Lecture 18Multicomponent Phase Equilibrium1 Theories of Solution The Gibbs energy of mixing is given by: And the chemical potential is: For ideal gases,

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


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


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


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


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


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:


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


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:


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


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.


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.


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.


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


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



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)



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


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.

Lecture 18Multicomponent Phase Equilibrium1 Theories of Solution The Gibbs energy of mixing is given by: And the chemical potential is: For ideal gases,

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