Supplemental Derivation

8/14/2019 Supplemental Derivation

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Solvation Thermochemistry

The treatment of solvent effects on the thermochemistry and kinetics of systems in

condensed phases has been studied theoretically and empirically in a number of fields.

However, often ambiguities present in the underlying assumptions and confusion with

regard to standard states and reference states can cause difficulty when attempting to use

the results correctly. An example of where confusion arrives could be in the typical

concentration-based equilibrium constant CK , which is a physical property of the

system. However, the free energy change of reaction that it is related to in

phenomenological thermodynamics is actually the standard state change oRxnG , since

the actual change in free energy at equilibrium must be zero. This means that there are an

infinite number ofo

RxnG values, since the standard state can be chosen arbitrarily. The

same problem is seen for gas phase systems, but it is less problematic since the nearly

universal standard state is an ideal gas at 1 atmosphere or bar of pressure. A simple

example derivation for a reaction in a condensed phase is given in terms of the typical

standard state thermodynamic quantities, as well as the pseudo-chemical potential

* *i ior G of Ben-Naim. The concepts of Gibbs free energy and chemical potential will

be used almost interchangeably here, with the knowledge that chemical potential is more

specifically a partial-molar Gibbs free energy.

The pseudo-chemical potential (PCP) as defined by Ben-Naim is the chemical potential

(CP) of a solute confined to a fixed position, and is denote by the * superscript. The

concentration dependence due to the translation is no longer present in a PCP; however,

any concentration dependence of the chemical potential due to non-ideal effects would

still be included. As will be seen below, one benefit of the PCP is that it will allow for a

direct correspondence with experimental measurements, without the need for standard

states. There are other subtleties to the PCP definition and use, and the reader is

encouraged to seek the aforementioned references for more information. Equation

relates the traditional CP to the PCP as defined by Ben-Naim. Equation is a useful

relationship that will be used to convert between the standard state CP and the PCP,


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where is the number density (or molar concentration) and 3 is the momentum partition

function. It is derived by equating the traditional and PCP-based expressions for the

chemical potential.

# * # 3lni i i ikT

#, #,

*# 3 *# 3

, ,ln ln ln lno o o oi ii i i i i i i io o

i i

kT kT or kT kT

In all equations, an o superscript will be used to signify the standard state, a + will be

used to signify an arbitrary reference state/behavior, a # will be used for the actual state

of the mixture, an will be used for the dilute-limit behavior reference, and a p will

be used for the pure i reference state. For example,*#

i is the PCP at the actual state of

the mixture, ando

i is the CP at the standard state conditions. The symbolxi will be used

for condensed phase mole fractions, and yi will used for gas phase mole fractions (orpi

for the partial pressure). Unless otherwise noted, the condensed phase fugacity reference

will be either dilute-limit behavior extrapolated to xi = 1, or Lewis-Randall behavior atxi

= 1. The analysis is not requisite on this assumption, but simplifies the notation

somewhat since we do not need a symbol for the mole fraction at which the reference was

defined. The terms,

i

or,

i

represent the correction that must be made to the

behavior at the state to achieve the behavior at state , and are known as fugacity

and activity coefficients. All activity coefficients discussed here will be based on the

mole fraction concentration scale, as is typical in chemical engineering thermodynamics

using the fugacity formalism.

Derivation of a Condensed-Phase Equilibrium ExpressionThe general method used to derive the equilibrium relationships shown in the paper will

be demonstrated here through a simple solution-phase equilibrium constant example. It

will be based on the use of the concept of fugacity as a means to relate the behavior of the

chemical potential in the actual state to some arbitrary reference behavior. This reference

behavior could theoretically be anything, but practically it is useful to define the


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reference fugacity as a straight line as a function of mole fraction. Typical examples of

this would be to assume the reference behavior as Henrys Law behavior or ideal solution

behavior over the entire mole fraction range. These reference lines are illustrated by

dashed lines in the figure below, which was adapted from the 3rd edition of

Thermodynamics and Its Applications by Tester and Modell. Henrys Law is the

extrapolation of the fugacity slope at low concentrations, and ideal solution behavior is a

straight line connecting zero and the pure component fugacity pif at a mole fraction of

one. First, the concentration-based equilibrium constant (KC) will be derived in terms of

the standard state free energy change of reaction. Then, an expression forKC in terms of

the PCP difference will be derived.

xi

#i

f

p

if

if

#,

i

#,p

i

##,

#

i

i

i i

f

x f

+

+=

#i

f

0 1

Standard State Methodology

The derivation begins with a typical expression for the chemical potential in terms of

fugacity in the condensed phase. A straightforward extension of this example to the gas

phase or liquid-vapor equilibrium is possible.

#

#

, ,

ln

o i

i liq i liq o

i liq

fRT

f


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In this expression,#

,i liq is the chemical potential in the actual state, ,o

i liq is the standard

state chemical potential,#

if is the fugacity in the actual state, and o

if is the fugacity in

the standard state. Since the goal is to derive an equilibrium relationship for the example

reaction A B , one must realize that the actual state chemical potentials ofA and B

must be equal at equilibrium. Writing equation for each species and setting the CPs

equal allows one to arrive at equation , dropping the explicit condensed phase notation. A

simple rearrangement of the terms allows one to extract the well-known activity-based

equilibrium constant,KA, which is defined in terms of the standard state CP difference or

the standard state Gibbs free energy change of reaction as shown in equation .

# #

ln ln

o oA BA Bo o

A B

f fRT RT f f

##

exp exp

o o o oB A Rxn B A

A o

B A

G f fK

RT RT f f

In order to simplify this relationship, one has to redefine the fugacity as a product of a

simple reference fugacity and a term accounting for the non-linear behavior as a function

of composition. The reference fugacity behavior will take the form of a line connecting

the zero fugacity point atxi = 0 with some arbitrary point at xi = 1. As alluded to earlier,

typical examples of this arbitrary point are the Henrys Law extrapolation or the pure

component fugacity. The common name for the second term is the activity coefficient,

which corrects the reference state fugacity at a given composition to the actual state

fugacity at that composition. In general, the activity coefficient is very complicated and

depends on the composition of the mixture, the reference behavior chosen, temperature,

and pressure. A general expression for the fugacity of a species in solution is givenbelow in equation .

# # #,

ii i i

i

ff x

x


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Here,#

ix is the mole fraction in the actual state,

if is the reference fugacity defined at

the mole fraction ix

on the arbitrary reference line betweenxi = 0 andxi = 1, and#,

i

is

the activity coefficient needed to convert from the reference fugacity to the actual state

fugacity. Since if

lies on the reference line between mole fractions of 0 and 1, it is

customary and logical to take 1ix , which simplifies the expression and results in

equation .

# # #,i i i if x f

Now, if

is the reference state fugacity atxi = 1, which means that the reference fugacity

at any mole fraction can be defined as#

i ix f . Essentially, equation says that the fugacity

in the actual state is equal to the reference fugacity at the same composition as the actual

state multiplied by the activity coefficient that corrects for deviations from the reference

state behavior. It is often helpful to see this graphically, and the reader is urged to

supplement this text with graphical representations of the fugacity coefficient that can be

found in most chemical engineering thermodynamics and physical chemistry textbooks.

Now that the fugacity has been defined, it can be used in the previously derived

equilibrium constant relationship. This definition replaces the actual state fugacity and

the standard state fugacity, and the result is shown in equation . To be explicitly clear,

reference state fugacities typically make an assumption about linear behavior across the

entire composition range; however, actual state and standard state fugacities and chemical

potentials should include all non-idealities.

# #, ,

, # #,exp

o o o

Rxn B B B A A AA o o

B B B A A A

G x f x f K

RT x f x f

Up to this point, very few assumptions have been made regarding the behavior or state of

the system, and it is desired that the assumptions be kept to a minimum to allow the

reader to see where all terms arise in the equilibrium expression. However, we will now

make one simplifying assumption, which is that the reference behavior of a given species


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is taken to be the same for the actual state and the standard state. This is a very logical

assumption that was implicitly made when equation was written with if

identical for

the actual state and standard state. This means that ratio of if

s in the equation will be

equal to one, resulting in the simplified equation . However, let it be clear that A Bf f

was notrequired to make the simplification, only that the standard state and actual state

fugacities for a given species were based upon the same reference behavior, such as

Henrys Law or ideal solution behavior. Different reference behaviors may be chosen for

each molecule as desired, and the choice will not affect the resulting equilibrium

constant, assuming any activity coefficient can be calculated to the same degree of

accuracy.

# #, ,

, # #,exp

o o o

Rxn B B A AA o o

B B A A

G x xK

RT x x

Equation is a general expression for the equilibrium constant and shows several

important aspects ofKA. First, it is dependent on the standard state, which should be

obvious given that it is defined by the standard state free energy change, and there are

standard state mole fractions in the definition as well. It can also be seen that the

reference behavior chosen in defining the fugacity is completely arbitrary, but this

arbitrariness is compensated for by the activity coefficient. Therefore, it does not affect

the equilibrium constant. For example, if you define your reference far from the behavior

of the actual system, then the actual state activity coefficient will be very large, and if you

define it close to the actual state behavior the opposite will be true. This brings to light

another important aspect, which is that the activity coefficients for the actual state and

standard state are completely independent because the standard state is arbitrary. In fact,

it is usually the case that the standard state and the actual state compositions are different

(otherwise 0o

RxnG ), which necessarily makes the activity coefficients different. It is

often possible to define the reference behavior beneficially, such that the activity

coefficient of the actual state or standard state is close to one. In many cases and

especially with computational chemistry calculations, the standard state is defined by the


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calculation or experiment that one can perform, and the reference state follows logically

from that. In a computational chemistry calculation, the state of the molecule is usually

an isolated molecule in a continuum, which essentially mimics Henrys Law behavior

where solute-solute interactions are negligible. If the standard state concentration is also

taken to be low, then the Henrys Law reference would be a logical choice because it

would allow one to assume that, 1oi

, simplifying the expression for the equilibrium

constant.

If one is attempting to construct detailed kinetics models, then it is necessary to have the

concentration-based equilibrium constant (KC) to calculate reverse rate constants. The

activity-based equilibrium constant can be related to KC with relatively little effort. In

order to do this, the solution phase mole fractions must be converted into concentrations,

by defining the mole fraction as the concentration of i iC divided by the total

concentration TC . Applying this definition, one can arrive at equation .

# #, # ,

# , # #,

#, # ,

# , #,

expo o o o

Rxn B B T T A AA o o o

T B B A A T

o o o

B T T A A

C o o oT B B A T

G C C C C K

RT C C C C

C C C

K C C C

One can quickly notice that the total solution concentration is not necessarily the same at

the actual state composition and the standard state composition. In this equimolar

reaction example, the total concentration factors would cancel as long as the standard

state was chosen to be a mixture of A and B in a solvent witho

AC and

o

BC being the

concentrations in solution. Note that one does not necessarily need the standard state

concentrations of A and B to be equivalent, though they usually are taken to be so.

However, for a non-equimolar reaction, one would always be left with a term of the form

#n

o

T TC C

. Therefore, a more useful condition would be when the total concentration

under the standard state conditions and actual state conditions are equal. If this is the

case, then all total concentration terms will cancel out and equation can be further


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simplified. Generally, this condition is usually not satisfied rigorously, but at relatively

low concentrations in aqueous solution, the error introduced by making this assumption

will be small. Using this assumption, one can arrive at equation , which relates the

concentration-based equilibrium constant to the standard state free energy change.

, #,

#, ,exp

o o o

Rxn B B AC o o

A B A

G CK

RT C

Equation can then be evaluated if one is able to estimate a standard state free energy

change, the activity coefficient of A and B in the actual system state, and the activity

coefficient of A and B in the standard state mixture. These are not necessarily easy tasks

to accomplish, and often further assumptions can be made based upon the system

conditions and the choice of standard and reference states to simplify the process.

It is useful to say a bit about what equations and actually mean from a physical

perspective. One starts with the free energy change under the standard state conditions.

The standard state activity coefficients in the equation serve to remove the non-idealities

at the standard state, essentially leaving a free energy change under the reference state

behavior at the standard state concentration. The ratio of standard state concentrations

addresses the simple concentration dependence of the free energy. The actual stateactivity coefficients then take the free energy change under the reference state behavior to

the behavior at the actual system conditions. It is very much like a thermochemical cycle

with three steps: (i) real behavior to reference behavior at the standard state composition,

(ii) correction for the composition difference between the standard state and actual state,

and (iii) reference behavior to real behavior at the actual state composition. It is not

exactly a cycle as written, but looks very much like one if the equation is solved for

o

RxnG

. This brings up an interesting point about what one needs to do when defining thestandard state free energy change. Although the work here assumes that

o

RxnG includes

all non-idealities, making it a true free energy change under the standard state

conditions; one should realize that this is not a necessity. As in the computational

chemistry calculations, it may be much easier to obtain energies under very dilute


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conditions, ignoring non-ideal effects. However, it is also possible to evaluate the

partition functions at a concentration of far from the dilute limit. If this is done, one is

left with a free energy estimate that ignores non-idealities, but is not evaluated under

conditions where non-idealities can be ignored. The result is an estimate that lies on the

reference behavior line, which would be the Henrys Law extrapolation in this dilute-

referenced example. If this type of standard state energy data is used, one must simply

throw out the standard state activity coefficients in equations and because you already

lie on the reference behavior line. Then steps (ii) and (iii) of the cycle are the same.

The point to be taken away here is that the standard state is arbitrary and can be defined

in a variety of ways; however, one needs to be sure of exactly what assumptions are

implicitly accounted for in the equations and data.

Pseudo-Chemical Potential Methodology

In this part, extensive use will be made of the previously-derived standard state

expressions. First, if one recalls equation for the condensed phase and writes the

standard state CP difference for the reaction, equation is the result. Equation was

derived by equating and , using equation to define the fugacity, and assuming that the

total concentration of the actual state and the standard state are equivalent # 0T TC C .

This assumption could be relaxed, but would result in more complicated notation andmore intimidating equations.

3 #, ,

*# *#

3 , #,

3 #, ,

3 , #,

ln ln

ln ln

o oo o B B B A

B A B A o o

A A B A

o oo B B B A

Rxn Rxn o o

A A B A

kT kT or

G G kT kT

Once equation has been established, the expressions in terms of the PCP difference

RxnG come rather naturally from the previous equations in terms of oRxnG . The

relationship between KA and the PCP difference is shown in equation and is found by

simply inserting equation into equation . The expression forKC can be found by

combining equations and , realizing that i and Ci are the same, and canceling terms.


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The simple result is shown in equation . The momentum partition function terms still

remain because the PCP is the CP without the momentum partition function. In a

unimolecular reaction or when examining phase equilibria for a single molecule, the

lambda ratio will be unity because of conservation of mass and the classical definition

used by Ben-Naim that is only a function of mass and temperature. One can also see that

equation can be derived directly by equating for A and B at equilibrium and solving for

the concentration ratio,KC.

3 #, ,

3 , #,exp

o o

Rxn A A B AA o o

B B B A

GK

RT

3

3expRxn A

C

B

GKRT

Although this example was for the specific case of unimolecular reaction equilibrium in a

condensed phase, it should be straightforward to extend this to non-equimolar reactions,

gaseous processes, and liquid-vapor equilibria. Many other equilibrium relationships can

be derived in a similar manner. In this derivation, an attempt was made to keep

assumptions to a minimum and to pedagogically explain them when implemented. It is

hoped that it was completed with enough clarity and rigor to allow others to evaluate

what assumptions hold for their application and to modify the given expressions as

necessary.

Supplemental Derivation

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