines). Experimental data (solid circles) are from Knapp et al. (234) and Wilson et al. (235).
CHF2C1 and HF + CF3CC13 systems without any binary parameters (dotted lines) and with one regressed binary parameter (solid line).
The main difficulty associated with the use of equations of state based on the chemical theory is the determination of purecomponent parameters because five parameters have to be fitted to vaporpressure and liquiddensity data and, most importantly, their values have to be physically meaningful. However, this difficulty is compensated by the simplicity of calculations for mixtures.
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4.6 E Q U A T I O N S O F S T A T E AS F U L L Y P R E D I C T I V E M O D E L S
Most equations of state contain some binary parameters that have to be determined from experimental data. Therefore; numerous methods have been developed to enhance the predictive capability of equations of state so that they can be used without having to regress experimental data. The simplest way to achieve this is to develop correlations for the binary parameter k o. for separate families of mixtures that contain one common component. A more general approach is to create equations of state with parameters calculated from group contributions. A third possible approach is to combine equations of state with existing predictive groupcontribution methods such as UNIFAC.
4.6.1 Correlations for Binary Parameters
Correlations for binary parameters in equations of state can be developed for homologous series of mixtures, i.e., mixtures containing one common component and a number of components that belong to a certain family. Not surprisingly, most such correlations have been developed for mixtures of hydrocarbons and a few selected nonhydrocarbons such as carbon dioxide or nitrogen. To develop such correlations, it is necessary to determine the binary parameters for individual binary pairs and find appropriate independent variables for the correlation.
For example, Valderrama et aL developed correlations for binary pairs containing hydrogen sulfide (203), carbon dioxide (204), hydrogen (205), nitrogen (206) and nonpolar compounds. They used a temperaturedependent function for the k,j parameter:
k o. = A  B / T w (4.131)
with the parameters A and B correlated with the acentric factor:
2 (4.132) A = A o + A~co j + A2co j
2 (4.133) B=B 0 +BI(.O j +B2(o j
In this correlation, the subscript j denotes the nonpolar component that forms a binary system with either H2S or CO2 or H2 or N2.
Kordas et al. (207) and Avlonitis et al. (208) developed similar, but somewhat more elaborate correlations for systems containing carbon dioxide and nitrogen, respectively. Correlations for CO2  containing mixtures were also developed by Kato et al. (209) and Moysan et al. (210). Nishiumi and Gotoh (211) obtained a correlation for systems containing hydrogen and Schulze (212) generalized the parameters for heliumcontaining mixtures. Another correlation was proposed by Gao et al. (213) for light hydrocarbon systems. More recently, Coutinho et al. (214) proposed a correlation for binary parameters on the basis of the theory of dispersion forces and applied it to systems containing CO2 and hydrocarbons.
It should be noted that such correlations can be obtained only when a single binary parameter is used for each binary pair. If two or more parameters are used, they usually become strongly correlated with each other which makes it practically impossible to correlate
117
them with parameters such as Tc and co. Therefore, correlations of this kind are used only in conjunction with the classical quadratic mixing rule with one binary parameter. In particular, they can be established when the oneparameter quadratic mixing rule is used for equations of state based on the chemical theory. This has been demonstrated by Anderko and Malanowski (194), who obtained correlations for homologous series of mixtures containing methanol and phenol.
4.6.2 GroupContribution Equations of State
A much more general approach is the development of equations of state in which all parameters are calculated from group contributions. Generalization of the classical quadratic mixing rules in terms of groups rather than components is relatively straightforward even though it requires largescale parameter regressions. For example, Pults et al. (215,216) presented a generalization of the classical quadratic mixing rule in terms of group contributions:
NcUc No~ a = Z Z X i X j Z Z V i m V j n q m q n a m n (4.134)
i j m n
Uc U~
b : Z x i Z v i m b m (4.135) i m
where the number of groups m in molecule i is given by Vim and the quantities Nc and NG represent the number of components and the number of groups, respectively. The quantity qm is a reduced area for group m, normalized to a value of 10 for methane. Pults et aL applied these mixing rules to a simplified version of the COR EOS (93) obtained by combining the COR repulsive term with a RedlichKwong attractive term. Since the mixing rule is a group contribution version of the classical quadratic mixing rule, the equation is limited to nonpolar components. Another equation of state for nonpolar fluids was proposed by Georgeton and Teja (217), who rewrote the PatelTeja EOS (16) in terms of group contributions.
A more elaborate groupcontribution EOS for hydrocarbons and nonpolar fluids was proposed by Abdoul et al. (218). Their model is based on the combination of a cubic equation of state with an excessGibbsenergy model. The excess Gibbs energy is defined at a constant packing fraction rl=b/v. The same packing fraction is used for pure components and mixtures so that the excess functions can be unambiguously calculated. Apart from the constant packingfraction assumption, the method of Abdoul et al. (218) is similar to the HuronVidal technique (161). To calculate the properties of mixtures, an excess Helmholtz energy is added to purecomponent Helmholtz energies calculated from an equation of state. The excess function is given by
A E (T,~l,x) = N R T ~ X i In ~ + E(T,x)~ (7/) (4.136) i
The first term results from mixing at the constant packing fraction and corresponds to a combinatorial contribution to the excess functions. The term ~(r/) is chosen in such a way that
118
the same PV relationship applies to mixtures and pure components. Since Abdoul et aL use a translated PengRobinson EOS, the term ~(r/) is equal to [ln(l+~,r/)]/?. The term E(T,x) introduces the actual excess function and is calculated from
E ( T , x ) = E 1 ( T , x ) + E 2 ( T , x ) (4.137)
where E1 follows from Guggenheim's quasilattice model and E2 is an empirical correction for chainlength differences. E1 is given by
b ~ ~ x i b i xjbj Eij (T) E1 = 2 i=l j=l b b
(4.138)
with
1 N N
E O. (T) = ~ Z Z (aik  {~jk )(ai,  ajl )Akl (T) k=l l=1
(4.139)
where a/k is the relative amount of group k in molecule i and Akl is the interaction parameter between groups k and l. The parameters Akt are calculated from temperaturedependent group contributions. In view of the composition dependence of El, this method is suitable only for mixtures containing nonpolar or weakly polar compounds. For these classes of compounds, this method is very accurate because it predicts not only phase equilibria, but also heats of mixing and volumetric properties. Following the original work of Abdoul et al., Le Roy et al. (219) expanded the group contribution table to improve the prediction for systems containing heavy hydrocarbons. Also, Fransson et al. (220) extended the method to chlorinated and fluorinated hydrocarbons.
A group contribution method that is not limited to mixtures of nonpolar and weakly polar components has been developed by SkjoldJorgensen (221,222). In this equation, the residual Helmholtz energy is calculated as a sum of a freevolume term and an attractive term. The freevolume term is obtained from the CarnahanStarling EOS with the BoublikMansoori mixing rules whereas the attractive term is a groupcontribution version of a densitydependent mixing rule with the NRTL equation used as the excessGibbsenergy model. The Skjold Jorgensen model was specifically designed for the calculation of gas solubilities in solvents of different chemical character. The original group contribution tables were later extended by Wolff et al. (223) and Pusch and Schmelzer (224).
The existing groupcontribution equations of state gained only limited popularity for two reasons. First, their group contribution tables are not as extensive as those of the existing excessGibbsenergy group contribution methods (i.e., UNIFAC and ASOG). Second, end users in industry tend to prefer to use the wellknown and extensively tested UNIFAC method. Since UNIFAC can be coupled with cubic equations of state in an essentially unchanged form, the resulting UNIFACbased equations tend to be more appealing. However, equations of state with independent group contributions may be advantageous for mixtures containing components with strongly differing volatilities, for which the UNIFAC parameters may be missing or less reliable.
4.6.3 Utilization of Predictive ExcessGibbsEnergy Models
119
Because of the great popularity of the UNIFAC groupcontribution model for chemical engineering calculations, several authors developed methods for combining UNIFAC with cubic equations of state so that the groupcontribution tables can be used unchanged. An early method for achieving this, called UNIWAALS, was proposed by Gupte et al. (225), who developed a technique for matching the excess Gibbs energy at zero pressure obtained from the van der Waals EOS with that from UNIFAC. The technique of Gupte et al. suffered from a problem that precluded it from generating the critical point. UNIWAALS was later improved by Gani et al. (226) to remove this deficiency. However, the resulting EOS was difficult to use because the pressure and volume were related through a differential equation rather than the customary cubic equation.
More recent procedures developed by Heidemann and Kokal (168), Michelsen (167) and Wong and Sandier (171) made it possible to incorporate UNIFAC into equations of state in a fully consistent way. These procedures have been described in Section 4.4.4. Their usefulness has been extensively demonstrated. In particular, Dahl and Michelsen (169) and Dahl et al. (227) used Michelsen's MHV2 mixing rule (167) and obtained good prediction of high pressure VLE. Holderbaum and Gmehling (228) applied a slightly modified MHV2. Orbey et al. (229) used the WongSandler mixing rule and obtained good predictions of both VLE and LLE. In all cases, the UNIFAC groupcontribution parameters were used unchanged.
Although these methods make it possible to use the existing UNIFAC parameters, it should be noted that new parameters have to be regressed for numerous systems of interest. This is due to the simple fact that components that occur in highpressure VLE do not necessarily occur in lowpressure VLE. For example, Holderbaum and Gmehling (228) had to regress additional groupcontribution parameters that were necessary for calculating gasgas and gasalkane phase equilibria. Another problem associated with this approach is the temperature range for which the UNIFAC parameters are valid. Since the UNIFAC parameters were determined from lowpressure VLE, this temperature range is usually very limited. As we go to highpressure VLE, the high pressures are frequently accompanied by high temperatures. Therefore, an extrapolation of the parameters is necessary. This extrapolation may be particularly doubtful if temperaturedependent UNIFAC parameters are used (i.e., the socalled Dortmund and Lyngby modifications of UNIFAC).
4.7 C L O S I N G R E M A R K S
Cubic equations of state are currently the most widely used models for a variety of practical applications, including chemicalprocess and reservoir simulation. Cubic equations for pure components have reached maturity and no substantial progress can be expected in this area. Reliable techniques have been established to represent vapor pressures of both nonpolar and polar or associating compounds. The representation of volumetric properties has been improved within the limits imposed by the cubic form. The only exception is the representation of properties of heavy hydrocarbons and illdefined compounds, which is still an area of active research.
In contrast to pure components, the representation of mixedfluid properties by using cubic equations of state is still in a state of flux. While the classical quadratic mixing rules are the method of choice for mixtures containing nonpolar or weakly polar components, several alternative methods are available for mixtures containing strongly polar or associating
120
components. The composition  dependent combining rules provide the simplest successful method. They are capable of satisfactorily representing the properties of binary mixtures, but may lose accuracy for multicomponent mixtures. This may be due either to their internal consistency problems (i.e., the MichelsenKistenmacher syndrome) or the simple fact that empirical parameters regressed for a multiparameter model may be strongly correlated with each other and, therefore, not always suitable for extrapolations to systems with a larger number of components. An alternative method is based on the combination of excessGibbs energy models with equations of state. Such combined models have essentially the same properties for multicomponent mixtures as the excessGibbsenergy models. Thus, they are usually fairly reliable for the prediction of multicomponent phase equilibria from binary data, except for the most strongly nonideal systems. An important advantage of the more recent models of this type (such as the MHV2 or WongSandler mixing rules) is the possibility of using the existing parameters of excessGibbsenergy models without having to regress them specifically for the combined EOS+G E model. This feature has been extensively utilized to extend the applicability of the UNIFAC groupcontribution method to high pressures and temperatures.
Substantial progress has been achieved in the generalized van der Waals equations of state. Their development has been influenced, in particular, by the progress in the statistical mechanics of chainlike molecules. It is becoming apparent that the more recent equations of this kind (such as SAFT or PHSC) are particularly suitable for polymer systems. Another area in which the generalized van der Waals equations may be somewhat superior to cubic equations is the modeling of illdefined mixtures, for which special parameterization techniques are necessary. However, cubic equations of state are well entrenched in the modeling of phase equilibria and thermodynamic properties for hydrocarbon processing and the manufacture of commodity chemicals. It is extremely unlikely that they will be superseded in these areas by models with a better theoretical background.
A lot of attention has been devoted in this review to methods that combine simple equations of state with chemical association models. This is justified by the fact that these methods significantly improve the performance of equations of state for associated mixtures without resorting to multiparameter, empirical mixing rules. Also, they are only moderately more computertime consuming than regular cubic equations of state. The only major shortcoming of the equations of state based on the chemical theory is the relatively cumbersome procedure for evaluating both the chemical and physical purecomponent parameters from the same experimental data (i.e. vapor pressures and densities). It should be noted that the equations of state based on the chemical theory face a strong competition from associatedmixture models based on the thermodynamic perturbation theory, which are reviewed in Chapter 12 of this book. The statistical mechanicsbased equations of state (such as the SAFT equation) have been proven to be very valuable for systems, in which the strength of association varies from weak hydrogen bonding to covalentlybound polymers. However, the models based on the chemical theory give very good results for several important systems, such as mixtures containing hydrogen fluoride or hydrocarbonwater reservoir systems.
At present, most researchers in this area believe that further progress will result from developments of statistical thermodynamics. At the same time, almost all calculations in industrial practice are performed using empirical, mostly cubic, equations. For practical applications, equations of state with a better theoretical background have been shown to be superior to the cubic equations only in a few niche areas such as polymer systems and electrolyte mixtures at high pressures and temperatures. It can be expected that the slow
121
process of merging theoretical concepts with empirical techniques will lead to improvements in the future.
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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 127
P E R T U R B A T I O N T H E O R Y
TomA~ Boublik
Faculty of Science Charles University of Prague 128~ 0 Prague, Czech Republic [email protected], cuni. cz
5.1. Introduction 5.2. Basic Concepts of Perturbation Theory 5.3. Perturbation Theories of Pure Simple Fluids
5.3.1 Van der Waals Equation of State 5.3.2 Equations of State and Radial Distribution Functions of Hard Spheres 5.3.3 Perturbation Expansion for Fluids with Infinitely Steep Repulsions 5.3.4 Second Order Perturbation Term 5.3.5 Perturbation Expansion for Fluids with Soft Repulsions 5.3.6 Quantum Effects
5.4. Mixtures of Simple Fluids 5.4.1 Mixtures of Hard Spheres 5.4.2 Perturbation Expansion for Mixtures with Infinitely Steep Repulsions 5.4.3 Perturbation Expansion for Mixtures with Soft Repulsions
5.5. Perturbation Theories of Molecular Fluids 5.5.1 Pair Potential of Molecular Fluids 5.5.2 Equations of State and Distribution Function of Hard Nonspherical Molecules 5.5.3 Perturbation Expansion for Pure Molecular Fluids
5.5.3.1 Fluids of Kihara Molecules 5.5.3.2 Fluids of Molecules with Multicenter Pair Potential 5.5.3.3 Fluids of Molecules with Electrostatic Interactions
5.5.4 Water 5.6 Mixtures of Molecular Fluids
5.6.1 Mixtures of Nonelectrolytes 5.6.2 Mixtures of Electrolytes and Aqueous Solutions
5.7. Conclusions References
128
5.1 I N T R O D U C T I O N
Perturbation theories of fluids belong to a class of modern statisticalthermodynamic theories in which the structure of a studied system is not a priori determined (e.g. by assuming some kind of a quasilattice) but is characterized by a set of distribution func tions. In a system with given intermolecular forces the distribution functions express average numbers of pairs, triplets etc. as functions of molecular distance(s), orientation coordinates and state conditions.
The basic relationship of statistical thermodynamics (2, 3, 4) is the expression relating the Helmholtz energy, A, to the canonical partition function, Q, of a thermodynamic system
A =  k B T In Q
where kB is the Boltzmann constant and T the absolute temperature. If Ej is the energy of quantum state j of the studied system, then
(5.1)
Q = ~ e x p (  E j / k B T ) (5.2)
and the probability of finding the system in quantum state i is equal to
eEi/kBT Pi  (5.3) Q
a s
In fluids composed of N identical molecules, the partition function Q can be written
1 ( v ) N Q = ~ . ZN (5.4)
where ZN  ~((V) "'" / eUN(rl'r2""rN)/kBT drldr2""drN
and the Helmholtz energy
1 ( v ) N A =  k B T In ~ ZN
(5.5)
Here V denotes the volume of the system, q the molecular partition function (containing contributions of translation, rotation, vibration, etc. of a single independent molecule), ZN the configuration integral and UN the potential energy of the studied system. In the simplest case ZN depends only on the coordinates rl,r2, ...,rN. It is obvious, that, if UN  O, then the configuration integral equals V N and Equation (5.6) reduces to the relation valid for the ideal gas. In the general case (UN ¢ 0) the determination of the configuration integral represents a formidable task.
The potential energy, UN, can be expressed as a sum of intermolecular pair potentials, uij, of all the couples i  j , triplet potentials (given by the difference of interaction energy of a triplet and three pair potentials), etc. Higher potentials than triplet ones are of negligible magnitude; quite often also triplet interactions are neglected and only pair potentials are considered (1). Then in terms of 'effective' pair potentials
(5.6)
129
= (5.7) j>i>l
If the pair potential depends only on the centercenter distance, rij, of molecules i and j, the system is called a simple fluid.
A great variety of pair potentials of simple fluids has been proposed to this date [see reference (5)]; in this chapter we will consider only the simplest ones, namely that of hard spheres defined as
× v :3
f
Figure 5.1 Hard sphere, squarewell and LennardJones pair potentials.
u(r) = oc for r < d
= 0 for r > d
the squareweU potential oc r K d
u ( r ) =  e for d < r K v d
0 r > v d
and the LennardJones 126 potential
u(r) : 4e
(5.s)
(5.9)
(where d stands for the hardsphere diameter, "y scales the width of the well and e and a are the energy and length parameters). The potentials are schematically depicted in Figure 5.1; parameters of the LJ pair potential for several substances are listed in Table 5.1.
In the following text dealing with simple fluids we will limit ourselves to these three model potentials, because (i) the use of more sophisticated potentials usually brings only slight additional problems in formulation of theoretical expressions, (ii) application
130
Table 5.1 Parameters of the LennardJones 126 potential.
Compound e/kB (K) a (rim) Ne Ar Kr Xe CH4 CF4
34.2 0.2860 119.8 0.3471 164.0 0.3827 222.3 0.4099 148.9 0.3783 151.5 0.4744
of the LJ potential often yields sufficiently accurate results and (iii) the use of complex potentials is connected with increasing amount of computational time and often results in only modest improvement of the calculated values.
2
1
0 3 4 5 6 7
r
Figure 5.2 Radial distribution function of simple fluids.
As already mentioned, modern theories of fluids do not consider a priori structure of fluids but describe the structure by a set of distribution functions. If we assume the potential energy UN to be pairwise additive [see Equation (5.7)] then the distribution function of the second order, p(2)(rt, r2), defined as
p(2)(rt r 2 ) = N ( N  1) f f
(where/~ = 1/kBT) and the radial distribution function, g(rt2), given by
(5.11)
g ( ~ ) = p(~)(~l, r~)/p ~ (5.12)
play the most important role in the theory of fluids. The product of the radial distribution function (rdf), bulk density (p) and volume element gives the average number of centres of
131
molecules in a given elementary volume (with distance r12 from the testparticle centre). The rdf as a function of distance is depicted in Figure 5.2
The knowledge of rdf's makes it possible to determine all the thermodynamic functions of pure fluids and mixtures. Thus, for pure monoatomic fluids (where only translation motion contributes to q) we can write the following relationship for the internal energy (3)
U 3 p f oo NksT = 2 + 2kBT .u u(r)g(r)4~rr2dr (5.13)
The compressibility factor is given by
PV = 1 2~p fo oo du(r) NksT  3kBT _ dr g(r) r3dr (5.14)
For the chemical potential it holds
# = In A3p + 4~p f01 f0 °°
0u(r, ~) kBT ~BT O~ 9(r, ~) r2dr d~ (5.15)
and the isothermal compressibility is
( Op ) = l + 47rp f [g(r) _ l]r2dr (5.16) kBT ffP T,V
Here A = (h2/2rmkBT) 1/2 is the de Broglie wave length and ~ a coupling parameter (2). Combinations of the above equations then give relations for all the other thermodynamic functions. Unfortunately, determination of distribution functions is not an easy matter even in the case of relatively simple pair potentials, as they depend (besides the explic itly given dependence on distance) also on density and temperature (and in the case of mixtures on composition). Actually, only the distribution functions for systems of hard spheres (and to some extent of the other hard bodies) and of LJ molecules are available (see Section 5.3.2.).
It is a great advantage of the perturbation theory that it allows us to exploit our rather limited knowledge of distribution functions (and equations of state) to describe the behavior of real systems, i.e. systems with more sophisticated intermolecular interactions.
5.2 BASIC C O N C E P T S OF P E R T U R B A T I O N T H E O R Y
Application of the perturbation technique is known from different fields of physics (e. g. quantum mechanics, astronomy) where some property of a studied system is expressed on the basis of the known property of a reference employing an expansion in a series; the expansion is usually limited to the first or second order.
In statistical thermodynamics we usually work within a canonical ensemble where the basic relation is that between the Helmholtz energy, A, and the partition function, Q, [cf. Equation (5.1)]. Therefore, the Helmholtz energy is the property for which perturbation expansion is considered first. In his pioneering work Zwanzig (6) assumed the potential
132
energy, UN, to be given by a sum of the potential energy of a reference, U0, and the (much smaller) perturbation energy, Up. Since the reference and the studied systems do not differ in number of molecules and molecular partition function, q, we can write the following expression for the difference in the Helmholtz energies, A and A °, of the studied and reference systems
A  A o
kBT 1
  ln_~Nf. . . feZ(Vo+Up)drldr2. . .dr N 
_ l n f . . . f e  ~ U p e~U° ZO drldr2...drN
Realizing the fact that the ratio eBUo
z o =7'0
equals the probability density of a configuration of the reference system, and expanding exp(/~Up) in a series we have
knT =  I n 1  ~(Up)o + ~. (U~)o + ... (5.18)
where the symbol 00 denotes properties averaged over all the configurations of the refer ence system. Expanding the logarithmic function in series one gets
A  A ~d A °  A ~d ( U p ) o (U~)o  (Up)o 2 = ~ + ... (5.19)
kBT kBT kBT 2!(kBT) 2
Here A id stands for the idealgas Helmholtz energy. We thus introduce residual functions of the studied and reference systems which are quantities often used in the thermodynamics of liquids, simply linked with equations of state.
Another way leading to Equation (5.19) consists in the use of a coupling parameter, ~, scaling the perturbation energy,
UN = U0 + (5.20)
Writing OA +...
( A  Aid)  (A °  Aid) + ~ 0 0
and letting ~  1, we find that the single derivatives are equal to the first, second, etc. perturbation terms:
1 At = (Up)o A2 = 2!(kBT)((U~)o  (Up)~) (5.22)
etc. It is, however, also possible to write (7)
A  A id A 0  A id f l (Up)~ NkBT  NkB~ + Jo NkBTd~ (5.23)
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express (Up)¢ in terms of the radial distribution function and consider an expansion of g(r, ~) in terms of ~. A slightly more involved perturbation method, the blipfunction expansion will be discussed later.
In the case of the pairwise additive perturbation energy we can express Up in Equation (5.22) in terms of the perturbation pair potentials, Up; after integration, N ( N  1)/2 equal contributions result. Therefore
A, N ( N  1) 1 : 2 f fu,(r12)Podr dr#r3...drN = ~ f fup(rx2)p(o2)(rl,r2)drldr2 (5.24)
Transformation of variables [rx, r2] + [r~, r12] and integration over rl gives
1 f0~ A1 = 2 NP up(r12)go(r12)47rr22dr12 (5.25)
(where the index 0 denotes properties of the reference). It is obvious, that the knowledge of the course of the reference rdf allows us to evaluate the firstorder term in the expan sion of the Helmholtz energy for any pair potential of simple fluids. Determination of the secondorder term is more difficult; we postpone the discussion of its evaluation to Section 5.3.4
5.3 P E R T U R B A T I O N THEORIES OF PURE SIMPLE FLUIDS
5.3.1 Van der Waals equation of State
Van der Waals' equation of state is a prototype of a class of cubic equations of state widely used in thermodynamics and chemical engineering. Here we will show its deriva tion within the perturbation approach under the condition of low density. Discussion of approximations used to develop the van der Waals equation may serve to put limits on its applications and also to point out goals of the theory.
In the derivation we will consider the firstorder perturbation expansion for the Helmholtz energy
A  A id Ao   A id
NkBT NkBT p O0
+ 2ksT fo Up(r)g°(r)4~rr2dr (5.26)
Let the intermolecular forces be characterized by the squarewell potential (which very roughly describes repulsive and attractive interactions). The obvious choice for the refer ence is the hardsphere (hs) system. Available equations of state (EOS) and methods to determine the rdf will be discussed in the next section. At low densities we can find the expression for ZN of hard spheres when we realize that the integrand in Equation (5.5) possesses two values  zero (in the case where at least two spheres overlap) and one (in all other cases); each molecular center can thus move in a 'free' volume, i.e. the volume of the system lessened by a small volume around each sphere, excluded for centers of the other particles. For each pair of molecules (with hs diameter, d) this excluded volume at
134
low density amounts to 4rda/3 so that the total free volume, V}, is
Vf = V  2~rNd3/3 = V  Nb
The residual Helmholtz energy of the reference is thus
kBT~ =  N In  =  N ln(1  pb)
(where p  N/V) . In the firstorder perturbation term, A~, the perturbation pair potential is
(5.27)
u p ( r ) =  e for d_<r_<Td
0 r > 7d (5.28)
and the rdf possesses at low densities two values; zero for r < d and one elsewhere. After substitution we have
Defining parameter a as
A1 2 e kBT =  3 7tWo (k~) d3(73  1) (5.29)
we have
Finally
2 3 a = ~Tred (9 ' 3  1) (5.30)
A1 pa = (5.31)
NkBT kBT
A  A id pa NkBT =  ln(1  pb) kBT (5.32)
The equation of state follows from the thermodynamic relation
O ( A  Aid) P Op = Z  1 (5.33)
where Z  P V / N k B T is the compressibility factor. By differentiating Equation (5.32) we obtain the van der Waals equation of state,
P V 1 pa = (5.34)
NkBT 1  pb kBT
From the derivation of Equation (5.34) one can see that the form of the van der Waals equation does not depend on details of the form of the assumed pair potential. It should be emphasized, that we obtained the EOS under the assumption of low density and with neglection of the second and higherorder perturbation terms. If these terms are taken into account the simple dependence of the attractive term on temperature would change. (Such a change is expected taking into consideration the temperature dependence of the
135
second virial coefficient). Moreover, the dependence of a and b on density would change. (For the density dependence of the attractive term, however, the dependence of the rdf on density is more important.)
As far as the repulsive term ( 1  pb) 1 is concerned, at higher densities the free volume decreases more slowly due to overlapping of the excluded volumes around neighbouring molecules. These arguments give some evidence in favor of modifications of the van der Waals equation of state (and other cubic EOS) in which a, b are considered as density and temperature dependent quantities. (See Figure 5.3)
J i
a a
i a
a S
e S
vdW ,, J
i
 1 , I i I
0.0 0.1 0.2
Figure 5.3 Compressibility factors from the van der Waals EOS and perturbation theory, •  simulation data.
From the brief outline of the formulation of the perturbaton expansion for the Helmholtz energy and consequent determination of the compressibility factor (or internal energy and chemical potential) it is evident that (i) a system of hard spheres represents a natural choice of the reference in the case of potentials with a harsh repulsive branch, (ii) fair knowledge of the equation of state  or an expression for another thermodynamic function  of the hardsphere system and its rdf is necessary in order to get good pre diction of the behavior of the fluids studied. In the case of the pair potential with soft repulsion a system of representative hard spheres for the given reference soft spheres is sought and thermodynamic functions are then determined from the hard sphere EOS.
As a consequence of the mentioned choices of the reference system, the zeroth and firstorder terms possess large (possitive and negative) values which yield relatively small residual thermodynamic .functions. It is thus obvious, that accurate knowledge of the hardsphere EOS and the corresponding rdf is essential for the perturbation approach to yield an accurate description of real systems. Therefore, we will briefly discuss proper ties of the hs system in the next section and those of nonspherical ones in the further part.
136
5.3.2 Equations of State and Radial Distribution Functions of Hard Spheres
The system of hard spheres (hs) is the simplest possible model of fluids in which only the repulsive forces a r e  in the simplest way  taken into account. Because of the form of the hs pair potential, Equations (5.13)  (5.15) simplify considerably: internal energy is given only by the contribution of translational motion, U  ~NkBT. Due to the fact that the derivative of the pair potential multiplied by exp(~u), [cf. Equation (5.43)], gives the Dirac function 5(r  d) the compressibility factor is
P v NkBT = 1 + ~rpd3g(d) = 1 + 4yg(d) (5.35)
where y stands for the packing fraction, y = rpd3/6, and g(d) is the value of the rdf at r  d. In the expression for the chemical potential a new coupling parameter, ¢, can be introduced which scales the size of the test particle (e.g., molecule 1); the lower and upper bounds of the first integral of Equation (5.15) are chosen in such a way that for ~  a full decoupling occurs and for ~  b the full size of the test particle is obtained, (i.e. that of the other hard spheres). Then
/. # = l n A a p + 4 r p g(¢,¢)¢2d¢ = l n A 3 p + 4 r p d 3 bG(x)x2dx (5.36) kBT
where G(~) stands for g(~, ¢). In the case of ~ = b, G(b) = g(d). It is obvious that the value of the rdf at r /d = 1 yields the corresponding EOS im
mediately; on the other hand knowledge of the Equation of state allows formulation or improvement of expressions for the rdf.
There are several sources of our knowledge of the radial distribution function and Equation of state of hard spheres:
• Pseudoexperimental (simulation) data. These have been determined either by the Monte Carlo or Molecular Dynamics technique. Both the methods represent very powerful tools of theory and the number of simulation studies of different physical phenomena is ever increasing. Simulation techniques are described in several mono graphs and textbooks (8, 9). Hardsphere simulation data published before 1986 were collected in references 10,11.
• Integrodifferential methods (see reference 12). When p(2)) in Equation (5.11) is differentiated with respect to the coupling parameter (Kirkwood) or the gradient of p(2) is taken (BBGY) differential Equations are obtained which  after rearrangement  can be integrated again. As a result, the rdf (of the second order) is expressed in terms of an integral containing g(3)(rl, r2, r3). It is possible to repeat the operation for g(3) and to obtain an expression for g(3) in terms of g(4) etc.; a hierarchy of integrodifferential equations results. Usually this hierarchy is broken after the first step assuming (13)
g(3) (rl, r2, r3)  g(r12)g(r13)g(r23)
Solutions of the integrodifferential equations with this 'superposition approxima tion' yield in both cases (Kirkwood, BBGY) values of the rdf, which differ con siderably from simulation data at high densities. If, however, the superposition
137
approximation is used for g(4) very accurate rdf values are obtained at the cost of extensive computing (14).
• Integral equations. These originate from the OrnsteinZernike (15) equation (OZ) for the total correlation function, h(r)  g(r)  1
f
h(r,2) = c(r,2) + p J c(r13)h(r23)dr3 (5.37)
where the direct correlation function, c(r), is defined by this relationship. The essence of introducing c(r) is in the fact that the range where c(r) is nonzero is similar to that of the pair potential; the direct correlation function can thus be easily a subject of simple approximations. These approximations and methods of solution are discussed in detail in Chapter 6. For perturbation methods the results found for the PercusYevick (BY) approximation (10,16) are of special importance. The solution of the OZ relation in the PY approximation (16,17) yields two forms of the equation of state, the vform [resulting from Equation (5.14)] and cform [from Equation (5.16)]:
P V 1 + y + y 2  c  form (5.38)
N k s T (1  y)a
and P V 1 + y + y2 _ 3ya
= v  form (5.39) N k s T (1  y)3
Moreover, the Laplace (and Fourier) transform, g(s), of the product xg(x), becomes
sL(s) (5.40) G(s) = 12y[L(s) + S(s)e 8]
Here L(s) and S(s) are simple polynomials in s. Performing an inverse transform one obtains values of g(x) at given reduced distances x = r/d. As the c and v forms lie above and below simulation data, a better EOS will result by taking some kind of average of Equations (5.38) and (5.39). If (2/3) of the c and (1/3) of the v form are considered, the widely used CarnahanStarling (CS) EOS (18) is obtained
P V _ l + y + y2  y3 (5.41) N k B T (1  y)3
While CS or Kolafa (see later) EOS's agree very well with simulation data at packing fractions y _ 0.5 (see Figure 5.4) but disagree at higher densities, the course of the PY rdf disagrees slightly with simulation data in the interval x E (1, 1.6) even for y < 0.5. Several approximations were proposed to get equal c and vforms of the EOS and improve the predicted behavior of the rdf. Recently, a modification of the Laplace transform [cf. Equation (5.40)] was proposed (19) with two coefficients adjusted to exact results at low densities. The modification gives the rdf vs distance dependence accurately and an EOS valid even at y > 0.5. In practical applications,
138
lO
8
I 6 n," > Q. II
rq 4
 A   I 
y Figure 5.4 Compressibility factor of hard spheres;  CS EOS, •  simulation data.
however, a semiempirical correction is employed (20) where g(x) is calculated at slightly lower density yc  y(1  y/16) and Ag for x E (1,1.6) is calculated from
= A 1) (5.42) X
• Scaled particle theory (SPT) (21). This theory employs the coupling parameter which continually changes the diameter of a test hs particle in Equation (5.36).
The distribution function G(() is expressed as a secondorder polynomial, three coefficients of which are determined from the conditions of continuity of G and its derivative at ~ = 1/2 and the condition (P/kBT) = pG(oc). The SPT EOS is identical with the PY cform. In spite of the fact that additional conditions on G(~) are available, which makes it possible to consider a higher order expression, attempts to improve the SPT EOS have not been successful. The SPT gives  besides EOS  only the contact value of the rdf.
• Background correlation method (22). From the solution of integrodifferential equa tions (and other sources) it follows that at low densities g(x) = exp[~u(x)]. It is thus useful to express the rdf as a product of the exponential function and the background correlation function, Y(x),
(5.4a)
(x = r/d). In the case ofhs , g(x) = r ( x ) f o r x >_ 1. In contrast to therdf , the background correlation function (of hs and other pair potentials with infinitely steep repulsions) is continuous at x = 1. Several exact relations were found (23, 24) allowing expression of Y(x) in terms of the residual chemical potential of pure hard spheres and that of a hard dumbell, with the sitesite distance x (see Section 5.5.2) infinitely diluted in hard spheres
139
In Y(z) = 2/~# hs  / ~ # ~ (5.44)
The residual chemical potential of hs can be determined from any accurate hs EOS and/3#~ from the EOS of a mixture of hard nonspherical bodies (see Section 5.5.2); finally
x 3 y2 3x 2 Y ( 7  6x + ~) + (15  lSx + 3x 3) In g(x)   l n ( 1  y) + (1  y) 2(1  y)2 ~
+ (1  y)3 ~ + ~) (5.45)
In the limit of low densities the exact expression on the interval x E (1,2) can be obtained
X 3 X 3 g(x) ~ exp[y(8  6x + ~)]  1 + y(8  6x + ~) (5.46)
Due to reasons connected with the geometry of the hard dumbbells, Equation (5.45) gives values of the rdf only for distances smaller than that of the first minimum. To extend the range of distances it was proposed recently (25) to use a formula for damped oscillations, i.e.
g ( x )  A e B(xxmin) cos ?r(x  Xmin)/C ( 5 . 4 7 )
where constants A and c are determined from the conditions of continuity of g(x) and its derivative at x = Xmin; B is a function of y. In Figure 5.5 a comparison is given of the rdf calculated from Equations (5.45) and (5.47) with simulation data; a fair agreement is seen.
• Virial coefficients. For hard spheres the first few virial coefficients, Bi, axe known either from theory ( B 2  B4) or numerical calculations ( B 5  Bin) (10). These coefficients allow one (i) to judge the accuracy of the approximate EOS, (ii) to formulate new EOS's. There are two routes to get the EOS: (i) to consider the socalled Pad~ approximant p(m, n)
P V 1 + a~y + ... (5.48) N k B T = p ( m , n) = 1 + bly + ...
with coefficients a,~, bn determined from the individual virial coefficients after expan sion of the last Equation, (ii) to resum all the virial coefficients whose approximate expression in terms of n (where n denotes the order of the virial coefficient) is avail able. Thus, Carnahan and Starling found (18) that the nine virial coefficients in the expansion of P V / N k B T in terms of y, see reference 10, can be approximately written (assuming rounded values)
Bi+l = i(i + 3) (5.49)
140
6
5 * p = 0.925
X "~3
2
1
15 i i f1.0 1. 2.0 2 .5 3.0
× Figure 5.5 Radial distribution function of hard spheres;  calculated, •  simulation data.
Resumming then the virial expansion
P V 1 + y + y2 _ y3 N k B T = 1 + ~ i(i + 3)y = (1  y)3 (5.50)
i > 0
i.e. the CS EOS is obtained. Kolafa (10) improved the approximation for Bi by taking
5 B i = ~ ( i 2 + 3 i  6 ) fo r i>_3 (5.51)
and obtained
P V 1 + y + y2 _ 2y3/3 _ 2y4/3
N k B T (1  y)3 (5.52)
This equation is in even better agreement with simulation data than the CS EOS.
The deficiency of these equations is that a singularity is found at y  1, whereas the closepacked value for the cubic lattice is ~ 0.74. Several hs EOS have been proposed which predict correctly the P   V   T behavior up to y ~ 0.74; because of their complicated form they have not been used in the perturbation approach.
5.3.3 P e r t u r b a t i o n E x p a n s i o n for Flu ids wi th Inf in i te ly S teep Repu l s ions
Systems of this type can be represented without loss of generality by the system of squarewell molecules. The choice of the reference is unambiguous  it is the hardsphere system with the hs diameter d. For the reference residual Helmholtz energy one can use the expression derived from the CS equation
141
A0  A id y ( 4  3y)
NkBT (1  y)2 (5.53)
(A slightly more complicated expression follows from the Kolafa EOS.) By differentiating Equation (5.53) with respect to p and multiplying by p (or equally well differentiating with respect to y and multiplying by y) one finds
Z 0  Z id   Z 0  1 = 4 y  2y 2 (1  y)a = 4ygo(d) (5.54)
where Z stands for the compressibility factor; the equation can be obtained directly from the CS EOS. Summation of Equations (5.53) and (5.54) yields the reduced residual Gibbs energy or reduced residual chemical potential of the reference
Go  G id A#0 8y  9y 2 + 3y 3 = = (5.55) NkBT kBT (1  y)3
Thus, all three functions of the reference system are available in simple analytic forms (the same is true also for the residual reduced entropy, the absolute value of which is equal to the residual Helmholtz energy due to the fact that the diameter and, therefore, the packing fraction are independent of temperature).
In the firstorder term, A1, it is usually advisable to express the rdf in terms of the total correlation function, h, i.e. g  1 + h, and write
A1 2rp o¢ 2rp fo¢ NkBT  kBT fd Up(r)g°(r)r2dr = [dJ~ Up(r)r2dr + fd up(r)h°(r)r2dr] (5.56) 
The former integral can be determined analytically while the latter (in the general case) by numerical integration.
In the case of the squarewell system, the perturbation potential is
After substitution we obtain
up(r)  e f o r d < r < T d
0 elsewhere (5.57)
A1 NkBT
2rpd3 [ (73  1 )+ 3fl'Yho(x)x2dxJ 3 T*
(5.58)
(Here T* = kaTie is the reduced temperature and x = r/d). To get some feeling about the integral in the last equation we can consider the lowdensity limit and substitute from Equation (5.46); then
A1 NkBT T* (./3 _ 1) + y(873  ~'), + ~'),  )
The firstorder contribution to the compressibility factor, Z1, is
(5.59)
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[ (5.60) OA1 =  4 ~ , ( 7 3  1 ) + 3 Y Oy Z1 = p Op
In our special case
1 0 _ ~ ] Z1 =  4 ~ , (,y3 _ 1)+ 2y(8"y 3  974 + ~7 ) (5.61)
It should be mentioned that the EOS obtained as a sum of the above expression for Z0 and the last equation is valid only at low densities. It represents an improvement over the statistical analogue of the van der Waals EOS. Comparison of the compressibility factor calculated from the van der Waals EOS with that obtained from Equations (5.53) and (5.60) vs y is given in Figure 5.3. As in the case of the reference, the firstorder pertur bation contribution to the residual chemical potential is given by the sum of expressions for A1 and Z1.
It is obvious that the same approach can in principle be used to determine A1 and Z1 in the case of systems with other pair potentials of the same type. Thus, if the Sutherland potential, defined by Equation (5.62) is considered
u(r) = c¢ f o r r < d
 e ( d / r ) 6 r > d (5.62)
the first part of Equation (5.56) is  4 y / T * . Generally, the remaining integral can and should be determined numerically.
There is, however, another route for determining A1, in which it is assumed that the function XUp(X) is the Laplace transform of some (for a moment unknown) function H(t). After substitution in the firstorder integral, one has
/0 /1 /0 /0 XUp(X)xgo(x)dx = e t*bl( t )xgo(x)dtdx = ld(t)~j(t)dt (5.63)
The Laplace transform, g (t) is available for the PY approximation of the OZ equation and the inverse transform of functions like x 6 is readily available, too. Numerical evaluation of the integral in the last equation should be performed stepwise because of the character of the function G(t) and corrections to PY rdf should be considered. In passing we note that the method can be used also for the direct determination of Z1.
5.3.4 SecondOrder Perturbat ion Term
The secondorder perturbation term in the expression for the Helmholtz energy, A2, is given by fluctuations of the perturbation energy due to fluctuations in configurations of the reference. Employing the grand canonical partition function Barker and Henderson (26) derived an exact relation
143
1 A~ = ~Npf vo(X,2)u~(X,2)d2 + Np ~ f f g~)(x,2,a)up(X,2)~(x,a)d2d3
f f[,0(')(1, 2, z, 4 )  go(X, 2)g0(3 , 4)]up(X, 2)Up(3, 4)d2d3d4
(where 1,2,... stand for rx,r2, ...). One can see that distribution functions g0 (4) and g0 (a) must be known in order to determine A2 exactly. At present, however, these functions are not available. We may employ the superposition approximation for g0 (4) and g~3) in terms of go; complicated integrals result (which form the basis of the ORPA and EXP expressions  see Section 5.3.5). Alternatively, we may turn to simple approximations containing only the rdf of the second order, go Such approximations were proposed by Barker and Henderson (26) who wrote Up as a sum of squarewell contributions, Up, and assumed that only fluctuations in the number of couples with the same distances are correlated and contribute to (U~)  (Up)~. From these considerations they obtained
A2 f0 °~ ~(~)~0(~):d~ (m.c.) (5.65) NknT   /~rP (Op/OP)° up
The derivative (Op/OP)o can be determined from any accurate EOS for hard spheres. Equation (5.65) is called the 'macroscopic compressibility' approximation (m.c.) as the relation for fluctuations in the macroscopic hs system is employed. The authors proposed also a slightly modified 'local compressibility' approximation (1.c.)
/0 (,c) Up A2 fllrp \ OP )o
(/3  1/knT). For the CS EOS the expression for kBT(Op/OP)o is
( Op ) = ( l  y ) 4 k n T \ z ~ 0 l + 4 y + 4 y 2  4 y 3+y4 (5.67)
It is apparent that Equation (5.67) tends to 1 at very low density, whereas, for high densities, it is of the order 10 2 . This behavior is in good qualitative agreement with the fact that the thirdorder polynomial (in the reciprocal temperature) is necessary in order to express the temperature dependence of the second virial coefficient with suffi cient accuracy, whereas at high densities the meanfield term is large but all higherorder terms small. The predicted A2 vs y dependence, however, differs from the pseudoexperi mental data. This difference affects considerably the accuracy of the secondorder terms in expansions of the compressibility factor and residual chemical potential. In order to improve on the BH approximation, Zhang (27) took into consideration also correlation of the number of molecules in different shells with the common central molecule. As a result, the secondorder term reads as
A2 NkBT = 131rp (Op/OP)o (1 + 2Ky 2) fo °~ U2p(r)go(r)r2dr (5.68)
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Better agreement with the simulation data of the secondorder term for the SW system was found.
Concluding this part we note that within the framework of BH approximation an ex pression for the thirdorder perturbation term was also derived (28); it is, however, of limited practical importance.
5.3.5 Perturbation Expansion for Fluids with Soft Repulsions
Pair potentials, such as the squarewell or Sutherland models, characterize repulsive interactions of molecules in an oversimplified way. Actual repulsive interactions are soft, expressed by the power law or exponential function of distance. The LennardJones pair potential with its r n terms represents the interactions of simple fluids well and is widely used. In this part, we will consider the LJ 126 model to show some details of the perturbation approach when dealing with systems of realistic pair potentials.
The first question is the choice of the reference system. It is known that repulsive interactions determine the 'volume' (and 'shape') of individual molecules and consequently the structure of fluids; it is thus of special importance to define properly the range of action of repulsive forces. At present, two definitions of this range  due to Barker and Henderson (BH)(29) and due to Weeks, Chandler, and Andersen (WCA) (30) are widely used. In the former case the reference pair potential is
uo(r) = 4e  for r _ a
0 r > a (5.69)
(where a is the distance where ULj  0). In the latter case the reference potential is given by the prescription
uo(r) = 4e  + e for r _~ rmin
0 r > rmin (5.70)
Here rmin stands for the distance of minimum on the potential function; for the LJ po tential rmin  21/6a.
Thermodynamic properties of the reference in both cases are determined from the EOS of the representative hard spheres. For the BH definition of the reference soft spheres, the diameter d of the representative hs is calculated from
dBH f01 = c = (1  e~U°(~))dx (5.71) (7
where x  r / a . In the WCA approach the diameter of the representative hard spheres follows from the
blipfunction expansion. The blip function, Ae, is a difference of the Boltzmann factors for the reference and representative (hs) potentials, u0 and u hs
145
Ae = e ~°  e ~hs (5.72)
We can write (31) e  ~ = e ~h" + ~Ae (5.73)
For ~ = 0 we get the Boltzmann factor of hs, for ~  1 that of the reference. Let us consider an expansion of the Helmholtz energy, A0, in ~ about ~  0
Ao=AhS+(O~~)¢=o + ... ( 5 . 7 4 )
When performing differentiation we obtain N(N 1)/2 terms which yield after integration equal contributions. Thus
OflA N ( N  1 ) f f[ff_~ e~U(~) e~U(~)e~UN drldr2dra..drN (5.75)
The derivative in brackets yields Ae; considering the definition of p(2) and realizing that exp (flu)g is the background correlation function, Y, we have
( OflA ~ _ Np yhs 2 (5.76/
Putting this term equal to zero, the Helmholtz energy of the reference is given (up to the secondorder term) by A hs of hard spheres whose diameter, dwcA, is given by the prescription
formln[e~UO__ et~uhs] yhs d r  0 (5.77)
The reference rdf is then
go(r)  e~°(~)YhS(r) (5.7s)
In the case of the BH variant the representative hs diameter depends only on temper ature, in the latter case both on temperature and density. In the hybrid approach the WCA definition of the range of repulsive forces is combined with an analogue of Equation (5.71); dhs depends only on temperature.
If the value of the hs diameter (or c) of the representative system is known, one can find its packing fraction yhs _ ~pc3a3 and determine the reference residual Helmholtz energy from Equation (5.53). For the WCA approach the formulas given above can be used with a slight modification. Due to the dependence of y on temperature and density, the consistent values of the reference compressibility factor and residual chemical poten tial for WCA are determined by numerical differentiation.
The perturbation pair potential is given by the difference of the original and reference potentials. In the case of the BH variant, up is zero for r < a and Up  U L j for r > a. In accordance with this definition and the fact that dBH < a we see that
146
A1 = 2r/~p I NkBT Ja Up(r)go(r)r2dr 27r/~p [~°°Up(r)r2dr ~ faCCUp(r)ho(r)r2dr] (5.79)
The first integral in brackets equals 8 a 6ea . When determining the remaining integral, we have to realize that ho depends on the variable r/d; the integral can be expressed in two different ways
oo( ) ( I ) 4ecaa 3 = 4ea 3 _x 10 _ x4ho_x cdx = I /c
After substitution one has
[(xc)  1 2  (xc)6]ho(x)x2dx (5.80)
NkBT = T* 9 + 4 (x 1°  x4)ho(x/c)dx (5.81)
The last integral is determined numerically either by calculation of single values of h0  h as on the short interval, e.g. x E (1,4), or employing the Laplace transform g(t) for the PY approximation.
The firstorder perturbation term in the expansion for the compressibility factor is obtained from the derivative of A1 with respect to p; the firstorder contribution to the residual chemical potential is given by a sum of expressions for A1/NkBT and Z1.
In the case of the WCA choice the perturbation potential is
Up(r)  e for r ~ rmin
ULj(r) r > rmin (5.82)
Then
31 __ /~p ~O°° Up(r)e_~uo(r)ghs(r)dr : ~~_ [_~ ~ormi'e_~6uoghSdr _[_ ~r~iC~ uLjghSdr] NkBT 2
(5.83) From Equation (5.77) we see that the first integral of the last equation equals
fie fd r=i~ yhSdr
One can thus write
A1 NkBT
21rpaa [ fc°° Up(X)x2dx "b /c°° Up(x)hhS(x/c)x2dx kBT
Performing integration in the first term one obtains
(5.84)
A1 2rpaa I 8_~92 c 3 1 ~ °° ] NkBT = T*    + 3 + e Up(x)hhS(x/c) x2dx (5.85)
The expression for the compressibility factor follows from the derivative of A1 with respect to p, usually performed numerically because of the dependence of the hs diameter or c
147
i
[
0
Figure 5.6 Compressibility factors of the LJ system at T* =1.15;  perturbation theory (WCA), •  simulation data.
 on density. In Figure 5.6 a comparison of the calculated compressibility factor of the LJ system with MC data is given.
The determination of the secondorder perturbation term in the case of the BH approximation does not bring any greater problem than that for fluids with infinitely steep repulsions. In the WCA approach the secondorder term is seldom considered because of its relatively small magnitude in comparison with the firstorder term at high densities. Instead, more general approaches  ORPA (optimized randomphase approximation) and EXP  were formulated (32,33). In these versions the terms of the perturbation expansion were expressed as sets of diagrams and the Fourier transforms of single classes (e.g. ring diagrams) resummed. Thus, to the pair potential of the studied system another potential ut with an infinitely steep repulsive branch is attached, such that
a t ( r )=oc f o r r _ d and ut ( r )=up f o r r > d
Because of the behavior of u as we can also write
u~(r) = u~S(r) + u~(r) (5.86)
Considering the Fourier transform of the potential up(x), fi(k), and denoting by S the structure factor [S = 1 + ph(k)], the Fourier transform of the renormalized potential, C(k), is
O(k) = 1 +
Its inverse transform, C, then appears in equations for the function g,(r) in the ORPA and EXP approximations
gt(r) = 9b~(r)  C(r) ORPA (5.88)
148
In ORPA the residual Helmholtz energy is
EXP (5.89)
A  A id A h s  A id
NkBT NkBT NkBTA1 + 4~pl f {ln[l+pShS(k)fl~(k)]_pShS(k)/~(k)}k2d k (5.90)
where the firstorder term is given by Equation (5.85). From Equation (5.86) for ut(r) it is obvious that the form of the potential up(r) for r E (0, d) is irrelevant for the behavior of ut; however, it affects the magnitude of the Fourier transform. It is therefore possible to optimize the form of Up on the interval r < d in such a way that the functional derivative
6(A/NkBT) 5%
= 0
(In EXP a further class of diagrams is considered and an additional term appears in the expression for A  Aid).
Thermodynamic functions calculated from these approximations agree very well with pseudoexperimental data, but the determination of state points is often quite time con suming. Both the mentioned methods are more suited for description of solutions of electrolytes.
5.3.6 Q u a n t u m Effects
So far we have assumed that the kinetic part of the total partition function can be separated from the configurational part and expressed via the partition function of the ideal gas. This is generally correct; however, for some compounds belonging to the class of simple fluids (such as light rare gases) this factorization is not fully justified even at temperatures close to their critical values and a correction term should be added to take into account these quantum effects. The theory (34) leads to a series expansion of the Helmholtz energy in powers of A 2, where A = (2~rmkBT/h2) 1. If the firstorder expansion is considered, then the correction term in the expression for the Helmholtz energy is
A 2 N f V2 481rkBT V g(r) u(r)dr
In the case of potentials with soft repulsions the correction for the quantum effect can be accounted for by using an effective hardsphere diameter slightly larger than the real one.
5.4 M I X T U R E S OF S I M P L E F L U I D S
5.4.1 M i x t u r e s of Hard Spheres
Sources of our knowledge of the behavior of hard sphere mixtures  i.e. systems of hard spheres of two (or several) different diameters, are similar to those for pure hs fluids. Within the scaled particle theory (21), and from the OZ equation in the PY approximation (35), the following equation of state (PYc) was derived
149
P V NkBT
whereas the PYv form is
1 3~1{2 3{ "a = 1  { } {o(1{)2 + { o ( 1 _ { ) 3 P Y  c (5.91)
Here
P V 1 3~2(~ + ~1) N k B T  1  ~ + ~0(1~)2 P Y  v (5.92)
~k = 6 p ~ x i d ~ k = 0 , 1 , 2 , 3 ~=~3 i
The extended CS equation was formulated by the present author (36) and Mansoori et al. (37). It is
P V 1 3~1~2 ~3(3  ~) NkBT = 1~ ÷ ~0(1  ~)2 + ~0(1  ~)3 BMLCS (5.93)
The extended Kolafa EOS (38) reads
2~2~ P V 1 3 ~ 2 ~ ( 3  ~  5 j (5.94)
N k B T = 1  ~ t ~0(1~)2 + ~0(1~)3
Both the BMSCL EOS and Equation (5.94) describe well the behavior of mixtures of hs that do not differ extremely in the ratio of their diameters. However, for the extreme case of the socalled colloidal limit (infinitely diluted mixture of a big molecule in small molecules) some corrections (39,40) have to be made.
Within the PY approximation expressions for the Laplace transform of rdf's of the binary mixture were also determined (35) which were more complicated than those of pure fluids; they will not be reproduced here. Also Equation (5.45) was extended to mixtures. For the rdf of a pair i  j in the mixture of the given composition it holds that
Y ~ * * *
In gij(x)   ln(1  y) + (1  y)(3Rmsm ÷ 3S~r~ + V~pm)
y2 _ _ , , , , , , y (2 y / 3 ) , , , ( 5 . 9 5 ) + 2(1  y)2 (3Qmsm ÷ 6S~sm ÷ 6V~rmsm) ÷ (1  y)3 VmSm
Here R*, Q~, S~, V,~ denote geometric properties of a hard dumbbell (see Section 5.5.2) and r~, s~, v~ geometric properties of a given mixture (calculated from the properties of pure components). Values of rdf's for distances larger than the distance of the first mini mum are determined from Equation (5.47). The method allows the determination of rdf's
$ * * * in mixtures with an arbitrary number of components, rm, sin, vm, p,~ remain unchanged for all the different pairs; R*, Q~, S~, V,~ are characteristics of the hard dumbbell composed of hard spheres i and j with the sitesite distance equal to 1 = xdij.
In the end we would like to stress the fact that all the discussed relations are valid for the socalled additive hard spheres where the cross diameter is given by the arithmetic mean, dij = (dii + djj)/2. Mixtures of nonadditive hs's are considered when dealing with metallic systems.
150
5.4.2 P e r t u r b a t i o n E x p a n s i o n for M i x t u r e s w i t h Infini tely Steep Repuls ions
In this part we will briefly discuss methods used to determine thermodynamic functions of mixtures, the molecules of which interact via a squarewell pair potential (as a prototype of potentials with an infinitely steep repulsive branch).
Usually it is assumed that the squarewell cross parameter, d i j  (dii ÷ djj)/2, so that the EOS and rdf formulas for additive hard spheres can be employed. The reference residual Helmholtz energy is then
 1 l n ( 1  ~ ) + ~ + ) 3~t~2 ~3 (5.96) Ao~ A~ d ( ~ ]
NkBT = \ ~ o ~ ~  ~ o ( t  ~ ) ~ o ~ ( 1  ~)~ The firstorder term is
Als NkBT
27rp fd °° , kBT ~i ~ • is UPij(r)g°ijr2dr
kBT ~i" ~" xixjeij i~ UPij(r)r2dr + i, UPij(r)hoijr2dr
= 27rp ~ y~ x i x j ~ (73  1) + hoijx2dx i j J1
(5.97)
Here xi stands for the mole fraction, u* = u/e and T/~ = kBT/eij; values of e12/kB and 712 are usually determined from some combining rules.
For the secondorder perturbation term the simplified relation
A2s _ _Tr ( Op ) ~ ~~xixj fa °° Upij(2 r)goij(r)r2dr (5.98) NkBT  ~ o '~
is the only one used although the exact extension of the BH approach was derived, as well. The derivative (Op/OP)o is available from the EOS for mixtures.
The compressibility factor follows from the derivative of A s  A~ d with respect to p; the leading part in the firstorder term can be written analytically. The same situation occurs when we determine internal energy from the derivative of A,  A~ with respect to temperature.
5.4.3 P e r t u r b a t i o n E x p a n s i o n for M i x t u r e s w i t h Soft Repuls ions
In analogy to pure fluids, the mixture of LJ molecules is a typical example of multi component systems with soft repulsions. Soft spheres with interactions defined either by the BH or WCA decompositions form the reference system. Its thermodynamic functions can be calculated from the equation of state for mixtures of additive hard spheres. In a binary system we have three pairs of parameters eij/kB and aij. For each pair of parame ters one can determine a value of D 0 (in the case of like pai rs representative diameters), either from the BH or WCA prescriptions. When employing an EOS for additive hard spheres we take du  Du and dij = (du + djj)/2.
The reference term in the perturbation expansion for A,  A ~ possesses the same form
151
as in the case of the squarewell potential. In the firstorder contribution, an additional term appears as a correction for the fact that the thermodynamic functions of the reference are determined from the EOS of additive hs's. Then
Als 27rp fa g N k B T = kBT ~ ~ xixjaij ghS(x/cij)Upij(X)x2dx + Aa c°r (5.99) ij
(here c~ij = 1 in the BH and c~ij = cij in the WCA variants). The correction term Aa c°r in the BH approach is
AaCOr 2 hs = 47rpxix jd i jg (dij)[dij  Dij] i ¢ j (5.100)
In the WCA approximation a similar term is usually neglected. Further thermodynamic functions, such as the compressibility factor, internal energy,
etc. follow from the corresponding derivatives. In passing we note that the ORPA version was also extended to mixtures and applied
to a special kind of solutions.
5.5 P E R T U R B A T I O N T H E O R I E S OF M O L E C U L A R F L U I D S
5.5.1 Pair Potent ia l s of Molecular Fluids
Molecular fluids are composed of molecules whose pair interactions are described by potentials of the form u  u(r, Wl, w2), where wi stands for a set of three angles (or, in the special case of axially symmetric molecules, of two angles: wi = 0i¢i; f dwi  1). Pair potentials of this type describe (i) repulsive and dispersion forces of nonspherical molecules, (ii) electrostatic interactions.
We focus here on the interaction of permanent multipole moments such as dipoles, quadrupoles, etc. Electrostatic pair potentials (ES) can be written in the general form (31,41)
u(rx2, x, 2) = ¢2) li=O ljO m
or u(r~2,w~,w2) = X r n O(w~,w2) (5.101)
In this general expression Sh,~(0i , ¢i) stands for spherical harmonics; the coefficients X contain values of two multipole moments, n possesses integral values and ~ is a function of angles 01, 02 and ¢12.
For two permanent dipole moments, #1 and #2,
X = #1#2, n = 3, • = 2 cos 01 cos 02  sin 01 sin 02 cos ¢12 (5.102)
for two permanent quadrupole moments, Q1 and Q2,
3 X = ~Q1Q2, n = 5,
(I) = 1  5(cos 2 01 + cos 2 02 + 3cos 2 01 COS 2 02) 2r 2(sin 01 sin 02 cos ¢12  4cos 01 cos 02) 2 (5.103)
152
for permanent dipole and quadrupole moments, #1 and Q2, resp.,
3 X = ~#lQ2, n = 4,
Z
(I)  cos 01 (3 cos 2 021) +cos 02(3 cos 2 011) 2 (cos 01cos 02) sin 01 sin 02 cos ¢12 (5.104)
Interaction of permanent and induced dipole moments can be expressed as
1 u(r, O)l,OJ2)= 2 r 6 [o~1~22(3 COS 2 02~ 1 )+ a2#~(3 cos 2 O1 ~ 1)]
and similarly for quadrupoles
(5.105)
O u(r, wx,w2) = ~r S [~1Q~(5 cos 4 02  2cos 2 02 + 1) + c~2Q~(5 cos 4 01  2cos 2 01 + 1)]
(5.106) (where a stands for polarizability). To the class of ES interaction models can be added the overlap potential (which describes interactions of slightly nonspherical molecules, charac terized by parameter 5)
u ( ~ , ~ , ~ ) = 4~(~ )~ (3 cos ~ o~ + aco~ ~ o~  2) (5.107) r 
and the Ganssianoverlap potential (based on the idea of the Gaussian distribution of interaction sites within two molecules, with the hs or LJ interactions between sites) (42) which reads
u(r,2, wl, w2)  4%  (5.108)
Here both eg and ag are functions of wl and w2. In the simplified version (GOCE) % is assumed to be orientationally independent; for ag(wl,w2) we obtain
°~(~" ~ ) 1  x + (5.109) o0 1 + ~(u , , u~) i : } ~ g :
(where ul, u2, T stand for three unit vectors characterizing orientation of main axes an the centrecentre distance). The parameter X is defined as X  ( a2  1)/( ~2 + 1); and a0 are characteristics of the spatial distribution of sites. For the special case of the hard Gaussianoverlap model [where u g ( r 1 2 , w l , w 2 )   oo for r12 _< ag(Wl,W2) and zero elsewhere] the GOCE interactions correspond closely to those of two hard ellipsoids of revolution with the short axis equal to ao and the axes ratio equal to n. In passing we note that the modified G aussianoverlap potential is used in simulation experiments to substitute for interactions of hard dumbbells.
Intermolecular interactions of nonpolar nonspherical molecules are characterized most often either by the multicenter model or by the Kihara generalized potential. In the former case the pair potential is assumed to be given by a sum of interactions between individual sites on the first and second molecules of the given pair:
153
?~(r12, ~.dl, ~d2)  Z Ua~(r~2~) (5.110)
where r~2 ~ stands for the distance of the c~site on molecule I from the flsite on molecule 2; ua~ can possess e.g. hard sphere or 126 functional forms. The multicenter potential model is close to the chemical view of structure of molecules and easy to apply in simulations of molecules with one or a few types of sites. On the other hand the derivation of the rdf faces considerable difficulties.
In the case of the Kihara pair potential a hard convex core is ascribed to given molecules; the pair potential can generally be written as
U(r12,031,O32) = U(8) (5.111)
where s is the shortest surfacesurface distance between the cores of two interacting molecules; u can again possess different functional forms; in case of the 12  6 form we have
(:/'] = =  ( 5 . 1 1 2 )
The Kihara generalized pair potential can be considered as some kind of average over
Table 5.2 Parameters of the Kihara pair potential for hydrocarbons.
System e/kB (K) a (rim) l(nm) ethane propane butane pentane hexane
366.76 0.3057 0.3098 471.26 0.3193 0.4309 532.50 0.3314 0.5376 641.20 0.3409 0.6469 709.87 0.3486 0.7586
different orientations of a couple of molecules; its advantage is good characterization of the repulsive forces by suitable choice of the size and shape of the ascribed hard convex core. The use of the Kihara potential brings difficulties in the simulation processes; on the other hand its use simplifies expressions in perturbation expansions. Parameters e/kB, a and length of the rodlike core, l, of several hydrocarbons are listed in Table 5.2.
Both the multicenter and Kihara models can be combined with the electrostatic con tributions to obtain potentials characterizing molecular interactions of polar (dipolar, quadrupolar, etc.) nonspherical molecules. A special case of such models is the Stock mayer potential which combines LJ and dipoledipole expressions to describe behavior of spherical molecules with the embedded dipole,
= + (5 .113)
Similar expressions can be written for nonspherical molecules with any of the ES inter actions.
154
5.5.2 Equations of State and Distribution Functions of Hard Nonspherical Molecules
A considerable amount of pseudoexperimental data has been published up to the present time on the equilibrium behavior of hard nonspherical molecules, especially of hard dumbells and other fusedhardsphere models (FHSM) and of prolate hard sphero cylinders. The data include the compressibility factor, chemical potential and distribution functions (often in the form of sphericalharmonic expansion coefficients). Also data for the second, third and fourth virial coefficients are at our disposal (10,11). All these data indicate the fact that the reduced virial coefficients (i. e. values divided by the correspond ing powers of volume) and consequently the compressibility factor depend considerably on the shape of the studied molecules.
Extension of the hardsphere equations of state to nonspherical hardbody systems represents a formidable task. At the same level of approximations as considered for hs only, the EOS for the hard convex body system was derived within the scaled particle theory (43,44). The SPT method employs the fact that the mean volume V~+b of two hard convex bodies a and b in contact is (45)
V.+b = y~ + s.P~ + &R. + Yb (5.114)
where Vi, Si, Pu are geometric characteristics of the given hard convex body (hcb), namely the volume, surface area and the meancurvature integral divided by 4r. If the dilatation coefficient (~) is used as the coupling factor, it is possible to determine from an expression similar to Equation (5.114) the mean volume Va+b even for negative values of ¢ and consequently the distribution function Ghcb(¢) for ~ E (1, 0). From the conditions of continuity of G hcb and its first derivative at ¢ = 0, and from the expression for Ghcb(¢ + O0) three parameters in the secondorder polynomial can be determined. Knowledge of G hcb allows one to write the hcb EOS
P V 1 3c~y 3c~2y 2  { ~ + ~ SPT (5.115)
NkBT 1  y (I  y)2 (i  y)3
where o~  P u S i / 3 V i is the parameter of nonsphericity. By modifying the expression for Gcontact in the same way as was done in the case of hard sphere mixtures, an extension of the CS equation has been formulated (47)
P V 1 3(~y c ~ 2 y 2 ( 3   y) = ~ + ~ + ISPT (5.116)
N k B T 1  y (i  y)2 (I  y)3
On the basis of the analysis of the viria] coefficients of hard convex bodies the present author (I0) proposed the expression
P V = 1 ~ 3c~y + y213c~2(1  2y) + 5c~y] MSPT (5.117) N k B T 1  y (1  y)2 (1  y)3
Comparison of the compressibility factor for hard prolate spherocylinders (7  2) calcu lated from MSPT with simulation data is given in Figure 5.7.
It was found that the third and higher virial coefficients for prolate and oblate hcb's
155
Prolate spherocylinders 7=2
/
0 , i . . . .
Y
Figure 5.7 Compressibility factors of hard prolate spherocylinders (~,=2);  calculated from MSPT, •  simulation data.
of the same parameter a differ slightly in their magnitude. The same is true for the com pressibility factors at the same y and a. Therefore, a second nonsphericity characteristic has been sought and further parameters have been introduced. None of the (correspond ing) EOS with 2 nonsphericity parameters has been used in the perturbation theory.
Whereas the characterization of the P  V  T behavior of hcb systems from the recent EOS is fairly accurate (see Figure 5.7), knowledge of the distribution functions is unsat isfactory. The most important function is the average correlation function, ghcb(s). It is defined as an average over all orientations of the product of the molecular distribution function and surface area, divided by the mean surface area. From the fact that hard spheres are a special case of hcb (with a = 1) and from semiempirical arguments, the following relation was proposed (48)
a(x 1) ) (5.118) hhCb(x) = O h hs 1 + 1 + Rci + Rcj
Here Rci stands for the mean curvature integral divided by 47r of the icore, @ = shcb/Seh~v and shay is the surface area of the "equivalent" hard sphere with its volume equal to that of the hcb considered.
To determine the distribution function of the hard convex body system, one can also apply a method originally developed for the FHSM fluids (see the section dealing with hard dumbbells and FHSM). Determination of the perturbation terms is, however, more difficult.
All the above relations can be extended readily to mixtures. From the SPT, it follows
P V 1 3a.sy 3flsy 2 = ~ + SPT (5.119)
N k B T 1  y (1  y)2 (1  y)3
156
where
and
Similarly
rs qs 2
a S = 3 p y ~ ' = 9 p y 2
y = p E z , y~, r = p E z , P~, s = p E z , S~, q= p E z , R~ i i i i
and
P V 1 3c~sy ]~sy2(3  y) = } ~ + ISPT (5.120)
N k B T 1  y (1  y)2 (1  y)3
P V 1 3a, y y213/~(1  2y) + 5a~y]  { + MSPT (5.121)
N k s T 1  y (1  y)2 (1  y)3
In the case of hard dumbbells and other FHSMs, the situation is less satisfactory. On the basis of simulation data, Tildesley and Street (49) suggested for the system of hard dumbells
P V 1 + (1 + UL + VL3)y + (1 + W L + X L 3 ) y 2  (1 + Y L + ZL3)y 3 N k B T = (1  y)3 (5.122)
where L  I /a , I is the sitesite distance and U  Z are empirical coefficients. The equation yields very accurate values of the compressibility factor for homonuclear dumbbells with L < 1. For systems of FHSM of arbitrary shape the ISPT variant of the hcb EOS is also employed; the BoublikNezbeda (BN) rule (50) to determine a is used
o 3  FHSM
where the hard convex body considered just envelopes the given FHSM. In passing we note that several EOS have been suggested to describe the P V T
behavior of chain molecules  models of polymers (5154). The compressibility factor of a hardchain of m segments is
[ OlnghS(d)] (5.123) Z hch = m Z hs  (m  1) 1 + y 0 ~
This EOS describing the behavior of associating fluids is often combined with van der Waals attractive term and the resulting SAFT equation used to desribe the behavior of alkanes, polymers, etc.. A similar equation of state also valid for the "pearlnecklace" chains was derived by Chiew (55) solving PYOZ equations for mixtures under conditions following from the pearlnecklace structure; within this approximation it holds
1 + y /2 (5.124) Z h¢h = m Z h s  ( m  1)(1  y)2
Both Equations (5.123) and (5.124) consider one kind of segments; extension to different segments in a chain was proposed e.g. in reference 54.
157
Two types of distribution functions can be used in perturbation expansions to charac terize the structure of fusedhardsphere models: (i) the molecular distribution function gFHSM(rx2, W~,W2) and (ii) the sitesite correlation function gss(ra~). For homonuclear dumbbells radial slices of the molecular distribution function at specific orientations (par allel, endtoend) can be calculated via the method proposed for the background cor relation function. More often, however, gFHSM is expressed in terms of the background correlation function,
gFHSM(r12, Wl, W2)  exp[~UFHSM(r~2, 031, 032)] YFHSM(r12,031,032) (5.125)
KShler (56) and Fischer (57) assumed that the effect of the nonsphericity of a FHSM is absorbed mainly in the exponential function, whereas the background function depends mainly on the centrecentre distance. Thus, we can substitute for YFHSM the background function for soft spheres with potential u'(r12, T*) obtained from
uS(r12, T*)     l n ( e u~"sM(r'2'~l'w2)/ksT\ /w,w2 (5.126)
kBT
Then, gFHSM(r12, Wl,W2) = e t~u~'ssM(r'2'~''~'2) YS(r12, T*) (5.127)
Values of y8 for the softsphere potential u 8 are obtained from the solution of the OZ equation in some approximation such as, e.g., PY. It is obvious that the method is limited to dumbbells (and linear FHSM) with small sitesite distances. The sitesite correlation functions of chain molecules are  as mentioned above  available from the Chiew method.
In passing we note that all the above expressions were suggested for isotropic systems; characterization of nematic, smectic, etc. fluids is beyond the scope of this Chapter.
5.5.3 Perturbat ion Expansion for Pure Molecular Fluids
5.5.3.1 Fluids of Kihara Molecules
In the general case, the pair potential and molecular distribution functions depend on a set of (at least three) angles in addition to the dependence on the centrecentre distance. Therefore, formulation of the perturbation expansion and evaluation of the single perturbation terms are more tedious than in the case of simple fluids.
The perturbation theory of the Kihara molecules (59) can be considered as an extension of the theory of simple fluids to systems where the molecular core is not a point but a rod, convex figure or convex body. In the simpler BHlike method the soft convex bodies  parallel bodies (e.g., bodies enveloping individual cores with the soft 'thickness') ascribed to the studied molecules  form the reference system. The pair potential, u0, is given by Equation (5.112) for s <_ a. If the same coupling parameter as in the BH method was assumed, it was found that
2~ = (1  e  ~uo ) dr (5.128)
158
where ~ stands for the thickness of the parallel hcb to the given core; knowledge of enables one to define the representative hard convex body. Its geometric characteristics axe
and
R4  Rci + ~, Si  Sci + 87rRci~ + 4 ~ 2, Vi  Vci + Sci~ P 4~rRci~ 2 + 4~~3/3
y = pVi a = P~Si/3Vi
The reference residual Helmholtz energy, A 0  Aid, can be calculated from the ISPT or MSPT EOS's. In the former case
in the lat ter case
Ao  A id = (a2 _ 1)ln(1  y) + (~2 + 3c~  3o~y)y N k B T (1  y)2
(5.129)
A0  A ia N k B T  (6~2  50~ 1)ln(1  y) + 2(1  y)2
(12c~ 2  4 a ) y  (15~ 2  9(~)y 2
The firstorder per turbat ion term for moiecular fluids is generally given as
(5.130)
At _ p oo N k B T  2kBT fo //up(r12,wl'w2)gO(r12'wl'w2)drdwldO)2 (5.131)
In the case of the Kihara molecules both the pair potential and the average correlation function depend only on the shortest surfacesurface distance s; for hcb it holds that
(5.132)
where $1+2+s is the mean surface area of convex bodies I and 2 with the constant surface surface distance s. This mean surface area can be written as
$1+2+8 = St+2 + 8rRt+2s + 41rs 2 (5.133)
where S1+2 = St + $2 + 8rR1R2 and Rt+2 = Rt + R2 (5.134)
After substi tution and rearrangement one obtains, for the pure Kihara fluid in the BHlike aproximation, (59)
A1
N k B T ~1 °° 8* [8,2 2rpaa Up(8*)ghCb() +4R's* + 2 ( 3 " / 4 r + R*2)]ds *
kBT c (5. 35)
where s* = s /a , R* = R / a , S* = S / a 2 and c = 2~/a. In the case of the LJ fluid R*  S*  0 and the BH relation for simple fluids is recovered. It is obvious that one can write ghcb _ 1 + h hcb and express the firstorder perturbat ion integral as a sum of two parts, the larger of which can be determined analytically by
159
NkBT = T* + ~R + ~~(S*/4r +
27rp* fo~ Up(S,)hhcb(S)[s, 2 + 4R's* + 2(S*/4r + R*2)]ds * (5.136)
Within the BHlike method, one can also determine the secondorder perturbation term in the m.c. approximation; the corresponding expression will not be given here.
For systems of Kihara nonspherical molecules one can consider also the WCAlike demarcation of the regions of repulsive and attractive forces. Determination of the temperature and densitydependent thickness from the extended version of Equation (5.77) is more difficult due to the form of ghcb and the fact that &+2+8 has to be consid ered in the corresponding integral for the determination of ~WCA. Quite often a 'hybrid' version is used instead of the WCA.
5.5.3.2 Fluids of Molecules with Multicenter Pair Potentials
Application of the perturbation theory to systems of diatomic molecules or, more generally, to molecules with a multicenter pair potential (mC) is less unambiguous than is the case for the Kihara fluid. In the first place the demarcation of the range of repulsive forces causes problems. For example, if the WCAlike division of the pair potential is considered, we have to realize that, at the distance of the potential minimum of the total pair potential, some of the contributing sitesite pairs will already reach distances where attractive forces prevail, whereas other pairs will contribute by strong repulsions. An analogy of the BH division would be, however, even more problematic. Thus, the WCAlike division is the only one used for systems with a multicentre type of molecular interaction; the reference system is defined by
Uo(rl2,031,032)  umC(rl2,031,032)  umC(rl2min, 031,032)
0 r12 > rl2mi n
The perturbation pair potential is then
for r12 _~ rl2min
(5.137)
=  ( 5 . 1 3 8 )
In the perturbation theory proposed for systems with multicentre pair potentials, the KShlerFischer approximation (56,57) for the molecular distribution function, Equation (5.127), would be used; the hs diameter of a site in a homonuclear molecule then follows from the condition
fo ° T . ~ r2  AeoYS°ft(r12, ) 12ar12 (5.139)
where Ae0 denotes the blip function for the reference and fusedhardsphere model and ysoft is the background correlation function of the softsphere multicenter model. Knowl edge of the hs diameter of the representative hard body (e.g., hard dumbbell) makes it possible to determine thermodynamic functions of the reference by employing, e.g., ISPT
160
EOS; for the residual reference Helmholtz energy, one then obtains Equation (5.129). The firstorder term is
A1 2rp oo N k B T = k B T f o //Up(r12,Wl,W2) eU°(r12'wl'w2)/kBTysoft(r12,T*)r212dr12d~1dw2
(5.140) The manifold integral is solved  after determining yso~ _ by a numerical method (e.g., that due to Conroy).
It is obvious that the determination of the firstorder term in this case is a time consuming procedure; quite often only a limited number of values is determined and A1 is expressed in the form of a polynomial. Further thermodynamic functions are then calculated from derivatives of this formula.
5. 5. 3. 3 Fluids of Molecules with Electrostatic Interactions
Fluids of this class contain spherical or nonspherical molecules with permanent and/or induced dipoles, quadrupoles, etc. An example of the spherical polar molecule is the model of H C1, of the latter group a model of ethyl chloride. Intermoleculax interactions in the former case axe characterized usually by the Stockmayer potential, in the latter case, e.g., by the twocenter LJ plus dipoledipole pair potential.
The Stockmayer potential is a prototype of potentials which can be written as
~zA(rl2, St)l, 502)  ~o(r12) + ~a(r12, 031, 032) (5.141) where ua denotes the anisotropic part of the potential; the reference potential, uo, is defined as (41)
uo(.1.) = f f (5.142/ Considering the Aexpansion one finds
(0u(ri2, We, w2) ) =0 (5.143)
Thus, the first nonzero contribution is given by the secondorder perturbation term which for multipolemultipole interactions reduces to the form
A2a ~ 2 p / 2 ,w2)>Wl•2 g0(r12)drl2 (5.144) =
NkBT 4 For electrostatic interactions it is necessary to consider at least the thirdorder perturba tion expansion. For A3a one obtains
A3a _ 1 (AaA + AaB) ~13~ ' f<ua(r,2,Wl,W2))~,o~2 go(r12) dr12
NkBT NkBT 3p2
/ / / (~a( i , 2)Ua(X, 3)Ua(2, 3))WlW2Wa g0( i, 2, 3)dridr2dr 3 (5.145) (In the expression for AaB, symbol 1 stands for rl ,wl, etc.). When substituting for ua the potential, Equation (5.101), one finds that only function (I) depends on the mutuaJ
161
orientation of two molecules. After averaging over all the orientations wlw2, a constant factor results in the second and AaA terms. Distribution functions go(r) and g0(123) correspond to the reference system of LJ molecules in the case of the Stockmayer potential and hs in the case of hard spheres with embedded multipole moment. Evaluation of the integral from the product go r  " is straightforward in case of the hs reference system; for the LJ reference, parametric expressions are available in the literature (58). To determine AaB, the superposition approximation is used (60).
In the case of electrostatic forces, the perturbation expansion converges slowly for larger values of the multipole moments; determination of perturbation terms higher than the third is, however, prohibitively difficult. To improve the quality of prediction, Stell (61) suggested writing the contribution of the anisotropic part of the potential in the form of Pad~ approximant
A  A0 A2 1 = (5.146)
NkBT NkBT (1  As) A2
Due to the factorization of the perturbation integrals, the differentation with respect to density or temperature is easy and one can quickly obtain the contribution to the compressibility factor and internal energy.
The perturbation theory for nonspherical dipolar, quadrupolar, etc. molecules follows ideas discussed for the case of spherical polar particles. Polar nonspherical molecules have been modelled as hard dumbbells, hard prolate spherocylinders, twocentre LJ or prolate Kihara models with permanent point dipoles, quadrupoles, etc. embedded in the centre of the molecule. For such models the firstorder ES perturbation terms are zero and the reference thermodynamic functions are determined either from the hard body EOS or from the perturbation expansion for the nonpolar part of the potential (e.g., the Kihara fluid).
In the secondorder perturbation term (62)
A2a   p X 2 /oC~r2n((I)(O)l 032)go(r, w1 w2))w,w2r2dr (5.147)
NkBT (knT) 2 ' '
however, the molecular distribution function, go, of the reference (for single orientations) is not available. In the beginning of this section the similarity of the hard gaussian overlap model and hard ellipsiods of revolution was mentioned; one can expect that such similarity holds also between Kihara general and GOCE molecules. This is the basis for the approximation
go( , where x in the GOCE expression is defined as x = r/a(wx,w2). Substituting for go and changing variables yields
Nk T f f (I)2dwld O2f 0 gGO(x)x2nX2dx (5.148)
Defining X~  X/kBTa~ it holds
162
A2a = ~r "~3X*2j~ ~0 c~ x2r*+2dx
The manifold integral, Jk, over angular coordinates depends only on a, as follows from Equation (5.109). The A3A term is
AaA = 1 .Pao3X~aKa(a:)foo=gGO(x)x_a,,,+2dx NkBT 3
(5. 50)
For given a, integrals Jk and Kk can be evaluated from Pad~ approximants available in the
6 N
5
4 I I 0 1 2 3
Q"2
Figure 5.8 Compressibility factor of the quadrupolar Kihara fluid of the reduced length l* = 0.8118 at temperature T*  1.1 and density p = 0.36.  theory, *  simulation data.
literature (62). For A3B, a semiempirical parametric equation is to be used. To calculate the total electrostatic contribution to thermodynamic functions, Pad6 approximants are formulated from the second and thirdorder terms. The dependence of the compressibility factor of quadrupolar hard dumbbells on the quadrupolar moment at constant density is compared with MD data in Figure 5.8.
In passing we note that, in principle, the approach discussed for nonpolar multicenter models of molecules can be used for electrostatic contributions, too. Thus, hard dumbbell or twocenter LJ molecules with embedded point dipoles or quadrupoles can be considered as models of dipolar and quadrupolar molecules; the approach of averaging the product of the Boltzmann factor and potential over all the orientations, which has been mentioned, is applicable as long as the sitesite distance is small.
Another way of evaluating the contribution of electrostatic forces is to consider an expansion of the molecular distribution function of the reference in spherical harmonics (an expansion similar to that given in the beginning of this section for pair potentials) and then evaluate individual terms; due to the properties of spherical harmonics simple expressions for these terms result. The lowest coefficient in the gexpansion equals rdf go. This very general method faces two problems: slow convergency and the fact that higher coefficients of the expansion are available only from numerical methods.
163
5 . 5 . 4 W a t e r
Due to the importance of water for life on earth the equilibrium behavior of water has been studied by a large number of scientists. It appears, that due to its ability to form hydrogen bonds, and due to substantial polar effects, water represents a special case of molecular fluids which shares the traits of both the polar molecule and associat ing molecule systems (see Chapter 12). To interpret the equilibrium behavior of water, several kinds of pair interaction potentials (some of them quite sophisticated) have been introduced and used in a large number of theoretical and simulation experiments (see Chapter 12.)
Applications of the perturbation theory to describe the behavior of water depend on the range of temperatures and pressures considered. It was found (63) that at high tem peratures and pressures (P ~ 100 MPa) the perturbation expansion of simple fluids yields a fair description of the system. At low pressures and room temperatures, however, such a method is inappropriate and one has to use a perturbation method in which the reference includes a model of hydrogen bonding also. The use of such a reference has been proposed by Kolafa and Nezbeda (64) who introduced the 'primitive model of water'. Within this model the water molecule is assumed to be a hard sphere with four tetrahedrally arranged interaction sites, namely two of the H and two of the etype. The squarewell potential describes the interaction between H and e sites (resulting in the formation of the hydro gen bond) whereas zero potential energy corresponds to interaction of pairs of like sites. Applying the results of the Wertheim theory of associating molecule fluids (52), one can express the compressibility factor (considering the CS EOS) as
Z P M w = ZhS .Jr ZaSS _ i + y + y2  y 3 2y(1+2c) ( O c ) (5.151)  ( 1  y ) a  ( 1 + c ) 2  Oyy T
where
{1 cl( i  y /2)  4"5c2y(i + Y) } ~ / 2 1 (5.152) c = ~ + 48y[e 1/T*  1] (1  y)3 2
and Cl and c2 are two constants. The Helmholtz energy follows from the corresponding integral. The distribution function for this primitive model of water is not available; thus, a van der Waals attractive term with an adjustable parameter is added to the reference term, instead of the correct perturbation term(s), to characterize the effect of the attractive forces in water.
5 .6 M I X T U R E S OF M O L E C U L A R F L U I D S
5 . 6 . 1 M i x t u r e s o f N o n  E l e c t r o l y t e s
The perturbation theory of mixtures of molecular fluids combines ideas and approx imations of the approaches for pure molecular fluids with those for mixtures of simple fluids. It is practical to consider first the mixtures of nonpolar nonspherical molecules and only then mixtures of polar molecules.
In the case of mixtures of Kihara molecules, the BHlike method of dividing the pair
164
potential into reference and perturbation parts and determining the respective hcb thick ness is generally used; alternatively the hybrid approximation is considered. In the WCA expression for thickness, ~, the background correlation function depends not only on tem perature and density but also on the relative size and shape of representative bodies.
Once the thicknesses of the representative hcb's are known the reference residual Helmholtz energy is calculated, e.g., from
A0,  Aid NkBT  (6~s  5c~s  1)ln(1  y) { 2(1  y)2
(12/~s 4c~s)y ( 1 5 ~  9a,)y 2 (5.153)
The firstorder perturbation term is
_ p f o o • • •
NkBT  2kBT ~ ~ xixja~ . . Up(S*)ghCb(s /c)S~+j+sds
pa 3 12 hcb * * (5.154)
where the asterisk denotes the reduced quantities. The average correlation function is available from Equation (5.118) where O is given by sums of surface areas for the pair of molecules considered. When determining the excess thermodynamic functions (which depend on subtle differences in expressions for a mixture and pure components) it is desirable to include the secondorder perturbation term. However, its rough form can lead, in some cases, to relatively larger differences between the calculated and experimental values.
Only the firstorder perturbation expansion is used in the case of mixtures of molecules interacting via multicenter pair potentials. Thus, binary mixtures of L J plus twocenter LJ molecules or mixtures of two twocenter LJ compounds were studied (65). The WCA like division of the pair potential was used and the firstorder integrals determined in a similar way to that used in the case of pure fluids. Excess thermodynamic functions of several binary mixtures of homonuclear molecules were determined. In spite of the fact that the (less well known) secondorder term is negligible in the WCAlike variant, the calculated values of the excess thermodynamic functions agreed with experimental data in the majority of cases quite well for mono and diatomic molecules of the moderate nonsphericity. This is apparent from Table 5.3. where some results of Bohn et al. (65) are listed.
Description of the equilibrium behavior of mixtures of polar nonspherical molecules represents a difficult task for the theory of fluids. For applications in chemical engineering an extension of the approach described previously for the pure fluid of polar Kihara (rod like) molecules was proposed (66). In this method the BHlike demarcation of the range of the predominant effect of repulsive forces is used and thicknesses of pure representative hcb's evaluated accordingly. The second and thirdorder electrostatic perturbation terms are determined after substitution of the GOCE distribution function for ghcb(r, Wl,W2) of the reference mixture. The corresponding GOCE bodies have the same volume and parameter of nonsphericity as the reference bodies. An arithmetic mean for a0 and ~ was assumed when evaluating the contribution of the unlike pairs. In the process of calculation
A1
165
Table 5.3 Excess thermodynamic functions (in J tool 1 and cm3mo11) from the perturbation theories for 2cLJ and Kihara (Kih) mixtures.
System expt Kih 2cLJ CH4N2(91 K) AG E AH E AV E
XeC2H4(ll6 K) AG E AH E AV E
O2N2(78 K) AG E AH E AV E
C2H6C2H4 (161 K) AG E AH E AV E
=0.960 ~=0.0.956 170 170 170 138 164 148
0.54 0.39 0.55
209 255
~=0.976 ~=0.967 209 209 181 219
0.46 0.49
~=1.003 ~=1.004 40 4O 4O 60 41 34
0.25 0.25 0.24
~=0.984 ~=0.984 99 99 99 193 132 143 0.16 0.10 0.12
of the thirdorder term the A3B contribution has been neglected because of problems with the distribution function g(a)(1, 2, 3) for hard nonspherical molecules of different size and shape.
In the case where the hardconvexbody mixture serves as a reference, one can write the secondorder ES term as
s
NkBT  pEE 3 [ 1
= xixja°ijXnijJkiJ(~iJ) 2 n  3 ~ 1 hs 2n+2 (5.155)
where X*ij stands for the reduced coefficient of the ES pair potential with r n dependence on distance; the integral in Equation (5.155) can easily be determined numerically.
5.6.2 M i x t u r e s of E l e c t r o l y t e s and A q u e o u s So lut ions
Description of the equilibrium behavior of aqueous solutions of salts or, more generally, solutions of strong electrolytes differs from that of nonelectrolytes, because, in the former case, the structure of the electrolyte systems depends more pronouncedly on the attractive electrostatic forces than in the case of nonelectrolytes. Due to this fact a special theory of electrolytes has been developed and applied to characterize the equilibrium behavior of ionic systems (see Chapter 16.). Only relatively recently has the general perturbation
166
theory been considered for the simplified models of aqueous solutions of electrolytes. The extended EXP variant, c.f. Equation (5.90), of the perturbation theory was used (67) to determine the behavior of the uniunivalent model of electrolyte solutions; fair agreement with simulation data was found. Another variant of the perturbation theory, in which resummation of terms of different orders is considered (similarly to the EXP variant), was proposed for mixtures of electrolytes by Henderson(68). The resummed terms contain only the perturbation pair potential (i.e., the product of 1 × Up where 1 comes from the expression for the reference distribution function, gO _ 1 ÷ h °) whereas contributions to the individual perturbation terms containing integrals from h ° ( r ) u p ( r )  which in contrast with the previously mentioned terms are convergent  are determined individually. This variant makes it possible to take into account the effect of properties of the solvent on the thermodynamic functions of ionic fluids.
Aqueous solutions of nonelectrolytes were studied by Nezbeda et al. (69) who deter mined the pressure of the reference from a sum of the EOS for a mixture of hard spheres of different diameters (or a similar EOS of hcb's) plus the associatingmolecule term for water. A van der Waals term for mixtures takes into account the effect of all kinds of attractive forces.
5.7 C O N C L U S I O N S
In this Chapter an outline of the perturbation theory and some applications have been given. We limited ourselves to simple and easily tractable methods that will allow chemical engineers to apply these expressions of statistical thermodynamics to calculate different characteristics of the phase equilibria of pure fluids and their mixtures. It was our aim to show that the use of expressions of the perturbation theory brings only a slight increase in the complexity of the procedures in computing programs (in comparison with those for cubic equations, most often used for these calculations); these changes are often marginal considering the present level of computing technique.
There are, of course, more fundamental methods available in the literature that allow very detailed characterization of the intermolecular interactions of molecules and sophisti cated methods for evaluating thermodynamic functions; these approaches are often quite timeconsuming, comparable in this respect with simulation experiments. Such methods (including those with corrections for quantum effects) axe of great importance for the systematic study of fluid behavior within physics, but are less suitable for chemical engi neering studies.
Perturbation theory yields very powerful and versatile approaches that enable the study of diverse problems of the thermodynamics of fluids and phase equilibria. A great ad vantage of this theory is the fact that any improvement in our knowledge of the model systems results in an improvement of the accuracy of perturbation theory for calculations of all the properties of real fluids.
Acknowledgement. The author is indebted to Professors U.K. Deiters and J. Fischer for materials and notes for this Chapter and to Dr. K. Aim for reading the manuscript.
167
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8. 9.
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38. 39.
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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 169
6 E Q U A T I O N S O F S T A T E F R O M A N A L Y T I C A L L Y S O L V A B L E I N T E G R A L E Q U A T I O N A P P R O X I M A T I O N S
Yu. V. Kalyuzhnyi ~b and E T. Cummings a
aDepartment of Chemical Engineering University of Tennessee Knoxville, TN 379962200, USA and Chemical Technology Division Oak Ridge National Laboratory Oak Ridge, TN 378316181, USA
blnstitute for Physics of Condensed Matter Svientsitskoho 1, 29001 Lviv, Ukraine
6.1
6.2
6.3
Introduction 6.1.10rnsteinZernike Equation and Closure Relations 6.1.2 Short Review of the Recent Progress in IntegralEquation Theory Baxter/Wertheim Factorization of the OrnsteinZernike Equation 6.2.1 OneComponent Monatomic Fluids 6.2.2 MultiComponent Monatomic Fluids 6.2.3 Remarks Equation of State for Analytically Solvable Models of Simple Fluids 6.3.1 HardSphere Fluid
6.3.1.1 Solution of the PYA for the HardSphere Fluid and Fluid Mixtures
6.3.1.2 Equation of State for HardSphere Fluid and Fluid Mixtures 6.3.1.3 Semiempirical Correction to the PYA: The Generalized MS A
6.3.2 Adhesive HardSphere Fluid 6.3.2.1 Solution of the PYA for the Adhesive HardSphere Fluid and
Fluid Mixtures 6.3.2.2 Equation of State for the Adhesive HardSphere Fluid and
Fluid Mixtures 6.3.2.3 PercusYevick Theory for the HardSphere Chain Fluid
6.3.3 HardCore Yukawa Fluid
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6.3.3.1 Solution of the MSA for the HardCore Yukawa Fluid and Fluid Mixtures
6.3.3.2 Equation of State for the HardCore Yukawa Fluid and Fluid Mixtures
6.3.4 Charged HardSphere Fluids 6.3.4.1 Solution of the MSA for the Primitive Model of Electrolyte
Solutions 6.3.4.2 Equation of State for the Primitive Model of Ionic Fluids and
Fluid Mixtures 6.3.4.3 Corrections to the MSA: the Generalized MSA and the
Associative MSA 6.4 Equation of State for AnalyticallySolvable Models of Molecular Fluids
6.4.1 InteractionSite Model Fluids 6.4.1.1 SiteSite OmsteinZemike Equation and Closure Conditions 6.4.1.2 SSOZPYA for the Fluid of Fused HardSphere Diatomic
Molecules 6.4.1.3 SSOZMSA for the Fluid of Fused Charged HardSphere
Diatomic Molecules 6.4.1.4 Corrections to the SSOZ Formalism: ChandlerSilbeyLadanyi
Equation 6.4.2 Hard Spheres with Dipolar Interactions
6.4.2.1 Analytic Solution of MSA for Dipolar HardSphere Fluid 6.5 Equation of State for Analytically Solvable Models of Associating Fluids
6.5.1 TwoDensity OZ Equation and Closure Conditions: Relation to the CSL equation
6.5.2 TwoDensity PYA for Dimerizing HardSphere Models 6.5.3 TwoDensity PYA for the SmithNezbeda Primitive Model of
Associating Fluids 6.5.4 TwoDensity MSA: RPM of Electrolytes and Dimerizing HCY Fluid
6.6 Conclusion Appendix: Summary of the Invariant Expansion Method for Linear
Molecules References
6.1 INTRODUCTION
Modern statisticalmechanical theories of liquids and dense fluids are based on the calculation of distribution functions, which describe the average structure
171
of the system. The major theoretical question to be answered is: Given the inter molecular potential between the molecules in the system, what will the structure of the system be at a given thermodynamic state point? Usually it is assumed that the intermolecular potential is pairwise additive, with the consequence that only the pair structure is needed to describe all the thermodynamic properties of inter est. Although the assumption of pairwise additivity is not valid at high densities and low temperatures, it proves to be a good approximation in most instances. Moreover, intermolecular pair potentials are frequently parameterized to take into account higherorder interactions.
In this chapter, we will focus on the integralequation approach to determining the structure of a fluid and therefore its thermodynamic properties and equation of state. For a fluid composed of spherically symmetric molecules, the princi pal quantity calculated by an integralequationbased theory is the radial distribu tion function, 90(r), which is proportional to the probability density of finding a species i and a species j molecule separated by distance r and is normalized so that 9ij(r) + 1 as r + ec. The radial distribution function is a statistical measure of the average structure in the fluid, and knowledge of this function permits all thermodynamic properties to be determined for a system interacting by pairwise potentials. The quantity pjgij(r), where pj is the number density of species j molecules, can be regarded as the local number density of species j molecules at a distance r from a species i molecule. For an ideal gas, 9ij(r) = 1 for all r; hence it makes sense to define a 'residual' function, called the total correlation function, given by hi j ( r ) = 9i j (r)  1. This function can be thought of as a measure of the correlation (nonrandomness) in the relative positions of two molecules.
There are three routes to thermodynamic properties from distributionfunction theories. We will quote these formulae here without proof, but note that a partic ularly transparent derivation is given by Baxter (1). The compressibility route to the thermodynamic properties of an mcomponent mixture is given by
] m / 1 0 P  1  ~ pk Cik(r)dg" (6.1)
kBT Opi T k=l
where P is the pressure, k B is Boltzmann's constant, Pm is the number density of a species m, T is absolute temperature, c~j(r) is the direct correlation function between a species i and a species j molecule separated by distance r defined in section 6.1.1 below. The virial route is given by
P 27rp m m foo pksT = 1 3ksT i~1 ~ xixj Jo 9ij(r) duij(r) r3dr (6.2)
• = j=l dr
172
where uij(r) is the pair potential between a species i and a species j molecule, p is the total number density and xi is the mole fraction of species i, i.e.,
m Pi P   ~ Pk , Xi    
k=l P
Finally, the energy route to the thermodynamic properties is given by
u res
NkBT 27rp m m f ~ ksT ~ ~ xixj Jo 9ij(r)uij(r)r2dr
i1 j1 (6.3)
where U res is the residual internal energy over the ideal gas at the same tem perature and composition. In general, there are three ways to obtain the pair correlation functions:
1. radiation scattering experiments, such as neutron and Xray scattering; 2. molecular simulations such as molecular dynamics and MonteCarlo
simulation; and 3. theoretical approaches such as integralequation methods and perturba
tion theory. The main advantage of the first two methods is that both of them are, in principle, exact: scattering experiments are exact for a particular substance while molecular simulation is exact for a particular model. However this immediately suggests their limitations. Since both of them are essentially experimental methods, the structural results obtained by the methods are restricted to a particular thermody namic state point at which they were measured. Thus these methods do not have the predictive ability of theory.
The integralequation approach is a theoretical method for the calculation of the distribution functions and is not subject to these limitations. However, unlike molecular simulation, integralequation theories necessarily involve approxima tions as noted below and consequently the results are not exact. For this reason, they are frequently referred to as integralequation approximations (IEAs). An important fact to bear in mind about integral equation approximations is that the approximations inherent in the correlation functions result in inconsistency be tween the thermodynamic properties predicted by the three different routes. This is especially important in considering the critical behavior of integralequation approximations. An additional source of uncertainties in the predictions of the IEA is insufficient information concerning the precise form of the intermolecular forces in real fluids. [The same problem arises when the results of molecular sim ulations are compared directly with experiment.] In the case of theoretical studies
173
the latter problem can be bypassed by assuming a certain reasonable potential model, which correctly reproduces the most characteristic features of a particular class of fluids. The accuracy of the results obtained with such a model is quite sufficient for most theoretical studies, which have as their aim understanding the origin of fluidphase properties. However, engineering applications frequently re quire a much higher degree of quantitative accuracy, so that the direct application of IEAs in such practical calculations is of limited use. In practice, the desired accuracy can be reached by using empirical formulae (in which highorder poly nomials are fitted to experimental data) or semiempirical methods (in which the molecular parameters in a theoretically derived equation of state are fitted to ex perimental data). The predictive capability of such methods strongly depends on the quality of the theoretical basis used. Purely empirical methods without any reasonable theoretical basis are little more than convenient interpolation schemes, the results of which cannot be used to extrapolate with confidence outside the in terpolation region. This was clearly demonstrated by Henderson (2) in his analysis of the semiempirical equations of state commonly used in chemical engineering calculations. A good example of a very accurate semiempirical equation of state (EOS) for the LennardJones (LJ) fluid developed on a sound theoretical basis is the EOS proposed recently by Kolafa and Nezbeda (3).
IEAs can play an important role in the formulation of semiempirical equa tions of state by providing the theoretical basis and functional form for equations of state which can then be used to correlate experimental data. In particular, ana lytically solvable IEAs are therefore of greatest interest for this purpose.
The goal of this Chapter is to introduce the reader to some of the recent ad vances in the analytic solution of IEAs for molecular fluids and to demonstrate their ability in predicting thermodynamic properties of physical systems focusing mainly on the results for the EOS. Analytic solutions are now available for a num ber of rather nontrivial Hamiltonian models of dense fluids and liquids which, in particular, can undergo the liquidgas and fluidfluid phase transitions. [A de scription of the solid phase, and hence fluidsolid transitions, is precluded from IEAs by the disordered fluid assumption, Equation (6.18).] We illustrate the re sults obtained for thermodynamic properties by presenting the liquidgas phase diagram. The paper is organized as follows. In the remainder of this section, we introduce the OrnsteinZernike equation and its closure relations and discuss briefly the recent progress achieved in integralequation theory. In Section 6.2, we review the Baxter WienerHopf technique for pure fluids and mixtures used to solve IEA analytically. Section 6.3 contains a review of applications of the Baxter technique to simple fluids. This includes sections on the hardsphere fluid and its
174
mixtures, adhesive hardsphere fluid and its mixtures, the hardcore Yukawa fluid and the charged hardsphere fluid. In Section 6.4, we review analytical solutions of IEAs for nonspherical molecules, including interactionsitemodel fluids and fluids composed of hard spheres with multipolar interactions. In the last section, Section 6.5, we discuss IEAs for associating fluids and present recently obtained analytic solutions for a number of simple models of associating fluids.
6.1.10rnsteinZernike Equation and Closure Relations
To calculate the total correlation function, Ornstein and Zernike introduced a second function via the equation that bears their name, the OrnsteinZernike (OZ) equation. This second function, called the direct correlation function, is less singular at the critical point (i.e., does not become longranged in the sense that the volume integral remains finite) than the true function of interest, the total correlation function. The OZ equation for an mcomponent mixture is given by (4)
m
hij(r)  cij(r) q ~ Pk /Cik(l~)hkj(I ~' ~)d~ (6.4) k1
This equation can be interpreted as follows: the total correlation function between species i and species j molecules at a distance r apart, h#(r), can be broken up into a direct part, e~j(r), and indirect part ~=~ Pk f c,k(]~)hkj(]~" ~)d~. The form of the latter term indicates that indirect correlations take place when the species i molecule, located at the origin, is directly correlated with a species k molecule, located at the position g, which in turn is totally correlated with a species j molecule located at ~'. For a pure (i.e., onecomponent) fluid, Equa tion (6.4) simplifies to
h(r)  c(r) 4 p f c( ] 8] ) h( ]~'  (6.5)
To have a closed mathematical problem, an independent relation between the direct and total correlation functions is required. A formally exact independent relationship can be written down using the language of graph theory, which is beyond the scope of this review. The total and direct correlation functions are expressed in terms of an infinite sum of graphs ordered by density, the direct correlationfunction graphs being a subset of the total correlation graphs, as is clear from the OZ equation. The difference between the two functions can there fore be expressed as a graphical expansion in which the graphs have well defined
175
topological features. The graphical definition of this difference amounts to the for mal, exact independent relationship between the two correlation functions. The reader interested in learning more about the graphical expansions is referred to the monograph by Hansen and McDonald (5) for a complete and very understand able account of the graph theoretical developments pioneered in rigorous form by Stell (6,7). The net result can be written as
hi j ( r )  ciy(r)  log[hi j (r ) + 1]~ k B T
B i j ( r ) (6.6)
where, because of their topological structure, the graphs in the function B i j ( r ) are often referred to as the bridge diagrams. Choosing B i j ( r )  0 yields the hypernetted chain approximation (HNCA) (5),
while rewriting Equation (6.6) as
#ij (r)  exp
k B T
u ij ~ r~ ] + B i j ( r ) + h i j ( r )  c i j (r)]
k B T J
,.~ exp m U ij ~ r~ ]
+ k B T j " '
(6.7)
g q ( r )  exp [  U ~ j ( r ) l [g~j(r)  (6.8)
An important correlation function from the point of view of solvent effects on chemical reaction is the cavitydistribution function, y i j ( r ) , defined by
Yij(r)  exp L k s T 9i j (r ) (6.9)
For systems without hard cores (that this is not the case for hardcore systems is a subtle point not often appreciated (6)), Equation (6.8) can be written as
(6.10)
setting B i j ( r )  O, and neglecting the higher order terms in the expansion of the exponential yields the PercusYevick approximation (PYA) (8,9)
176
For hardcore systems (i.e., systems for w h i c h uij(r)   ~ for r < aij), all the IEAs satisfy the exact core condition 9ij(r) = 0 for r < ai3. In addition, we have the exact asymptotic result for the direct correlation function given by (10)
For hardcore systems, we can combine the exact core condition with the assump tion that the asymptotic form holds for all intermolecular separations outside the hard core. That is, we define the closure
hij(r)   1 r < aij (6.11) cij(r)   u i j ( r ) / k B T r > aij
which is known as the mean spherical approximation (MSA) (9). The advantage of the MSA is that it is analytically solvable for a broad class of pure fluids and mixtures, as we shall describe below.
Many other integralequation approximations have been defined, and it is be yond the scope of this Chapter to attempt to review them here. The monograph by Hansen and McDonald (5) contains a comprehensive discussion of more re cent developments. We mention two others briefly here: the soft mean spherical approximation (SMSA) can be defined for systems without hard cores by break ing the intermolecular potential up into a reference part, u°j(r), and an attractive perturbation part, ui~(r ), and is given by (11)
eli(r)  9ij(r) 1  exp u°j(r) UiJ (6.12) kBT kBT
The hybrid MSA (HMSA) (12) is defined as a combination of the SMSA and the HNCA, and is defined by
1 In 1  s + sg i j ( r ) exp [ k s T k s T (6.13)
where s is an adjustable parameter. In the limit s = 1, Equation (6.13) reduces to the HNCA while for s = 0 it yields the SMSA. The parameter s is adjusted to yield consistency between the energy and compressibility routes to the thermodynamic properties. The application of the SMSA and HMSA, as well as other IEAs, to supercritical fluids was recently reviewed by Cummings (13).
6.1.2 Short Review of the Recent Progress in IntegralEquation Theory
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The OZ equation, Equation (6.4), together with the relation Equation (6.6) is a formally exact recipe whereby the structure and thermodynamics of a fluid can, in principle, be calculated if the intermolecular interaction potentials are known. In fact, however, exact calculations have been carried out for only a very limited number of highlysimplified model potentials. One example is the one dimensional hardsphere system (14,15), for which the PYA, Equation (6.8), ap pears to be exact (16). The study of most systems of practical interest requires the use of a variety of different approximations. The success of a given lEA depends on its ability to reflect the most specific features of the intermolecular interactions as well as on its ease of implementation.
In general the different parts of the total potential cause different physical effects. The shortrange steeply repulsive part of the potential gives rise to the excludedvolume effects and provides the main contribution to the entropic and structural properties of the system. The longrange interactions, such as attractive dispersion forces and multipolar forces, vary with the distance more smoothly and contribute mainly to the energetic properties. The longrange Coulombic in teractions cause the collective behavior of charged molecules which results in screening. Short and longranged effects differ fundamentally and thus require different approximations in their description. Although this idea was formulated by van der Waals a long time ago, significant progress in its application became possible only in the last three decades primarily due to the development of the PYA and HNCA integralequation theories. The PYA appears to be quite accurate in describing excludedvolume effects (17), while the HNCA has proven to be rel atively successful in accounting for the longrange part of the potential (1822). Finally these developments result in the implementation of the van der Waals' idea in the form of the reference hypernettedchain approximation (RHNCA) of Lado (2326), which is perhaps the most accurate integralequation theory for simple fluids. Recent progress extending Lado's ideas can be found in the work of Labik et al. (2729) and Pospi~il et al. (30,31) who developed an accurate description of the molecular reference system for fluids with nonspherical hardcore interaction, and of Lomba et al. (32) and Lombardero et al. (33,34) who used the RHNCA to obtain results for the properties of the LJ and polar molecular fluids.
An alternative approach, which can be also viewed as an implementation of the van der Waals' idea of using different approximations for different parts of the potential, has been initiated by Rowlinson (35) and developed in later studies by Hutchinson and Conkie (36), Rogers and Young (37), and Zerah and Hansen (38). This is based on the idea of constructing a closure which interpolates between two different closure conditions, for example PYA for shortrange and HNCA for
178
longrange (37). A mixing parameter in the closure is defined to enforce equality of the virial and compressibility equations of state. An important advantage of this approach is the elimination of the intrinsic thermodynamic inconsistency of the traditional IEAs.
The difficulty encountered in use of the above discussed IEAs is the need for numerical methods for the solution of both the RHNCA and the selfconsistent theories. However, the numerical methods are much less time consuming than the corresponding computersimulation calculations. One can expect significant progress in the future through the idea of selfconsistently combining different ap proximations for different parts of the potential in the context of analytical tech niques discussed in this review.
All the IEAs discussed above are generally applicable only to the case of a pair potential with attractive interactions of moderate strength (i.e., characteristic attractive energies of the order of several kBT at most). Attempts to apply them in the case of associating fluids have met with limited success. The latter class of fluids is characterized by strong shortrange attraction with the absolute value of the attractive potential well varying from 3  5kBT up to 1023kBT. This short ranged attractive part of the potential causes the quite different physical effect of clustering of particles and thus requires the use of an appropriate approximate description. The regular IEAs are closely related to the Mayer density expansion, or pexpansion. For strongly associating liquids, there is a severe problem with this expansion: an inf ini te number of terms must be included to reproduce the correct lowdensity limit (39,40). This can be demonstrated by using the following simple argument (40). Consider the virial expansion for the pressure. In the low density limit, the power series in the activity z for the pressure P and density p may be terminated at a second order to give
P = z + b2z 2, p  z + 2b2z 2, (6.14)
k B T
where b2 is the second virial coefficient. Elimination of z from these two equations yields
P 1 (1 + 8b2p) 1 / 2  1 =  + . (6.15)
k B T p 2 8b2p
The more familiar corresponding virial equation of state is obtained from Equa tion (6.15) by expanding the square root in the power series of p to obtain
P = 1  b2p + 462p 2  20b~p 3 + .... (6.16)
k B T p
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For systems with weak interactions, b2 is finite and both Equations (6.15) and (6.16) reproduce the ideal gas EOS, P / k B T p = 1, in the p + 0 limit. However in the limit of strong attraction between pairs, b2 ~ cx~ and the correct equa
l follows only from Equa tion of state for an ideal gas of dimers, P / k B T p  ~, tion (6.15), and cannot be reproduced by Equation (6.16) with a finite number of terms. Hence, the usual OZ closures, with their connection to the pexpansion, are not designed to treat strongly associating systems, and cannot be expected to systematically produce good results.
Inspired by early attempts to model association effects using the (4144) PYA, Wertheim developed an integralequation theory for strongly associating fluids (4548) which was subsequently extended by others (40,49,50). This the ory is based on the multidensity formalism, in which the descriptions in terms of the activity and density expansions are combined. Such a combined description allows one to introduce different approximations for different parts of the poten tial. For the shortranged strongly attractive part an activity expansion is used as a starting point, while the rest of the potential is taken into account by means of the standard Mayer pexpansion. Thus the shortranged repulsive and longranged parts of the potential are described using approximations similar to those used in standard integralequation theory, while for the shortranged attractive part new approximations, based on the multidensity descriptions, have been proposed. The theory appears to be quite successful in the description of a number of different models of associating fluids (40,50,5157). Details of the theory and its applica tion in the case of analytically solvable models are discussed in Section 6.5 of this review.
We conclude this section by noting that the above short discussion cannot be considered a comprehensive review of recent progress in the field of integral equation theory. We discuss here only those recent developments which, in our opinion, are the most promising for the future development of equations of state. In all of these theories, a reasonable compromise between the accuracy and the simplicity of the approximation used is achieved on the basis of van der Waals' concept of utilizing different approximations for different parts of the potential.
6.2 BAXTER/WERTHEIM FACTORIZATION OF THE ORNSTEINZERNIKE EQUATION
Historically, the first IEA to be solved analytically was the PYA for the hard sphere fluid (58,59) just under forty years ago. It is difficult to overstate the im portance of this analytic solution in the subsequent development of the statistical
180
mechanical description of fluid systems. For example, the development of pertur bation theories relied on the availability of analytic expressions for the structural and thermodynamic properties of the hardsphere fluid which served as the refer ence system (5).
Wertheim (60) and Baxter (61) subsequently developed general techniques for the analytic solution of IEAs based on factorization of the OZ equation in the Laplace domain (in Wertheim's case) and in the Fourier domain (in Baxter's case). Baxter's factorization, based on the WeinerHopf method (62), has proved to be the technique most frequently used by later authors for solving IEAs analytically. The basic rationale behind the WeinerHopf technique is to examine the integral equation in Fourier kspace, and to seek a factorization of the transform of one of the required functions into either a sum or product of functions which have identifiable analyticity properties in separable regions of the complex kplane. The exact manner in which this is done is very much dependent on the specific problem; however, the general technique is a standard one, which can be found in any advanced textbook on integral equations (6365). The key contribution of Baxter (61) was in recognizing that the WeinerHopf techniques developed for analyzing onedimensional equations similar to the OZ equation (for example, (66)) could be successfully carried over into three dimensions. Essentially, this stems from the observation that for n odd, the ndimensional Fourier transform of a function f(r') (where g' is an ndimensional vector) which depends only on Itq, reduces to a onedimensional Fourier transform after performing the angle integrations. Once the problem is cast in the language of onedimensional Fourier transforms, the factorization proceeds in a standard way, a fact pointed out by Baxter (61).
6.2.1 OneComponent Monatomic Fluids
The Baxter WeinerHopf factorization (which we shall simply refer to as the Baxter factorization) is based on only two assumptions. First, the direct correla tion function is assumed to be finite ranged,
c(r)=O, r > R (6.17)
This is the case for the PYA and the MSA applied to a finiteranged potential (u(r) = 0 for r > R). The hardsphere fluid, described below in Section 6.3.1, is clearly a special case (R = a) of this general assumption. Second, the fluid is assumed to be disordered. This means that the total correlation function is not
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longranged (as it is at a liquidgas critical point or in the solid phase), i.e.,
f h(r)d~' < (6.18) cx:).
Multiplying both sides of the OZ equation, Equation (6.5), by exp(ik • ~) and integrating with respect to ~' over all space, we find that,
h(k)  ~(k) + p]z(k)~(k) (6.19)
where ]z(k) and ~(k) are the threedimensional Fourier transforms of h(r) and c(r), depending only on the magnitude k of the vector k. Equation (6.19) may be rearranged into the form
[1 + ph(k)][1  p(?(k)] 1 (6.20)
which is more suitable for factorization. It is possible to prove, by using Equations (6.17) and (6.18) and standard prop
erties of integrals in the complex plane, that 1  pO(k) can be factorized into a product of the form (61)
1  p~(k)  Q ( k ) Q (  k ) (6.21)
where Q(k) is regular, has no zeros in the upper half plane and approaches unity as Ik[ + ec. These properties permit one to show that Q(k) is essentially the onedimensional Fourier transform of a realspace function q(r), defined by
1 o~ 27rpq(r)  ~ f ~ e ik~ [ 1  Q(k)] dk (6.22)
with the following properties
q ( r )  O for r < 0 and r > R (6.23)
One then writes the OZ equation, Equation (6.19), as a set of two equations
1  p5(k)  Q ( k ) Q (  k ) (6.24)
Q(k) [1 + p i t ( k ) ]  [(~(k)] 1 (6.25)
and uses the properties of (~(k) and ~)(k) in the respective half planes to perform the onedimensional Fourier inversion of these two equations, obtaining
rc(r)  q ' ( r ) + 27rp q ' ( t ) q ( t  r)dt (6.26)
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fo R rh(r)  q ' ( r ) + 27rp q(t)(r  t )h( r  t l)dt (6.27)
As a result the initial OZ equation, Equation (6.19), has been rewritten as a pair of equations in which the functions h(r) and c(r) are decoupled. These equations are the principal results obtained by Baxter (61) and yield a particularly useful form for the OZ equation. First they indicate that, if c(r) is finiteranged, knowledge of c(r) and h(r) on that range alone is sufficient to determine h(r) for all r. This result does not follow from the OZ equation itself, and is a simplification that comes out of the WeinerHopf analysis, Equations (6.21)(6.27). Second, it is also clear that computing the Baxter qfunction on the range (0, R) is sufficient to determine h (r) for all r.
A similar factorization can be carried out for the multicomponent version of the OZ equation, Equation (6.4), as will be demonstrated in the next section.
6.2.2 MultiComponent Monatomic Fluids
The factorization, described above, was extended to mixtures by Baxter (67). We assume, as in Section 6.2.1, that the c4j(r) are finiteranged, that is
(Ri + Rj) (6.28) c # ( r )  O, r > R~j = 2
where R/, i = 1 , . . . , m are some additive range parameters. Under this assump tion, Baxter (67) was able to argue that a WeinerHopf factorization of the OZ equation, Equation (6.4), should be possible. The result of the factorization is
m
rcij(r)  q~j(r) + 27r ~ Pk f qki(t)q'kj(r + t)dt k=l
(6.29)
rhij(r)  q~j(r) + 27r ~ Pk qik(t)(r  t)hkj(Ir  t l)dt (6.30) k= l ik
_ 1 (Ri  Rj) the integration in Equation (6.29) is over the range where Sij ~
Ski < t < min[Rki, Rkj  /'] (6.31)
and the Baxter qfunctions satisfy
qij(r) = O, r < Sij, r >_ Rij (6.32)
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Hence the OZ equation for mixtures has been transformed into a form which de couples the hij(r) and cij(r) functions. In addition, for Sij < r < Rij , Equa tion (6.30) involves each function hkj(s) only over the range (0, Rkj) . Thus, as in the case of a onecomponent system, knowledge of the Baxter qfunction in the range (0, Rij) enables one to calculate hij(r) in the whole range of r.
6.2.3 Remarks
As we shall see below, the factorization method has permitted the analytic so lution of IEAs for many important model fluids, beginning with the hardsphere fluid. In some cases, the analytic solution is in closed form, with expressions for all the quantities of interest given explicitly in terms of the thermodynamic state variables, temperature, density and composition. In other cases, the analytic so lution reduces to a set of simultaneous nonlinear algebraic equations which must be solved numerically. In the latter case, the calculation of thermodynamic prop erties on the liquid side of the gasliquid phase transition can be difficult, if not impossible, due to limitations on integrating through the twophase region which typically contains a region of no real solutions to the algebraic equations. Finally we comment that, in all cases studied to date, a variation on Perram's method (68) permits calculation of the radial distribution function. In some cases, such as the hardsphere fluid in the PYA, the radial distribution function is known analytically in closed form (69). In this review, we shall regard an integralequation approx imation as solved analytically if an explicit analytic expression for the Baxter qfunction can be obtained.
6.3 EQUATION OF STATE FOR ANALYTICALLY SOLVABLE MODELS OF SIMPLE FLUIDS
In this section, we shall consider simple fluids (i.e., composed of spherically symmetric molecules) for which the PYA or the MSA is analytically solvable. The PYA is solved analytically for the hardsphere (section 6.3.1 and adhesive hard sphere (section 6.3.2) fluids, while the MSA is solved analytically for the hard core Yukawa fluid (section 6.3.3) and the charged hardsphere fluid (section 6.3.4).
6.3.1 HardSphere Fluid
The hardsphere fluid (HSF) plays a central role in the modern molecular the ory of fluids because of its use as the reference system in many perturbation the ories. The intermolecular pair potential for the pure HSF is given by
u(r) = ~ r < cr = 0 r > cr (6.33)
184
where cr is the diameter of the hard sphere. For mixtures, the HSF is defined by
ui j ( r )  c~ r < aij (6.34) = 0 r > O'ij
where aij is the distance of closest approach of the centers of a species i and a species j hard sphere, and so is related to the individual diameters ai and aj by
ai + aj (6.35) O'ij = 2
The analytic solution of the PYA for the HSF and HSF mixtures led to the devel opment of the highly accurate analytic expressions for the HSF structure (70) as well as an accurate equation of state for HSF mixture (71). In the following, we derive the analytic solution of the PYA for the pure HSF, for HSF mixtures, and their thermodynamic properties and discuss corrections to the PYA HSF structure and thermodynamics.
6.3.1.1 Solution o f the PYA f o r the Hard [email protected] Fluid and Flu id Mixtures
For the PYA applied to the onecomponent hardsphere fluid, we obtain
h(r )   1 r < a
c ( r )  0 r >
(6.36)
(6.37)
so that c(r) is finiteranged and the factorization of Section 6.2.1 may be used. Substitution of Equation (6.37) into Equation (6.27) yields a Fredholm integral equation for q(r) of the first kind with degenerate kernel. Hence, using continuity of q(r) at r = R = a, we obtain (61)
where
q(r)  ~ a ( r 2  cr 2) + b(r  or), 0 < r < cr (6.38)
f0 O" t" a  1  27rp q ( t )d t  Q(O)
b  27rp tq ( t )d t
(6.39)
(6.40)
185
Equations (6.39) and (6.40), when combined with Equation (6.38), yield two lin ear equations for a and b, which may be solved easily to obtain
(1 + 2r/) 3r/o a = b = (6.41)
(1  r/) 2 ' 2 ( 1  r/) 2
where ~7, the volume fraction occupied by the spheres, is given by
7I" 2 ~7 ~pcr (6.42)
This completes the solution of the PYA for the onecomponent hardsphere sys tem.
The PYA for hard spheres was first solved by Wertheim (58,72) and Thiele (59) by Laplace transform techniques. Their methods, and also Bax ter's transformation, (73), of the OZ equation, formally gives a solution for h(r) through its Fourier transform. Baxter's factorization, on the other hand, yields a realspace equation for h(r) , Equation (6.27), which, for a given value of r, re quires only the values of h(t) on the range r  cr < t < r. This permits a simple numerical calculation of the total correlation functions for arbitrary separations via the method due to Perram (68). This numerical method supplements the exact formulae for 0 < r < 5(r obtained by Smith and Henderson (69) by inversion of the Laplace transform of 9(r).
The PYA for a mixture is given by
c ~ ( r ) = [1  e x p ( u ~ ( r ) / k B T ) ] g ~ ( r ) (6.43)
For a mixture of M species of hard spheres with additive diameters, the PYA, first solved by Lebowitz (74) by Laplace transform methods, may be written as
h ~ ( r ) =  1 O < r < a ~ (6.44)
c~ (~ ) = o r > ~
where a ~ _= R ~ = (cr~ + a~) /2 and u ~  S ~ = ( a ~  a~)/2. The conditions for Baxter's factorization are now met, and, analogously to the singlecomponent case, we find that (67)
2 (6.45) q~z(r)  a~(r 2  cruZ) + b~(r  cruZ), u~Z < r < cruz
where 1  ~3 + 30a~2
as = (1  ~3) 2 (6.46)
186
I0
• , , ' i . . . . i . . . . i . . . . i . . . .
. . . . . P~n_v I / / /
. . . . . . . . . /A.;:,
.../Z...ff ;"
0 0.0 0.1 0.2 0.3 0.4 0.5
Figure 6.1 The equation of state of the hardsphere fluid obtained from the PYA compressibility (labelled PYAC), PYA virial (PYAV), HNCA compressibility (HNCA C) and HNCA virial (HNCAV) equations of state compared with the essentially exact CarnahanStarling equation (CS). After Hansen and McDonald (5).
and
O2 b , ~ =  3 ~2 (6.47)
2(1  ¢ 3 ) ~
M 7/
~j  ~ ~ p.ycr~ (6.48) "),=1
Perram's algorithm (68) may also be generalized to mixtures enabling the compu tation of the hardsphere correlation functions 9~(r).
6.3.1.2 Equation of State for HardSphere Fluid and Fluid Mixtures
Combining Equation (6.39) with Equation (6.1), and integrating with respect to p, we find that
pc 1 + r /+ 7? 2  (6.49)
pksT (1  ~7) 3
where pc denotes the pressure obtained from the compressibility equation of state. The virial equation of state is given by
pkBT P = 1 3kBT27rp r3 d ) g(r)dr  1 + if y(r)dr (6.50)
187
where f ( r )  e x p [  u ( r ) / k u T ]  1 is the Mayer ffunction and y(r), the cavity distribution function given by Equation (6.9), is continuous. For hard spheres, d f ( r ) / d r  6(r  or), so the virial pressure is given by
pv = 1 + 4qg(a +)  1 + 2q + 3q 2 (6.51) pkBT (1  q)2
where, from Equation (6.27),
b 1 + rl/2 g(cr +)  h(cr +) + 1  a +   (6.52)
cr (1  q)2
As is evident from Figure 6.1, the compressibility pressure overestimates the true pressure while the virial pressure underestimates it, with the former being the more accurate of the two. The need for an accurate equation of state for hard spheres was met partially by the ReeHoover (75) Pad6 approximant (76). In recent years, however, the more popular equation of state for hard spheres is that developed by Carnahan and Starling (77). These authors fitted an integervalued function B(n) to the known nth virial coefficients, then extrapolated the series to an infinite number of terms to obtain
P 1 ] 7]] 7] 2 [ 7] 3 = (6.53)
pkBT (1  q)3
which is comparable in accuracy to the ReeHoover Padd approximant. The CarnahanStarling (CS) pressure (denoted by p c s ) was subsequently found to be simply related to the PYA compressibility (pc) and virial (pv) pressures (cf. Equations (6.49) and (6.51) by (78)
pCS 2pc 1pv  + (6.54)
3 3
This is consistent with the observation that the compressibility and virial pressures bracket the computer simulation values.
In the case of the multicomponent hardsphere system the compressibility pressure pc is given by
OP = 1  4 7 r
Op~
which, on solving for pc, yields
p c _ _ T r ( ~0
kBT 6 1  ~3
M
E p, fo #=1
(6.55)
3~'1~'2 3~'2 + (6.56) ( 1  [3) 2 ( 1  [3) 3
188
and the virial pressure, pv, is given by (79)
kBTPV M ( 27r ) Pc 18 ~3~3 = ~1 p~ + ~ PZ9~Z(cr~+~)  (6.57) = Z=I kBT 7r (1  ~ 3 ) 3
The same observations regarding the PYA for singlecomponent hard spheres may be made for mixtures: the machinesimulation pressures are bracketed by P: and pv. The CamahanStarling equation of state (Equations (6.53) and (6.54) has been generalized to mixtures as the MansooriCarnahanStarlingLeland (MCSL) (80) equation of state, whose form is the same as that given by Equation (6.54).
6.3.1.2 Semiempirical Correction to the PYA: The Generalized MSA
The main deficiencies in the PYA 9(r) are the lowered first peak which, from Equation (6.50), will yield an underestimation of the vifial pressure, and the fact that the PYA 9(r) is slightly out of phase with respect to the simulation results, causing an overestimate of the main peak of the structure factor. The need for accurate correlation functions for the hardsphere fluid has been met by the semi empirical correction to the PYA 9(r) due to Verlet and Weis (70). Verlet and Weis corrected the underestimation of the first peak by adding a very shortranged contribution ("~ 0 for r > 1.6a); the correlation function is adjusted to be in phase with the simulation results by using the PYA radial distribution function at a slightly modified density chosen to minimize the squared difference between the simulation and theory over the range 1.6or < r < 3or.
A em(r_~) [m(r or)] (6.58) _ gp (_,r _ c o s 
o r
where the parameters a and m are adjusted to yield the CS pressure and compress ibility and ~7'  6P[ 1  ~r/] 3 The corresponding semiempirical expression for the direct correlation function was obtained by Grundke and Henderson (81).
c(r) = Cpv(r)[1  0.127(po3) 2] r < o (6.59)
c ( r ) _ _B e2O(1_~) r > a (6.60) r
where parameter B is chosen from the condition imposed on the jump discontinu ity of c(r) at r = a, viz. c(a +)  c(a) = 9(cr+).
Finally Waisman (82) introduced the concept of the generalized MSA (GMSA) for hard spheres by combining the hardcore condition of h(r) with a
189
Yukawa form for c ( r ) outside the core, the latter being an ansatz for the correct hardsphere c(r). Thus, the GMSA is defined by the closure
h ( r )   1 r <
(6.61) ~(,)  K ~  ~ / , /   ~ ) / ( ~ / . ) , >
Note, that the latter equation has a form similar to that assumed by Henderson and Grundke. The parameters K and ~ are now regarded as densitydependent parameters whose values are set by the following two conditions,
b 1  q / 2 g ( ~ + )  a +  + 9 d ~ = ( 6 . 6 2 ) o (1  q)2
a2 = (1 + 2q) 2  @ ( 4  q) (6.63) ( 1  ~ ) 4
where a, b, fl and d are parameters defined in Section 6.3.3 below. Equation (6.62) requires that the vifial pressure in the GMSA be equal to the CamahanStafling pressure, and Equation (6.63)requires that the compressibility in the GMSA match that of the CamahanStarling equation. Thus, the two equations demand thermodynamic selfconsistency in the GMSA (equivalence of compressibility and vifial pressures) plus agreement with the CamahanStarling equation of state.
An analytical solution of the closure of the type Equation (6.61) is obtained in Section 6.3.3. Here we only note that GMSA for most densities pGa _~ 0.7 is in error by less than 1%, and at worst is in error by no more than than 3% (83). This is a significant improvement over the PYA.
Thus, by combining the CamahanStarling equation for thermodynamics with the VerletWeis or GMSAcoGected PYA correlation functions one obtains a fully analytic and very accurate theory of the hardsphere fluid. The availability of these analytic expressions has made a tremendous impact on the theory of fluids, since most perturbation theories ultimately require easilycalculated ha[dsphere properties to make them computationally feasible.
6.3.2 Adhesive HardSphere Fluid
So far the models considered in previous sections take into account only the ef fects of excluded volume, the main contribution to structural and thermodynamic properties of dense fluids. However, in addition to shortranged stronglyrepulsive
190
interactions, the pair potential for any real fluid generally includes attractive in teractions. A simplified model for a real fluid, which includes both types of inter action, is the hardsphere system with a squarewell attraction. This potential can be defined by
u s w F ( r ) = ~ r < ~ l
=  e al < r < a (6.64) = 0 r > a
which has the Mayer ffunction given by
f ( r )   1 r < c r l = e x p ( e / k s T )  1 or1 < r < a (6.65) = 0 r > c r
Attempts to solve the MSA and PYA for this potential analytically using the fac torization technique have not been successful. Baxter proposed an analytically tractable limit (al + a  while keeping the second virial coefficient fixed) of the squarewell fluid (84), the socalled adhesivehardsphere fluid (AHSF). The AHSF is defined by its Mayer ffunction
f ( r )   l + ( a / 1 2 z ) 5 ( r  a  ) r < cr
= 0 r > a (6.66)
The parameter r is a dimensionless measure of temperature and 5(x) is a Dirac delta function (85). The actual connection between r and temperature for a model of a given fluid can be established by equating the second virial coefficient for the AHSF with that of a squarewell fluid (86). The PYA for the AHSF was initially solved by Baxter (84).
6.3.2.1 Solution o f the PYA f o r the Adhesive HardSphere Fluid and Fluid
Mixtures
Within the PYA for the AHSF, c(r)  0 for r > cr so that Baxter's factoriza tion, Equations (6.26) and (6.27), is applicable with R  a. We require h(r) on the range (0, a), and from the cluster expansion for h(r) we expect that
~(~ ~), h(r)   1 + ~ 0 < r < a (6.67)
191
where A is an, as yet unknown, dimensionless parameter. Substituting Equa tion (6.67) into Equation (6.27), we obtain
Aa2[1  0(r  a)] , q(r)  a(r 2  o r 2) + b(r  cr) + 0 < r < c ~ (6.68)
where a  Q ( O )  ( l + 2 r /  p )
( 1  r/) 2
where O(x) is the Heaviside function,
b  3 ~ + # ,  = (6.69)
o 2(1  r/) 2
O ( z )  0 z < 0 = 1 x > l
and #  At/(1  r/)
The parameter A follows from the quadratic equation
(6.70)
0 (I~) ~ A+ (I~)~
which is obtained from the PYAclosure conditions written in the form
9 ( r )  [1 + f ( r ) ]y ( r ) (6.72)
where y(r) = 9(r)  c(r) (6.73)
The analytic solution of the PYA for the AHSF mixture, developed indepen dently by Perram and Smith (87) and Barboy (88), is a straightforward general ization of the two analyses given here and in Section 6.3.1.1. The AHSFmixture model assumes that the Mayer ffunctions for the mixture are given by
f ~ ( ~ )   1 + ( ~ o ~ / 1 2 ~ . ~ ) 6 ( ~  < ~ ) ~ < ~
= 0 r > cr~/~ (6.74)
where c ~ is defined in Section 6.3.1 and ~~ is both a measure of the temperature and the relative strength of the c~/9 attraction. [In fact, one can make the identifi cation ~~  e ~  , where ~ is the dimensionless temperature of the mixture and
192
c ~ is related to the welldepth of the aft interaction.] Correspondingly, the total correlation functions will be given by
 h ~ z ( r )   1 + 12 r < cr,~/~ (6.75)
where the A~ have yet to be determined. The condition on c~(r) is the same as that for hard spheres given in Equation (6.44). The mixture factorization can then be applied and Equation (6.30) yields
a q~z(r)  ~(r 2  a2Z) + ba(r  cruz) f A~Za2Z12  ~ , u,~ < r < o,~ (6.76)
where 1  ~c3 + 3a~2 X~
 (6.77) as = (1  ~3) 2 1  ~a
2 3cr a ~2 aaXa b~ =  ÷ (6.78)
2 ( 1  2 ( 1 
M 71"
y1 (6.79)
The parameters A~ are the solutions of coupled quadratic equations obtained, as above, by application of the PYA closure:
A.~.~  a . + + ~ z_, Pr (6.80) Oafl 7=1 O'a#
6.3.2.2 Equation of State for the Adhesive HardSphere Fluid and Fluid Mixtures
The compressibility pressure is computed using the variational principle de rived by Baxter (89) and is given by (84)
pc 1 + 7 + 72  p(1 + r//2) + #3/367
pkBT (1  7) 3
The virial pressure is given by
(6.81)
p v
pkBT
[ II 1 1(2 ÷ 107)# ÷ #2 ( 1 _ 7 ) 2 1 + 2 7 +372 ~
193
Table 6.1 Critical constants from the compressibility and energy equations of state for the adhesive hardsphere fluid. From Barboy and Tenne (90).
Property Compressibili ty EOS Energy EOS Tc 0.5858 0.7110 ~/c 0.1213 0.32
Pc/kBTcpc 0.3794 0.32
#3 1 _ r/ 1  5 r /  5r/2 + 6 r / #  #2 + ~ (6.82) 3T(1  ~7) 3
In the case of the energy EOS, it should be noted that it includes the hardsphere pressure pHS as the arbitrary constant of integration and thus the pressure isotherm is not uniquely defined since it depends on the choice of the expression for the hardsphere pressure.
R e _ p g S 6 ~ [ l n )~T(1  r/) 4 r /  1 pkBT = ( l  r / 2 ) ~ 3(2+~7) + 2 r/ f()~,r/)] (6.83)
where
f0 :~ d)~ f()~, r])  )~  12~7 + 18 + 12(1  r/)/r/ (6.84)
The latter integral can be evaluated analytically (90). Similar expressions for the EOS are also available for the mult icomponent case (90).
An important feature of the PYA for the AHSF is its prediction of a liquidgas phase transition. After analyzing the quadratic equation, Equation (6.71) for A
^
and studying the locus of the points where a  Q(k = 0) = 0, which defines the spinodal curve, one can identify two temperatures, 71 and Tc, which have the following properties: (i) for 7 > 71, there are real solutions of Equation (6.71) for the whole range of the density, while for 7 < T1 there is a range of r] for which there are no real solutions. There is a density ~7~ = ~TG(T~) = ~]L(T1) where two possible real solutions from the lowdensity side ~TG and highdensity side r/L merge; (ii) for 7 > ~~, a > 0 for all ~7 and at 7 = ~~ the equation a = 0 has a double root in density, r/~.
These two temperatures and densities coincide ~1 = 7~ = (2  v /2) /6 = 0.0976, ~7~ = ~7c. Thus the point (7~, r/~) can be identified with the critical point and its coincidence with (T1, r/i) means that the compressibili ty equation of state
194
1.2 . . . . i . . . . i , ' , , I . . . .
l[ ..... Energycompressibility [ 1.0 t ] " , .
~/'rc 0.60"8 ,' , • ",
0.4
0.2 ",\
0.0 ' ' ' ' ' . . . . ' . . . . ' ' " ' ' '
0 1 2 3 4 P/P~
Figure 6.2 Vaporliquid phase envelopes for the AHSF from the compressibility and energy equations of state. Note that the temperatures and densities are given in reduced units. After Barboy (90).
for the AHSF yields classical critical exponents (91), although as pointed out by Baxter (84) and other authors (92), the compressibility equation of state is not of the classicalscaling form near the critical point. Because of thermodynamic inconsistency the energy EOS, Equation (6.83), gives different values of the crit ical parameters Tc and qc (see Table 6.1). The energy EOS also yields classical exponents but the EOS is of the classicalscaling form near its critical point.
In Figure 6.2, we present the phase diagram for the AHSF obtained by using the Maxwell construction

#(~, ~G)  #(7, 7/L) (6.85)
where r/G and qL denote the equilibrium densities of gas and liquid, respectively, and # is the chemical potential. It should be noted that the # used in this equation is thermodynamically consistent with the corresponding volumetric EOS (i.e., ei ther the compressibility or energy expressions for both P and # are used in the equations to obtain the phase envelopes shown in Figure 6.2).
One cannot obtain solutions to Equation (6.85) from the virial EOS, Equa tion (6.82). Solutions to Equation (6.85) can be found for the compressibility
195
EOS, but only in the range of temperatures 0.292rc < 7 < re, while solutions to the energy EOS can be obtained for all r < r~. The limitations on the compress ibility and virial coexistence regions come about due to intersections between the coexistence region and the region inside which no real solutions exist.
6.3.2.3 PercusYevick Theory for the [email protected] Chain Fluid
An interesting application of the AHS fluid model for the description of a fluid composed of athermal freelyjointed tangent hardsphere chains has been pro posed by Chiew (9395). This is based on the "particleparticle" concept (96,97) and on the solution of the PYA for the AHSF.
The hardsphere chain fluid consisting of Nc chains each composed of n hard sphere units is recovered by the Ncomponent (N = nNc) AHSF with the ap propriate constraints imposed on the adhesive (frequently referred to as "sticky") interaction between different species. All N species are separated into Nc groups each consisting of n species, and adhesive interaction is allowed only between the hard spheres belonging to the same group. In addition, the adhesive interactions in the group are valid only between the particles of the first and second, second and third ..... n  1 and nth species of the group. The stickiness parameter, A~, Equation (6.75), is chosen from the condition that the number of nearest neighbors of a certain species bonded to the particle of the corresponding species is equal to unity
47rp~ fo ~ r2 g~(r) d r  2~c~3~]3(~) 3  1.
where r/~  (Tr/6)p~. Moreover, to ensure that only linear chains are formed in the system, each species is represented by only one particle. The corresponding set of OZ equations, together with the PYA closure conditions can be solved in a closed form yielding an expression for the qfunction similar to that presented in Section 6.3.2.1. In the simplest case of the system of onecomponent homonuclear chain molecules the compressibility EOS becomes
1 P 1 3q 3q 2 n  1 1 + ~q = + + ] (1  r/) 2 (6.86) pkBT [1  r/ (1  q)2 (1  r/) 3 n
One can easily see that the terms in the square brackets represent the PYA com pressibility pressure for the hardsphere system, while the last term represents the contribution to the pressure due to the effects of bonding. Similar expressions were derived also for the heteronuclear chain fluids, for mixtures of homonuclear
196
and heteronuclear chains, and for homonuclear chains in a hardsphere solvent. In the case of homonuclear chains of length n  4, 8, 16, the results of the present theory have been compared with computer simulation results (98). The agreement between theory and simulation appears to be very good.
6.3.3 HardCore Yukawa Fluid
The hardcore Yukawa fluid (HCYF) is a model for a simple fluid that consists of a hardsphere repulsion and a sum of tails of Yukawa form. The pair potential for the simplest onetail onecomponent version of the model is given by
u H c y F ( r )   ~ r < cr
(6.87) =  D e  z ( r ~) / r r > a
Thus, unlike the case of AHSF, the attractive part of the pair potential for HCYF has two parameters, one defining the depth of the potential (D) and the other defining its range (z).
Although there is no real system which has the interparticle potential of the HCYF form, it takes into account the basic elements of the interparticle interaction in real potentials, i.e., shortrange repulsion and longrange attraction. Therefore in the case of systems with strong attractive interactions, the HCYF is obviously a better candidate than the hardsphere system for the role of the reference system in a perturbation theory (99). This, together with the availability of the analytic description via MSA, are the main reasons for the considerable interest in the HCYF.
6.3.3.1 Solution o f the MSA for the HardCore Yukawa Fluid and Fluid Mixtures
The MSA for the potential model, Equation (6.87), yields the following con ditions on the total and direct correlation functions
h(r)   1 r < cr
c(r)  LeZ(~~)lr r > a
(6.88)
where L = D / k B T . The OZ equation with these closure conditions was originally solved by Waisman (82) using Laplacetransform methods to reduce the problem to a set of four nonlinear algebraic equations. Waisman used this analysis to de velop the generalized MSA for the hardsphere fluid described in Section 6.3.1.3.
197
On the basis of a graphical analysis of the MSA, Henderson et al. (100) de veloped lowdensity and hightemperature series expansions for the solutions to Waisman's equations, thus making it possible to distinguish the physically correct solution. HCye and Stell (101) simplified Waisman's equations significantly and identified the physically correct solution. HCye and Blum (102) studied the MSA for the HCYF using the Baxter factorization technique and reduced the problem to the solution of a single quartic equation. Cummings and Smith (103,104) also used the Baxter factorization approach, but were able to find a simplified quar tic equation which can be easily analyzed to determine the physically meaningful root. In this section, we follow the analysis of Cummings and Smith.
The Baxter qfunction, introduced via the factorization Equation (6.21) and relation Equation (6.22), is found to be similar in functional form to re (r ) in Equation (6.88), i.e.,
q(r)  qo(r) + a e z (r a) (6.89)
where qo(r)  0 for r > cy and the expression for the parameter, a
L a  z Q ( i z ) (6.90)
which defines the longrange asymptotic form of q(r), is obtained using the long range behavior of the direct correlation function c(r), Equation (6.88), and ana lytical properties of the function (~(k). The factorized version of the OZ equation in rspace now reads
rc ( r )   q ' ( r ) + 27rp q ' ( t ) q ( t  r ) d t r > 0 (6.91)
/o r h ( r )   q ' ( r ) + 27rp q ( t ) ( r  t ) h ( I r  t l ) d t r > 0 (6.92)
To determine the functional form of q0 (r), we consider the latter of these equations on the domain 0 < r < or, which gives
qo(r)  ~(r 2  cr 2) + b(r  or) + ad[1  e  z ( r  a ) ] (6.93)
where
fO ° a  Q(O)  1  27rp q o ( t ) d t  27rpaeZ~
Z
fO ° b  27rp tqo( t )d t +
21rpae z~
Z 2
(6.94)
(6.95)
198
27rp d = 1  .~(z) (6.96)
z
and .~(s) is the Laplace transform of rg ( r ) ,
{7(s)  fo ~ r g ( r ) e  ~ r d r
The set of equations (6.90), (6.946.96) defines the unknown parameters a, a, b and d. An additional equation for these parameters follows from Equation (6.92) after rearranging it in terms of the function 9(r ) = h ( r )  1,
/or r g ( r )  ar + b + adzeZ(r~)27rp q ( t ) ( r  t ) g ( r  t ) d t (6.97)
and, performing the Laplace transform of both sides of this equation, we obtain
r(a _jr_ 80")]'s 2b ad] ezo O(z)  tV'I J 1  27rp~(z) (6.98)
Finally, after performing the integration in Equations (6.94) and (6.95) and some algebraic manipulations, the set of equations (6.90), (6.94), (6.95) and (6.98) re duces to the following quartic equation
36r/2/34 + X ~ 3  12~7K/32 + K D f l  K 2 = 0 (6.99)
where/3 = a / a 2 and
18~ 2 q
[2  ¢  2 ~  ~  ¢~~] 6r/ [2  ~2 _ 2 e  ~ _ 2~e~]~2( 1 _ r/) 2 D
72r/2 [ 2  ~  2 e  ' ]  216r/3 2 [ 4  4 ~ + ~ 2  4 e ~ +~2e  ' ] e  6'qzr {2(1 _ r/) ~ 4 ( 1   77)
72~72 [2 _ ~ _ 4e¢ + 2e2¢ + @2¢] F  6 (1  e~)2~7 + {2(1 _ 77)
216~73 [4  4{ + {2  8~¢ + 2{:~ ¢ + 4~ ~¢ + 4{~ :~ + {:~:¢]
~ ~ 4 / 1 ~7i 2
36~72 [2  2 {  2~~ + {:~¢] X  6~7{e {  {~ (1  77)
 108r/~ [2  ~ "  2 e  ~  ~ 9 e  ~ ] , ~ ( 1  , ~ ) ~. I. J
199
with ~ = zcr. The solutions of this quartic equation and their relationship to phase behavior are described qualitatively in section 6.3.3.2 below. Equation (6.92) can be written
fo G rh(r)  27rp qo(t)(r  t)dt + 27rpaez(rG)eZSsh(s)ds
27rpc~eZ(rG) +adze z(rG) +
z 2
which permits calculation of the total correlation function using an extension of Perram's algorithm (83).
The solution of the MSA for the Yukawa fluid can be extended to the case in which the potential outside the hardcore region is given by a sum of Yukawa terms and to the multicomponent case. The solution of the MSA for a one component system with two Yukawa tails was obtained by HCye, Stell and Wais man (105) using Laplacetransform methods. For the general case of a multi component hardsphere system with MSA closure involving n Yukawa terms, the formal solution via the Baxter factorization method was developed by HCye and Blum (106108) and Cummings (83). We will briefly outline the HCyeBlum so lution here. We begin with the MSA closure conditions
h~m(r ) =  1 r < cr~ m
n . ( i ) _ z i ( r _ a ~ Z ) / r r > (Tam c.m(r)  Ei=l L.me (6.100)
which extend Equation (6.88) to the multicomponent multiYukawa case. Here L~ m = D~m/kBT. An analysis similar to that carried out for the onecomponent case leads to the following expression for the Baxter qfunctions,
q~m(r)_ ,~(o) k 1 (i) z~(r~.~) ~ (r) +  % ~ e i=1 zi
(6.101)
where
q(O) 1 n ~(i)
~m(r ) _ _~a~m(r_o. m)2+b.m(r_G.m)+ y~. ~ (e z { ( r ~ .~)_ l ) i . m < r < o.m i=1 Zi
(6.102) q(0) ( r )  0 r > cr~ m (6.103) am
,~ (i) ~(i) satisfy a The unknown coefficients of the q~m(r), i.e. a~ m, b~ m, ~ m and ~ m ' set of nonlinear algebraic equations, which can be reduced to a set of equations
200
,, (i) ~(i) involving only the unknowns ~ and Note the slightly different functional ~ " O~/~ '
form for q(0) ~ (r), which corresponds to a minor redefinition of b~. The slightly different form for the Baxter qfunction has some advantages for more complex closures such as the multicomponent multiYukawa system considered here. This cumbersome set of equations is not reproduced here and the reader is referred to the original publications (106108) for details, because it is beyond the scope of this review to report further on these developments. We shall complete this section by presenting briefly the solution for the model of a mixture which exhibits phase separation.
The model, which was introduced by Waisman (109), consists of an equimolar binary mixture (px = p2 = p/2) of species 1 and 2 molecules which are equal diameter and have pair potentials
U i j ( T )   CX:) r < (71
= (  1 ) i + j + l D e  Z ( r  a ) / r r > O
(6.104)
1[hll (r) + hl2(r)] , h ( ~ ) 
h(r) ~l[hll (r)  h12 (r)],
1 c ( r )   ~[Cl1(7") ~ C12(T)]
1  klX( ) 
we find that h(r) and c(r) satisfy the usual purefluid OZ equation with closure given by Equation (6.37). Thus, h(r) and c(r) are the same as for the hardsphere fluid in the PYA at number density p. The h(r) and e(r) also satisfy the usual purefluid OZ equation but with closure
h(r)  0 r < cr
~(r)  Le Z ( r ~ ) l r r > cr
(6.105)
This is a coreless version of the HCYF, and the same analysis as that given in Section 6.3.3 remains valid, except that the a and b terms in the q(r) function are zero, which yields
qo(r) =/3dcr 2 [ 1  e¢(r/~l)] , 0 < r < o (6.106)
so that like species interact via an attractive Yukawa potential and unlike species via a repulsive Yukawa potential. Introducing the quantities
201
where/3 is the solution of the quartic equation
36~2fl 4  [email protected]~fl 3 + 12r/K/32  K~fl + K 2  0 (6.107)
The solutions of this quartic equation are described qualitatively in section 6.3.3.2 below.
6.3.3.2 Equation of State for the HardCore Yukawa Fluid and Fluid Mixtures
The compressibility pressure of the HCYF in the MSA is calculated by inte grating the equation
1 Opc[  a 2 (6.108) kBT Op ]T
The virial and energy routes to the thermodynamic properties follow from the expressions derived in the case of MSA by HCye and Stell (110)
pv = 1 + 4~Tg(cr +) + J (6.109)
pkBT pe
= 1 + 2r] r _ _ /g2 (c r+)   '~[gHs(cr+)]2) k J (6.110) pkBT
where f oo j _ 27cp ra du(r)
3kBT dr g(r)dr
and the H S subscript refers to hardsphere quantities. For the case of the Yukawa fluid, J can be written in terms of ~(s) as
J4~TKe~[zd{7(s) I  ~(z)] (6.111) ds s:z
Similar expressions can be written also for the general case of a multicomponent multiYukawa system. Here the inverse isothermal compressibility is defined by
1 Op c aa~ 2  ~ x~(~ ) (6.112)
kBT Op y
while the virial and energy pressure follow from the corresponding expressions of HCye and Stell (108,110), i.e.,
pv 2~ x~x~%~g~(%~) + J (6.113) 1 + =p ~" a +
pkT = 5 "    " az
202
R e
pkBT 3 = pUS + 3PEX~Xza~z{[g~z(a+Z)]2 [g~H~(a~Z)]2} + J
a~
where x~ is the mole fraction of the species a,
(6.114)
271
a ~ i s   z i
The thermodynamic properties of the Yukawa fluid have been discussed at length by Henderson et al. (111), Cummings and coworkers (103,104) and Arri etta and coworkers (112,113), and its critical properties by Cummings and Stell (91). From the expressions for the pressure, Equations (6.108) (6.114), it follows that the thermodynamic properties of the HCYF are defined by the parameters in the Baxter qfunction. These parameters are determined by a set of nonlin ear algebraic equations, which in simple cases can be fully analyzed, providing some insight into the peculiarities of the thermodynamic behavior of the model. In the case of a onecomponent oneYukawa fluid, the set of equations reduces to a single quartic equation, Equation (6.99). For K > 0 (attractive potentials), this equation has two real and two complex roots. To leading order in density, the two real roots are given by (103,104)
K ~e ~ 91  ~ "ll 0(77), /~2 =   ~ + 0(7/0). (6.115)
Clearly, ill, the smaller of the two roots, is the physically meaningful root since this is the root which goes to a finite value as r/+ 0 and to zero as T ~ oo, thus yielding the correct lowdensity and hardsphere limits respectively. This is strik ingly similar to the behavior of the two roots of the quadratic equation obtained for the AHSF in the PYA described in Section 6.3.2. Thus, as in that section, we can compute the locus of two lines: L1, the line along which a + 0 and the compress ibility diverges; and L2, the line along which the two real roots of Equation (6.99) are equal. We can also compute, for a fixed value of ~, the dimensionless critical temperature (lIKe) of the HCYF as the temperature below which there is a range of r/for which a < 0. Likewise, l /K1 is a value of the dimensionless inverse tem perature below which there is a range of ~/where Equation (6.99) has no real roots. As in Section 6.3.2, there are corresponding densities, r/c and ~71. Unlike the case of the AHSF, the critical temperature and density (1/Kc, r/c) do not coincide with
203
1 . 4 . . . . i . . . . ! . . . . ! . . . . i . . . . i . . . . ! . . . . i . . . .
1"3 I
1.2 "" T*
1.1
1.0 N
O. 9 "~,
0.8
0.7 0.0 O.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
p*
Figure 6.3 The liquidvapor phase envelope for the LennardJones (LJ) model fluid calculated from the MSA for the HCYF (solid line), from the PYA for the LJ fluid using the energy EOS (dashed line) (114), from the RHNCA approximation (dotted line) (115), and from MonteCarlo simulation (diamonds) for the LJ fluid (116118).
(1//£1, ~]1). The consequence is that for the HCYF the thermodynamic behavior near the critical point of the compressibility equation of state yields spherical model exponents (7  2, 6  5) rather than classical critical exponents (91). A similar analysis can be carried out for the case of the twocomponent HCYF con sidered in the previous section. Again, for Equation (6.107) there is a locus of
t . .
points in (/3, r/) space along which Q0(0)  0. Thus, as one increases the den m
sity along an isotherm for this fluid, one reaches a spinodal point at which h(r) becomes long ranged, and the fluid phase separates. Since there is no phase be havior associated with h(r) and c(r) (they are simply hardsphere quantities) the separation can be regarded as a fluidfluid phase separation.
There is a sizable literature dedicated to the investigation of the analytical structure of the MSA for HCYF (99,101,103105,112,119127) and to its appli cation to the description of different systems and physical phenomena, (128,129). However, it is beyond the scope of our study to discuss and analyze all possible applications of the HCYF. Rather, we will conclude this section by demonstrat ing the possibilities of the HCYF and MSA in the description of a system with a realistic potential. In Figure 6.3, we present the liquidgas phase diagram for
204
the LJ fluid (116118). We also present the RHNCA and PYA results obtained from the energy route for the same LJ fluid (114,115) and results of the MSA for the corresponding HCYF (130). The parameters of the HCYF model have been chosen to approximate the LJ potential. The agreement between the MSA results and the results of the MC simulation is in general good and is comparable to that for the PYA and RHNCA.
6.3.4 Charged HardSphere Fluids
An important model, which belongs to the class of analytically solvable mod els, is the primitive model for ionic fluids. It is defined as an ncomponent mixture of charged hard spheres of species 1, ..., c~, ..., n with number densities p~ and the following pair potentials
u ~ ( r ) = cc r < cr~
 r > o  ~ ( 6 . 1 1 6 ) er
where e is the elementary charge, Z~ is the charge of the ion of a species in electron units and c is the dielectric constant of the continuum. In spite of its sim plicity this model has been widely used to study ionic solutions (in which case represents the solvent dielectric constant), molten salts and liquid metals. Unlike the pointcharge model used in the classical DebyeHtickel theory of electrolytes, the primitive model takes into account the effects of excluded volume. This, to gether with the availability of an analytical solution in the MSA (which properly describes the hardsphere repulsion), are the main reasons for the considerable in terest in the primitive model, Equation (6.116). The MSA for the primitive model satisfies exact Onsager bounds (131) on the Helmholtz energy and on the internal energy, as well as the StillingerLovett moment conditions (132).
In this section we present the analytic solution of the MSA for the primitive model (PM) of electrolytes and the EOS which follows from it. At the end of this section we briefly discuss corrections to the MSA which extend its applicability to ionic systems under the conditions of high coupling (low temperature or low dielectric constant) and low density.
6.3.4.1 Solution of the MSA for the Primitive Model of Electrolyte Solutions
We consider first an analytical solution of the MSA for the restricted primitive model (RPM) of electrolytes. This model consists of a mixture of two species
205
of equal diameter, equal charge ( q  Z e , where Z is the valence and e is the electronic charge) hard spheres. In order to solve the MSA for this model, we begin by defining a slightly different model, v i z
~ ( ~ )  ~ ,. <
= (  1 ) ~ + Z q 2 e  Z ( ~  ~ ) / c r r > cr (6.117)
so that the RPM is recovered in the limit z + 0. Just as in Section 6.3.3.1, we define the correlation functions h ( r ) , c(r), h ( r ) and e(r) and find that the mixture OZ equations separate into two OZ equations, one for h ( r )  c(r), the other for _ R
h ( r )  ~(r). The h ( r ) and c ( r ) are given by the PYA for hard spheres; the h ( r )
and ~(r) satisfy the usual OZ equation with closure
(6.118)
m
h ( r )  0 r < cr
q2  z ( r  a ) / r r > (r  e ( r ) =  ~ k ~ e
We are interested in the limit z + 0, in which case Equation (6.107) becomes
  0 ~ / 3   6r/ (6 .119)
where q 2
K  (6.120) c k B T ( y
Equation (6.119) leads to expressions for cij(r) that agree with those obtained by Waisman and Lebowitz (133,134) using Laplace transform techniques. Here the Baxter qfunction, ~0(r), has the form
qo(r)  I t~ ( r  or) (6.121)
where
~2 _ 4 7 r p K , 27rpI  l + c r F
2For  (1 + 2ha)1 _ 1
The analytic solution of the MSA has been extended by Blum (135) by ap plying the Baxter factorization technique to the general case of any numbers of components with arbitrary charges (subject to electroneutrality) and hardsphere
206
diameters. In this general case it is not possible to decouple the initial set of the OZ equations into independent sets of equations by using functions defined as in section 6.3.3.1. The method of solution is quite similar to that used to solve the MSA for the multicomponent HCYF. For the general case the MSA closure conditions are
h a z ( r )   1 , r < a~Z
e2ZaZ~ e z(ra~) r > cr~/~, (6.122) c.~(r)  eksT
where we consider the limit z + 0. The system satisfies the condition of charge neutrality
n
~_, p~Z~  0 (6.123) a = l
The final solution of the problem can be expressed in terms of only one parameter, F, which is the solution of the algebraic equation
4F 2  a ~ p~ OL=I 2A(1 J
(6.124)
where rc p~a a l ~ p ~ a ~ Z ~ ~  1 +
Pn  ~ 1 + a . F ' ~ 1 + a . F '
6 ' ckBT
The Baxter qfunctions are given by
12(r/~r/z)½q~z(r)  12(rl~r/z)½q(°)(r) Z~az e ~('~"~) af~ (6.125)
(o) l ( r )2q~
" (1 + q ~ 
q ~  ~ 2 a ~ + a~a~2 ~  ~ Da~a~,
4F 2 a 2 2AZ~  rcPna~ D  a 2 , a s  2F 2 A ( l + a ~ r )
in the limit (# + 0).
207
6.3.4.2 Equation of State for the Primitive Model of lonic Fluids and Fluid Mixtures
The thermodynamic properties of the primitive model of ionic fluids are calcu lated using the scheme proposed in (136,137). In the case of a restricted primitive model, Equation (6.117), the virial and energy routes to the thermodynamic prop erties yield
pv _ PI~S 1 E  E id
pkBT 3 pkBTV
pe __ Pus
pkBT
(6.126)
F3 = (6.127)
37rp
where E  E id is the residual internal energy
E  E id q2 F
pkBTV kBTe 1 + crF
In the general case of the unrestricted primitive model, Equation (6.116), we have, (136,137), that
p v _ PIYlS 1 E  E id =  (6.128)
pkBT 3 pkBTV
where
p e PHS F3 o~2 p2
pkBT 37rp 8p A2 (6.129)
_ _ _ n 71" 2 E E id e2 ~ 1 PaZ2 Jc" kBrV  ekBT [r 1 + o~r  ~ aPx]
The compressibility pressure follows from the integration of the expression
kBT
n OP ~ _ _ 1 Q2
Op r 4~o 7r2~=1 ~ p~ ~ (6.130)
where ~2~ [1 + (2o~ ~ ~ 1 Q~  + ~a~P~]
OZ 2 71"
~° = 2 r (1 + ~ r [z~  P . ~ ~£]
The thermodynamic and structural properties of the primitive models, both re stricted and unrestricted, of ionic systems predicted by the MSA have been studied
208
1 . 4 , , , , , , , i , , , , , , , , , , , , , , , , , , ,
1 . 3
1 . 2
1 . 1
1 . 0
0 . 9
0 . 8
0 . 7
0 . 6
0 . 0
11 s a l t
1 2 S a l t
13 S a l t
/ f H N C A
• M S A
, , , I , , , I , , , I , , , I , , , I , , , I , , ,
0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4
c ( m o l / l i t )
Figure 6.4 Comparison of the osmotic coefficient of the primitive model calculated from the MSA (solid circles) and the HNCA (lines). For the 11 and 13 salts, the cationic and anionic diameters are equal (3.6~ and 2.14 respectively). For the 12 salt, the cationic diameter is 3.6~ and the anionic diameter is 5.76~. After Blum (138).
in detail (139149). The results of these analyses show that the MSA gives accu rate predictions for the thermodynamics of systems with low values of the inverse temperature/3*  o~21Z+Z  ] and/or at high densities and is unsatisfactory at high values of/3* and low densities, which corresponds to high charge, low tempera ture, low ~ or any combination of these conditions. It is also found that the energy EOS provides the best predictions for thermodynamic properties. In Figure 6.4, the osmotic coefficients for 1  1, 1  2 and 1 ,3 va lence salt solutions obtained from the energy EOS (138) are presented and compared with HNCA results, the latter being in good agreement with MonteCarlo simulation results. One can eas ily see that the agreement becomes less satisfactory with increasing charge. In the case of 2  2 and highervalence electrolyte solutions, the results are rather poor as is evident from Figure 6.5.
In Figure 6.6, we compare the phase envelope of the restricted primitive model obtained from the MSA using the energy EOS with that obtained by molecular simulation (154,155). There is clear evidence that the MSA result differs signif icantly from the simulation result with a critical temperature which is almost a factor of two larger than the simulation result. This is not surprising given that
209
1.0
0.9
0.8
0.7
0.6
0.5
. . . . i . . . . i . . . . i . . . .
~ , . . • M o n t e C a r l o
" ~ , . ,.  . . . . M S A
" ~ , _ ~ ,. ~ A M S A
"  . . . ~ •      H N C A n d e r s o n
• ~ ~t d
I , ~ i i I , i , , I , , i
3 2 1 log(c)
Figure 6.5 Osmotic coefficient of the restricted primitive model at/3* = 6.8116. The solid line refers to ASMA results obtained from the energy equation. After Kalyuzhnyi and Holovko (55).
the phase envelope of the restricted primitive model is at very low temperature and density, precisely the domain on which, as we have already noted, the MSA becomes quite inaccurate.
Because of the rather high accuracy of the MSA for 1  1 and 1  2 electrolyte solutions, a number of researchers have modeled the properties of real electrolyte solutions (144147,156158) using the MSA as the basis for the thermodynamic properties.
In general, unlike the case for thermodynamics, the MS A does not appear to be a reliable source for structural properties. Comparison of MSA radial distribu tion functions with MonteCarlo simulation results yields rather poor agreement. In particular, the pairdistribution function for the equally charged ions becomes negative at short distances, which is clearly an unphysical result. This behavior is a consequence of the linearity of the MSA.
6.3.4.3 Corrections to the MSA: the Generalized MSA and the Associative MSA
As was already noted in the previous section, the predictions of the MSA be come rather poor for high values of/3* and low concentration. Several different schemes have been proposed to extend its range of applicability. To correct the
210
0.08
0.07
0.06
T*
0.05
0.04
• ' ' 1 ' ' ' 1 ' ' 1 ' ' " 1 ' • ' 1 ' ' ' 1 ' '
ss
 . S#S s ~ /s tl i 
1E0()7 1E005 0.001 0.1 Reduced density
Figure 6.6 Phase envelope of the RPM as predicted by the AMSA (solid line), MSA (dashed line 1), MSA3 (dashed line 2), the version of the FisherLevin theory (150) with the consistent account for the hard core contribution (dashed line 3) and by MonteCarlo simulation (symbols) where 7* = 1//3". AMSA results are after Kalyuzhnyi (151), MSA3 results are after Stell et. al (152) and results of the FisherLevin theory are af ter Guillot and Guissani (153). MC results are after Caillol (154) (solid diamonds) and Panagiotopoulos (155) (open cirles).
MSA for structural properties and for thermodynamic consistency, the concept of the GMSA (82,159), has been used by several authors (160162). An impor tant feature of this approach is that the GMSA preserves the fulfillment of the StillingerLovett moment conditions (132). Another possibility for the correc tion of the MSA is the EXP approximation of Andersen and Chandler (163,164). Although the EXP approximation leads to an overall improvement for the struc ture results, it violates the StillingerLovett moment conditions. In addition, t h e thermodynamics of the EXP approximation cannot be expressed in analytic form. Larsen et al. (165) proposed the socalled truncatedgamma secondorder approx imation (TG2A) in which the MSA Helmholtz energy is corrected by adding the secondorder chain diagram. This leads to an analytical expression for the osmotic coefficient,
~TG ~___ (~MSA __ St (6.131)
211
where s '  ~(1+ ~) X
12s(1 + 2~)
{20z 2  12z + 3 + [4(4z  1)  sin(2z)  (8z  1)cos(2z)] e 2z }
and z  oF. This approximation extends the application of the MSA over a relatively large region of/3* and concentrations.
The poor performance of the MSA is manifested in systems with large val ues of/~*, which means that such systems are characterized by an appreciable degree of ionic association. Thus, one mechanism for extending the applicabil ity of the MSA is to explicitly account for association effects. Several studies have been initiated with this aim. These studies fall into three general categories. In the first category, the corrections to MSA are introduced by adding an addi tional adhesive term into the closure conditions (see reference 166 and references therein). Following the concept of the GMSA, this adhesive term can be chosen to reproduce additional information about the system, such as known thermo dynamic behavior. However, as was already noted in Section 6.1.2, the regular OZ closure approximations, with their close relation to the Mayer pexpansion in terms of a density, are not well suited to treat highly associating systems. In the second category, a description in terms of both the density and the activity is used. This type of approach was initiated by Bjerrum (167) who combined the law of mass action with the DebyeHtickel theory of electrolytes. Several [168, 169, 170, 171,172, 173, 174, 175, 152] authors (152,168175)extended this concept of correcting the DebyeHtickel theory for the effects of ionic association but used the MSA solution in place of the DebyeHtickel theory.
As noted in Section 6.1.2, a consistent integralequation theory for associat ing liquids has recently been proposed (4548). This is based on the multidensity formalism in which the description in terms of activity and density expansions are combined. In references 40, 49 and 50, the multidensity formalism was re formulated and presented in a form suitable for the study of association effects in ionic fluids; this is the third category of approaches to incorporating associ ation. The simplest twodensity version of the latter theory has been applied to correct the MSA predictions for thermodynamic properties of the RPM of elec trolytes (55) and for its liquidgas phase diagram (151). An analytic solution for the twodensity MSA (or associating MSA (AMSA)) (53,176) was obtained and closed form analytical expressions for the thermodynamic properties has been derived (151,177,178). We will present here only the final results for the pres sure of the 22 electrolyte solution and liquidgas phase diagram of the RPM as
212
predicted by the AMSA. These are shown in Figures 6.5 and 6.6 where the re suits of the theory(55,151,178) are compared with predictions of the computer simulations, regular MSA and other currently available theories. The agreement between AMSA, the cluster theory of Tani and Henderson (169) and the Monte Carlo results in the case of the osmotic coeficcient is reasonably good, especially at lower concentrations, while the MSA results are rather poor. The prediction of the AMSA for the coexistence curve is more accurate than those of the MSA and pairing MSA3 of Zhou et al.(152) and is of the same order of accuracy as that of Fisher and Levin with a consistent account of the hard core contribution (153). Thus, use of the AMSA enables the range of applicability of analytic theory to be substantially increased.
6.4 EQUATION OF STATE FOR ANALYTICALLYSOLVABLE MODELS OF MOLECULAR FLUIDS
Thus far in this chapter, we have considered IEAs for monatomic fluids i.e., fluids for which the intermolecular pair potential is spherically symmetric and thus depends only on the distance separating the two molecules. More complex models are required to describe molecular fluids. In general, each polyatomic molecule has 3n degrees of freedom where n is the number of atoms in each molecule. These include the translational, rotational and intemal degrees of freedom. In most cases of interest, the intramolecular interactions are much stronger than in termolecular interactions and, to a high degree of accuracy, one can neglect the intramolecular degrees of freedom and assume that the molecules are rigid. The pair potentials, and thus the paircorrelation functions, for rigid nonspherical par ticles are functions of the coordinates which specify the positions and orientations of the two molecules.
There are two formalisms for describing the structure and thermodynamic properties of molecular fluids. The first, the socalled molecular formalism, is based on the extension of the OZ equation and closures used for monatomic fluids by explicitly taking into account the orientational dependence of the pair potential and correlation functions. Although such an extension is formally rather straight forward, the resulting IEAs are much more difficult to handle both numerically and analytically. In addition, to perform calculations in the molecular formalism, one needs to expand the potential and correlation functions using the invariant expansion (see the Appendix). Truncation of the invariant expansion at a finite number of terms leads to additional approximations beyond those of the IEAs themselves. The second formalism, the socalled interactionsite formalism (ISF),
213
was developed by Chandler and coworkers(179,180). Some important limitations of early versions of the ISF are discussed by Cummings and coworkers (96,181 183). Within the ISF it is assumed that the intermolecular pair potential can be ex pressed as a sum of sphericallysymmetric sitesite potentials and that the structure and thermodynamic properties are described in terms of the sitesite correlation functions. In spite of the limitations imposed by these assumptions in the model, this approach is quite general and can be applied to a wide range of molecular fluids. The sites are usually associated with the atoms comprising the molecules. Because of the sphericallysymmetric character of the sitesite interactions and thus the sitesite correlation functions  the corresponding IEAs are much eas ier to handle than their molecularformalism counterparts. However, on the other hand, the sitesite correlation functions contain less information about the struc ture of the system than the molecularformalism correlation functions. Knowledge of the molecularformalism correlation functions allows one to calculate any site site correlation function, while it is not possible to construct molecularformalism correlation functions rigorously from sitesite correlation functions.
In Section 6.4.1 below, we discuss analytically solvable models for molecular fluids within the ISF, while in Section 6.4.2 models described by the molecular IEA will be considered. For the sake of simplicity, in each case the formal results will be presented for the onecomponent case, though extensions to mixtures are possible.
Several reviews have appeared on the statistical mechanics of molecular fluids in general (184) and on interactionsite fluids in particular (185) Accordingly, our presentation of formal results will be very brief and we will focus mainly on the recent developments in this field.
6.4.1 InteractionSite Model Fluids
As was already noted above, within the ISF one assumes that the intermolec ular pair potential u(12) can be represented as a sum of the sitesite spherically symmetric potentials u~z(r~z)
u(12) = ~ u~z(r~z) (6.132) az
where 1 and 2 stand for the spatial and orientational coordinates of molecules 1 and 2, r ~ is the distance between the site a in molecule 1 and site/3 in molecule 2. The structural properties of the model are described by the sitesite pair distribution functions 9~(r), which are proportional to the probability density
214
of finding a site a of one molecule and site/3 of another molecule separated by the distance, r, regardless of the orientations of the two molecules. Thus the site site correlation functions contain less information on the structure of the system than the molecularformalism correlation functions. However, unlike the latter, they can be compared directly with the results of Xray and neutronscattering experiments.
The thermodynamic properties of the system can be obtained from the sitesite pair correlation functions using the energy route, i.e.,
ures m f0cx ~ N = 27rp~ 9~( r )u~( r ) r 2 dr (6.133)
where U res is the residual internal energy, m is the number of sites in the molecule and p is the molecular number density, or the compressibility route
Op f0 ~ kBT ~  1 ÷ 47rp [ 9 ~ ( r )  1]r 2 dr (6.134) V,T
where a and/3 denote any pair of molecular sites. Note that Equation (6.134) im plies that the volume integral of sitesite correlation functions h~z(r) = g~z(r) 1 is independent of both a and/3. This independence, known as the compressibility theorem, causes singular behavior in the sitesite direct correlation functions de fined by the sitesite OZ equation given below and lies at the heart of many of the difficulties associated with the ISF (96,181183).
The virial route to the pressure cannot be expressed in terms of sitesite cor relation functions only. Additional information is needed concerning the orienta tional structure of the system.
6.4.1.1 SiteSite OrnsteinZernike Equation and Closure Conditions
An integralequation theory for calculating the sitesite correlation functions has been proposed by Chandler and Andersen (186). It is based on the socalled sitesite OrnsteinZernike (SSOZ) equation, which can be derived using the same arguments as those utilized in the case of the OZ equation in Section 6.1.1. The total sitesite correlation function h~(r) = g~(r)  1 can be decomposed into a sum of direct correlations c~z (r) and all possible indirect correlations presented by the convolution integral
h~z(r)  ~ /w~(r~.y)c~(r~)w~z(rsz) d~'~d~'5+ ,y5
215
+p ~ f w~(r~)c~6(r~6)h6z(r6z) d ~ d ~6 (6.135)
Here, unlike the case of the regular OZ equation, the correlation is propagating also via the intramolecular distribution function
1 w~z(r)  47rL~ 6(r L~Z) (6.136)
where L ~ is the distance between the sites a and /3 in the same molecule. The same SSOZ equation (6.135) can be obtained using different arguments (96,187,188)
The closures typically used for the SSOZ equation (6.135) are straightforward extensions of the PYA, HNCA and MSA (186,189194). They can be presented in terms of the sitesite correlation functions as follows
g~(r)  [9~(r)  c~(r)]e ~(r)/kBT, (PYA) (6.137)
g~(r)  e  u ~ / k B T + h ~ ( r )  % ~ ( r ) , (HNCA) (6.138)
h~z(r)1, r < a~Z
c~z(r) u~z(r)/kBT, r > cruZ, (MSA) (6.139)
In the case of the MSA, it is assumed that the sitesite potential involves the hard sphere sitesite interaction.
These IEAs appear to be relatively successful in predicting the shortrange structure of a number of different sitesite models. However there are several im portant defects in the SSOZ formalism. As in the case of the usual OZ equation, the SSOZ equation, equation (6.135), can be viewed as a rigorous equation which defines the sitesite direct correlation functions c~z(r). However such a direct correlation function does not have the same status as that of the direct correla tion function introduced in Section 6.1.1. The latter represents a specific subset of the diagrams contributing to the total paircorrelation function which is quite different from the former case. The result is that the exact sitesite direct corre lation functions, as defined by (6.135), do not in general, satisfy, the longranged asymptotic relation c~(r) + u~(r)/kBT as r + oc as is embodied in the closure conditions (6.137)(6.139) (96,183,184). The use of the PYA, HNCA and MSA in the SSOZ formalism can lead to serious errors in the prediction of longranged orientational quantities such as the dielectric constant and the Kerr constant (96,183,184). In addition, the sitesite correlation functions as predicted
216
by the SSOZ theory depend on the presence of 'auxiliary' sites, which do not contribute to the sitesite potential (181,195).
Analytical solutions of the SSOZ equation have been obtained for a number of different simple models. The SSOZPYA has been solved for the symmetric natomic fusedhardsphere fluids (in which each molecule consists of n equal diameter fused hard spheres in geometries of maximum symmetry such as equi lateral triangles and regular tetrahedrons) (196200) and their mixtures with the hardsphere molecules (201). The SSOZMSA was solved for the polar version of the natomic symmetric fusedhardsphere molecules (194), for their mixtures with hardsphere ions (202), and for the fusedhardsphere diatomics with Yukawa attraction (203205). A general analysis of the class of analytically solvable site site models is provided by Cummings and Stell (96).
In the next two sections we will briefly present the results of the analytical so lutions of the SSOZPYA and SSOZMSA for the symmetric fusedhardsphere nonpolar and polar diatomic molecules. Analytical solutions for these models were recently reviewed in reference (185) and we refer the readers to this publi cation and to the original papers for more details.
6.4.1.2 SSOZPYA for the Fluid of Fused HardSphere Diatomic Molecules
Let us consider a fluid consisting of diatomic molecules with a hardsphere sitesite interaction of diameter R = 1 for each site and intramolecular distance between sites L. Since the two sites are equivalent the SSOZ equation (6.135) for this system can be written as a single scalar equation
h(k)  (1 + cb(k))~(k)(1 + cb(k)+ 2ph(k)) (6.140)
where h(k) and ~(k) are the Fourier transforms of the total and direct correlation functions respectively, cb(k)  sin kL /kL and p is the number density of the molecules. The SSOZPYA for the present system takes the form
h(r) =  1 , r < 1
c(r) = 0, r > 1 (6.141)
where for convenience the hardsphere diameter has been taken to be unity. The Baxter qfunction can be introduced in a way similar to that discussed in Sec tion 6.2.1. The final result for the case, L  ½, is
1 q(r)  ~ (anr + ~an  bn) q Re(Poeit)+ 
n=O
217
(9O + Z R~{[~
eiA,~ ei,kn r 1 dn(iAn + e l ix")] l~_ )~, 0 < r <
717 n1 1
q(r)  Z (a~r + ~a~ + bn) + R ~ ( i P o ~  ~ r ) + nO
oo i)~n ei)'" Ca 1  i/~ne½ ix" 1 + n=lE/~C{[ ~ =~n 71"/9 t( 1  /~n2 )dn]e i )~nr} , 2 < r < 1
oo q ( r )  ~pl n~ Re(~ne_iX.~ ), r > 1 (6.142)
where ~,~  i[©'(k)Q(An)] 1, Q(k)  1  27rp f ~ q(r)e ikr dr, An is the solution of the equation 1 + &(An)  0 and
an  4/~C(/~n , bn  41i~e[~nn (1 n I i)kn)e_,iA'~],
/oo d n   4 R e ~ rg(r)e ix'~ dr
The parameters a0, b0, P0, dn and ~ are obtained from the solution of a set of nonlinear algebraic equations (196). To utilize this solution one needs to truncate the infinite sums in the above expressions. The investigation of the influence of the number of terms included in Equation (6.142) on the solution of the SSOZ, equation (6.140), has been carried out (198200). It was demonstrated that, in the region of liquidstate densities, the simplest socalled zeropole approximation (ZPA), in which all the terms with n > 0 are neglected, yields accurate predictions for structural properties. The solution has also been applied to the case of different values of L and for models consisting of three or four hard spheres fused at a distance L to form an equilateral triangle or a regular tetragon.
The EOS for these models can be obtained by integrating equation (6.134) or equivalently from the expression for the bulk modulus
1 OP  1  p ~ f c~z(r ) d F  2[Q(k  0)] 2 (6.143)
k sT Op T aZ
The thermodynamic properties of the present model, as predicted from the compressibility route by SSOZPYA, have been studied by Lowden and Chan dler (190). In general the results for the pressure do not agree satisfactorily with the corresponding MonteCarlo simulation results. This conclusion is valid also
218
for the LJ diatomics studied using the SSOZPYA (206) and the SSOZHNC ap proximation (207). In contrast, the energy route, equation (6.133), to the SSOZ PYA pressure for the LJ diatomics gives reasonably good agreement at liquid state densities (208). Thus, for the present model, the SSOZPYA, while being relatively accurate in predicting the structural properties, is not a reliable method for calculating thermodynamic properties. This situation is different in the case of the SSOZMSA which will be considered in the next section.
6.4.1.3 SSOZMSA for the Fluid of Fused Charged [email protected] Diatomic Molecules
For the polar molecular fluid with molecules composed of two equaldiameter oppositelycharged hard spheres (the polar homonuclear harddumbbell fluid), the SSOZ equation, Equation (6.135), can be decoupled into two independent scalar equations by introducing the sum and difference correlation functions
1 h~(r)  ~(h++(r) + h+_(r)),
1 hd(r)  ~(h++(r) h+_(r)) (6.144)
The corresponding SSOZ equation and closure conditions for the sum correla tion functions coincide with the SSOZ equation and the SSOZPYA considered in the previous section, while the SSOZ equation and closure conditions for the difference correlation functions are
[1  & ( k ) + 2phd(k)][(1  & ( k ) )  1  2pOd(k)] = 1 (6.145)
and hal(r) = O, r < 1
q2 c d ( r )  kBTr' r > 1 (6.146)
where the hardsphere diameter is taken to be unity. The factorization of this equation has been carried out by Morriss and Perram (194). The final expression
1 for the qfunction for the system with L  ~ is
q(r) = 2ZJo + p cosr + v sin r, 1
O < r <  2
Z q(r) = 2ZJo + 27rp
1 1  p s i n ( r  ~) + v cos(r  ~),
1  < r < l 2
219
q(r) = 0, r > 1 (6.147)
where z = 6 / L 2 + 87rp/32q 2 and the parameter Jo = f ~ rhd(r) dr is the solution of the quadratic equation
z 576~72z(1C)J~ + 1 2 ~ 7 ( 2 z ( S + 3 C  3 ) + ( 2  C ) ] J o +  ~ ( 5  3 S  4 C ) + ( C  1 )  0
(6.148) 1 1 where ~  7rp/3, C  cos 7, S  sin 7 and expressions for the remaining param
eters are presented in reference 194. Here the analogue of the ZPA discussed in the previous section is used. The thermodynamic properties of this model have been analyzed using the energy route by Morriss and Isbister (209,210). The residual internal electrostatic energy, U res, is related to the parameter J0 by ures /NkBT = 24r//3J0q 2. Using this expression and the 'charging' procedure for calculating the residual Helmholtz energy (209,210), the following expression for the pressure has been derived
pe Po 1 12~flq 2 = + G'(z) + G(zo)  G(z)] (6.149)
pkBT pkBT 24r/(1  C) [ z
where
Z 2 2 1C)} (C1z +Co) 3/2 G(z)    ~  ( 3 C  3 + S )  z(1  ~ ~ 
C1   ( C  2 ) ( 1  S ) _71_ C 2 C o  (1  1 , 7C) 2 and/90 is the pressure of the corresponding uncharged version of the model. This equation, together with the corresponding expression for the Helmholtz energy, has been used to predict the vaporliquid phase separation which appears due to the dipole interaction (209) (see Figure 6.7). The analysis shows that this phase separation occurs at tem peratures much lower than that of the corresponding nonpolar LJ diatomics. In addition the analytical solution of the SSOZMSA was also used to study the liquidliquid phase separation (210) which occurs in a mixture of polar and non polar hard dumbbells.
6.4.1.4 Corrections to the SSOZ Formalism: ChandlerSilbeyLadanyi Equation
The success of the SSOZ formalism is mixed. In spite of its ability to give reasonably accurate predictions for the shortranged structure, it fails to describe certain angular correlations (such as the Kerr constant and dielectric properties). There are other deficiences of the SSOZ theory, such as the dependence of the
T * x l 0
1 .30 . . . . . . . . . . . . . . , . . . . , . . . .
1 .10
1 .20 , Coexistence
   Spinodal • CPDHS
i
1 . 00
0 . 9 0
0 . 8 0
0 . 7 0
220
, , , , I , , , , ! , , , , i , , , , i , , , ,
0 . 0 5 0 .1 0 . 1 5 0 . 2 0 . 2 5
11
Figure 6.7 The phase envelope for the dipolar harddumbbell fluid with elongation L = 0.Scr. The full line is the coexistence curve and the dashed line is the spinodal curve. The point above the curves is the critical point of the dipolar hardsphere fluid with charge separation 0.Set. After Morriss and Isbister (209).
results on the presence of 'auxiliary' sites, and the thermodynamic inconsistency between different routes to the EOS, although the latter problem is shared by all IEAs except those which are specifically designed to eliminate this problem.
A number of attempts have been made to correct the defects in the theory. These developments follow two alternative lines. The first one, initiated by Cum mings and Stell (96) and utilized in later studies (211,212), is based on the idea that SSOZ equation, Equation (6.135), is a definition of the direct sitesite correla tion functions, for which more appropriate closures should be used. In particular, the analysis carried out in references 182, 183 and 213 shows that, to correct the results for the dielectric properties, it is necessary to assume asymptotic behavior for the direct correlation functions which is different from that which follows from the regular SSOZ closure conditions, Equations (6.137)(6.139). Unfortunately, this asymptotic form depends on the particular geometry of the sitesite model used and quite often, due to its complicated functional form, is extremely difficult to treat as a direct correlation between the particles. This is in direct contrast to the case for simple fluids or to the case for the molecular formalism where the short range and simple asymptotic behavior of the direct correlation functions is
221
the key to the success of the IEAs away from the critical point. The alternate possibility is to correct the SSOZ equation itself, rather than
to use more sophisticated closure conditions. This possibility can be realized by constructing an OZlike equation in which the direct sitesite correlation functions consist of a subset of diagrams from the set contributing to the total sitesite pair correlation function. Such a modified version of the SSOZ equation, the socalled ChandlerSilbeyLadanyi (CSL) equation has been proposed by Chandler et al. (214,215). We will discuss the CSL equation using the notation of Rossky and Chiles (216) who reformulated this equation and presented it in a convenient form. In this formalism the total sitesite correlation function h ~ ( r ) is written as a sum of four terms
hoL~(r)  ho~alOB (r) + hola~ s ( r ) + hoLs~ a (r ) + hoLstOs ( r ) (6.150)
The subscripts s or a indicate whether or not the indirect correlation of the corre sponding site starts with the intramolecular distribution function. The correspond ing sitesite direct correlation functions are defined via the following OZlike in tegral equation
(6.151)
which is a slightly modified version of the CSL equation presented by Rossky and Chiles (216). Here
li~  (h~°~a
S~ __(^1 a2a~
hoLs~s" ' Ca~  CoLs~a Co~s~s '
o) 1 ' P ~  p 0 ' c o ~  k L ~
and Equation (6.151) is written in the Fourier kspace. The analogues of the regular HNCA, PYA and MSA, which couple the direct
and total sitesite correlation functions are (i) HNCA (216)
90la~a (r )  hoLal~a (r ) + 1  ~ I~U~13(r)jltc~al~a(r)
has l3 a ( r )   haal3 s ( r )  ga a~ a (r)taaZ~ (r),
h~z~ (r) = g~a"o ( r ) ( t~ ,~ (r) + t~oz ~ (r)t~,Za (r)) (6.152)
222
(ii) PYA (216) COta~ a ( r ) _7_ faz(r)[1 + tOla~ a (r)]
C . ~ (r)  c.~Z. (r)  f . z ( r ) t . ~ a ( r ) , ¢c~.~. ( r )   f . z ( r ) t . . z . (r)
(iii) MSA (53)
(6.153)
= r <
ca,f ly(r)(~ia(~jaflUa~(r) r > R~Z (6.154)
The PYA and HNCA for a fluid of LJ homonuclear diatomics have been solved numerically by Rossky and Chiles (216). Although the results for the structure appear to be qualitatively correct, they are less accurate than the corresponding results of the SSOZPYA. Recently, taking into account the bridge diagrams, the HNCA for the same LJ model has been solved by Attard (217). In general the results obtained are in better agreement with computersimulation results than those of Rossky and Chiles.
Analytical solutions of the PYA and MSA for a number of simple models with hardcore short,range interactions are possible within the CSL formalism. How ever most of the solutions were obtained within more general integralequation formalism for associating fluids, for which the present CSL theory is a limiting case of complete association. The only exception is the solution of the PYA for the multicomponent molecular fluid with hardsphere sitesite interaction (218). Nevertheless, the general scheme of the solution is quite similar to that of the associative PYA for the multicomponent dimerizing hardsphere system (57). Ac cordingly, we postpone the discussion of these solutions to Section 6.5, in which the integralequation theory for associating fluids is presented. We only note here that the solution obtained in reference 218 was extended to the case of flexible molecules and a closed form analytical expression for the compressibility pres sure was derived. In the case of the linear flexiblechain model polymer system, this equation of state coincides with the equation of state derived by Chiew (93) (see Section 6.3.2.3 of the present review).
6.4.2 Hard Spheres with Dipolar Interactions
Despite the difficulties associated with the molecular formalism noted above, it has nonetheless proved to be a useful approach to certain classes of molecular fluids. In particular, fluids which can be modeled as spherical in shape with the anisotropy in the pair interaction resulting from multipolar forces (such as dipole
223
and/or quadrupole forces) have proved amenable to the molecular formalism, and many technologically important substances, such as water, can be approximated by such a model. In fact, a large measure of our current understanding of the origin of the static dielectric properties of water at ambient conditions comes from the numerical solution of the HNCA for water modeled as a LennardJones sphere with a polarizable dipole and tetrahedralquadrupole interactions (219).
The number of molecularformalism models for which IEAs are analytically solvable is limited, but includes models for polar fluids, such as dipolar hard spheres (220,221) and hard spheres with arbitrary multipole moments (222224), and simple models for water (225,226). To give some indication of how an IEA is solved analytically within the molecular formalism, we present the analytic solu tion of the MSA for dipolar hard spheres.
The microscopic structure of dipolar hard spheres (DHS) is given in terms of the molecular pairdistribution function 9(12) which is proportional to the proba bility density of finding a molecule of type i with center of mass located at ~'1 and with orientation ~1 [equal to (01, ¢1) for a linear molecule and equal to the Euler angles (01, ¢1, X1) for a nonlinear molecule] and a molecule of type j with center of mass located at ~'2 and with orientation ~72. [We use the numeric symbol, z, to symbolize collectively ~'z and ~z, z = 1, 2, etc.] The molecular pair distribution function is related to the molecular total correlation function by the usual relation ship h(12)  9(12)  1 and the molecular direct correlation function c(12) is then defined by the molecular OZ equation
h(12)  c (12)+ ~~p f c(13)h(32)d73d~3 (6.155)
where ~  f dw  4~ for linear molecules such as DHS, 87v 2 for nonlinear molecules. For DHS with hardcore diameter a, the intermolecular potential u is given by
u(12)  ~ r~2 <
(6.156) =  >
I+'12[ 3 le'1215
Here, g~ is the unit vector lying along the direction of the dipole in molecule i and rq2 is the vector joining the centers of mass of molecules 1 and 2.
6.4.2.1 Analytic Solution of MSA for Dipolar HardSphere Fluid
224
For the dipolar hard spheres (DHS), the MSA yields
h(12) =  1 r12 < cr
(6.157) C ( 1 2 )   ~t2 [ 8"1 "8'2 38"1"r'12 8'2 "r'12 ]
    k B T 11~'121 a   1~121 s r12 > O
The OZ equation, Equation (6.155), combined with the closure Equation (5.157), was first solved analytically by Wertheim (220). We present here the analytic solution obtained using the technique of Blum and coworkers (222224). In contrast to the solution of Wertheim, which exploits particular simplifications peculiar to the dipoledipole interaction, the method employed here is readily generalized to include higherorder multipole and nonspherical overlap interac tions (221,226,227).
In terms of the rotational invariants defined in the Appendix, the pair potential for DHS is given by
u(12)  .000~.000 ~112~i)112 '/Z WOO ((.dlCd2&Jr)n t =:00 ((M1Cd2(Mr) (6.158)
where u°° ° ( r )  c o r12 < c~
= 0 r12 > cr (6.159) u l 1 2 ( r )     ~ i ~ r / ~ 3 r12 > a
where # is the dipole moment of the DHS. Consequently, there are only three expansion coefficients of h(12) and c(12) which are nonzero in the MSA [mnl
= 000, 110 and 112] and the closures on them are given by
]~°°°(r)  1 r < cr ~oOO(r)  0 r >
h 1 1 ° ( r )   0 /" < (y
~11°(r)  0 r > (7
h ~ l ~ ( r )  0 r <
o l 1 2 ( r )     k B T r 3 r > cr
(6.160)
Of the functions . ~ n(r) (9 r = J or S) defined in the Appendix, only three are distinct and nonzero" .~0°(r), 0vii(r) and 9r111 (r) = .~.1] (r). Thus, according to equation (A12), we obtain three distinct OZ equations:
HO0(k)  CO0(k) + pH°°(k)C°°(k) H l l ( k )  Vi i (k) [ pHl l ( k )V l l ( k ) HI l (k)  Vi i (k) Jr pH11(k)CIl(k)
(6.161)
225
To proceed further, we need to obtain closures on J~n(r) and S2n(r ) in real space. From Equation (6.232) in the Appendix, we have
m n S x (r)  0 r > o for rnnx  000,110,111 (6.162)
and from Equation (6.230) we obtain
m n m n ~ m n 2 J~ (r)  /3~, o + Px,2 r r < a for m n x  000,110,111 (6.163)
where
00o  booo, ° =
~x1,%   ( _ ) x [ ( 1 1 0 )b l l0 1 ( 1 1 2)b~12 ] X  X 0  2 X  X 0 (6.164)
/31,1 __ (__)x3_ ( 1 1 2)5112 2 X  X 0
In these equations,
 27r foo ~ ]~mnt(r)rlPdr (6.165) b ; n*
We now note that each of the three OZ equations given in Equation (6.161) and the closures given in Equation (6.164) have the same genetic form given by
subject to the closure
H(k)  C(k) + pH(k)C(k) (6.166)
J(r)  /3o+/32r 2 r < c r
S(~)  o ~ >
(6.167)
However, there is a crucial difference between the mn = 00 problem and the mn  11 problems: in the former case, 132 is known explicitly (= 70 whereas in the latter case (for X = 0 and 1)/32 will be determined by the closure relation. Despite this difference, it is expedient to perform the Baxter factorization on the problem as stated in Equations (6.166) and (6.167) and then to consider the specific cases.
Solution of generic problem. Since the S~n(r) are finite ranged, implying that C~n(k) is the Fourier transform of a finiteranged function, the OZ equation, Equation (6.166), can be factorized in the usual fashion yielding.
1  pC(k)  Q(k )Q( k )
1 + pH(k)  [ Q ( k ) Q (  k ) ] 1 (6.168)
226
fo (y Q ( k )  1  p q( t ) e x p ( i k t ) d t (6.169)
The realspace function q(r) satisfies q(r)  0 for r < 0 and r ___ or. One should note that the symmetry of the harmonic coefficients used in the dipolar hardsphere mixture problem (m  n for all coefficients) implies the following relationships between the H, C, J and S quantities"
H ~ n ( k )  2 f ~ c o s ( k r ) J ~ n ( r ) d r
(6.170) c ~ n ( k ) = 2 f ~ c o s ( k r ) S ~ n ( r ) d r
These equations are special cases of Equations (6.229 and (5.231). Inverting Equation (5.158) to real space yields (see the Appendix)
J ( r )  q(r) + J ( l r  t l ) q ( t ) d t (6.171)
Substituting Equation (6.167) into Equation (6.171) yields
1 q(r)  ~a(r  a ) r + b(r  a) (6.172)
Substituting Equations (6.167) and (6.172) into Equation (6.171) and solving for a and b, we obtain
2/32 (1 + 3eaa/32) b /32 (6.173) a ~     
Solu t ion o f the m = n = O, X = 0 problem. In this case, the/32 = ~ is known explicitly yielding
2~r (1 + 2r/) b 7r a  (1  ~7) 2 ~ = (1  r/) (6.174)
where ~7  (Tr/6)a~. Taking into account the slight differences between the def initions of q(r ) , a and b used here and in Baxter's solution of the PercusYevick approximation for hard spheres, Equations (6.38) and (6.39), it is readily seen that the results for a and b are consistent. Hence, the m = n = X = 0 problem reduces to the PercusYevick approximation for a hardsphere fluid as it should.
Solut ion o f the m = n  1, X  O, 1 problems . We begin by noting that
90,2   271"~, 91,2  Tr~ (6.175)
227
where
Following the general analysis above, we find that
(6.176)
11 1 qx (r)  ~ax(r  cr)r + bx(r  or) (6.177)
where 47r~ (1 + 4 ~ ) 27c~ (1  2 ~ ) / (6.178)
a o = (12~/~) 2 a l   ( l+ r ]~ )
Hence, the solution of the problem reduces to finding ~ which comes from consid ering the closure o n O l 1 2 ( r ) for r > o. From Equation (6.235) of the Appendix, for r > cr we obtain
c l 1 2 ( r )   ~ / ~ [ f o o a ~ 1 1 ( r l ) d r 1  f o ~ l l l ( r 1 ) d r l ] r ~ 27r ( 6 . 1 7 9 )
upon evaluation of the 3  j symbols. Combining this equation with Equa tion (6.160) we obtain after some manipulation
( )2 3y  ao _ a l = (1 + 4r/~) 2 _ (1  2r/~) 2 27r~ (1  2T]~) 4 (1 1 T]~) 4
(6.180)
where 47~p# 2
Y  9kBT
The residual internal energy per molecule, ures/N, is given by (138)
(6.181)
ures 1 m~nl ()l f°e~tmnl ~mnl N = ~47rp 2l + 1 (r) (r)r2dr (6.182)
which simplifies in the present case to
u res
N = 3y~ (6.183)
This completes the analytic solution of the MSA for DHS. The energy equation of state for DHS in the MSA predicts a liquidgas phase transition. This is a some what artificial system since the sole source of the attraction giving rise to the phase
228
transition is the dipoledipole interaction. Nevertheless, the MSA provides an an alytic expression for the state dependence of the dipoledipole contribution to the internal energy via Equation (6.183). Cummings and Blum (221) extended this analysis to dipolar hardsphere mixtures and performed MonteCarlo simulations to compare with the MSA results. They found that the MSA underpredicts the magnitude of both the dielectric constant and the residual energy, U res. However, the importance of the analytic solution of the MSA for DHS cannot be under stated: It inspired much of the subsequent work leading to microscopic theories of the dielectric constant and provided the basis for more accurate equations of state for DHS. Many of these developments are reviewed by Stell et al. (228).
6.5 EQUATION OF STATE FOR ANALYTICALLY SOLVABLE MODELS OF ASSOCIATING FLUIDS
During the last decade, significant progress has been achieved in the devel opment of statisticalmechanical theories for associating fluids. Associating flu ids are characterized by the formation of longlived clusters of particles due to strong shortranged, frequently directional, attractive interactions, such as hydro gen bonding and ionpairing. Theories for associating fluids date back to the work of Bjerrum (167) and include an important study by Ebeling and Grigo (168) using the 'chemical approach' to treat ionic association as a chemical reaction. Impor tant milestones in the development of the statisticalmechanical theory of associ ating fluids are Andersen's work on the cluster expansions for hydrogenbonded fluids (229231) and the subsequent general formalism developed by Chandler and Pratt for chemically associating fluids (232). However, these developments do not readily lend themselves to the accurate description of the structure via integralequation approximations.
Two alternative versions of the integralequation formalism for associating fluids have recently been proposed. The first one, which was initiated by Cum mings and Stell (41,42), consists of utilizing the regular integralequation closure approximations for studying model systems with strong attractive interactions. The shieldedstickyshell (SSS) versions of this type of model are considered in (41,171,233) and is defined by the twocomponent mixture of hard spheres of equal diameter cr with hardsphere interaction between the particles of the same species and hardsphere and sticky interaction (of the same functional form as the attractive part of the adhesivehardsphere fluid interaction) between the particles of different species. The sticky interaction is placed inside the hardcore region and is effective at a distance L < ~r between the particles. For L _< cr/2 only
229
dimers can be formed in the system, while for L > a/2 linear and branched poly merization is possible (171,233). The theory was applied to several similar model systems (43,44,211,234,235).
However, since the regular IEAs are not well suited for treating strongly asso ciating fluids, this approach has several deficiencies. In particular, it is believed that the IEAs used are not faithful to the law of mass action in the lowdensity limit (236). In this study and in the later studies of Zhoe and Stell (237,238), the theory was corrected for this deficiency and leads to results of reasonable accu racy (239241). However, the corrected version of the theory requires, as an input, knowledge of thermodynamic properties of the system. Since the purpose of this review is to demonstrate the abilities of the IEA in predicting thermodynamic properties, we will not discuss this approach here any more.
The second version of integralequation theory for associating fluids was ini tially developed by Wertheim (4548). The theory is based on a combined de scription, in terms of activity and density expansions, leading to the multiden sity integralequation formalism. There is a close structural similarity between the multidensity integralequation theory and the regular singledensity integral equation theory, thus permitting the derivation of an OZlike integral equation and analogues of widelyused closure conditions. Originally, the theory was de veloped for associating fluids modeled by hard spheres with offcenter sites in teracting via a strongly attractive shortranged potential responsible for associa tion. More recently, the multidensity theory has been reformulated and extended to the case of associative potentials of any symmetry (including the spherically symmetric case) by Kalyuzhnyi et al. (40,49). Analytical solutions of the sim plest twodensity versions of the IEA for both theories have been obtained for a number of simple models of associating fluids. These will be discussed in the succeeding sections. It is important to note here that Wertheim's formalism for associating fluids (4548) forms the theoretical basis for a highly successful engi neering equation of state, the statistical associatingfluidtheory (SAFT) equation of state (242244).
6.5.1 TwoDensity OZ Equation and Closure Conditions:Relation to the CSL equation
The twodensity version of the theory begins by expressing the pairwise po tential energy uab(12) as a sum of two terms
U~b(12)   " (n)(12)}~ (a) ~b ~b (12) (6.184)
230
where 1 and 2 stand for the spatial and orientational coordinates of the two par ticles, ~ (a) and ~ (n) ~ab ~ab are the associative and nonassociative parts of the total po tential respectively, and the subscripts a and b denote the particle species. Fol lowing the separation of the potential into associative and nonassociative parts, Equation (6.184), the total number density of the system p~ is separated into two contributions, the density of 'nonassociated' particles p~ ('monomers') and the density of 'associated' particles p~
Pa  P'~ + P~ (6.185)
and the paircorrelation function h~b now becomes a sum of four terms
hab(12)  h~b(12)+ xah~0b(12)+ xBh~(12)+ XaXBh~(12) (6.186)
where xa  P~/Pa, and the lower subscripts in the functions hi~b(r) indicate the degree of association of the corresponding particles.
The twodensity analogue of the OZ equation is
hab = Cab + E Cad * Pet * hdb (6.187) d
where • denotes the convolution and
ho0 ii , Ca %0o c10 p~ 0
Here the c ab functions are analogues of the direct correlation functions. To complete the theory one needs to assume an approximate closure condition,
~b and densities p~ and p~. which provides the additional relation between h~ b, c~j In the present theory, there are two approximations. The first is similar to those used in the usual onedensity integralequation theory and follows from the ap proximation for the 'bridge function.' The second approximation is specific to the multidensity description and is related to the choice of the maximum number of particles which can be simultaneously bonded by the associative part of the poten tial. Assuming the simplest 'dimer' approximation, in which not more than two particles can be bonded simultaneously, and making use of the close similarity between the structure of the multidensity and of the usual onedensity integral equation theories, we can derive the twodensity analogues of the widelyused singledensity IEAs, the HNCA (46)
90abo  eu(~) /kBT+t~bo,
231
g ~ ~ _ab.ab ab _ab.ab 900~01, glo   gOO~lO,
_ ÷ab÷ab ~(a) g ~ goabo(t~ t~01~10 + Jab ) (6.188)
ab ab ab ab ab the PYA (46) where 9ij   h i j + 5oi~oj and t i j   h i j  cij ,
ab ab ab Yij   gij  Cij (6.189)
where the analogues of the regular cavity correlation functions, ~b y~j, are defined by
ab __ eU(ab ) /kBT ab ,~ g(a) ab gij (Yij + (~il YO0)" UjlJ ab
and the MSA (53,55)
ab dli(~lj f a b ( a ) (n) Cij   (~Oi(~OjUab / k s T , for r > Crab ,
ab _ hij 5oi5oy, for r < aab, (6.190)
where 5ij is the Kronecker delta, aab = (cr~ + 08)/2 and era is the hardsphere diameter.
Within the same 'dimer' approximation, the relation between the 'monomer' and total densities is given by
Pa  P~ + 47Cp~o ~ pbo f go~(12) r(a) Jab (12)d l (6.191) B
where g~b _ hi? + (~0i50 j and ~e(a) u(a)/k  e ~b / B~ _ 1 The latter relation can be , J ab viewed as the analogue of the massaction law.
In the more general case of 'trimer' and higherorder approximations an infi nite number of terms involving the higherorder correlation functions will appear in the closure conditions. However, there are several models for which the 'dimer' approximation appears to be either exact or quite accurate.
Since the pair potential, Equation (6.184), depends on the mutual orientation of the two particles, all the partial correlation functions, involved in the OZ equa tion, Equation (6.187), are also orientationally dependent. This makes the solution of the OZ equation, Equation (6.187), rather difficult. However, in most applica tions it is assumed that only the associative part of the potential is orientationally dependent, while the nonassociative part is spherically symmetric. This, together with the specific structure of the OZ equation, Equation (6.187), makes it possible to orientationally average Equations (6.187), (6.191) and the closure conditions, Equations (6.188)(6.190). The orientationallyaveraged version of the OZlike
232
equation with closure conditions is equivalent to Equations (6.187)(6.190), in which the orientationallyaveraged correlation functions ~f (r) and h~ f (r) replace the original correlation functions c~b(1 2) and h~b(12)
Thus far, the solution of the OZ equation, Equation (6.187), has been ob tained only for its orientationallyaveraged form. In addition, we note that the twodensity version of the theory, developed in references 40 and 49 to study the association effects in fluids with sphericallysymmetric potentials, is equivalent to the orientationallyaveraged version of the present twodensity theory.
Finally, it is interesting to note that in the completeassociation limit the present orientationallyaveraged twodensity version of the integralequation the ory reduces to the CSL equation discussed in Section 6.4.1.4 of this review. This can be easily demonstrated by considering the models with associative interaction presented by the sticky interaction. For the sake of simplicity we will show this for the onecomponent system. In this case the Mayer function for the associative po tential will be substituted by the deltafunction term f(a)(r) = BS(r  L), where B is the stickiness parameter and L is the distance at which bonding between par ticles occurs. The direct contribution from the deltafunction, which appears in the total and direct correlation functions, hij(r) and c~j(r), can be separated out
hij(r) = Hij(r)+SiXSj lByoo(L)5(rL) , cij(r) = Ci j (r)+Si lSj1Byoo(L)5(rL)
and the OZ equation, Equation (6.187), can be renormalized with respect to this contribution. This yields the following renormalized OZ equation written in ma trix form in Fourier kspace
_ ~ r l ~ r T + ~v~(~Vfl (6.192)
where I=I and (~ are the matrices containing the elements which are the Fourier transforms of the correlation functions H~j(r) and C~j(r) and
I?Vij  ~ij + ~ilSjO47rL2Bpo sin kL kL Yoo(L) (6.193)
The relation between the densities, Equation (6.191), now takes the form
P  Po + 47rL2p~Byoo(L) (6.194)
In the completeassociation limit B + ee and from Equation (6.194) we have
p2oB  P (6.195) 47rL2Yoo(L)
233
Substituting this into Equations (6.193) and (6.192) and multiplying both sides by the matrix with the elements Xij  5io5jo + 5i15jlpo/p we finally arrive at the CSL equation
I=I  W ( ~ W T + ~q(~p~2I (6.196)
where ~ r  X W X  1 , ~6  X lpX 1 and the elements of the matrices ISt  XI2IX and (~  X(~X are the partial correlation functions as defined by the CSL formalism (214,216). From Equations (6.188)(6.190)one can easily see that the closure conditions for the functions Cij(r) and Hij(r) coincide with the corre sponding closure conditions, Equations (6.152)(6.154) for the CSL equation. A similar analysis can be carried out for 'trimer' and higherorder approximations. In this case, the twodensity OZ equation, Equation (6.187), reduces to the CSL equation for the molecular system with the corresponding number of sites in the molecules (218).
As in the case of the onedensity IEA, analytical solutions for the present two density IEA have been obtained for simple models with the nonassociative poten tial presented by the hardsphere, chargedhardsphere or hardcore Yukawa fluid models with the sticky potential for the associative interactions. This includes the onecomponent and multicomponent dimerizing hardsphere models (51,57), dimerizing RPM and PM of electrolytes (53,176), dimerizing HCYF model (245), and the SmithNezbeda (SN) primitive model for associating fluids (54). Recently, analytical solutions where obtained for polymerizing models (130,246251) using the multidensity formalism with the number of densities higher than two (47 49) and their thermodynamic properties, in particular equation of state, has been analysed (252,253). However, due to the lack of space we will not discuss these solutions here, refering the reader to the original publications.
6.5.2 TwoDensity PYA for Dimerizing HardSphere Models
The two dimerizing hardsphere models we will consider in more detail are the shieldedstickyshell (SSS) (41) and shieldedstickypoint (SSP) (45,236) models of associating fluids. The former model, which was discussed at the beginning of this section, consists of the twocomponent mixture of hard spheres of equal diameter cr and with sticky interaction between unlike species effective at a dis tance L < cr between the two particles. For the values of L from the interval 0 < L < ~r/2 only dimers can be formed in the system. The SSP model can be viewed as a limiting case of the SSS model, in which the sticky area is re duced to a single sticky point (171). For this model the associative interaction is
234
orientationally dependent
~tSSP a ab (12)  uHS(r) + u.b(12 ) (6.197)
and bonding between the particles occurs at a distance L if they have an appropri ate mutual orientation. This, together with the hardcore repulsion, provides the saturation of the associative potential on the dimer level at any values of L. The unshielded onecomponent version of this model (L = (aa + OB)/2 , where era is the hardsphere diameter) has been used by Wertheim (45,51), while the shielded version of both models was proposed by Cummings and Stell (41) (SSS) and Stell and Zhou (236).
Analytical solutions of the twodensity PYA have been obtained for both SSS and SSP versions of the dimerizing hardsphere models (51,240,254,255). In the latter case an orientationallyaveraged version of the twodensity PYA was uti lized. Since there is close similarity in the solutions for both types of the model we will present here only the solution for the SSP model.
An orientationallyaveraged version of the twodensity PYA for the one component SSP model can be written in the form
hij(r) = 5oiSojBVoo(L)5(r L), r < (7
cij(r)  0, r > o (6.198)
The corresponding twodensity OZ equation can be factorized using the Baxter factorization method. The resulting qfunctions are
1 r2 qio(r)  ~ai + bir + Ci
qii (r)  D}n)r 3 + Aln)r 2 + B}n)r 4 C~ n),
where In denotes the following intervals
1L], 1],
I1 = [0, 1  L], I 2 = [ 1  L , L ] , I3 = [L, 1],
for r C In (6.199)
o" f o r L < 
2 o"
f o r L >  2
the constants D} n), AI '~) and B} n) are functions of the unknowns ai, b,, c,, C~ n) and Yoo(L), which follow from a set of algebraic equations. For the monomer density, P0, we have
 1 + (1 + 167rByoo(L)L2p) 1/2 Po  87rByoo ( L) L 2 (6.200)
235
The set of algebraic equations for the solution parameters can be solved in a closed form for L  cr (51)
1 + 2r/ 3r/o. o .2 ao = (I  r/) 2' bo  2(1  ~])2' Co  2(1  q) '
27rpod11 7rpod11 1 } 1 a l = b l  ~ , c 1  0 , y o o ( l + ) _ gr/
r /  1 ' 1  r/ (1  r/) 2'
P o   1 + (1 + 167rpd11) 1/2
87rdll
where r /  7rp~73/6 and dll  o.B9oo(a+). The compressibility pressure for the onecomponent SSP model follows from
the twodensity analogue of the PYA compressibility pressure (256) derived by Wertheim (51 )
p c
kBT = ~ Pijq~}(O)  27r y~ ~ Pijqmj(O)ptmq~m(O)+ ij i j l m
4 +57c2 ~ Y~. ~ Pijqlj(O)ptmqkt(O)Pknqin(O)  27rL2Bp2y2o(L) (6.201)
i j l m k n
The expression for the virial pressure (240,254) follows from the standard relation, Equation (6.2),
p v
kBT 2 2
= p + ~lrp 9(1 +)  7rp~L2B{3yoo(L)+ Lyoo(L)} (6.202)
where the expression for y/~0(L) is given in reference 240. Again, for L  cr the resulting EOS can be obtained explicitly as (51),
pc _ _ 1 + r / + r/2 _ (1 + lr/)(1  x) (6.203) pkBT (1  r/) 3 2(1  7/) 2
pv 1 + 2 r /+ 3~ 2 (1 + 2q)(1  x) q(1  x) 2 1 1 pkBT (1  q)2 2(1  r/)(1 + gq) 4(1 + 7q)
(6.204)
where x  p0/p. These expressions are valid for any degree of association, including the limit
of complete association, B + ec. As was argued in Section 6.5.1, in this limiting
236
P/PdkB T
2 0 . 0 . . . . . . . . . . . . . . . . . . . . . . . t~ , , , , , ]
o TS, L=a i I /1   PYC, L=o I 15.0 . . . . . PYV, L=cr ~ / "
[] MC, L0.6o I ee ,, • PYC, L=0.6o I • tz ,,"
......... PYV, L=0.6cr , ' . . " / /
1 0 . 0 / / , .....
~D • . ..."
le, • [] . ' " ,~ • . .
5 0 ,,~'" . . ...... q
4
0 . 0 . . . . ' ' . . . . . . . . ' . . . . ' . . . . ~ . . . . .
0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6
p °.3
Figure 6.8 Pressure for the homonuclear harddumbbell fluid calculated from the completeassociation limit of the SSP model for two bond lengths, L  0.6or and L  or. The compressibility (labelled PYC) and virial (PYV) pressures are labelled accordingly. For L = a, the results are compared to the TildesleyStreet (TS) equation of state while for L  0.6a comparison is made with MonteCarlo (MC) results.
case the present solution of the twodensity PYA reduces to the solution of the CSL equation supplemented by the corresponding PYAlike closure conditions.
In Figure 6.8 we present the comparison of the results for the compressibility and virial pressure of the SSP model in the complete association limit at L  0.60 (254) and L  cr (51) with MonteCarlo and TildesleyStreett empirical EOS results (257). A comparison of the results for the pressure as predicted by the twodensity PYA and MonteCarlo simulation results for the SSS model at L  0.42a and L = 0.3a under the condition of partial association (K0 = 4 7 r L 2 B =
33) (240) is presented in Figure 6.9. Predictions of the twodensity PYA for both models are in good agreement with the simulation results as long as the density of the system is relatively low or, at high densities, when the hardcore volume of a dimer is not substantially less than that of the two free monomers from which it is formed.
The above method of solution has been extended and applied to the general case of the ncomponent SSP model in which bonding occurs at the contact dis tance between the hard spheres of species a and b (57) (i .e. , Lab = crab).
237
7 . 0 , , , , , , , , , , , , , , , , , , , , , , ,
6 . 0
5 . 0
P c Y 3 / k B T 4 . 0
3 . 0
2 . 0
1 . 0
0 . 0
0 . 0 1.2
. . . . . A P Y  V , L0.42~
     A P Y  C , L0.42~
o MC, L0.42~
. . . . . . . . . A P Y  V , L = 0 . 3 ~ f /
~ A P Y  C , L0.3~ / o •
[] M C , L=0.3c~ o ~ ' o • f • •
/. ": ,1..s, •
i , i , , , , , , , i , , , i i , , ,
0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
p (~3
Figure 6.9 Pressure for the SSS model for the homonuclear harddumbbell fluid cal culated for two bond lengths, L  0.42cr and L = 0.3a, under the condition of partial association (K0 = 47rL2B  33) (240). The results for the APY compressibility (la belled APYC) and APY virial (APYV) pressures are labelled accordingly. The results are compared to MonteCarlo (MC) simulation results.
6.5.3 TwoDensity PYA for the SmithNezbeda Primitive Model of Associating Fluids
An interesting extension of the twodensity formalism has been proposed by Wertheim (52) in his study of the SmithNezbeda (SN) primitive model of associ ating fluids (258). The model consists of hard spheres, each having one attractive site located on the surface. The attraction acts only between the site of one sphere and the center of another sphere, and is given by a squarewell potential. It thus may be considered to mimic the behavior of a hydrogen bond. The pair potential of this model is
u(12)  UHS(r) + u01(12)+ Ulo(12) (6.205)
where U H S is the hardsphere potential of diameter cr = 1 and U 0 1 and ul0 are the centersite and sitecenter potentials respectively. The appropriate choices for the parameters of the squarewell potential guarantee that saturation conditions, in which the site can be bonded to only one center, are satisfied. Unlike the dimer ization case, the present model allows polymerization. However it appears that
238
the twodensity theory can be modified to include the present SN model (52). The theory results in the OZlike integral equation which formally coincides with Equation (6.187). As in Equation (6.187), the lower indices denote the bonded states of the particle. However, now the bonded states are defined only for the sites, while for the centers the theory does not distinguish between bonded and un bonded states. This results in the following orientationallyaveraged twodensity PYA
< 1
cij(r)  (1Sij)goo(r) f(a) (r)+SilSjl {291o(r) f(a) (r)goo(r)[f(a) (r)] 2} (6.206)
and relation between the densities fl/2+A
P  P0 + 47rp0 [pgoo(r) + pogol(r)]f(a)(r)r 2 dr (6.207) dl where A is the width of the squarewell potential, F (a) is the angle average of the Mayer function, fox, corresponding to the hydrogen bond. These relations are quite different from the corresponding twodensity PYA expressions for the dimerizing fluid.
To make the present PYA analytically solvable, the original SN model must be modified by substituting the Baxter sticky interaction for the squarewell interac tion. Applying the Baxter factorization method to this model (54), we can derive the Baxter qfunction given by
qij(r)  (~air 2 + bir)5oj + dij (6.208)
The unknown parameters of the problem ai, b~, dij and P0 follow from the solution of a set of eight linear equations and one nonlinear equation (54). As in the case of the dimerizing hardsphere system, the analogue of the Baxter expression for the compressibility pressure can be obtained (52,256). However only the virial EOS was analyzed using the analytical solution, (54). The latter was calculated from the following expression
4 fl/2+~ /3PV = 1 + ~Trpg(1 +) + ~Trp0 [g00(r) + ~glo(r)][f(~)(r)]'r 3 dr (6.209) P
In Table 6.2, we present a comparison between the pressure calculated from Equa tion (6.209) and that obtained using MonteCarlo simulation for the original SN model with finite squarewell associating potential. The largest discrepancy be tween theory and simulation is at high densities and approaches 10%, suggesting that the agreement between theory and simulation is quite satisfactory.
239
Table 6.2 The compressibility factor, ffp~T, of the SmithNezbeda primitive model of associating liquids (54). For each temperature the first row is the analytical result from the virial EOS, the second row is the simulation value. Temperature and density are given here in units of e/lc and 7ro3/6 where e is the depth of the squarewell associative
kBT/e 7rpo3/6 0.2 0.3 0.4
0.5 2.107 3.262 5.313 2.05 3.39 5.85
0.3 1.695 2.699 4.629 1.61 2.79 5.18
0.2 1.314 2.320 4.272 1.32 2.68 4.92
0.15 1.200 2.222 4.189 0.97 2.55 4.75
potential.
6.5.4 TwoDensity MSA: RPM of Electrolytes and Dimerizing HCY Fluid
The analytical solution of the twodensity MSA has been obtained for the dimerizing RPM of electrolytes, defined as a charged hardsphere version of the twocomponent SSP model at L = R (53) and for the same dimerizing RPM of electrolytes with an additional Yukawa interaction (55).
The latter model is more general and therefore we will briefly discuss its so lution within the twodensity MSA (or associative MSA (55)). In the closure conditions, Equation (6.190), the nonassociative part of the potential is defined to be the sum of the Coulomb and Yukawa potentials. Similar to the case of the dimerizing hardsphere system, the Mayer function for the associative potential in Equation (6.190) is modeled by the Dirac delta function. The method of so lution of the present AMSA is similar to that of GMSA for the primitive model of electrolytes. The solution results in an analytical expression for the Baxter q functions with parameters determined by a set of nonlinear algebraic equations. We will not present the analytical expressions here for either the qfunctions or the set of algebraic equations because the resulting expressions are too tedious: We refer the interested reader to the original publications (55). We present here
240
the expressions for the EOS obtained from the virial route
p v v
N k B T
where
1  527rP[9~°s) (or+) + 2x9;~) (a+) + x29~1s)(or+)] + 3 UNkBT Uid
U  V id 9 VkBT = 2 z*(go + x J1),
1
Ji = pJoi + poJ l i ,
P = P + + P  ,
1 +_
po  P+o + Po, x  x+ + x_
and the energy route
(6.210)
0 N ( P e  PHS)  p(1  p4)[A~ + A~2]
op  (6.211)
where A (a) 1 1
V k B T = p ln x  ~po + ~p,
A (n) is the Helmholtz energy of the system interacting only via the nonassociative
part, u~b ), of the total potential. The parameters, Ji, follow from a set of nonlinear algebraic equations. The above solution of the twodensity MSA was used to take into account the effects of association in the RPM model for a 22 electrolyte solution. With this aim the total potential of the model was split into associative and nonassociative contributions,
    + (6.212)
where UHS and u ~ ) are the hardsphere and Coulombic potentials, respectively, and o (a) and° (n) U'ab ~ab are the associative and nonassociative parts of the total potential, respectively, and the subscripts a and b denote the particle species.
To obtain an analytical solution, the associative part of the potential is chosen as a sum of Yukawa terms (55) with the parameters calculated from an extremum condition on the Helmholtz energy. In addition, the Mayer function which arises for this associative potential is substituted by a delta function. An analysis of the results obtained is contained in Section 6.3.4.3.
241
1 . 5 , , , i , , , i , , , i , , , i ,
f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
1.4 ,." "'"'..
13 I / ' " . . . . . .. "'"'.... ":. / I ~ ~ , ".,.
T* i / <> ~ ": I N :.,
1 1 t
' ,,.,. 1 0
0 9
0.8 0.0 0.2 0.4 0.6 0.8
p*
Figure 6.10 Liquidgas phase envelope for the dimerizing hardcore Yukawa fluid and the hardcore Yukawa fluid (HCYF). The solid line is the phase envelope for the HCYF from the MSA. The dashed and dotted lines are the phase envelopes for the dimerizing HCYF obtained from the AMSA when the squarewell depth is ten times (dashed line) and fifteen times (dotted line) the well depth of the Yukawa interaction. The symbols are Gibbs ensemble MonteCarlo simulation results for the phase envelope of the HCYF given by Smit (259).
Another application of the present solution is related to the study of the ther modynamic properties of the dimerizing hardcore Yukawa fluid. The solution of the twodensity MSA for this model can be obtained as a particular case of the solution discussed above by setting the ionic charge to zero. In Figure 6.10, we present the liquidgas phase diagram for the dimerizing hardsphere Yukawa fluid as predicted by the twodensity MSA from the energy route to thermody namics (55). Comparison with the corresponding phase diagram for the nonasso ciating hardsphere Yukawa fluid shows that the liquidgas phase transition for the dimerizing hardsphere Yukawa fluid occurs at higher temperatures.
6.6 CONCLUSION
It is just mder forty years since the first analytic solution of an integral equation approximation for a nontrivial model of fluids (the PYA for the hard sphere fluid) was reported. Since then, analytic solutions of integralequation
242
approximations have been used to develop equations of state for simple, molec ular and polymeric fluids, for systems undergoing chemical reaction (including polymerization) and for electrolyte solutions. They have also been used to de velop molecular expressions for adsorption isotherms in nonuniform systems, a topic not covered in this review. Furthermore, they have provided the input to perturbation theories of fluids and led to the development of the modem theory of the dielectric constant. In short, despite their limitations, integral equations have played an invaluable role in furthering our knowledge of the molecular ba sis for thermodynamic behavior and providing the insight and mathematical ex pressions necessary to develop accurate equations of state. Recent developments in associatingfluid models and their relationship to polymeric fluids suggest that integralequation approximations, and the analytic solutions thereof, will continue to make important contributions in the coming decades.
ACKNOWLEDGMENTS
The authors are grateful to their many colleagues with whom they have had the pleasure of interacting over several decades in developing analytic so lutions to integralequation approximations, including Lesser Blum, Miroslav Holovko, Gary E Morriss, Ivo Nezbeda, Edgar R. Smith, George Stell and Michael Wertheim. The authors gratefully acknowledge support of this research and the preparation of this review by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy. The authors are especially grateful to Ariel Chialvo for his helpful comments on an earlier draft of this chapter.
APPENDIX: SUMMARY OF THE INVARIANT EXPANSION METHOD FOR LINEAR MOLECULES
In this Appendix, we briefly summarize the invariant expansion introduced by Blum (222224) as a generalization of the method employed by Wertheim (220) in solving the MSA for dipolar hard spheres. In general, the invariant expansions of the functions h(12) and c(12) would be given in terms of rotational invariants
I" m n l q?u~" (W1W2Wr) defined by "" m n l
[(2m+1)(2n+l)] 1/2 Z ( m rt l ) m u'~"~" #' U' A' D~'u'(w1)D~'~'(w2)Dt°)"(Wr) (6.213)
243
m ( ) In this equation, Du,(w ) is a Wigner rotation matrix, m n l is a 3  j p u A
symbol in the notation of Edmunds (260) and ~1, ~22 and Wr are, respectively, the Euler angles specifying the orientation with respect to an arbitrary set of axes of molecules 1 and 2 and of the vector g'12 joining their centers of mass. For linear molecules, however, we require only # = u = 0 terms in the rotational invari ants (222), so in all the succeeding equations the pu subscript on the expansion coefficients of the correlation functions will be suppressed on the understanding that this implies #u = 00. Thus, we define the invariant expansions of h(12) and c(12) to be given by
h(12)  ~ ]~mnt ^mnZ (r) ¢b00 (~Zl~Z2~Zr) (6.214) mnl
C(12)  Z cmnl(r)~r~onl(('dl('d2OJr) (6.215) toni
The rotational invariant expansion is derived in reference 233. The Fourier trans form f(12) of the general function f(12) [= h(12) or c(12)] is given by
f(12)  ~ fmnl(k)( f )mnl :tO0 (Cd1032~k) (6.216) mnl
where fm~t(k)  47ri t fo~Jt(kr)fmnZ(r)r2dr (6.217)
where jr(x) is the lth order spherical Bessel function. In Equation (6.216), cok is the orientation of the vector k in the Fourier transform
37(12)  f exp(if~. 7~2)f(12)d~'12. (6.218)
Expansions (6.214), (6.215) and (6.216) are referred to as laboratoryframe (LF) expansions by Gray and Gubbins (184); the gframe (RF) (also known as intermolecularframe) expansion of f(12) (f = h, c) corresponds to the LF expan sion when the zaxis of the laboratory frame is chosen to lie along the vector ~2 joining the centers of mass of the two molecules, thus yielding
f ( 1 2 )  E()x f~(r)[(2m + 1)(2n + 1)]I/2D~(wl)Donx(w2) m n
x
(6.219)
where m+n
Z t=lmnl
m
x n
)(. l fmnl o) (r) (6.220)
244
This equation is derived by noting that, for this choice of axes, Or  0 so that Dl0:~ (wr)  3t0 where 3ij is the Kronecker delta function. From the orthogonality properties of the 3  j symbols, it follows that (222224) an "inverse" relation to Equation (6.220) exists and can be written in the form
inf(m,n) front(r)(21+ 1) E (_)x ( m n l ) mn
x=_inf(m,n) ~ X 0 fx (r) (6.221)
We can similarly write down a k~frame (KF) expansion for f(12) by choosing the zaxis of the coordinate system to lie along the k:
f(12)   E()xF~n(k)[(2m + 1 ) ( 2 n + 1)]'/2D~(w1)Don_x(W2) mre X
(6.222)
where
m+n
Z l=lmnl X X 0
(6.223)
m + n ( ) f o C X ~ = ( _ ) x 4 ~ ~ m n l it jt(kr)fmnt(r)r2dr l=lmnl X  X 0
(6.224)
and
inf(m,n) Z o (k) (6.225)
Xinf(m,n) ~  ~ 0 
Note that w1 and w2 in Equations (6.214), (6.220) and (6.222) are all different: in Equation (6.214), Wx and w2 are measured with respect to an arbitrary laboratory frame; in Equation (6.220), they are measured with respect to the ~'frame and in Equation (6.222) with respect to the kframe.
The utility of the KF expansion results from the fact that the molecular OZ equation can be written in a particularly simple form in terms of the KF harmonic expansion coefficients as (184,222224)
mn mn n 1 H x (k)  C x (k) + p ~ X[[mnl ( k ) C ' ~ (k) nl
(6.226)
245
This equation bears a striking resemblance to the OZ equation for a mixture of spherically symmetric molecules; this similarity is exploited in Section 6.4.2.1 to solve the MSA analytically for dipolar hard spheres.
The final result which is required is the equation relating fmnl(r) to Uxn(r), the realspace function whose Fourier transform is F~xn(k ). From Blum and Tor ruella (222), we have
fm"(r) = ()z fn~Z(r). (6.227)
The lth order spherical Bessel function can be related to Pl (x), the Legendre poly nomial of order l, by
1 fo I [eikrt )l ikrt]d t jt(kr)  ~ P~(t) + (  e (6.228)
By substituting this relation into Equation (6.224), one obtains
/7 Hxn(k)  [eikrJxn(r ) + eikrJxm(r)]dr (6.229)
where
m+n ) fr (< ) jxn(r) _ (_)x2r c ~ ( m n 1 ~ r £mr~t (6.230) l=lm,~l X X 0 Pt (rl)rldrl.
Similarly, for the direct correlation function, one finds
/7 mn _ _ mn S X (r)]dr (6.231)
where m+n
Sxn ( r )_ (_ )x2r r ~ ( m n lIrahi X X
1 oo T O)fr P l ( < ) ~mnl(rl)rldrl" (6.232)
The functions F~n(k) (F = C, H) satisfy the symmetry (224)
(6.233)
The "inverse" of Equations (6.229) through (6.232) is given by (224)
fmnl(r )  2 l + 1 inf(m'n) ( ) f0 ~
X=inf(m,n) X X 0 (  ~ (rl)
(6.234) 1 1 p , , _
X { !(51(rr re) 7~P/(1)5(r r l )+ r3 l ( ~ ) O(r rl)} drl (6.235)
where (f, 9 r)  (h, J) or (c, S) and 0(x) is the Heaviside function.
246
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Sci.) 2, 20 (1993). 256. R.J. Baxter, J. Chem. Phys. 47, 4855 (1967). 257. D.J. Tildesley and W. B. Streett, Mol. Phys. 41, 85 (1980). 258. W.R. Smith and I. Nezbeda, J. Chem. Phys. 81, 3694 (1984). 259. B. Smit, Simulation of Phase Coexistence: From Atoms to Surfactants,
PhD thesis, University of Utrecht (1990). 260. A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton
University Press (1957).
Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 255
7 Q U A S I L A T T I C E E Q U A T I O N S O F S T A T E F O R M O L E C U L A R
FLUIDS
N.A. Smirnova and A.I.Victorov
Department of Chemistry St.Petersburg State University St.Petersburg, 198904 Russia
7.1 Introduction 7.2 Basic Features of Lattice Theories; Structure of a Quasilattice EOS 7.3 Influence of Molecular Size and Shape 7.4 ContactSite Models for Fluids with Strong Directional Attractive Interactions 7.5 Results of Thermodynamic Modeling by the Quasilattice EOS 7.6 Conclusions References
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7.1 INTRODUCTION
Lattice theories of fluids can be traced back to the nineteenthirties (14) when Eyring, Frenkel, Lennard Jones, and Devonshire showed that lattice models can serve to obtain an equation of state (EOS) and to describe the thermodynamic behavior of a dense fluid composed of similar sized spherical molecules. For a long time, however, the lattice models belonged rather to the realm of physics, and their application for practical purposes had not been started in chemical engineering. The wide practical use of quasilattice models was first due to the works of Flory (5,6), Guggenheim (7), Barker (8), and others on the description of the excess thermodynamic functions of liquid solutions. These models allowed a description of the entropy effects in their relation to the differences in molecular size. The approaches also accounted for the contribution of intermolecular attractions to the excess thermodynamic properties of a solution. However these approaches were incapable of describing the volume effects, nor did they lead to an EOS, because a model of a rigid, incompressible lattice had been utilized. Thus, until the nineteenfillies the existing quasilattice EOS's were restricted to systems of nearly spherical molecules and found little practical application. The interest in the quasilattice EOS's was animated after the work of Prigogine (9), in which the ideas of the lattice approach were combined with the correspondingstates theory, and an EOS was constructed for solutions containing chain molecules. Further development of this approach is due to Flory et al. (10,11), and to some more recent studies (1215). The fragments of the Prigogine and Flory theory are present in many contemporary EOS's, e.g., PHCT(16), APACT (17) and some other (18). However the Prigogine approach is originally designed to model the liquid, not the gas and, hence, does not allow a description of liquidgas equilibria.
The practical application of quasilattice theories for modeling of molecular fluids, both in the liquid and in the gas states, was started by the works of Sanchez and Lacombe (19,20), who first derived a lattice gas EOS for a fluid composed of molecules of arbitrary size on the basis of Flory's statistics for mixtures of rmers (5). Among the numerous modifications of the EOS proposed later, the most general versions are related to the implementation of the molecular contactsites approach (2124) and the quasichemical approximation (21,2325). These made it possible to describe the orientation effects and the association and to formulate a group contribution version of a latticegas EOS. At present the applicability range of latticegas EOS's is rather wide. Certainly they should be considered as semiempirical, the lack of rigor in their theoretical derivations being to a large extent compensated by the introduction of various adjustable parameters to perform calculations. The merit of lattice models is the simple way they provide to reflect the molecular characteristics of the system, in particular, strong interactions such as repulsion or hydrogen bonding.
A common weak point of all these models (leading, e.g., to their inability to describe the compressibility factor of some model fluids correctly) is embedded in their discretespace provenance. Recently, new continuousspace versions of the latticegas EOS's were developed (26,27) which give a quantitative description of computer experiments on long chain (and short chain) molecular fluids. This work has inspired the appearance of a new family of latticegas EOS's, mainly for hardchain fluids (2831).
In what follows we shall discuss the basic features of the holelattice approach to an EOS. A variety of latticegas EOS's of practical importance will be surveyed. The influence of the molecular size and shape, the orientation effects and association between molecules will be considered in various approximations. For associating fluids a comparison with "chemical
257
equilibrium theories" (32) and with the Statistical Associated Fluid Theory (SAFT) (33) is drawn. The results on fluidphase equilibria modeling with the aid of different EOS's are given. The applicability of the EOS's and the trends and perspectives in their further development are discussed. In the present paper we will leave apart the consideration of polymer systems. A separate chapter is devoted to this large and very important field. Critical phenomena are also beyond the scope of our discussion, though many useful results have been obtained in this area with the aid of the lattice approach.
7.2 BASIC FEATURES OF LATTICE THEORIES; STRUCTURE OF A QUASILATTICE EOS
Lattice models attribute a quasicrystalline structure to fluids. This implies that the system volume is divided into cells, whose size is of the order of intermolecular distances in a liquid. In the socalled cell theories every cell is occupied by a molecule, whereas in hole (or lattice gas) theories some vacant cells are allowed. Consequently, the latticegas theories are capable of modeling both liquid and gas states of a fluid, and the cell theories apply exclusively to the liquid state.
The quasilattice approach is used to evaluate the interaction energy and the configurational partition function. It is not concerned with the thermal motions of molecules and, hence, does not impose any restrictions on their movements throughout the system volume. The evaluation of the configurational partition function reduces to the estimation of the interaction energy for different ways of distributing the molecules over a perfectly ordered array of the lattice cells, and, to the calculation of the configurational integral of a single molecule within the volume of a cell, which is called free volume. Since the number of ways to distribute the molecules over the discrete array of cells becomes countable, it seems to be much easier to estimate the probability of any particular configuration of cells, and, hence, to evaluate the partition function, rather than taking the configurational integral over the continuous set of spatial coordinates. However reasonable these arguments might seem, our ability to get rigorous results, even for this system with countable states, still remains essentially limited to what is called the twodimensional Ising model (34), and we shall not consider these theoretically refined approaches. The shortcomings of a quasicrystalline approach in attributing too much order to a fluid structure are obvious. The quasilattice models consider only a part of the continuous configurational space of a real fluid and their accuracy is determined by how completely they represent the configurations giving significant contributions to the statistical averages. Nevertheless, the approach served to establish the approximate relations between the characteristics of intermolecular interactions and the thermodynamic behavior of fluids and has proved impressively fruitful in the derivation of various simplified equations of state for complex molecular fluids.
The main goal of the cell theories of liquids is to find the free volume, and it is evaluated by writing a meanfield interaction energy between a particle and the environment. The dependence of the free volume on temperature and density determines the equation of state of a fluid. In the case of chain molecules their segments are accommodated in the cells, each having an effective number of external degrees of freedom, which is less than three to account for the connectivity of the molecule. There has been substantial progress in the development of cell lattice EOS's, in particular, new versions of them have been created, which can be applied to associated solutions (13,15). Nevertheless, the leadership in the modeling of thermodynamic
258
properties and phase equilibria for practical applications belongs to latticegas EOS's, and we turn to discuss these equations now.
In the classical versions of latticegas models, all the molecules are located at the lattice sites (there is no freevolume term, the models are not concerned with the displacements of molecules within the cells), and thus the thermodynamic properties of a fluid are determined solely by the lattice statistics. To bring into consideration the density of the system, a certain fraction of lattice sites is allowed to be unoccupied (occupied by holes), which leads to the dependence of thermodynamic properties upon the volume, and, consequently, to the EOS. If we write the system volume, V, as
n
V = vh(N 0 + Z r i N i ) (7.1) i=1
where n is the number of components, N O is the number of holes, N i is the number of
molecules of type i occupying r i lattice sites each, and assume that the volume per lattice site, • v h , is constant, then, for the chemical potential of the component i, we get:
/ y / )   IAi  =~.l i  ril.t o ( i , j = 1,2,...,n) (7.2) T , V , N j¢ i
with
la k = ( k , l = 0,1,2,...,n) (7.3) T, N t , k
Here index 0 refers to holes and A is the Helmholtz energy of the system. Equations (7.2) and (7.3) relate the thermodynamic functions of an ncomponent system to the properties of a lattice fluid having n+l components, one of which represents the vacancies. The system pressure, P, is connected to the quantity/Y0 for holes:
~t o =  V h P (7.4)
and the condition of equilibrium between phases (~) and (13) can be expressed as:
fi!a) = fi}/3) (i=0,1,.. . ,n) (7.5)
which includes the condition of mechanical equilibrium of the lattice fluid (p (a ) = p(I3)) .
Thus, for any rigid lattice model (i.e., one without vacant lattice sites), a corresponding lattice gas EOS can be immediately generated by means of Equations (7.3) and (7.4).
The most typical approximation in the existing versions of the latticegas theories is that intermolecular repulsion and attraction are treated independently. The repulsion is connected to the molecular size (represented by the number of lattice sites, r i, needed to accommodate a
259
molecule without overlap with the others), and is reflected in the lattice combinatorics. The attractive contribution is embedded in the residual terms (normally only the nearestneighbor interactions are taken into account). This leads to a configurational partition function Q which can be written as a product of two terms:
Q = QcombQres (7.6)
and to the EOS in the form:
P = Prep + eres (7.7)
where
Prep:k(6° In Qc°mb t gV T, Ni
/Ores=kT(O In Qres~ (7.8) ' OV JT, Ni
Here Prep is the repulsion part of the pressure; this basic structure of the formulae corresponds to what is called the generalized van der Waals model in the nonlattice theories of fluids.
Further approximations are introduced to perform a calculation of the combinatorial and residual terms, in which the molecular nature of a fluid should be specifically reflected. In the first place, due consideration of the molecular sizes and shapes affects the expressions for Qcomb, while the existence of strong preferential attractions (specific interactions, like hydrogen bonding) must be accounted for in Qres. For a system of equalsized molecules occupying one lattice site each, the derivation of Qcomb becomes trivial and corresponds to the entropy of an ideal solution. Even with this simplest combinatorial (repulsion) term in the EOS, and assuming complete randomness in the distribution of the nearest neighbors for the calculation of Qres (i .e. , no correlation between interacting pairs, the socalled BraggWilliams or molecular field approximation), it has proved possible to reproduce basic features of liquid gas phase behavior for a binary mixture, including liquidgas and liquidliquid equilibria, positive and negative azeotropy, and gasgas equilibria of the first or second kind (35). In this simplest version of the hole model the type of phase behavior is completely determined by the relative values of nearestneighbor interaction energies between like and unlike molecules (Ui j ) , and the various phase diagrams have been classified in terms of a conventional lattice theory parameter, the interchange energy: co o. = U U (Uii .[Ujj)/2. If a neighboring site is
vacant (occupied by a hole), no interaction is assumed, so for the holemolecule interchange energy: COoi =  U i i / 2 . Too much symmetry introduced into the model by the equality of the
molecular sizes and by the requirement of complete randomness prevents it from being used for a quantitative description of real mixtures. The advantages of the quasilattice approach were much more recognized in connection with the mixtures of molecules differing strongly in size, after the pioneering work by Flory (36).
7.3 INFLUENCE OF MOLECULAR SIZE AND SHAPE
The central quantity in the Flory model for a mixture of flexible and nonattracting rmers with monomers (36) is the probability, that an rmer molecule may be inserted without overlap
260
into a lattice occupied partly (and randomly) by other rmers (26). This insertion probability, Pr , can be related in a straightforward manner to the chemical potential (37) and to the lattice gas EOS (26):
&.v, I f 1
[1 l n P r ( f , r ) ] + j ln    P r ( f ' , r ) d f ' R T r r o
(7.9)
where f is the lattice occupation fraction and v*= N A V h is the molar value of volume per
lattice site and N A is Avogadro's number. The Flory approximation (36) treats the occupied sites as if they were scattered randomly
over the lattice and disregards completely the chain molecule structure. This yields a very simple expression for the insertion probability:
P r ( f , r) = [ P r ( f ,1)] r = (1 f )r (7.10)
and corresponds to a random walk on a lattice with infinite coordination number, since it is assumed that there is always a vacant neighboring lattice site available to accommodate a consecutive segment of a rmer chain, with the probability being proportional to the average fraction of holes. The extension to mixtures of the chains of varying chain lengths is straightforward. From Equations (7.10) and (7.9) we get the repulsion part of the EOS:
repV" (r )   I n ( 1  f )  f 1  1 (7 .11) R T
The Flory approach has been used by Sanchez and Lacombe (19,20) to obtain the first practicallysound, holelatticetheory EOS (often called the latticefluid theory, LF). LF has the repulsion part given by Equation (7.11) and the residual part obtained in the BraggWilliams approximation. This equation has become rather popular since it has been successfully applied for the description of phase equilibria in pure and mixed fluids containing light and heavy compounds (20,38), and polymers (20,39). It can describe liquidvapor equilibrium and liquid liquid miscibility gaps with a lower or an upper critical consolute point, or with both. The main disadvantage of this EOS is its inability to describe systems with strongly polar compounds, which is obviously due to the fact that its residual contribution has been obtained under the assumption of complete randomness. Another EOS, which also has the Flory repulsion part, but takes into account the nonrandomness by means of an empirical correction parameter, is the KK (Kleintjens and Koningsveld) model (40). The KK EOS is rather flexible and is presently among the best; however, the correction brings too much empirism into the EOS. Less empirical assumptions to account for strong attractive interactions will be considered in the next section.
The Flory approximation to evaluate the combinatorics seems itself to be a very rough one. A certain improvement over this is attained in the FloryHuggins theory (41), which corresponds to a firstorder Markovian process of placing chain segments onto a lattice with a finite coordination number. The insertion probabilities are estimated then by:
P r ( f , r) = (1 f )[P(k I k  1)] r1
261
(7.12)
where P(k I k  1) is the probability, that a randomly chosen lattice site, k, is vacant, given that
the adjacent site, k1, is also vacant. This implies a difference in the joint probabilities if a site, k1, is occupied by an rmer segment, or is vacant. The probabilities are estimated taking into account the difference between the available lattice sites next to the terminal and to the internal segments of an rmer chain. Essentially the same lattice statistics have been considered by Chang, Miller, and Guggenheim (7). The repulsion part of the EOS takes the following form:
PrepV • z I qr 1  I n ( l  f ) + l n l + f (7.13) RT 2 r
Here z is the lattice coordination number, and q is an rmer surface area parameter such that zq is the number of lattice sites neighboring an rmer molecule. The generalization of the approach to the multicomponent case of flexible rmers of arbitrary chain lengths has been made by Guggenheim (7). Equation (7.13), unlike Equation (7.11), incorporates the parameter q, which depends on the structure of a chain molecule. However, the MillerGuggenheim Huggins statistics does not apply when some of the rmers contain closed rings. An empirical extension of this approach for more general molecular geometry, including both chainlike and ringlike molecules, has been developed by Staverman (42). His expressions contain a shape parameter, the socalled bulkiness factor 1:
I~D Z ~(q  r)  ( r  1 ) (7.14)
The Staverman combinatorics leads to an EOS of the form:
repV* Z I q r ] '  I n ( l  f ) + In l + f + f (7.15) RT 2 r r
which reduces back to Equation (7.13), when there is no ringlike structures in the system ( l = 0). Equations (7.13) and (7.15) have been incorporated into some practically sound EOS's (2325,4346) to account for the molecular shape.
The basic deficiency of the EOS's considered so far is that all of them give logarithmic dependency of the athermal (repulsive) compressibility factor on the occupation fraction, this dependency being much too weak in comparison with the results of computer simulation for nonattracting flexible rmer chains (26,27). The lattice gas EOS's dramatically underestimate the compressibility factor for a hardsphere fluid (r=l) and also give too small an effect of chain length on the EOS (Figure 7.1). This major drawback of the latticegas EOS's has been overcome (26,27) by the invention of a continuousspace generalization of the Flory method. The molecular chain structure is taken into account by exclusion volumes, saying that because of the overlap of the rmer segments somewhat less space is needed to accommodate a segment of an rmer, than a monomer, in a fluid. In this approach, called the Generalized Flory Dimer EOS (GFD) (27), the insertion probability for an rmer is expressed as:
262
3 0 i I i t
O . (1)
20
10
0.0 0.0
r=4 ° r=2 /
 [] / / / 
/ r = l
[ ]
~ I"/*/
i I i I t
0.2 0.4 O.B f
Figure 7.1 Compressibility factor Zre p v e r s u s the reduced density (occupation fraction) for hardchain fluids of various chain length, r. Computer simulation results: asterisks, triangles (124) and open squares (26). The LCT EOS (52) [Equation (7.18)]: full squares. The lattice gas EOS [Equation (7.13)]: dotted lines. The SAFT EOS: dashed lines. The 'quasichemical dimerization' approach [Equations (7.24), (7.25)]: solid line.
Pr(f,r)= p~(f)Yr p2 (f)Yr +1 (7.16)
where P1 and P2 are the probabilities to insert a monomer into a fluid of monomers and a dimer into a fluid of dimers, Yr is the quantity related to the exclusion rmer volume and determined by purely geometric arguments (27). The EOS written in terms of the compressibility factors has the following form:
ZGFD(f, r) = (Yr + 1)Z(f,2) YrZ(f,1) (7.17)
where Z ( f , 2 ) , Z(f,1) are the compressibility factors for fluids composed of dimers and of
monomers, respectively, for which the TildesleyStreett EOS and the CamahanStarling EOS are used in this continuous version of the theory (27). General recursion formulae have also been derived (27), which relate the EOS for an rmer to the EOS's for (rl) and (r2)mers. The GFD EOS gives excellent agreement with the computer simulation results for rmers of various chain lengths, up to very long ones (r=201) (47). Definitely, these continuous generalized formulations opened a new period in the development of the quasilattice equations, and much of the effort has been directed to creating and testing new versions of them (2831). However, there is some evidence that the approach does not provide correct values for the insertion probabilities, and that the success of the EOS is merely due to a cancellation of errors (48).
263
Not only the chain length and the shape of the molecules, but also their rigidity affect the fluid structure and thermodynamic behavior. It is well known, that anisotropic hardcore molecules (such as rods, disks, etc.) can under certain conditions form nematic liquid crystals, that is the fluids having longrange orientational ordering. The Flory method has been utilized by Di Marzio (49) to describe the statistics of an anisotropic fluid formed by rigid rmer rods. A useful extension of this method was elaborated recently for a multicomponent lattice gas of hard rectangular parallelepipeds (50). The approach proved to be successful in explaining the effects of molecular shape on phase diagrams for mesogen  nonmesogen systems, and in studying the influence of molecular biaxiality on phase equilibria for binary liquid crystal mixtures (51 ).
Recently a new method for considering lattice statistics has been proposed which gives formally exact results (5254). The characteristic feature of this socalled Lattice Cluster Theory (LCT) is that, unlike other meanfield methods considered so far, it describes the multiparticle (twoparticle, threeparticle, fourparticle, and so on) correlations. The LCT provides a systematic method for evaluating corrections for many body correlations to the zeroth order uncorrelated meanfield approximation. The molecular shape and rigidity are also taken into account. The theory reduces, under certain conditions, to the Flory statistics for fully flexible rmers, and to the Di Marzio formulae for the case of rigid rods (52). It has been applied to describe a system of hard rigid rectangular mesogens, which may exhibit isotropic liquid and nematic, and also discotic nematic liquid crystalline phases (54). The LCT has been used to model polymer systems at low pressures (55) and to describe liquidliquid equilibrium (56). In both cases good agreement with experimental data has been achieved.
Because the LCT gives formally correct results, it can serve to illustrate the principle limitations of lattice gas EOS's. From the LCT corrections to the Flory meanfield Helmholtz energy (52) one can obtain the pressure of a lattice fluid as a power series in the occupation fraction and the space dimensionality, d :
 l n ( 1  f )  f   R r ~
4
 3 7  4a '~
_ ( ~ 3 2 (2 r 1 )2 (d 1 ) f4+o( f sd_3 ) 24d 3
(7.18)
At the same time, expanding the second logarithmic term in the meanfield EOS, Equation (7.13), we get:
R T  I n ( l  f )  f  f  ~2(@/3 4f~3 + O(f 4) (7.19)
z 3z 2
Comparing Equations (7.19) and (7.18) for a threedimensional (d=3) cubic lattice with z=6 we find that they are identical up to the terms includingf . This implies: (1) that the Guggenheim MillerHuggins EOS is a real improvement over the Flory EOS, Equation (7.11), and (2) even the exact results for a lattice fluid deviate significantly (Figure 7.1) from the computer simulation data for continuous fluids of nonattracting rmers (26). Thus, the main limitation in describing the repulsive part of an EOS is caused not by poor lattice statistics, but rather by the introduction of the lattice itself. Nevertheless, as we will see later, the quasilattice EOS can be
264
successfully applied for phase equilibria modeling, apparently because of a cancellation of errors in the repulsive and the attractive parts.
fortunate
7.4 CONTACTSITE MODELS FOR FLUIDS WITH STRONG DIRECTIONAL ATTRACTIVE INTERACTIONS
Today there are three basic classes of EOS's for associated fluids: the chemicaltheory EOS (combination of the law of mass action with an EOS selected to model physical interactions between the molecular species present in the fluid (17,57)), the SAFT (33,58,59) (an approach based on the perturbation theory expansion for the Helmholtz energy of an associated fluid (60), see Chapter 12 of this volume), and the lattice quasichemical EOS (21,2325,44,45). The two latter approaches utilize a model according to which a molecule has different contact sites (functional groups). Some particular kinds of contact sites can participate in strong shortrange attractions leading to the formation of aggregates, and some cannot. In this way the preferential molecular orientations due to association are reflected. The quasichemical approximation is one of the theories most commonly used to account for nonrandomness in lattice models.
The quasichemical approach was originally proposed by Guggenheim (7) and generalized later by Barker (61) and Kehiaian et al. (62) for the case of molecules composed of an arbitrary number of different functional groups. The expression for the residual contribution to the configurational partition function of a lattice fluid, Ores, reflects the effects of nonrandomness and can be written as:
Ores = (zA, s)! , (zA, t / 2)!
1
S s , t
(7.20)
where U c is the configurational energy of the fluid, zA,t is the number of pairs between groups of kind s and of kind t in a totally disordered random state, and zA, t is the most probable number of pairs for the fluid of interest, this number corresponding to the maximum term of the configurational partition function, Equation (7.6). The equations to find these latter numbers have the following form:
12ostl = 4 exp  (7.21) (zAsszAtt ) R T
This resembles the equation of the law of mass action for a 'chemical reaction' involving the formation of st contacts from ss and tt contacts, wherefrom the term 'quasichemical' originates. Typically, the condition of conservation of the numbers of contacting lattice sites applies for various lattice models, which allows one to rewrite Equation (7.21) in a more convenient form:
265
X~ ~ a,Xt r/s, = 1 (7.22) t
Here (~t is the surfacearea fraction of groups of type t in a fluid, ~st exp(cos,/RT), and X i is
the target variable connected to the numbers of pairs by:
Ast = AasatXtX~ fist (s :/: t) As, = A (a,X~ )2 /2 (7.23)
where A is the total surface area of all the species in the fluid. The quasichemical approximation has been carefully analyzed (6365) and criticized for
giving poor pair probabilities for energetically strongly unfavorable contacts (large positive COst); however, it gives reasonable results in the case of predominantly associating pairs, as we will see below, and still remains one of the most popular and useful tools to account for specific interactions. A limiting case of a very strong association has been considered by applying the quasichemical approximation to model covalent bonds between segments of an r mer by an infinitely intensive attraction between corresponding contact sites of monomers (66). Thus a monomer + rmer binary mixture has been treated as a fluid of (r+l) monomeric particles with associating contact sites. This approach leads to the same expressions for the activity coefficients as for the original monomer + rmer binary mixture (66), provided that the Guggenheim approach to the combinatorics is used for the athermal terms. The same method can serve to obtain the corrections to the EOS resulting from the connectivity of the molecules. If we consider, for example, a lattice fluid of monomeric segments associating to form dimer molecules, but otherwise athermal, then the system compressibility factor, Z, can be expressed a s l
Z = 2Zseg + Zbond (7.24)
where Zseg is the compressibility factor of the fluid of nonattracting monomers, and Zbond is
the correction for chemical bonding, which can be expressed via Qres, the noncombinatorial entropy term in the quasichemical approximation:
L ,z i,z } 1 Zbond " 2 f In( 1  f ) + f In f +     f In f  (7.25) z z z z z
Using Equation (7.25) the compressibility factor has been calculated for the system of dimers and compared with computer simulations, (Figure 7.1). The CamahanStarling expression has been used for Zseg , because, as we have already pointed out, the latticegas expressions would
dramatically underestimate the compressibility factor for the fluid of monomers. As can be seen from Figure 7.1, the quasichemical approximation gives a reasonable correction for the entropy loss due to chemical bonding, particularly at high densities. Another argument strongly supporting the usefulness of the quasichemical approach is that it gives, as we will show later, quite reasonable predictions of structural properties of associated fluids.
266
The first attempt to incorporate the quasichemical routine into a latticegas EOS of a chainmolecule fluid was made by Panayiotou and Vera (21); somewhat earlier the quasichemical approximation was employed in a celllattice EOS taking into account effects of molecular sizes (13), and in a latticegas theory of simple fluids composed of spherical molecules (71). Later, quite a few modifications of latticegas quasichemical EOS's have been proposed for practical applications (2325,44,45). To the best of our knowledge one of the most general versions of a quasichemical latticegas EOS has been developed by Victorov and Smirnova (23,72). This equation, denoted hereafter as the HM (Hole Model), combines the GuggenheimStaverman repulsion term, Equation (7.15), with a quasichemical residual contribution, which has a simple form:
eres V * _
R T lnX o (7.26)
Here X o is the solution of the set of quasichemical equations (7.22) for holes. Under various conditions this EOS reduces to the wellknown lattice gas EOS's considered above, e.g., for a completely random rmer fluid (corresponding formally to the case of infinite coordination number) the HM transforms into the LF EOS by Sanchez and Lacombe (19). If the interaction parameters COst for the groups of specific kinds are kept constant, regardless of the types of molecules these functional groups belong to, then the groupcontribution EOS follows. This possesses enhanced predictive abilities. The HM has been applied to a great variety of fluids and fluid mixtures (23,6769,7275). In the next section we shall discuss the application of the various quasilattice and related EOS's in more detail.
The solution of the nonlinear set of quasichemical equations (7.22) cannot be performed analytically, and hence the quasichemical EOS's are doomed to be implicit in their general form. However, a special technique has been proposed (76) which expresses the solution of the equations as a series expansion, thus leading to explicit EOS's (25,44,45,77). Such an expanded version of the HM has been developed (44,45), which is well suited for practical purposes. The first explicit version of a quasichemical latticegas EOS is apparently due to Kumar et aL (25). This equation is written in a rather convenient way and it has been extensively used for practical applications (25,78,79). The original EOS of Kumar et al. mainly differs from other quasichemical equations proposed later in that it assumes the complete randomness of the distribution of the vacant lattice sites.
Besides the quasichemical versions of latticegas EOS's there is the 'chemical theory' EOS (80), which combines the SanchezLacombe LF EOS (19) with association equilibrium constants. This theory has been used for modeling of phase equilibria in alkanealcohol mixtures and gave good results (81), though they are somewhat inferior to those obtained by the HM (67).
An intermediate between a purely 'chemical' and a purely 'quasichemical' (i.e., related to the Guggenheim original approach (7) developed specifically for lattice models) version of an EOS has been proposed by Panayiotou and Sanchez (22) (Hydrogen Bonding Lattice Fluid, LFHB EOS). As in the chemical theories the thermodynamic functions are separated into physical and chemical parts. The original version of LF (22), or SAFT (82), is employed to describe the physical interaction between the molecules, while for the chemical part statistical mechanical arguments are used, rather than phenomenological empirical values of association constants. This statisticalmechanical approach (originally due to Veytsman (83)) considers the probabilities of hydrogenbond formation between acceptors and donors as if they applied to
267
independent events. A quasichemicaltype expression is obtained for the number of hydrogen bonds in a fluid (22) [a set of quadratic equations, similar to Equations (7.22) for the contact pairs]. The theory gives good results for associated fluids at supercritical and subcritical conditions (22,82,84,85); however, it is difficult to say whether it presents any improvement over quasichemical models, since no direct comparison has been made. However, the weak point of the LFHB and similar EOS (85) is the arbitrariness in the separation of the 'physical' and 'chemical' parts of the interactions, a flaw present intrinsically in any 'chemicaltheory' EOS.
One of the possible routs to account for the effects of nonrandomness is provided by the local composition concept, which found a most extensive application in constructing GE models of liquid solutions. This concept has recently been employed in the frame of latticegas EOS's (46,86,87). The relation of the concept with the quasichemical approximation to nonrandomness is well established in the literature.
The analytic relation between the 'chemical theories' and the quasichemical lattice model has been discussed in its general form (88). Early consideration was carried out for two particular types of association (89): (I) the formation of mixed AB associates, and (11) the consecutive association of monomers A to form imers A i (i=2,3,...). Let us denote the specific contact sites participating in strong interactions by O and H. If the consecutive association (type II) takes place in an A+B binary mixture, both specific contact sites belong to the A molecule. To model the association of type I assume that A molecules have one Ocontact site each, and B molecules have one Hcontact site each. The numbers of different molecular species can be expressed via the law of mass action and also can be obtained from the number of contact pairs (the latter being estimated in the quasichemical approximation). For an association of type I, the number of associates is simply equal to the number of contact pairs NOH. For type H, the most probable set of the numbers of different molecular species A i , NAi, is determined by the maximum number of ways this set can be generated on a lattice at given values of NOH, NA, and NB. Using the Guggenheim lattice statistics for the mixture of chain molecules A i (and the Lagrange multipliers method), one finds the following distribution for i mers:
NA i = NAfl2(I_ fl)i1 (i = 1,2,...) (7.27)
where/3 = (N A  NOH ) / N A .
The nonideality of the mixture was taken into account in the Guggenheim athermal and nonathermal approximations using the law of mass action (via the chemical potentials of molecular species). It was found that the quasichemical approach and the law of mass action give similar monomer and associates concentrations, the association constant, K, and the interchange energy for the specific interaction being related approximately as:
K = exp( c0 OH/RT) (7.28)
with
268
K . _
X~ (xA , QA ' + xB, QB ' + x ~ Q ~ )
XA 1XB l oo
XA, (xBIQBI + ~ XA, QAi ) i=1
XA 1 XAi. !
(model I)
(model II)
(7.29)
Here x~ is the mole fraction of molecular species a, and Q~ is the number of contact sites for
molecular species c~ (a=A1, B 1, etc.) The difference between the two approaches is due to the fact that in the quasichemical
approach we deal with the numbers of pairs of different types and in the theories of association equilibria with the numbers of molecular species (monomers, dimers, etc.). Although, as has been shown, the approaches give similar results for the same association model, in general they differ in their capability to describe different types of association. The most serious restriction of contactsite models is their inability to describe the dependence of interaction parameters on the structure of associates (thus, the energy of the OH bond is supposed to be the same for associates AB and AB 2, for A 2 and A3, A4, etc.). Some types of association (e.g. formation of only trimers or only tetramers) cannot be described by contact site models, yet theories of association equilibria do not have any difficulties in this respect. But there are situations when the contactsite models work well, whereas the approach using the law of mass action can hardly be suitable (e.g., systems where a network of strong bonds is formed).
An alternative approach for associated fluids, the SAFT (33,58,59), is today the most theoretically rigorous and it also became the most popular one. It employs the contactsite model and in this respect is similar to the quasichemical approaches considered above. In its essence, however, the SAFT is a perturbation method (33,60), giving the correction for the Helmholtz energy of a fluid of nonbonded segments (monomers) due to strong shortrange attractions between specific sites (in the limiting case, of infinitely strong attractions due to the chemical bonding between segments). We merely give a very short outline of the theory, because Chapter 12 of this book discusses it in every detail. The difference, Abona, between the Helmholtz energy of the associating mixture with multiple associating sites and the Helmholtz energy of the reference fluid of nonassociating components is given by (33):
AR°n; "~i XiI~A~illnSiAS~2 I"t~l (7.30)
where x/ is the mole fraction of the component i, n i is the number of associating sites on the
molecule of component i, the second sum is over all such sites on the molecule. Xf is the
fraction of molecules of component i not bonded at site A. These quantities can be found as the solution of the nonlinear equations:
xA II+ i L j B ~ j tj
269
(7.31)
Here p is the total number density and A AB is defined by:
AB f ref A0 = Jgu (12)f7 AB (12)d(12) (7.32)
f/jAB =exp  k T )  I is the Mayer ffunction, g~ef is the referencefluid paircorrelation
function, and the integration is performed over all the orientations and separations of molecules 1 and 2. Thus knowledge of the pair distribution function of the reference fluid is sufficient to calculate the properties of the associated mixture. Often the distribution function for a hard sphere fluid is used together with a simplified expression for A~. (90):
AB .__ 4~gref (o')gAB exp  1 A 0. (7.33)
where VAB is the association volume parameter (the integral that measures the volume
available for bonding to sites A and B), gref(o.) is the contact value of the pair distribution
function, CAB is the association potential (the squarewell depth characterizing the specific AB
attraction). The EOS is obtained from the above equations by a standard routine. For instance, in the case of hard spheres with infinitely strong attractions, which make them stick into hard flexible chains, one can write:
Zchai n = ZHS + Zbond (7 .34 )
where ZHS is the compressibility factor of a reference fluid of free hardsphere segments, and
Zbond is the contribution due to bonding, which corresponds to Equation (7.30) and infinitely
large CAB. From this the wellknown SAFT EOS for athermal flexible rmers (33,90), which is in excellent agreement with MonteCarlo simulation results, follows. In a more general case of associating chain molecules interacting as well via dispersive attraction forces the compressibility factor consists of all these contributions:
Ztota 1 = Zchai n + Zasso c + Zdisp (7.35)
where Zasso c is, again, calculated with the aid of Equation (7.30), and the contribution of the dispersive interactions is accounted for by standard perturbation terms (91). The SAFT EOS has been applied to a great variety of pure fluids (58) and fluid mixtures (59,92,93). The generalization of this equation for the case of chains built by spheres of dissimilar sizes is
270
available (94). A special recurrent procedure to correct the EOS for long chains and to modify it for polymers has been proposed (95). Definitely, in the last few years the SAFT has become the most intensively used and rapidly developing method which is taking over quasilattice EOS's in the field of thermodynamic modeling of complex fluids for practical purposes. However, the SAFT is also based on certain approximations, some of them being similar to those of the quasichemical contactsite quasilattice approach. One of the most important simplifications of both methods is that the particular location of specific sites within a molecule is not reflected: thus, isomers can not be naturally distinguished. Nevertheless the effect of the location of contact sites can be very important. For instance, a two dimensional lattice system of squares, each having a couple of specific (attracting) edges, shows different types of orientational ordering, depending on whether the specific edges are adjacent or opposite (96). None of the existing molecular models, however, takes such an effect into account.
The relation between all three approaches for associating systems: the 'chemical theory' EOS's, the SAFT, and the quasichemical lattice models has recently been discussed (88). The analysis was performed using the Flory equalreactivity principle (the assumption that the probability of bond formation between any two sites is independent on the actual shape and size of the particular clusters on which these sites lie). It was shown that the equations describing association within all three approaches are quite similar. The same conclusion was made earlier (32) for several particular association schemes. The calculation showed that all three approaches lead to quite similar results for association, the difference being mainly due to different physicalinteraction terms in the EOS's. The fraction of nonbonded molecules is expressed by similar formulae, and the numerical evaluation for associating pure and mixed hardsphere fluids gives similar results for all three EOS's and agrees quite well with computer simulation (32).
7.5 RESULTS OF THERMODYNAMIC MODELING BY THE QUASILATTICE EOS
The quasilattice EOS's have been extensively applied for modeling phase equilibria and calculating thermodynamic properties for molecular fluids. Liquidgas (20,23,25,44 46,73,80,81,85,86,97100,102104), gasgas (103,106), liquidliquid (20,101,103), solidliquid (25,97) equilibria and equilibria with more than two coexisting phases (69,101) have been studied. Supercritical fluids and the solubility of solids in supercritical solvents were considered as well (77,84,98100,104,105,107). The performance of the lattice gas EOS's has been compared with that of nonlattice equations, e.g., cubic EOS's (38,74,75,105), chemical theory EOS's (32,67,68) and the SAFT (32,68,69,82).
Various chemical classes of pure compounds and different types of fluid mixtures have been described by lattice gas EOS's. To the most extensive studies belong those reported in references (4446,73,78,79,97100,102,107). The EOS's served to model many systems of practical importance. Apart from systems containing polymers (20,25,86,87,98,102,108) there are mixtures of typical components for the petroleum industry (46,98,99,101,103), solutions with near critical solvents (69,98,99), and fluids containing associating compounds (46,67,73,81,82,85,97). Special attention has been paid to aqueous  organic mixtures (68,98,101).
Among the different versions of the latticegas quasichemical equations the HM (23) and the EOS (45,46,98) are, perhaps, the most extensively developed for practical applications. They have been thoroughly tested and their parameters have been determined for a variety of
271
substances. Parameters are reported for 211 pure components (44) and for approximately 40 functional groups (46,73,9799), which can be built up into even more substances representing saturated and unsaturated hydrocarbons, aromatic compounds, alkanols, ethers, acetic acid, water, carbon dioxide, hydrogen sulfide, etc. The LF model has also originally been applied for many systems (38), however, as it is not capable of handling systems with strongly polar components properly, it has been overtaken by the extended versions of quasilattice EOS's. The KK latticegas EOS gives an accurate description of various substances in both the supercritical and the subcritical region (with 56 parameters for a pure component alone). This EOS is easy to use, and it has given really impressive results for diverse fluids [systems under extremely high pressures (100), supercritical fluids (100,107), polymers(102), watercontaining systems(101,109), and mixtures with electrolytes (109)]. A systematic study has been made to obtain correlations for two parameters of this EOS with molecular volumes and surface area derived using Bondi's groupcontribution method (102). However the empirical basis for the entropic correction parameter of this EOS is obvious, which leaves some doubts concerning the KK EOS predictive abilities. This is particularly true in the case of multicomponent systems. For these the HM is perhaps the most thoroughly tested among the quasilattice EOS's. Their performance for phase equilibria modeling will be illustrated mainly by the results obtained with the aid of the HM.
Table 7.1 and Figures 7.2, 7.14 contain the results for selected pure components; Figures 7.37.13, 7.15, 7.16 give examples of calculated phase diagrams for mixtures. In Table 7.1 Ap and AP are the absolute average deviations between the calculated and experimental values for saturated liquid density and saturated vapor pressure. The components are chosen to span quite different molecular characteristics: from nonpolar low boiling substances to strongly polar ones having small or long chain molecules. Comparison with other EOS's is given. These include the SAFT, the APACT (17) [one of the most highly developed versions of the chemicaltheory EOS, which reduces for unpolar compounds to the PSCT (110)], and a latticefluid EOS, LFAS
Table 7.1 Description of selected pure components by the HM EOS, a groupcontribution quasichemical model and several other EOS's.
Substance T(K) A9, % AP, % Model*) CO 2 216298 1.7 0.3 HM (68)
216298 0.8 3.0 APACT (68) 218288 0.86 2.8 SAFT
Methane 91188 5.4 1.5 HM (75) 91180 0.81 0.62 APACT (111) 92180 0.35 1.4 SAFT
Ethane 150300 5.1 1.6 HM (75) 150305 0.62 1.4 APACT (112) 160300 1.6 1.8 SAFT
Propane 193367 2.7 1.1 HM (75) 190360 1.8 2.1 SAFT
Butane 213423 2.2 1.6 HM 213423 2.8 0.63 APACT 220420 2.6 2.3 SAFT
Pentane 253470 3.1 2.4 HM
272
253470 3.3 0.60 APACT 200430 1.0 4.4 LFAS 233450 3.0 1.9 SAFT
Hexane 273507 2.3 3.0 HM 273503 2.1 0.44 APACT 220430 0.42 4.1 LFAS 243433 3.5 2.3 SAFT
Heptane 273537 3.6 2.3 HM 273537 2.1 1.1 APACT 240470 0.16 4.0 LFAS 273523 3.4 1.8 SAFT
Decane 273433 0.5 4.1 HM (74) 278525 1.4 0.89 APACT (111) 313573 3.5 2.2 SAFT
Water 273646.9 2.3 1.7 HM 319643 1.1 0.96 HM
273646.9 3.5 3.7 APACT 319643 3.3 3.2 APACT 283613 3.2 1.3 SAFT
473(single phase region) 46. HM 473(single phase region) 25 APACT
Methanol 270502 1.0 1.3 HM (68) 270502 1.5 3.2 APACT (68) 273487 0.88 0.83 SAFT
Ethanol 253513 0.85 0.19 HM 253513 1.9 1.4 APACT 290450 4.2 2.5 LFAS 302483 0.86 0.83 SAFT
1Propanol 293536 1.6 2.3 HM 273513 1.2 1.2 APACT (68) 290460 2.3 1.2 LFAS 293493 1.2 0.16 SAFT
1Butanol 373562 0.56 1.5 HM 293476 1.0 0.5 APACT (68) 320500 1.6 7.0 LFAS 313493 0.23 1.0 SAFT
1Dodecanol 393513 0.9 3.8 HM (68) 393513 0.4 2.9 APACT (68)
*) Unless otherwise stated the data are taken from the following references" HM and APACT (67), SAFT (58), LFAS (81). For CO 2 and alkanes the APACT EOS reduces to PSCT (110).
(80). The results for some of these EOS's were taken from original works in which, unfortunately, the considered temperature intervals were not the same, and, accordingly, they differ somewhat for different EOS's in Table 7.1. As far as the HM is concerned, very wide temperature intervals are taken extending almost over the whole liquidstate domain up to temperatures very close to the critical. The description of the singlephase states is illustrated for water by an isotherm at 473 K between 25 and 1000 bar. As can be seen from Table 7.1, the accuracy of the description of pure components with the aid of the HM is quite satisfactory for
273
practical purposes. Both the HM EOS and the SAFT EOS are density implicit and require the sets of nonlinear (quadratic) equations [Equations (7.22) and (7.31), respectively] to be solved in the course of calculations. In this respect the explicit expanded version of the model (45, 98), the local composition latticegas EOS (46,86,87), and the APACT EOS are somewhat easier to apply. The HM and several other latticegas models (46,86,98) are groupcontribution approaches and as is the case with any groupcontribution method the specification of groups and their parameter estimation is a difficult task. Once the parameters have been estimated, however, the models can be recommended for use, especially when there are no experimental data available for a given system.
A characteristic feature of the latticegas EOS's is their lower accuracy for the volatile nonpolar substances such as propane, ethane, methane or nitrogen. In this seemingly simpler case, the lattice gas EOS usually gives worse results than for longerchain compounds (Table 7.1) and is often inferior to the PSCT, SAFT and modem versions of cubic EOS's, which can be made impressively accurate by means of an empirical technique (113). For longer chain alkanes the quality of description by the HM improves somewhat, though it is still not as good as that given by the PSCT, SAFT or by purely empirical correlations (113). The situation changes for strongly associated fluids. In this case the HM becomes quite accurate and competes well with other physically meaningful EOS's. It should be noted also, that even though the HM somewhat overshoots the critical temperature (as do many other classical EOS's), it gives a fairly nice shape to the coexistence curve for strongly polar substances (Figure 7.2) and works reasonably well at temperatures only several degrees below the critical point [e.g. for pure ethanol the deviation of the predicted critical temperature from the experimental one is about 8 K (67)]. The results become somewhat worse in the case of nonpolar components, particularly low boiling ones.
One of the shortcomings of the HM, as has been already mentioned is that it is density implicit. As one can infer from reference (44,98), the expansion of quasichemical terms, leading to an explicit EOS, basically does not spoil the quality of the description of pure components, even in the case of strongly polar ones (Figure 7.2). The local composition versions of the lattice gas EOS's (46,86,87) are also explicit. We note that these latter can provide certain improvement over HM in the description of low boiling substances (46).
For mixtures composed of light nonpolar compounds simple cubic EOS's often work better than the HM, particularly in the neighborhood of the critical states, Figure 7.3. The latticegas EOS's become more useful for the mixtures containing a longchain component (Figures 7.4, 7.5). The capabilities of quasichemical EOS's are, again, unveiled to the full extent for systems with associating components, for which they give very good predictions of phase equilibria over a wide range of temperature and pressure, and are often superior to other EOS's. The examples of liquidgas equilibrium diagrams for such systems are given in Figures 7.67.10. Reasonable results are obtained up to extremely high pressures reflecting the existence of the double homogeneous point and of the gasgas equilibrium (Figure 7.10). The description of liquidliquid equilibrium at normal pressures is quite good (Figures 7.11, 7.12). Multiphase equilibria can be modeled as well by the quasichemical EOS's (Figure 7.13). Serious problems were encountered however, in predicting the dependence of liquidliquid equilibrium upon pressure: the dependence given by the HM EOS was far too weak (69). The performance of the lattice gas EOS in calculating caloric properties of pure fluids and mixtures is a field not very
274
1 O0 I I
co I, E o
0 3 v
> , ,
or) c" (1) n
0.80
0.60
0.40
0.20
000 *
250
] 350 450 550
Temperature (K)
Figure 7.2 Liquidvapor saturation line for pure ethanol calculated by latticegas EOS's. Solid curve: the HM. Dashed curve: the expanded version of the quasichemical latticegas model (44). Dotted line: the random hole model by Kumar et al. (79), given after reference 44. Points: experimental data (128).
13..
v
(1)
o') o')
13_
_ i i i i i i i i i
J _ . , ~ 8 K
j ~ z ~
• / *
2 ~ " * ' ~ * f 253 K
,~~" ,..~ ~ , _ _ _ _ _   
n a  r   7   r , , , , , , 0 0.2 0.4 0.6 0.8 1 . 0
Mole fraction of 002
Figure 7.3 The results of the liquidgas equilibria calculations for propane + carbon dioxide mixture at temperatures below and above the critical point of carbon dioxide. Solid curves: the HM (75). Dashed curves: the PengRobinson EOS (75). Points: experimental data (126,127).
t~ n
v
03
CO O0 03
13..
15
10
0 0.0 1.0
1 I I I
I 1 I I
0.2 0.4 0.6 0.8
Mole fraction of CO 2
275
Figure 7.4 Carbon dioxide + decane liquidvapor coexistence curves predicted by the KK EOS. Points: experimental data (139). From reference 101.
352 K
0.0 0.2 0.4 0.6 0.8 1.0
Mole fraction of H2S
Figure 7.5 Description of the vaporliquid equilibria in the hydrogen sulfide + heptane system by means of the expanded version of the quasichemical EOS by Kumar et al. (curves). Points: experimental data (141). From reference 140.
276
13_ v
(1) x,_,
x . . .
13..
0.20
0.15
0.10
0.05
I ' I ~ I ~ I '
363 K
343 K
323 K
0 . 0 0 1 , I ~ I , I t I , I
0.0 0.2 0.4 0.6 0.8 1.0
Mole fraction of ethanol
Figure 7.6 Description of the vaporliquid equilibria in the ethanol + water system by means of the expanded version of the quasichemical EOS by Kumar et al. (curves). Points: experimental data (142). From reference 140.
I I I I
6.5 T523 K 
5.5
4.5
~ 3 . 5 ) . T=473 K 
~ 2.5
1.5
0.00 ' '
0 1 Mole fraction of acetone
Figure 7.7 Vapor  liquid equilibrium for acetone + water system calculated by the HM (73) (curves). Points: experimental data (129).
3
2
© 2
1
0 0.0
l I I I I 1 473 K
448 K
0.2 0.4 0.6 0.8 1.0 Mole fraction of hexane
277
Figure 7.8 Prediction of the vaporliquid equilibrium for hexane + ethanol by the HM EOS in its groupcontribution version (solid curves) and by the APACT EOS (dashed curves) (67). Points: experimental data (130).
cl
v
©
3 o3 if)
cl
80
60
40
20
~ i l v J I I i ~ i ~ l l l l 1
i i i I i i i 1 [ i i i i I I i 1 ] i 1 i i i i i i
1 10 100 1000 Hexanol Concentration (10 3 g'c m3)o
Figure 7.9 The vaporliquid equilibria of the binary hexanol + sulfur hexafluoride at 362.8K predicted by the HBLF EOS without any adjustable mixture parameters (curves). Points: experimental data (137). From reference 84.
278
400 . . . . . . . . .
1 2 3 k ~43 2 1
300
• 200
lOO
O0 0.2 0.4 0.6 0.8 1.0
Mole fract ion of CO 2
Figure 7.10 Phase diagram for water + carbon dioxide system at elevated temperatures: 473K (1), 533 K (2), 538 K (3), 539 K (4), 541 K (5), 573 K (6). Solid curves: calculations by the HM EOS (103). Dashed curves: experimental data (131).
340
335 
330 
¢ 325 
.~ 3 2 0  0
E 315  © 1
310 
305 
3OO 0.4
nm
• • •
t I I I I I /
0.5 0.6 0.7 0.8 0.9 1.0
Mole fract ion of acet ic acid
Figure 7.11 Liquidliquid equilibrium in the acetic acid + dodecane binary mixture calculated (curve) with the aid of the groupcontribution quasichemical EOS by High and Danner (24) and experimental data (points). From reference 138.
MeOH
a)
279
nC 7 H20 C6OH
b)
nC 7 H20 04E1
c)
H20 nClo
Figure 7.12 Liquid miscibility gaps in temary aqueous mixtures predicted by the HM EOS (solid lines) (69). Points and dashed lines: experimental data. (a), (b): water + nheptane+ methanol and water+nheptane+hexanol mixtures, respectively, at T = 298 K, P = 0.1 MPa. Composition is given in mole fractions, the experimental data are after reference 132. (c): water + ndecane + butoxyethanol mixture at T  313 K, P  20 MPa. Composition is given in weight fractions, the experimental data are after 133.
280
Ace tone
10  ~ ,,
8  EL.
d.)
00
(~ 6  n
4  C O 2 H20
Figure 7.13 The phase diagram for the carbon dioxide + acetone + water system at 313 K. Experimental data (134,135) are represented by solid tie lines with stars at their ends. Solid curves and dashed lines are the HM predictions (69). The triangles in the lower pressure crosssections of the pressurecomposition prism correspond to threephase regions.
well explored, however some examples of quite reasonable results can be found in the literature (23,97,136).
The structural properties of associated fluids are reflected quite well by quasichemical quasilattice methods, in particular, this is true for the fraction of nonbonded molecules (32,67 69), when compared with computer simulation for model systems and spectroscopic measurements. Figure 7.14 is an example of such a prediction for saturated liquid ethanol. In Figure 7.15 a comparison is given with computer simulation results on binary mixtures of LennardJones molecules having conical specific sites (70). Figure 7.16 depicts the results for the carbon dioxide + hexanol mixture. The fractions of nonbonded molecules calculated by the HM agree better with the experimental data than those given by the SAFT and APACT.
For binary and temary mixtures many examples of phaseequilibrium calculations with other quasilattice EOS's are available in the literature (24,25,39,45,46,7881,8487,98101,104 107,136), therefore a general comparison can be made which shows basically the same trends for the EOS's with the effects of nonrandomness taken into account.
Thus the obvious merit of the lattice gas EOS's is their capability in describing more complex mixtures. This might be very important for practical applications of the EOS's. For instance, the quasilattice EOS's are not superior to the cubic ones for mixtures of typical oil and gas components. However, giving similar accuracy to these systems, the quasilattice EOS's
281
permit modeling of aqueous mixtures of these components, which essentially enlarges their range of applicability in comparison with the cubic EOS's. Another attractive feature of the quasilattice EOS's is that typically they give quite reasonable predictions over a wide range of temperature and pressure. This is due to the fact that these EOS's are related to a molecular
1.0 ' I ~ 1 ' I
c O .m O t~ , x._
0 . 5  4,
0 / / / , c j ~ . . / . O 
_ J / ~ / ......i    ' ' /
0.0 , ~ %    i " , , 2()0 300 400 500
Temperature (K)
Figure 7.14 The fraction of monomer molecules in saturated liquid ethanol. Spectroscopic data (125) and prediction by quasichemical quasilattice theories: the HM EOS (solid) and the LFHB theory (84) (dashed). APACT (67): dotted.
model, and the success of their application is gained not merely by fitting parameters for specific ranges of conditions, as is the case with some more empirical methods (113). The basic validity of this simple molecular model is confirmed by the good predictions it gives for fractions of nonbonded molecules in a fluid. Very important for practical usage is the group contribution approach which is easy to implement for the quasilatticecontactsite EOS's. Groupcontribution equations, such as the HM, or those proposed by High and Danner (24,138), Yoo et al. (98), Mattedi et al. (46) become rather convenient for mixtures formed by a very large number of homologues (103), in particular in combination with the continuous thermodynamics treatment (114).
It seems that the results given by quasilattice EOS's for fluids composed by nonpolar chainlike molecules can be improved. For these systems the most promising are, perhaps, the quasilattice continuous versions of the GFD family of the EOS's, as can be concluded from comparison with computer simulation results (2732,47). However a better approximation applied separately for the repulsion or for the residual contribution does not necessarily guarantee better overall performance of the EOS. Thus a version of a quasichemical lattice gas EOS combined with the SAFT repulsion part did not show significant improvement over the original HM in the description of phase equilibria (74). Similar results have been obtained by trying several repulsion parts with a quasilattice residual term for a number of real fluids (115): the best description has been obtained with a lattice gas repulsion contribution.
282
As has been pointed out by several authors (74,104,116), a problem is encountered when trying to apply an EOS to phase equilibrium and to single phase (supercritical) states. It has been shown that it is very difficult to describe simultaneously the whole phase diagram of a pure substance unless two different sets of the model parameters are used for the sub and supercritical states, the parameterversustemperature plots show an obvious discontinuity around the critical point (74,104). This peculiar feature does not seem to be an exceptional property of quasilattice models, it is more likely to be connected with some basic limitations of classical EOS's in reproducing the whole phase diagram including both the singlephase and the twophase states.
a) b) 1.0
cl
E 0.8 o o
N.,.
o c 0.6 0
o ~J X. 4 0.4 ©
E 0 , 0 . 2  o
i i •
0 . 0 I I I I
0.0 0.2 0.4 0.6 0.8 1.0 Mole fraction of comp. 1
E 0.8 O o
N.
O c 0.6 0
o ~J
~ 0 4 ~ °
©
E 0 c 0.2 0
0.0 0.0 0.2 0.4 0.6 0.8 1.0
Mole fraction of comp. 1
Figure 7.15 Monomer fractions for component 1 in coexisting vapor (upper curves) and liquid (lower curves) phases for model binary systems with selfassociation (a) and crossassociation (b). Points: computersimulation data (70). Curves: prediction by the HM EOS (69) using the interaction energies between specific and nonspecific sites after reference (70), see below.
(a): U b°na = 8Ull , U22   1.2Ull, f,2 = (UllU22) 0"5, T*=kT/Ull=I.1. (b): U b°nd = 10U, UI~  U22 = U12 = U, T*=I.0.
In the both cases r 1 = r 2 = 1.
7.6 CONCLUSIONS
Basically it can be concluded that the quasilattice EOS's play an important role in the modeling of phase equilibria in fluid systems and that they are especially useful for studies of complex systems with components differing in size, polarity, and shape, and for modeling associated mixtures. The advanced versions of quasilattice models (2225,4446,98,107) can be successfully applied for the description of such systems and for the prediction of the behavior of multicomponent mixtures over a wide range of conditions and can be listed among the best modem EOS's in this area. Groupcontribution formulations of quasilattice EOS's provide
283
remarkably good predictive abilities for systems containing various chainmolecule homologues. However, the quasilattice approach is not superior to the other modem EOS's for systems composed of small nonpolar and slightly polar molecules, in which case new modifications of the empirical cubic EOS's (113), or EOS's having more theoretical grounds (e.g. PSCT (110), SAFT(33,59)) give somewhat better results.
17
n 13 v
©
a..
' " " it I • : \\\ I I / " /; •
, d . ,\H II l ~ '/ / . p J I I • / //
• I I I
i I
II 1
5 
, ,
0.0 0.22 0.4 0.6 0.8 1.0 Hexanol monomer fraction
Figure 7.16 Comparison of the experimental and calculated (68) fractions of hexanol monomeric molecules in vapor and liquid phases for carbon dioxide + hexanol mixture at 402K. Solid curves: the HM EOS. Dashed curves: the SAFT EOS. Dotted curves: the APACT EOS. Points: the experimental data (137).
Certainly the role of the lattice EOS's and their further development are influenced by the impressive results obtained for model systems by means of techniques which are more rigorous than the quasilattice approach. First of all the description of fluids of flexible rmers should be addressed, for which the traditional FloryHugginsMillerGuggenheim approximations presently give way to new equations showing excellent agreement with the behavior of the fluids studied by computer simulation. Such a family of quasilattice EOS's based on the empirical generalization to a continuous case is under development (30,31,47).
The consideration of strong directional interactions with the aid of the contactsite model in the quasichemical approximation still remains among the attractive features of the lattice models. A general mathematical form for equations for various types of association, the possibility of considering systems with spatial networks of associates, and satisfactory accuracy are the stimulating benefits of the quasichemical approach. It has somewhat better capabilities than methods based on the law of mass action and is simpler compared to SAFT. Of interest are the attempts to combine the quasichemical approach (for the description of strong attractions) with rather accurate expressions available for the contribution of repulsive and dispersive interactions into the EOS's (117,74,115).
284
One of the general trends of the last several years is that more rigorous methods, such as SAFT (33,91), are used very intensively in practical applications (59,92,95) somewhat diminishing the interest in the lattice EOS's in their traditional domain. The applications of quasilattice models shifts to other areas, such as modeling of nonuniform molecular systems, the structure of interfacial regions, micelles, microemulsions, and phase equilibria in such systems. As has already been mentioned above, a fluid of rmers can be studied applying a formalism for a fluid of spherical segments with strongly associating contact sites to mimic the chemical bonds of the rmers. A kind of sphereandbond formalism has been applied within the framework of a lattice model to study the concentration and orientation profiles of solutions near a flat surface (66). The way to extend the formalism for the case of a latticegas fluid with curved interface is quite clear and we are about to carry out the modeling of such systems. A lattice model can be an effective tool to apply the sphereandbond formalism in the study of local ordering in amphiphilic chain fluids (118,119). From the criterion of the system stability with respect to fluctuations of segmental concentrations the microphase separation can be revealed on the phase diagram (119). Another new application of the quasilattice EOS's is their combination with the methods of continuous thermodynamics to describe polydisperse polymer systems (120).
As concerns the further development of EOS's for bulk fluid phases, the description of the behavior of fluids in the nearcritical region appears to be of major importance. A general shortcoming of the existing quasilattice versions of EOS's, is that nearly all of them have been derived in a meanfield approximation. This assumption leads to an analytic EOS incapable of giving the correct asymptotic behavior of a fluid in the near critical region, and is typically present in most of the EOS's derived for molecular fluids. It is likely, that the quasilattice approach will play an essential role in the search for nonanalytic EOS's. Thus, a practical nonanalytic approach, the renormalizationgroup theory, has been, to a great extent, developed for lattice models (121). Another possibility for describing correctly the asymptotic critical behavior is provided by decorated lattice models (122). As these are isomorphic to the Ising model, one can apply the exact results known for the latter. An alternative approach for practical applications is an empirical search for correlations, which are in good agreement with computersimulation data for nonathermal lattice systems. A correlation of this kind recently yielded a very good description of the whole liquidliquid coexistence curve (123) including the upper and the lower consolution points. The results are much more accurate than those obtained with the aid of the LCT or quasichemical approximation. The correlation, designed for modeling of liquidliquid equilibrium in a noncompressible twocomponent lattice fluid, may work as well for liquidgas equilibrium in a onecomponent lattice gas. The extension to the multicomponent case is ambiguous, though it is a challenge to solve this problem.
Acknowledgments
The authors are indebted to the Russian Foundation for Basic Research for financial support (projects 960333992a and 961597399).
285
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l~quations oJ ~'tate Jor 1~ luwls and P luid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 289
8 THE CORRESPONDINGSTATES PRINCIPLE
James F. Ely and Isabel M. F. Marrucho
Chemical Engineering and Petroleum Refining Department Colorado School of Mines Golden, CO 804011887, U.S.A.
8.1 Introduction 8.2 Theoretical Considerations 8.3 Determination of Shape Factors
8.3.1 Other Reference Fluids 8.3.2 Exact Shape Factors 8.3.3 Shape Factors from Generalized Equations of State
8.4 Mixtures 8.4.1 Van der Waals OneFluid Theory 8.4.2 Mixture CorrespondingStates Equations
8.5 Applications of the Extended CorrespondingStates Theory 8.6 Conclusions References
290
8.1 INTRODUCTION
During the past 25 years there have been great advances in the theory of dense fluids and in the application of these theories to complex molecular systems. These advances which include both integralequation and statisticalmechanical perturbation theory have been brought about primarily by the advent of faster, cheaper computers. As pointed out by Kreglewski (1), however, these results seem hopelessly complex for a chemical engineer who is generally interested in simple, practical solutions. With this complexity in mind, the most powerful tool available today Oust as 25 years ago) for making highly accurate, yet mathematically simple, predictions of the thermophysical properties of fluids and fluid mixtures is the correspondingstates principle. The power of the correspondingstates principle is that it allows the prediction of fluid properties with a minimum amount of information for the system of interest, given a detailed knowledge of few reference systems. The principle is well founded in molecular theory but certainly is not new. Its fundamentals and applications to purefluids and mixtures have been reviewed in almost all of the recently published thermodynamic and statistical mechanics books (25). As discussed by Leland and Chappelear (6) in their review of the correspondingstates principle 25 years ago, the basic concept of corresponding states is to apply dimensional analysis to the configurational portion of the statisticalmechanical partition function. The end result of this analysis is the expression of residual thermodynamic properties in terms of dimensionless groups. On an empirical basis, the correspondingstates principle was originally proposed by van der Waals who observed that the reduced form of his equation of state could be written for all fluids. All modem generalized engineering equations of state are examples of applications of this principle.
The original, twoparameter correspondingstates principle leads to an equation of state which expresses the residual compressibility factor in terms of a universal function of the dimensionless temperature and molar volume (or density):
Z r ~ P~VI = F(V*,T*) (8.1) RT
where p is the pressure, R is the gas constant and V* and 2"* are the dimensionless volume and temperature, respectively. Starting from a molecular basis, V* would be identical to V/N~ where ~ is the intermolecularpotential distance parameter and T* would be given by kT/c where k is Boltzmann's constant and c is the intermolecularpotential well depth. If one invokes the stability criteria for a purefluid critical point, the dimensionless volume and temperature would be given by V/Vc and T/Tc, respectively, where the subscript c denotes a value at the critical point. We note that this twoparameter correspondingstates principle can be applied to any polynomial equation of state which has a liquidvapor critical point (6). Since Equation (8.1) says that all substances obey the same reduced equations of state we can make a slight transformation of this result to directly relate the properties of one fluid to another. For two fluids j and 0 which obey the simple correspondingstates principle we can write from Equation (8.1)
Z~ (V*, T*) = Zo(V*,T*)= F(V*,T*) (8.2)
o r
zs( ,rj)= Zo(ro,r o) (8.3) where Vo and 7'o are related to their corresponding values in the jfluid by
V o = Vj ~hi and T o = Tj /f j (8.4)
291
where hj =VjC IV oc =oj3/00 3 and fs. = Tf/T0 ~ = e j / c 0 . The quantities £ and hj are known as
equivalentsubstance reducing ratios. Experimental evidence has shown that the twoparameter correspondingstates equation is
obeyed only by the heavy noble gases (Ar, Kr and Xe) and nearly spherical molecules such as methane, nitrogen and oxygen. In order to extend the correspondingstates theory to a larger spectrum of fluids, additional characterization parameters have been introduced into the basic twoparameter correspondingstates equation (8.1). Two main approaches have been followed in this parameterization. The first is to introduce the additional characterization parameters and then perform a multiparameter firstorder Taylor series expansion of the compressibility factor about the parameters. Mathematically,
• " "I ~ i ( 8 . 5 ) Z(V*,T*,/~I,/~2,...,/~n) Z(V*,T*) {;~. :0} i : l {~:0}
where the {Li}are the characterization parameters. The derivatives appearing in this equation are typically evaluated by making finitedifference approximations using reference fluids that differ in the parameter of interest. A good example is the use of a single additional characterization parameter is the Pitzer acentric factor co (7,8). In this case
(c3Z~ = Z(T*, V*,co = col)Z(T*, V*,co = 0) (8.6)
t,&o) co =0 col
and the correspondingstates model becomes
z ( r * , v* , co = co, )  z ( r * , v* , co = o) Z(T*,V*,co) = Z(T*,V*,co = O)+
(01 co (8.7)
Examples of this approach and its generalizations include those of Pitzer (710), Lee and Kesler (1113), Teja and coworkers (1419) and, more recently, Johnson and Rowley (20,21) and Malanowski and Anderko (22). Both of the latter two references present concise overviews of the different techniques used in corresponding states, giving special attention to the development of fourparameter models which are capable of describing fluids that exhibit large deviations from the simple twoparameter correspondingstates principle due to size/shape and polarity/association effects. In addition, Reid, Prausnitz and Poling (23) summarize a large number of correspondingstateslike correlations that use this approach.
The second approach is to extend the simple twoparameter correspondingstates principle at its molecular origin. This is accomplished by making the intermolecular potential parameters functions of the additional characterization parameters {2;} and the thermodynamic state, e.g., the temperature T. This can be justified theoretically on the basis of results obtained by performing angle averaging on a nonspherical model potential. The net result of this substitution is a correspondingstates model that has the same mathematical form as the simple twoparameter model, but the definitions of the dimensionless volume and temperature are more complex. In particular the dimensionless volume becomes
V V V* = = (8.8)
No 3 (T, {2 i }) Vc(P(T, {2 i })
and the dimensionless temperature takes the form
292
k T T T* = = (8.9)
E(T, {~'i }) LO( T' {/~'i })
The quantities 8 and q) which appear in these equations are referred to as shape factors and we refer to this method as the extended correspondingstates theory (ECST). Several review papers have been published that focus on this approach, the most extensive of which are those of Leland and Chappelear (6), Rowlinson and Watson (24) and Mentzer et al. (25). Applications of this method are presented in Section 8.5.
Thus far we have only introduced the purefluid correspondingstates principle which, as mentioned above, has a rigorous basis in molecular theory. The extension of this theory to mixtures cannot, however, be made without further approximation and the problem of rigorous, yet tractable, prediction of mixture properties remains unsolved. These approximations take the form of mixing rules which are a topic of another chapter in this volume. We will only discuss mixing rules from an illustrative basis to show problems that can arise in the implementation of a correspondingstates model. In that regard, we will focus our mixture discussions on the onefluid theories~primarily the van der Waals onefluid theory proposed by Leland et al. (26, 27) The essence of this model is that the properties of a mixture are first equated to those of a hypothetical purefluid whose properties are then evaluated via the purefluid correspondingstates principle. The onefluid theory is capable of providing highly accurate results, especially when the extended purefluid corresponding states formalism is used.
In the remainder of this review we have chosen to focus on the correspondingstates nature of generalized engineering equations of state and the more general shapefactorbased extended correspondingstates principle. We start by reviewing the basic molecular theory of purefluid corresponding states and then describe in some detail the extended corresponding states principle. Some time will be spent discussing the methods of calculating and/or predicting shape factors. Part of this discussion will be spent examining common engineering equations to extract the dependence of their shape factors on temperature, volume and other characterization parameters. We will also try to illustrate the dependence of the 'experimental' shape factors on temperature, volume and for example, dipole moment by studying highly accurate equations of state for both nonpolar (e.g., hydrocarbons) and polar materials (e.g., refrigerants and water). Finally we discuss the implementation of the extended corresponding states principle for mixture calculations and demonstrate some of the successes and difficulties encountered in the application of the model.
8.2 THEORETICAL CONSIDERATIONS
The direct calculation of thermophysical properties from statistical mechanics involves not only extraordinary mathematical complexity, but also detailed knowledge of the interactions between the molecules. Although considerable progress has been made in developing molecularly based predictive methods, most of the results have been obtained using more or less drastic approximations and are computationally complex. Also, most of these correlations are developed for specific types of fluids in certain regions of the phase diagram. Examples include the statisticalmechanical perturbation theories and integralequation theories. The correspondingstates theory provides an alternative route to calculate thermophysical properties since it uses experimentally measured properties of one or more reference fluids to represent the solution of the configurational part of the partition function. In this section we briefly review the molecular basis of the correspondingstates theory.
293
As a detailed derivation can be found in the literature (2,4,7), only the basic assumptions of simple corresponding states are stated here: 1. The partition function can be factorized into a densityindependent intramolecular
contribution and a densitydependent configurational contribution. 2. The configurational contribution can be treated using classical statistical mechanics. 3. The intermolecular pair potential may be written as a product of an energy parameter and a
universal function f, which depends only on the distance between the molecules, r, e.g., u (r)= ~f(r/cr) where 6 is the potential well depth and ois the collision diameter.
The first assumption is generally valid at low densities and even at high densities for simple fluids but it does not necessarily apply to polyatomic fluids or associating molecules. The second assumption excludes fluids that exhibit quantum behavior. The third assumption is the most restrictive since it excludes all nonspherically symmetric molecules.
Given these assumptions the correspondingstates principle may be easily derived from scaling arguments applied to the residual canonicalensemble partition function, Qr = Q/Qideal. Using standard notation (28)
1 _ O "3N Qr(N,V,T) = ~~ f... feUN(ru)/kTdr N vN f"" feUN(r*N)/kTdr *N  e F(V*'T*) ' (8.10)
where UN is the configurational energy. Since the Helmholtz energy A is given by  kT In Q(N, V, T), this equation reduces to
i r
= F(V" r ' ) (8.11) 6"T*
Applying this relationship to two conformal fluids (fluids which obey assumptions 13 given above) results in our basic working relation in correspondingstates theory, namely that if two fluids are conformal, their dimensionless residual Helmholtz energies are identical when evaluated at equivalent conditions
(Vj, L)= U A; (Vo, to)= L.4; / (8.12)
where we have used the definition of the equivalentsubstance reducing ratios j~ and hj presented in the previous section. Relations between other thermodynamic properties of the two conformal fluids can be obtained by straightforward differentiation. For example, for the pressure one finds
£ f ,
(vj /hj, / f j) Pj (Vj, Tj) = ~j Po (Vo, To) =   Po (8.13)
In order to extend the simple molecular correspondingstates principle to nonspherical fluids, two approaches are possible. The first simply amounts to introducing models for the non spherical interactions into the intermolecular potential. For example, the intermolecular potential between two axially symmetric molecules whose electrostatic interactions can be represented as point dipoles and quadrupoles can be modeled as (4)
2 02 H(F, 01,02,~12)  ocL (F/ 0") "~  ~ fg (01,02,~I2) [ 7 fo (01'02'~12)
r r (8.14)
294
where 01, 02 and 012 are angles describing the relative orientation of the two molecules and ¢t and ® are the dipole and quadrupole moments, respectively. When this potential is used in the evaluation of the configurational energy and the residual partition function is made dimensionless one finds
A r  F(V*,T*;p*2,® .2) (8.15)
sT*
and the correspondingstates working equation becomes
ol/ r fl'lj'Oj)= fJf~ ( V ° ' T ° ' [ d ° '  hj ' f j f~ hj' f jh~J (8.16)
From a theoretical view point this approach is perfectly viable. However, from a practical point of view neither theory nor experiment have provided quantitative details about how the equation of state depends on the multipole moments. Thus Equation (8.16) cannot be used directly in any correspondingstates predictions.
The second approach to extending the molecular correspondingstates principle to non spherical molecules was suggested by the work on angleaveraged potentials by Rushbrooke (29), Pople (30, 31) and Cook and Rowlinson (32). For example, if the spherical portion of the potential of an axial dipolar molecule can be represented by the LennardJones (126) model
(8.17)
and the dipole can be modeled as a point dipole as in Equation(8.14), Boltzmann averaging of the potential yields an effective spherical LennardJones potential for which the parameters are temperature dependent and are given by / 4)2 / ),J6
c(T) = s o 1 + and o(T) = 1 + (8.18) lZkTsocro 6 °'o lZkTc0cr06
Allowing the potential parameters to be temperature dependent retains the form of the simple, twoparameter correspondingstates relationship except that the equivalentsubstance reducing ratios become functions of temperature. In particular,
A)(Vj, Tj ) = f j Aro(Vo, To)=f j Aro(Vj /hj(T),Tj / f j(T) (8.19)
The temperature dependence of the equivalentsubstance reducing ratios somewhat complicates the thermodynamics of the thermal properties calculated from this model. For example, the internal energy of the fluid j is related to that of the reference fluid 0 by the equation
[?~= 1  ~ , O T j J ~ o  ~  ~ ) Z o (8.20)
295
In applying this formalism one must have knowledge of the reference fluid properties (as denoted by a 0 subscript) and the equivalentsubstance reducing ratios. These reducing ratios are typically expressed in terms of the molecular shape factors 0 and (p that are defined as
c V.~    _ _ 2 2   _ _ 2 2 Tj 0(Tj,ps,Oj,.. .) and hs J qg(Tj,ps,Os,...) (8.21)
fJ To c V o
8.3 DETERMINATION OF SHAPE FACTORS
The theoretical basis for the molecular shape factors was derived in the previous section. That analysis, which led to temperaturedependent shape factors represents an idealized case where the nonspherical potential parameters may be incorporated with the spherical parameters through angle averaging. Although that approach is correct in certain circumstances, it is of limited practical use since the intermolecular potential function for real fluids is not known precisely. Hence, one is forced to use macroscopic thermodynamic measurements to determine the shape factors and then try to develop a generalized correlation for them which depends on known molecular parameters. We shall refer to the shape factors determined from experimental data as the apparent or exact shape factors and their generalized correlation as the correlated shape factors. The shape factors are weak functions of temperature and, in principle, density and can be visualized as distorting scales that force the two fluids to conformality. Although there is no direct theoretical evidence for the density dependence of the shape factors, mathematical solutions for exact shape factors found by equating the dimensionless residual compressibility factor and Helmholtz energy of two pure fluids exhibit a weak density dependence.
The first attempt to find exact shape factors is due to Leach (33), who equated the residual compressibility factor and fugacity coefficient of two fluids, viz.
z) (vj, rj)= z;(V / h,, r /f j)
Oj , ) = Oo / , / £ ) (8.22)
and solved for the equivalentsubstance reducing ratios, fj and hj. In these equations the superscript r denotes a residual (real minus ideal) property evaluated at the temperature and molar volume of the system. For example, ~b r =f/pRT wheref is the fugacity of the fluid. At low density these equations become identical and the apparent shape factors were found by simultaneously solving correspondingstates relationships for the second and third virial coefficients
Bj(Ts )= hj Bo(Tj / f j ) (8.23)
Cs(Ts )= hJ Co(rs / f s)
As emphasized by Leland and Chappelear (6), the shape factors determined from the solutions to Equation (8.22) depend on both density and temperature and, as such, cannot be related to any sort of intermolecular pair potential. More recently, Massih and Mansoori have discussed the statistical mechanical basis of the shape factors (34).
Using methane as reference and a large number of pure normal hydrocarbons from C1C15, Leach (35) obtained solutions to these systems of equations and empirically fitted the results in
296
terms of the acentric factor and the critical parameters. The set of correlated shape factors that were obtained is given by:
0 = l + ( c o j  c o 0) a,  a21n T ] + a 3 
(8.24)
Z°~/1 + (coj (.00)[j~ 1 (V;  ~2 )lnT; ] J~3 (Vf  J~4 )]} q~ = Z~. t J
where 0 and cp are the shape factors, co the acentric factor, T* is the temperature,/I* the volume and Z ~ is the critical compressibility factor. The subscripts j and 0 indicate the fluid of interest and the reference fluid (methane), respectively. The values of the parameters reported by Leach et al. (33) for the 0 shape factor were al = 0.0892, a2 = 0.8493, a3 = 0.3063 and a4 = 0.4506 with the cp parameters being fll =  0.9462, f12 = 0.7663,/33 = 0.3903 and/34 = 1.0177. When Vj.* >2.0 its value is set equal to 2.0; when Vj*< 0.5 its value is set equal to 0.5. These limits correspond to the virial region and dense liquid regions, respectively, where the apparent shape factors are independent of density. For other values of Vj* between these limits, the shape factors were found to be density dependent. Note that if the apparent shape factors are defined according to Equation (8.24), two parameters in addition to T: and Y~ are introduced,
co s and Z~, giving rise to a fourparameter correspondingstates model.
In 1981 Ely and Hanley (36) developed an extended correspondingstates theory for the viscosity of hydrocarbon mixtures. In conjunction with that work they developed a wide range referencefluid equation of state for methane and a new set of correlation parameters for the apparent shape factors. The functional form of that correlation was the same as determined by Leach et al., Equation (8.24), but the parameters were somewhat different owing to the expanded range of temperature. The values of the parameters reported by Ely and Hanley for the 0 shape factor were al = 0.090569, a2 = 0.862762, a3 = 0.316636 and a4 = 0.465684 with the ~o parameters being/31 =0.93281,/32 = 0.754639, ,83 = 0.394901 and/34 = 1.023545.
8.3.1 Other Reference Fluids
A disadvantage of the original Leach shapefactor approach is that the reference fluid is fixed as being methane. This introduces errors in the method when using it to calculate properties of fluids which have very different properties from methane or when the reduced temperature of the fluid of interest (target fluid) is less than the triplepoint temperature of methane. Ely and Hanley attempted to overcome this latter problem by developing an equation of state for methane which had a fluid region extrapolated down to 40 K (T* = 0.21). Another solution to this problem was adopted Leach (35) and later by Rowlinson (24)who showed that it is possible to convert shape factors relative to one reference fluid into shape factors relative to another reference fluid, thereby allowing two different reference fluids to be used in the correspondingstates calculations. For example, Leach et aL (33) used methane and pentane in their original extended correspondingstates model. In developing the transformation equations we assume that we know the shape factors or the equivalent substance reducing ratios of fluids i and j relative to some reference fluid which we shall denote as 0. The two fluids are at an equivalent corresponding state w h e n / ] / f o = I ] / ~ o = To and/I,. / hio = Vj. / hjo = Vo. Note that we have added a second subscript to make the reference fluid clear. Rewriting these equations we obtain relationships between the state parameters of
297
the i fluid with respect to fluid j, e.g., Ti =f0 Tj/fj0 and V,. = hio Vj/hjo. Finally, if we define the equivalentsubstance reducing ratios for i relative to j as f j = f0 /fj0 and h,j = hio / hjo we obtain the same functional mathematical relationships between the state points of the i and j fluids as we started with for the i and j fluids relative to the reference 0. The difference, however, is thatfj, and h a involve the state points of both fluids, rather than just the state point of fluid i. The shape factors which are associated with fj. and h a were called relative shape factors by Leach et al. and are given mathematically by
Oi°(Ti*'Wi*)  Oj°(Ti~*'Wi*) (8.25) 00" (Ti" V/" )  Ojo(T; , V;)  OJ° (~ Oj°Tii "  (pj°Vi"
and
e,o(~*, v,*) _ ejo(~*, v,*) (P ° ( Ti * ' Vi * ) = (P j o ( T ; , V ; )  (/3 J 0 I O j ° Ti * (/3 j ° Vii * i 0 ' ( / 9 io
(8.26)
These equations are nonlinear and must be solved numerically. In 1987 Younglove and Ely (37) reported a widerange equation of state for propane
based on the functional form of the 32term modified BenedictWebbRubin equation (MBWR32) which was proposed by Jacobsen and Stewart (38) for nitrogen. The advantage of that equation of state was that its range of fluid states included the triple point of propane which is 85 K (e.g., the reduced triple point is 0.22). This development eliminated the need to use two reference fluids or to use an artificially extrapolated reference fluid as in the work of Ely and Hanley (36). To avoid having to use the shapefactor transformation formulas given above, Ely redetermined the apparent shape factors relative to the propane reference. In determining the apparent shape factors, a slightly different method than that used by Leach et al. was incorporated. In particular, at subcritical conditions the procedure suggested by Cullick and Ely (39) was used. In this procedure the vapor pressures and saturatedliquid densities of the target and reference fluids are equated
P~t(Tj) = pSat (Tj / L ' ) f j / h j (8.27)
p~at(Tj) p~at(Tj/ f j ) /h j
and solved simultaneously for J) and hj.. At supercritical conditions the virial method given in Equation (8.23) was used. This procedure, unlike that used in previous studies, generates apparent shape factors which only depend on temperature. Their correlation gave the following results:
0 = 1 + (coj COo)[0.052020.74981 lnTj* 1
(,o = Z~ {1 +(coj COo)[0.14359 + 0.28215 lnT;l } (8.28)
298
More recently, Marrucho and Ely (40) developed a saturation boundary based method to evaluate the shape factors that is easily transferable between reference fluids. The method is based on the FrostKalkwarf vaporpressure equation (41) and the Rackett equation (42) for saturatedliquid densities. Using these relations and Equations (8.27) one finds for the 0 shape factor
1C o + 2(1 Tj*)2/V ln(Zj /Zo )  ~ * + AC* lnTj* + B; /Tf 0 = (8.29)
1C o + Bo/Tf and for
(Zj)(1Tj *)2/7 (8.30) (~'(Zo)(lZ;/O)2/7
In deriving these equations we have assumed that 0 is close to one and therefore In 0 _=_ 0  1 and have defined AB* B )  B* o and AC*= C )  C O and neglected the D* term in the Frost
Kalkwarf equation:
lnp"~at=B* 1 +C*lnr*+D' t  ~ 1 (8.31)
Since the reference fluid parameters appear explicitly in the shapefactor expressions, this formulation is easily transferable between reference fluids. In the supercritical region, Marrucho and Ely proposed a method of calculating the shape factors assuming that
(pj = Z o /Z~ and that isochores were nearly linear, resulting in an expression for the 0 shape
factor
o=lh°(p'T~Y')+(h°y'y°)TjlTf'po yoTo To (8.32)
The superscript cr indicates the isochore which intersects the reference fluid saturation
boundary at Po = pho, ho = ZoPo/Z~p~ and y  (c~ / cT/')p, y c for the target fluid may be
obtained from the FrostKalkwarf equation as
Y~i = a P ~i ( C~i  B~i  2 D* ) /(1 D* ) T~ (8.33)
As discussed in the next section, Mollerup (43) analyzed common cubic equations of state, such as the SoaveRedlichKwong (44) and PengRobinson (45) equations in terms of separating them into a shapefactor correlation and an equation for the pure reference fluid. The shape factors obtained from the cubic equations of state have the advantage of being similar to the ones developed by the more complex correlation schemes outlined here, but are simpler to use, since they are density independent and can be used with any reference fluid.
8.3.2 Exact Shape Factors From the late 1970's to the early 1980's, an increasing number of highaccuracy, analytic,
widerange equations of state started to appear in the literature. The availability of these highly accurate equations of state allows one to 'exactly' (although numerically) solve the twoparameter correspondingstates relationship for the apparent molecular shape factors. The possibility of density dependence in the shape factors does, however, complicate the
299
thermodynamic description of the target fluid in terms of the reference fluid and the resulting solution for the shape factors themselves. The basic equation from the scaling of the partition function remains the same,
a~ (Vj, Tj) = a; (V 0 , T 0) = a~ (Vj / hi, Tj / fj ) (8.34)
where we have introduced a more compact notation which denotes a dimensionless residual property by a lower case letter with a superscript r to emphasize that the property represents the difference between the real and ideal values evaluated at the same volume and temperature. For example, in the equation above a r   (A(V,T)  A idea! g a s ( g , T ) ) / RT. Using the thermodynamic relationship
r r
da r =  z    d v  U   d r (8.35) V T
and the chain rule, we find the following thermodynamic relations between the target and reference fluids:
r r r
zj : Z o [ 1  H v ]  u o F v
u~ = U o [ 1  F r ]  z r H r (8.36) r r r r
sj = s o  z o H r  u o F r
The enthalpy and Gibbs energy can be constructed from their thermodynamic definitions and the relations given above. In the correspondingstates relations summarized in Equation (8.36), dimensionless derivatives of the equivalentsubstance reducing ratios must be known. These derivatives are defined as
T ( a f j / and Fv= V [0 f j ] Fv =  f f ~ , O T ) v ff ~,~V) r (8.37)
with similar expressions for the derivatives of h i . Given the thermodynamic relations summarized above, it is not possible to solve for the equivalentsubstance reducing ratios without making some other assumption, i.e., the set of equations given in Equation (8.36) are underdetermined since a knowledge of both the values and derivatives of J) and hj is required.
r r The simplest assumption is to choose the solution for which zj = z0, which requires the
relationship zroHv =  uroFv (8.38)
Figures 8.18.6 illustrate the results of the apparentshapefactor calculations obtained using this technique with propane as a reference. Figure 8.1 shows the shape factors for methane obtained using the MBWR32 equation of state of Younglove and Ely (37). Methane differs from propane in both size and molecular shape but both are nonpolar. The shape factors reflect this in their weak temperature and volume dependencies. Figure 8.2 shows the same results obtained using the newer methane equation developed by Friend et aL (46). The results are similar with the differences in the shape factors being only a fraction of one per cent. We conclude that the volume dependence of the shape factors, although weak, is real and not due to artifacts of the equations of state used in the calculation.
300 1.08
1.04
9 1.00
0.96
0.92
' I i I ' I ' I i
p*=2.7 C H 4 
! 
p*=O.6
I I, l I I I t I I,,
1.12 ' I i I ' I i I i
CH 4
1.08
1.04
1.00
0.96
0.92 I 0.40 0.80 1.20 1.60 2.00 2.40
Figure 8.1 0 and ¢ shape factors for methane relative to a MBWR32 propane reference using the MBWR32 equation of state for methane (37).
Figures 8.38.5 show the apparentshapefactors of carbon dioxide, 1,1,1,2tetra fluoroethane (R134a) and water relative to propane. Carbon dioxide (47) was chosen for illustration because of its large quadrupole moment which would lead to an intermolecular potential which is substantially different from that of propane. In this case, the shape factors are nearly independent of volume and weak functions of temperature. R134a has a large dipole moment and exhibits a stronger volume dependence in its shape factors relative to propane than is observed with CO2. Finally water is highly polar and associating and exhibits stronger temperature and volume dependence than that observed in the other fluids. These
1.04
1.00
9 0.96
0.92
0.88
' I ' I ' I ' I l
 p*=2.? C H,,,, _
_ _i
p*=l.
9*=0.6
p* =0.3
I , I i I , I ,
i I ' I * I i 1 '
301
1.12 i
C H 4
1.08
1.04
o
1.00
0.96
0.92 , , I 0.40 0.80 1.20 1.60 2.00 2.40
, T
Figure 8.2 0 and rp shape factors for methane relative to a MBWR32 propane reference using the SchmidtWagner of state for methane (78).
figures illustrate that the extended correspondingstates approach is extremely powerful in that it can be used to make any fluid, regardless of intermolecular potential, conformal to a selected reference fluid. One would hope that by studying the shape factors of various families of fluids (e.g., refrigerants, alcohols, etc.), relative to a fixed reference fluid, behavioral trends could be identified and correlated with known molecular parameters. Figure 8.6 illustrates this type of relationship for the shape factors of various polar compounds as a function of acentric factor.
302 1.02
1.00
0.98
0.96
0.94
0.92
1.04
1.02  
1 . 0 0
a i ' I ' I '
_ • C O 2 
1.7
p*=2
L  
0 . 7
_ m
_
=0.3 _
i 1 I I t I ,
' I ' I ' I l
C O 2
p*=2.5
~ * = O . 3 ~ 0.7 ~
0.98 I I I I I ! I ...... 0.40 0.80 1.20 1.60 2.00
T
Figure 8.3 0 and (p shape factors for carbon dioxide relative to a MBWR32 propane reference using the SchmidtWagner of state for carbon dioxide (47).
8.3.3 Shape Factors from Generalized Equations of State
Thus far we have discussed the determination of shape factors from equations of state which have a high degree of accuracy in representing the properties of the pure fluids. Mollerup (43) observed that, although this procedure offers a very high accuracy in the determination of the shape factors, it can be very time consuming in terms of evaluating equations of state. This led him to examine 'simple' generalized equations of state which are commonly used in engineering calculations. Examples include the RedlichKwong Soave (RKS) equation (44), the PengRobinson (PR) equation (45)and others. In our analysis we have also included the CarnahanStarling de Santis (CSD) equation (48) because it has a temperaturedependent volume parameter and a more complex volume dependence. Although we do not normally think of these modified van der Waals types of equations of state in terms
303
1.30
1.20
9 1.10
1.00
0.90
1.12
1.08
1.04
1.00
0 . 9 6
' I i I ' I
R134a
• = .
m
_ p * = 0 . 3
p * = 3 . 0
I I I I I I
' I ' I ' I
* = 3 . 0
p * = 0 . 3
R134a 
0 9 2 I I ~ I t I i 0 4 0 0 8 0 120 160 2 0 0
T
Figure 8.4 0and q~ shape factors for R134a relative to a MBWR32 propane reference using the MBWR32 equation of state for R134a (123).
of shape factors, they all contain a prescription for their determination. To illustrate this, consider the RKS equation given below
RT a(T*, co) p =   (8.39)
v  b v(v +b)
where V is the molar volume, b is a volume parameter given by E2bRTZ/p c which is independent of temperature in this model and a is a parameter which is given by
304
1.30
1.20
1.10
9
1.00
0.90
0.80 1.12
1.08
1.04
1.00
0.96
I ' I 1 I ' I ' l
J _ ~ H20
 p * = 1.8
2.2 _ p *  3 . o 2 .
. ' I ' I ' I i I ' ]
p* = I .~
 .
I I w I i I i I t 0.40 0.60 0.80 1.00 1.20 1.40
T*
Figure 8.5 8 and q~ shape factors for water relative to a MBWR32 propane reference using the Saul Wagner equation of state for water (92).
where
a(T*,co) ~'~a (RTC)2 = a(T*, co) (8.40) pC
(8.41)
305
0.0004
0.0002
' I ' I ' I '
N2 © • CH4
0 2 0
0.0000
0.0002
C2H 4
R12 [ ] ACO 2 R l 1 [ ]
• [ ] R22 C3H 6
C4H10 •
[ ] R32
R125 i 1 R134a I']
R123 [ ] A [ ] H20 _
R124
C5H12 •
 o . o o o 4 , I , I , I ,  0.20 0.10 0.00 0.10 0.20
6oo30
Figure 8.6 Comparison of the slope of the temperature dependent term in the 0 shape factor (see Equation (8.28)) as a function of acentric factor difference for various fluids.
The parameters Qa and 62b are universal for this equation of state and m(co) is a simple quadratic function of the acentric factor. The residual compressibility factor and dimensionless residual Helmholtz energy are given by
and
z~ (Vj, Tj.) = bj a(T~, coj) (8.42) vj  R S (vj + )
Aj In  In RTj Vj bjRTj ( Vj
(8.43)
An examination of the correspondingstates relations, Equations (8.29) and (8.31), show that if the equivalentsubstance reducing ratios are given by
hi= bj and f j = ajb° (8.44) bo a0b0
the mathematical correspondingstates principle is obeyed. For the shape factors, these
relations imply that cpj  Z o / Z~ and
_ a(T],coj) _ { l + m(coj) + Im(COo)  m ( c o j ) ] ~ } 2 (8.45)
Oj  a(To,, COo )  1 + m(COo)
306
where we have used the fact that T0* = Tj.*/0j in deriving the second relationship. As an example of a more complex modified van der Waalstype model, we have also
made shape factor calculations with the generalized CarnahanStarlingDe Santis (GCSD) equation which in dimensionless residual form is given by
4 yj + 2 y 2 4a(T])yj r . _ _ _ _ (8.46)
zj (1 yj)3 RTjb(T~)(14yj) where y is the packing fraction b(T*)/4 V and the parameters a(T*) and b(T*) are given by
a(T*) = ~")a ( R r c ) 2 elal(1T*)+a2(1T*)2] p~
(8.47) RT c
[1 + b a (1  T*) + b z (1  T*) 2 ] 7 Again the parameters if2 a and Oh are universal for this equation of state with values of 0.461883 and 0.104999, respectively. At temperatures above the critical temperature, bl and b2 are set equal to zero. In this case (or any case in which the volume parameter has temperature dependence), we find a set of coupled equations for the shape factors which cannot be solved in closed form. For the GCSD equation
e[a'(1Tf')+a2(1T;)2] / 1 + b a (1 To* ) + b 2 (1To) 2 ]
Oj = e [a,(1To)+a20To)z ] \ 1 + b, (1  Tj) + b E (1  T ; ) 2 ) (8.48)
~pj = Z__~ ( I + bl(1 T;) + bz(1 T;)Z i Z~ 1+ ba (1 T0*) + b2 (1 T0*) 2
where To* = Tj /0 j . In Figure 8.7 we have compared the PengRobinson shapefactor relations, Equation (8.45) and the CSD relations, Equation (8.48) for methane, relative to a propane reference, with the shape factors calculated using the MBWR32 equations described above. This figure also shows results obtained using the generalized shapefactor correlation, Equation (8.24). These comparisons show a relatively good agreement at subcritical conditions but fairly substantial differences in the supercritical region. The RKS model predicts a q~ shape factor which has a constant value of 0.965, as compared to 'exact' values which range between 0.92 and 1.02, as shown in Figure 8.7.
8.4 MIXTURES
In order to apply the simple correspondingstates theory to mixtures of conformal molecules, approximations must be made concerning the microscopic interactions and resulting structure of a mixture. A basic problem arises in this extension because the configurational energy in a mixture is a function of the position of the molecules and the species of the molecule located at these positions. This should be contrasted to the purefluid case where the molecules are indistinguishable and the energy of the system is only a function of molecular positions. Thus, for a mixture, the scaling arguments that led to Equation (8.11) for pure fluids, do not apply, even if the intermolecular potentials for all the mixture components are conformal.
I ' I ' I '
A
& A A , ~ A A S " " ' " " "
"" 0.6
0.3
0 . 8 8 ~ I I , I l I 0 .4 0 .8 1.2 1 .s 2 . 0
,
7
{ i
C H 4  
I
2 .4
1.12  1 ' ' t ' 1" '
t 1 08
1.04 O
1.00
0.92 ~ 1 , I , 1
307
0.4 0.8 1.2 1.6 2.0 2.4
T*
Figure 8.7 Methane shape factors calculated via several equations of state and correlations. "exact" values calculated with MBWR32 equations of state; © Cullick and Ely saturation boundary method; A Generalized correlation, Equation 8.28; * CarnahanStarlingDeSantis equation and II Peng Robinson equation of state.
The earliest theoretically based attempt to deal with this problem was to average the configurational energy of a mixture over all possible random assignments of species to a given position (49). This random mixing concept leads to an effective, hypothetical, purefluid potential (equivalentsubstance potential) which can be used in the formalism developed for purefluids. Consideration of the explicit form of the conformal potential (e.g., LennardJones potential) leads to mixing rules for the equivalentsubstance reducing ratios of the hypothetical purefluid, i.e.,
and
where the subscript x denotes the hypothetical purefluid and for example fj =e 0/So where ij denotes the interaction of an 0" pair in the mixture. The term {x•} indicates the explicit dependence upon composition. For example, for the LennardJones 126 potential
308
N N N N
fxh2 = Z Z xixjfiJ h2 and fxh4x = ~' ~_~ xixjfijh; ( 8 . 5 0 ) i=1 j=l i=1 j=l
Even though the randommixing theory played an important role in the development of mixture theories, its predictions are not reliable due to its unrealistic physical basis (random assignment of molecules). However, the onefluid concept, i.e., that the properties of a mixture can somehow be equated to those of a hypothetical purefluid whose properties can be evaluated from correspondingstates, has persisted and forms the basis for what are currently the most accurate correspondingstates models for mixtures.
8.4.1 Van der Waals OneFluid Theory
The most successful correspondingstates theory for mixtures is called the van der Waals onefluid theory. This theory was developed on a molecular basis by Leland and coworkers (26, 27, 50) and follows from an expansion of the properties of a system about those of a hard sphere system. A hardsphere system is one whose molecules only have repulsive intermolecular potentials with no attractive contributions. The starting equation for the development of the van der Waals onefluid (VDW1) theory is a rigorous statistical mechanical result for the equation of state of a mixture of pairwiseadditive, spherically symmetric molecules.
Z = I _ 2 Z p N N 3kT Z Z XiX j ~u~ (r)g~i (r; p 'T' { xk } ; { ek } ' { crk } )r2dr (8.51)
i=1 j=l
In this equation Z is the compressibility factor pV/RT, go is the radial distribution function which gives the probability of finding a molecule of type i at a distance r from a central molecule of type j, u o. is the intermolecular potential whose parameters are c,j and or, 7, the prime denotes differentiation with respect to distance r, k is Boltzmann's constant and p is the number density. In the development of the VDW1 theory the intermolecular potential is assumed to be composed of a hardsphere term plus a longrange attraction, i.e.,
HS uij(r)=u o. + ruF*(r/cr~) (8.52)
where HS foO r < c r
U~/ = ~ 0 r > O " (8.53)
F(r) is a longrange attraction contribution to the potential, such as C6/r 6. Before one can proceed, assumptions concerning the radial distribution functions of mixture pairs must be made. In the development of this model, Leland proposed the meandensity approximation. This approximation amotmts to saying that the radial distribution function of the/j" pair is identical to that of a purefluid evaluated at the reduced conditions of the pair and a mean number density. Mathematically
gu (r; p,T, {xk } ; {sk }, {crk } ) = go (r / crij; p'Cr3x,kT / c~i ) (8.54)
where the subscript 0 denotes a purefluid distribution function. The next step is accomplished using the expansion techniques developed by Kirkwood et al. (51) for the distribution function of a real fluid in terms of that of a hardsphere fluid, namely
309
. ' 3 : (r/o.O.,iOO.x).lt_Z[kr) go(r /Cro,PCrx,kT/s#) gHoS . ' 3 ~ij k~ln(iO'O.3 ) ( 8 . 5 5 )
n=l
where the Tn are complicated integrals over the hard sphere radial distribution function. Substituting these results into Equation (8.51) we find a temperature expansion for the mixture compressibility factor
r 2 ~ P £ Z Z X i X j £0" 3 ' 3 Zm~x  3k r .=o i=1 j=l [ kT ) O'~iL~ln p O'x (8.56)
The analogous result for a hypothetical purefluid at the same reduced number density is n
2ap S, ( a',j "~ o.3T (p,O.x3) (8.57) zr = 3 k V ,zTd=o ~.kf ) o ,
Subtracting, one finds
r r n 3  = (p x) (s.ss) Zx Zmix ~ =0 i=l j=l xixJ~io'iJ
Thus, we have at our disposal an infinite set of terms (coefficients of T ~) from which we can choose two for the determination of the potential parameters of the hypothetical purefluid. In the van der Waals onefluid model the first two members of the series are chosen, giving
N N N N 3 3 O~xO" x : Z ZXiXjoC(jO'~ and Ox : Z Z x i x j o 3 (8.59)
i=0 j=0 i=0 j=0
Dividing through by the reference fluid parameters we obtain N N N N
fxhx = Z Zxixjf i jhij and h x = Z Zxix jho " i=0 j : 0 i=0 j=0
(8.60)
These are exactly the relations that van der Waals assumed in the generalization of his equation of state to mixtures. Within the meandensity approximation we see that these mixing rules are correct to order T 2 in the compressibility factor; terms that have higherorder temperature dependence in the reference fluid are not correctly mapped by the van der Waals mixing rules.
8.4.2 Mixture CorrespondingStates Relations
The working equations for the mixture extended correspondingstates theory are exactly the same as in the case of a purefluid, Equation(8.36). The expressions for the derivatives of the equivalentsubstance reducing ratios in terms of the component ratios are, however, somewhat complex. In particular, application of the formulas given above to mixtures requires derivatives offx and hx with respect to temperature, density and composition. An inspection of the mixing rules and the definitions of the equivalentsubstance reducing ratios shows that the arguments of the shape factors are the effective temperatures and densities of components in the mixture. These, in fact, do not correspond to the temperature and density of the mixture unless the shape factors are identically unity. Thus, in a mixture, the arguments of the shape factors are themselves functions offx and hx. The dependence of these is nominally given by
310
Tj. =Tx~/fx and Vj. = Vxhj/hx. Differentiating these relations with respect to Tx one obtains two equations:
Fj(Tx) Fj(Tj)[1 + Fj (Tx) F x (Tx) ] + F s (Vj)[Hj(Tx) g x (Tx) ] (8.61)
Hj (rx) = Hj (Tj)[I + ~(rx)Fx(rx)]+ Hj(Vj)[Hj(rx)Hx(rx)] Differentiation of the mixing rules with respect to temperature, Tx, yields
Jxnx1N g I ] Fx(rx)+ Hx(r~)=Wi~_~_xixjfjh U F~(Tx)+ qcH,(Tx) (8.62) i=1 j=l qo
and I N
Hx(Tx) = ~x .~l xixjhij qLi Hi(Tx) (8.63) "= qo"
where qi = hi lz and qi = (qi + qy2. Simultaneous solution of Equations (8.55) for ~.(Tz) and Hj(Tx) and substitution into Equations (8.56) and (8.57) and subsequent solution for Fx(T~ and Hx(Tx) yields Fx(Tx) = 1  $7/R and Hx(T~ = $6/R. Similar procedures yield for the volume derivatives
Fx(Vx) =($2+$4$7) and Hx(Vx) = (S6 + S1 S3 + Dx) (8.64) Dx Dx
In order to perform phaseequilibrium calculations, fugacity coefficients can be calculated from the thermodynamic relationship
ln ~i " g; nt U;Fx(l'li)nt" zoHx(n i )  ln Z (8.65)
and the necessary composition derivatives of the equivalentsubstance reducing ratios are given by
Fx(nk) =S}k)S7'S}k)(S2+S4) and Hx(nk) =S~k)(SI+S3)S}k)S6 (8.66) D D
where k denotes component k in the mixture and Dx = (S1 + $3) $7  ($2 + $4) $6. The definitions of the sums Sm that appear in these results are given in Table 8.1.
8.5 APPLICATIONS OF THE E X T E N D E D C O R R E S P O N D I N G  S T A T E S T H E O R Y
Historically, applications of the extended correspondingstates theory have been limited by the scarcity of highaccuracy thermodynamic data and widerange equations of state. As this data situation has improved over the past twenty years, refinements have been made in how the model is applied. In its original form, the extended correspondingstates model proposed by Leland et al. (6) incorporated a combined reference fluid of methane in the majority of PVT space and npentane in the lowtemperature, highdensity region. The shape factors used in this model were described in Section 3. Leland and coworkers originally applied this model to predicting vaporliquid equilibrium in nonpolar mixtures with a reported accuracy of about ten percent in the equilibrium Kvalues. Later, Fisher and Leland applied the model to predicting enthalpies, compressibility factors and fugacities in systems that did not deviate greatly from ideality (52).
311
Table 8.1 Sums required for evaluation of mixture correspondingstates properties
where
1 N N
Sl =~xhx ~"~Zxgxjfijh~i[1 Hi(Vi)]ni i=l j=l
1 U N
82 = ~xhx ZEXixjfijhijfi(Vi)Di i=1 j=l
1 N N
S 3 : ~xhxZEXixjfijhi j qLini(Ti)Oi i=1 j=l qo
1 N N
84 =~xhxZZxixjfijh~ i qi [ I _ F i ( T / ) I D i i=1 j=l
~ ) _ 2x~ gy )..xgLhik J x~x i=1
I N N S 6 h~xZZXixjho. qini(Ti)Oi
i=1 j=l qu 1 N U
84 =~x ZZxixjhO'i=l j=l ~ijqi [1 F/(T/)]Di
S~k) 2x~ N = 2hx Zxihiki=l
D ; 1 = [1 H,. (V/)] [1  Fi (T/) ]  Fi (V/)H/(T/)
Starting in 1969, Rowlinson and coworkers published a series of papers applying the extended correspondingstates model to a series of fluids and properties. In their application they used the Leach shape factors, but an extended methane equation of state that incorporated the low temperature data of Vennix (53,54) was used as the reference fluid. In the first paper, Watson and Rowlinson (55) predicted bubblepoint temperatures and vapor compositions of argon+nitrogen+oxygen temary mixtures with satisfactory accuracy. Gunning and Rowlinson (56) calculated compression factors, enthalpies, JouleThomson coefficients and VLE for various systems and concluded that the extended correspondingstates principle had the advantage of requiting relatively little starting information and could be successfully applied to a wide variety of properties and fluids. Its primary disadvantage was pointed out to be a high degree of numerical complexity. In 1972, Teja and Rowlinson (57) applied the method to the prediction of critical and azeotropic states finding quantitative agreement for mixtures with one liquid phase and at least qualitative agreement in systems that had multiple liquid phases. Teja and Kropholler (58) and Teja (59) extended this study in 1975 by investigating azeotropie behavior in the mixture critical region. In the second study azeotrope formation and saturatedliquid densities in the CO2+ethane system were predicted with excellent results. In 1976, Teja and Rice (60) measured densities of benzene + alkane mixtures and compared their measurements to extended correspondingstates predictions using the Leach shape factors and Bender's methane equation of state (61) as the reference fluid. The average absolute percentage differences found between the measurements and predictions were less than two percent for alkanes from hexane to hexadecane. Teja (14) also applied the extended correspondingstates method to mixtures containing polar components such as ammonia and hydrogen sulfide.
In 1974 Goodwin published a very highaccuracy, widerange equation of state for methane (62) which was capable of providing complete thermodynamic data from the triple point to 70 MPa and 500 K. This referencefluid equation of state, along with the Leach shape factors, was used in a series of studies of liquefiednaturalgas (LNG) properties by Mollerup and coworkers. In the first study (63) Mollemp and Rowlinson found that it was possible to reproduce LNG densities to within 0.2%, even down to reduced temperatures of 0.3. In 1975, Mollemp (64) continued his study of LNG properties and reported results for phase equilibria, densities and enthalpies in both the critical and normalfluid regions. The method was also
312
applied to natural gas, liquefied petroleum gas (LPG) and related mixtures in a 1978 investigation (65, 66). Mixtures studied included methane through pentane and common inorganics such as N2, CO, CO2 and H2S. The paper reported density predictions to within ±0.2%, dew and bubblepoint errors "not exceeding those of good experimental data" and errors in liquidphase enthalpies which were less than +2 kJ/kg.
During the mid1970s, researchers at the National Institute of Standards and Technology (NIST) at Boulder, Colorado undertook a series of projects with the objectives of measuring and predicting the properties of LNG and related mixtures. One result from this project was an extended correspondingstates model for LNG densities developed by McCarty (67). That implementation used a 32term, modified BenedictWebbRubin equation of state for methane as the reference fluid and shape factors which had the same functional form as those proposed by Leach, but which had been refit to LNG density data. The model reproduced the available LNG density data to within +0.1%. Eaton et al. (68) used McCarty's methane equation to predict critical lines and VLE in methane+ethane mixtures. Another part of the NIST study focused on the development of predictive extended correspondingstates models for transport properties (36, 6974). That work has been reviewed recently (75) and will not be included here. As mentioned in Section 8.4, however, the transport property work produced another referencefluid equation of state for methane that was extrapolated to 40 K so as to avoid problems with the relatively high triple point of methane. That equation was later used by Romig and Hanley (76) as the referencefluid equation to predict Type III phase equilibria in nitrogen+ethane mixtures.
In addition to the studies mentioned here, Mentzer et al. (77) summarized extended correspondingstates results for phase equilibriumwespecially for systems containing hydrogen and common inorganics. They found accurate purefluid predictions for nonpolar compounds up to about C7. For mixtures they found a strong dependence on the binary interaction parameters but once those parameters were optimized for phase equilibrium, they could be used to represent a variety of properties accurately, without further optimization.
An overriding conclusion from all of these studies is that, while extended corresponding states calculations are very accurate for systems of similar molecules, the predictions tend to decrease in accuracy as the system of interest deviates in size and shape from methane. More generally, as the components in a mixture become more dissimilar both in size and polarity, there is a marked decrease in the accuracy of the predictions. With this situation in mind, a research program was initiated by Ely and coworkers in the mid1980's to investigate the possibility of improving the extended correspondingstates model in three areas: 1) Development of more realistic reference fluids, 2) Development of better shapefactor generalizations and 3) Development of improved mixing rules. In a sense, the most important part of this work has been the development of a base of highaccuracy equations of state that can be used as reference fluids or which can be used to generate 'exact' shape factors as described in Section 8.3. The equations have been reported by Younglove and Ely (37) and Ely and coworkers (46,47,7882). In addition, during the same time frame, highaccuracy equations have been reported by Jacobsen, Pennoncello and Beyerlein and coworkers at the University of Idaho (38,8390), Wagner and coworkers at Bochum (9195) and deReuck and coworkers at Imperial College (9698). Other work includes the studies of Haar and Gallagher on ammonia (99), Haar, Gallagher, and Kell on water (100) and equations summarized by Younglove (101). Most of the recent work in this area has focused on the development of highaccuracy equations of state for alternate refrigerants (102107).
Examples of applications that have incorporated these new equations are diverse and are only briefly summarized here. Parrish (108, 109) performed measurements of ethane+propane
313
and propane+butane mixtures and compared those measurements to extended corresponding states calculations using the 32term MBWR equations for methane, ethane, propane and butane. Ely and coworkers have applied the extended correspondingstates principle with exact shape factors to a series of carbondioxiderich mixtures including (0.98 CO2 + 0.02 N2) (110), (0.98 CO2 + 0.02 CH4) (111), (0.96 CO2 + 0.02 CH4 + 0.02 N2) (80), (0.99 CO2 q 0.01 C2H6) (112), (0.96 CO2 + 0.02 CH4 + 0.01 N2 + 0.01 CzH6) (80) and (0.90 CO2 + 0.10 CH4) (80). The average absolute percent deviation in density prediction for all of these systems, which include approximately 500 data points, was 0.29% with a rootmeansquare percent error of 0.43%. Figure 8.8 illustrates some of those results. Friend and Ely (113) also investigated the methane+ethane system over a wide range of conditions.
' ° I , , , t , r , 1 , 01  co,, ,,, !
0
0.0
1.0
2.0 0.00 10.00 20.00 30.00 40.00
X
CL _.._ 0 m (13 r J
(3_ v 0 C ) . , e 
2.0
1.0
0.0
i .0
2.0 0.00
' I ' I ' I '
C02+ C2H 6
0 0 0
P ° l , I , I , lo.oo 20.00 30.00 40.00
2.0 f ~ t l t I t ] '
C02+ CH 4 1.0
©
o.o o c ~ o o o o
1.0 L ~ I
2.0 I u ~ l ~ , ~ l ~ , ~ 0.00 10.00 20.00 30.00
Pressure, MPa 40.00
Figure 8.8 Comparison of extended corresponding states predicted and measured densities of carbon dioxide rich mixtures (80).
314
2 . 0 0 / ' " ' ' ' ' " I ' ' ' ' ' ' " I ' ' ' ' ' ' " I
0.00 / ( ~ ~  ~ ' ~ ~ ~ ' ~   ~ > ~ 
©© o
l , , , , J , , l , , i , l l , , l i I J , , , i , l  2 . 0 0
2 . 0 0
0.00
 2 . 0 0
I I I I I I ] ' r B l a n k e
I I I I I I I I
' ' ' ' ' " ' i ' ' ' ' ' " ' i ' ' ' ' ' " ' I ' ' ' ' ' " ' Kosolov
, , , , , , , , I , , , , , , , , I ©
I I I I l l l l i ,, I I I I I I I I
E 0 . m
G) r~ > ,
¢/} ¢..
rt
2 . 0 0
0 . 0 0
 2 . 0 0
I I I I I I I i I
I I I 1 I I l i d I
' ' ' ' ' " ' I ' ' " ' ' ' " ' I ' ' " ' ' " ' I © Michels
, , , , , , , I , , , , , , , , I , , , , , , , , I
2 . 0 0
0.00
' ' ' ''"'I ' ' ' ''"'I , , , ,a,,,I ' ' ' ''"
Romberg
 2 . 0 0 I I , I l l l l l I I I l l t i t l t , I I l l l J l
0.01 0 . 1 0 1 .00 1 0 . 0 0 Pressure, MPa
I I I I I I I I
1 0 0 . 0 0
Figure 8.9 Comparison of ECST calculated and experimental densities for air.
Air and air component mixtures (e.g., nitrogen, oxygen, argon and carbon dioxide) have been studied by Ely (114) and Clarke et al. (115,116). This system is especially well suited to the van der Waals onefluid theory since its components are very similar in size. Figure 8.9 compares some typical predictions of ECST for the density of air.
In 1987, Gallagher et al. (117,118) used the extended correspondingstates formulation to generate tables of thermodynamics properties and phase equilibria for an isobutane + isopentane mixture for use as a working fluid in a geothermal power cycle. They concluded that predictions of the singlephase mixture pVT surface and the dewbubble curve were generally better than 0.5% (usually 0.2%) except in the extended critical region. More
315
recently Levelt Sengers and coworkers (119) have applied the extended correspondingstates model to aqueous systems. In one study (120) a model was developed for carbon dioxide in water at nearcritical and supercritical conditions. This study found that the model could be used reliably for the prediction of thermodynamic properties and phase boundaries at the waterrich end in a large region around the critical point of water. Figure 8.10 compares the predicted and experimental critical line, while Figure 8.11 shows the apparent molar volumes and excess volumes. The agreement in all cases is very satisfactory. A second study investigated the nitrogen+water system (121). For this system it was found that the Henry's constants near the purewater vaporpressure curve were well represented, but the mixture critical line and phase boundaries could not be accurately reproduced. An interesting observation from this study was that the phase behavior of generalized correspondingstates models has not been systematically explored and that such an exploration seems urgently needed. In a more recent study Gallagher et al. (122) used the extended corresponding states theory to predict critical lines in aqueous mixtures.
As a final application of the extended correspondingstates theory, Huber and Ely (123) developed a predictive correspondingstates model for refrigerant mixtures using R134a as the reference fluid and determining shape factors by the saturationboundary method described in section 3. Generally the results were very good with deviations of the predictions being less than 1.0% for these highly polar mixtures..
3 0 0 1 .... [ 1 1 1
a_ 200
(b ::::3 o9
G) ,. 1 0 0 cl
tO I
, A I !
!
,., 6 i
/ £x
_ 1 1 1 . . . . . . . t
0 0.1 0.2 0.3 0.4
X C 0 2
Figure 8.10 Comparison of calculated and experimental critical line for the carbon dioxide+water system (120).
316
0 E
03
E
:> 49
2.0
1.5
1.0
0.5
I I 1 1 1 1 l
 p = 28 M P a
_ 0.0015 _
y  / / / ~ .  ~  . ~ M Pa 
0 .1 1 1 ,1, 1, 1 I I
640 660 680 700 720
Temperature / K
Figure 8.11 Comparison of calculated and experimental apparent molar volumes in carbon dioxide+water mixtures along three supercritical isobars (120).
8.6 CONCLUSIONS
The extended correspondingstates theory incorporating shape factors can predict thermodynamic properties of mixtures very accurately, especially if the exact or saturation boundary methods are used. Due to the higher level of complexity of this method, recent applications have been limited to studies where very high accuracy is of importance. Advances in the development of highaccuracy equations of state for a wide variety of substances (e.g., refrigerants and other polar compounds such as water and ammonia) has enabled researchers to extend the ECST methodology to a wide variety of systems and this trend will continue in the future.
Previous studies have shown that the approach is not very suited for the prediction of excess properties especially if there are substantial differences in molecular size. Since the pure correspondence is exact, this failure can be attributed to a failure of the van der Waals onefluid mixing rules to correctly map the composition dependence of higherorder temperature terms in the referencefluid equation of state. Efforts are Underway to develop
317
Acknowledgements J. F. Ely was supported by the U. S. Department of Energy, Basic Energy Sciences under grant number FG0395ER41568 and I. M. F. Marrucho was supported by JINCT Programa Ciencia BD\1534\91RM.
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