Lecture 18Multicomponent Phase Equilibrium1 Theories of Solution The Gibbs energy of mixing is given...
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Transcript of Lecture 18Multicomponent Phase Equilibrium1 Theories of Solution The Gibbs energy of mixing is given...
Lecture 18 Multicomponent Phase Equilibrium 1
Theories of Solution
0
00 lnlni
iiiii
P
PkTakT
P Pi i niRT
Vi
Xi PiP
ai
ii
iM aXRTG ln
ii
iM XXRTG ln
HM 0
SM R Xi lni Xi
GM TSM
G i
T
P,n j
Si Si '
0
Gi / T 1 /T
P,n j
Hi Hi'
0
The Gibbs energy of mixing is given by:
And the chemical potential is:
For ideal gases, the partial pressures are given by:
And the activity coefficients are:
Substitution 4 into 1 gives;
Thus:
G i
P
P,n j
Vi Vi '
0 VM 0
1- Ideal Gas Mixtures
1
2
3
4
Lecture 18 Multicomponent Phase Equilibrium 2
Ideal Gas Mixtures
GM
0
GA
GB
XB 1
For systems with zero enthalpies of mixing, which we generally call ideal mixtures, the entropy of mixing completely determines G of mixing.
SM
0
S ASB
XB 1
HM
0
H A H B
XB 1
HM 0SM R Xi lni Xi GM TSM
Lecture 18 Multicomponent Phase Equilibrium 3
Ideal Gas Entropy of Mixing
SM R Xi lni Xi
SM RXi ln Xi R(1 Xi ) ln(1 Xi )
dSMdXi
R ln Xi R ln(1 Xi )
For a binary ideal gas mixture we canplot the entropy as a function of composition as shown on the right (inunits of R).
The ideal entropy of mixing:
In the case of a binary becomes:
The slope of the entropy of mixing curve is given by:
This implies that it is impossible to completely purify a material!
S and dS/dX versus X
-8
-6
-4
-2
0
2
4
6
8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X
S o
r dS/d
X
Lecture 18 Multicomponent Phase Equilibrium 4
Raoult’s Law
Pi XiPi0
ai PiPi0
XiPi0
Pi0 Xi
HM 0
SM R Xi lni Xi
If upon forming a mixture the partial pressures of the vapor in equilibrium above the mixture is such that:
Then the pressure of the vapor will be a weighted sum ofthe partial pressures, where the weights are the mole fractions of each component:
P Pii XiPi
0
i
The solution is thus said to be Raoutian or Ideal and P(Xi) for aRaoutian solution is plotted on the right. Also note that for a Raoutian solution:
Since the activity and mole fraction are equal, we have the same thermodynamics as the ideal gas mixture. That is:
GM RT Xi lni Xi TSM
V V
S S
PA0
0 XB 1
PB0
P
Ideal
ai>1
ai<1
Lecture 18 Multicomponent Phase Equilibrium 5
Dilute Solutions
For a dilute solution, the Xi of one component is very small, while Xi for the other component is nearly one. In this case, the activity coefficient i of the dilute component should be composition independent since the component’s environment is constant (it is surrounded by the other component).
As the dilute component is added, the probability of it having a like neighbor is small and so its activity isconstant over a range of dilute concentrations.
PA bXA
aA PAPA0
bXAPA0 A
0XA
For this case, the partial pressure of the dilute componentis proportional to the amountof that component. This is knownas Henry’s Law: 1
0 XB 1
a
A0 1
B0 1
XAdGA XBdGB 0
dG i RTd ln ai
XAd ln aA XBd ln aB 0
Lecture 18 Multicomponent Phase Equilibrium 6
Dilute Solutions
Note that using the Gibbs Duhem equation for the partial molar G, it can be shownthat when B obeys Henry’s law, A obeys Raoult’s Law.
dG i RTd ln ai
XAd ln aA XBd ln aB 0
aB B0XB
d lnaA XBXA
d ln XB d lnB0
d ln aA d ln XA
aA XA
Henry’s Law for B dilute
1
0 XB 1
a
A0 1
B0 1
aA XA
For A-B binary
The infinitesimal change in thepartial molar Gibbs free energy
d lnaA XBXAd lnaB
Raoult’s Law for A:
Lecture 18 Multicomponent Phase Equilibrium 7
Excess Functions
i kT lnai kT ln i Xi kT lni kT ln XiRemember that:
For an ideal solution: i 1
iideal kT ln Xi
ix i i
ideal kT lni
Gi x Nix RT ln i
G' Mx G' M G' M
ideal
GMx RT XA ln A XB lnB
Mix Mi Mi
idealM x M M ideal
We define the excess function as thedifference between the actual valueof the mixture and the value for anideal mixture:
HMideal 0 HM
x HM
VMideal 0 VM
x VM
Lecture 18 Multicomponent Phase Equilibrium 8
Excess Functions
GM HM TSM
GMideal GM
x HMideal HM
x T SMideal SM
x GM
ideal GMx HM
x T SMideal SM
x
GMideal TSM
ideal
GMx HM
x TSMx
The entropy of mixing is usually assumedto be ideal so that the excess Gibbs free energyof mixing is the excess enthalpy of mixing
GMx HM
x
GMideal GM
x HMx TSM
idealThe Gibbs free energy of mixing is then The excess enthalpy of mixing minus Ttimes the ideal entropy of mixing.
Let’s take a closer look at the Gibbs free energy of mixing usingthe concept of excess mixing functions:
Lecture 18 Multicomponent Phase Equilibrium 9
Regular Solutions
The Regular Solution Model is a simple example of a non-ideal solution.
Recall that for a mixture:
GM XiG ii GM RT Xi ln ai
iGi RT ln ai
The partial molar Gibbs free energyof mixing (the difference betweencomponent i’s contribution to G in the mixture versus pure i) is relatedto the activity.
The Gibbs free energy of mixing is the weighted sum of thecontributions from each component.
The Gibbs free energy of mixingis then related to the activities asshown.
In the ideal case the activities were just the mole fractions:
GM RT Xi ln Xii
The excess Gibbs free energy of mixing is the difference between the non-ideal and ideal G of mixing:
GMx RT Xi ln ai
i RT Xi ln Xi
i RT Xi ln i
i
Lecture 18 Multicomponent Phase Equilibrium 10
The Regular Solution Model
The Regular Binary Solution is defined as one which has the following form for the activitycoefficients:
ln A RTXB2
Of course because the mole fraction of component A is just one minus the mole fraction of B we have:
And substituting the activity relationships for the Regular solution gives:
This can be manipulated to find:
GMx RT Xi ln i
i RTXA lnA RTXB ln B
ln A RTXB2
RTxB (1 XA )
RT
(1 XA )2
ln B RTXA2
GMx XAXB
2 XBXA2
The excess Gibbs free energy of mixing is:
GMx XAXB The excess Gibbs free energy of mixing
of the Regular Binary Solution.
Lecture 18 Multicomponent Phase Equilibrium 11
Regular Solutions
And substituting the Regular Solution excess G of mixing:
Notice that the first two terms are the negative ideal entropy of mixing multiplied by T:
GM GMideal GM
x RT Xi ln Xii RT Xi ln i
i
The Gibbs free energy of mixing is the sum of the excess and ideal Gibbs free energies of mixing :
GM RTXA ln XA RTXB ln XB XAXB
SMideal RXA ln XA RXB ln XB
Thus, the last term is the enthalpy of mixing (and also the excess enthalpy of mixing since the idealenthalpy of mixing is just zero):
HM HMx XAXB
Enthlapy of Mixing of the Regular Solution
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(B)
Series1
The enthlapy of mixingof the Regular Binary Solutionwith = 10 J/mol.
Lecture 18 Multicomponent Phase Equilibrium 12
G of Mixing Regular Solution versus T
-1000
-500
0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X
delt
a G
0K
100K
200K
300K
400K
500K
Regular Solutions
Regular Solutions with =10000J/mol.
Lecture 18 Multicomponent Phase Equilibrium 13
Regular Solutions
Regular Solutions at T=300K
G of Mixing Regular Solution vs. Omega
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X
delt
a G
20000J/Mol
15000 J/Mol
10000 J/Mol
5000 J/Mol
0 J/Mol
-5000 J/Mol
Lecture 18 Multicomponent Phase Equilibrium 14
Regular Solutions: Atomistic Interpretation
The enthalpy of mixing is related to the interactions between the atoms that make up the mixture. If the solid has bond energies as follows:
EAA
EAB
EBB
The enthalpy of mixing is given by:
HM ZNA2
2NTEAA
NB2
2NTEBB
NANBNT
EAB NA2EAA
NB2EBB
Z is the coordination number
NT the total number of atomsNA the number of A atoms
Lecture 18 Multicomponent Phase Equilibrium 15
Regular Solutions: Atomistic Interpretation
The enthalpy of mixing is determined as the sum of the total interactions between the like andunlike atoms in the mixture:
EAA
EAB
EBB
Then the enthalpy of mixing is:
HM ZNA2
2NTEAA
NB2
2NTEBB
NANBNT
EAB NA2EAA
NB2EBB
Z is the coordination numberNT the total number of atomsNA the number of A atoms
HM ZNA2
NANT
EAA
NB2
NBNT
EBB NA
NBNT
EAB
NA2EAA
NB2EBB
HM ZNA2
NANT
EAA
NB2
NBNT
EBB NA
NBNT
EAB
NA2
NANTA
EAA
NB2
NBNTB
EBB
A B B A(in A) B(in B)
Lecture 18 Multicomponent Phase Equilibrium 16
Regular Solutions: Atomistic Interpretation
HM ZNTXAXB EAB 12EAA EBB
Notice that the number of A atoms divided by the total number of atoms is just the mole fraction of A. Substituting in the mole fractions gives:
This is the correct form for the Regular solution where we make the definition:
ZNT EAB 12EAA EBB
The enthalpy of mixing of the Regular Binary Solution is determined by the difference between the AB bond energyand the average of the AA and BB bond energies.
Various more complex models of solutions have been developed with more complicatedexpressions for the enthalpy of mixing, including: next nearest neighbor interactions, non-ideal entropies of mixing, etc.
E
R
AA
BB
AB
Lecture 18 Multicomponent Phase Equilibrium 17
Solution of Defects
We could extend the principles of the thermodynamics of mixtures between atoms to mixtures between atoms and defects, such as vacancies
Vacancies increase energy because they result in broken bonds and the decrease in energy due to the entropy they contribute from the uncertainty of their placement in the solid.