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5.60 Thermodynamics & Kinetics

Spring 2008

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5.60 Spring 2008 Lecture #14 page 1

Multicomponent Systems, Partial Molar Quantities,

and the Chemical Potential

So far weve worked with fundamental equations for a closed (nomass change) system with no composition change.

dU TdSpdV dA SdT = = pdVdH =TdS+Vdp dG = SdT +Vdp

How does this change if we allow the composition of the system to

change? Like in a chemical reaction or a biochemical process?

Consider Gibbs free energy of a 2-component system ( , , 1, 2 )G T p n n

dG =

T

G , , dT +

G

p

dp+

G

n1

dn1 +

n

G dn2

p n n 2 1 2 , ,1 2 , , 2 T p n, , 1 T n n T p n S V 1 2

We define i G

as the chemical potentialof species iniT p n, , i

( , , ) is an intensive variablei T p nThis gives a new set of fundamental equations for open systems

(mass can flow in and out, composition can change)

dG = SdT +Vdp+ idnii

dH =TdS+Vdp+ idnii

dU TdSpdV + i i= dni

dA= SdT pdV + idnii

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5.60 Spring 2008 Lecture #14 page 2

G H U Ai = = = =

S p n , T V n,niT p n, , ji ni , , ji ni S V n, ji ni , ji

At equilibrium, the chemical potential of a species is the same

everywhere in the system

Lets show this in a system that has one component and two parts,

(for example a solid and a liquid phase, or for the case of a cell

placed in salt water, the water in the cell versus the water out of

the cell in the salt water)

Consider moving an infinitesimal amount dn1 of component #1 from

phase a to phase b at constant T,p. Lets write the change in state.

( , , phase a) = dn T p ( , ,phase b)dn T p 1 1adG = 1

( )b 1( ) dn1

a1( )b < 1( ) dG < 0 spontaneous conversion from (a) to (b)b1

( )a < 1( ) dG > 0 spontaneous conversion from (b) to (a)

At equilibrium there cannot be any spontaneous processes, so

b1( )a = 1

( ) at equilibrium

e.g. liquid water and ice in equilibrium

waterice ice( , ) = water( , )T p T p

at coexistence equilibrium

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5.60 Spring 2008 Lecture #14 page 3

water cell T p ( , andFor the cell in a salt water solution, ( ) , > water solution) T p( ) ( )the cell dies as the water flows from the cell to the solution (this is

what we call osmotic pressure)

The chemical potential and its downhill drive to equilibrium will be

the guiding principle for our study of phase transitions, chemical

reactions, and biochemical processes

Partial molar quantities

i is the Gibbs free energy per mole of component i, i.e. the

partial molar Gibbs free energy

G = i = Gi ni T p n , , j iG = n + n + + n = n n G"1 1 2 2 i i i i = i i

i iLets prove this, using the fact that Gis extensive.

G T p( , , n1, n2 ) = ( , , 1, 2 )G T p n n

( , , 1, n ) ( , , 1,dG

T p n 2 = G T p n n 2 )d

G

( )n

1

G

(

n2 )

+ = G n n n2 , ,n ( )1 T p, , ( ) T pT pn 2 , , 1 1, , 2 T p n

n1 n2

n1 1 + n22 = G

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__________________________________________________

5.60 Spring 2008 Lecture #14 page 4

We can define other partial molar quantities similarly.

A

= Ai A= n A+ n A +"+ n A = n A1 1 2 2 i i i i

i ni , , j iT p n

partial molar Helmholtz free energy

note what is kept constant not to be confused with

n

A

= i iT V n , , j i

H

n = Hi H = n H + n H +"+ n H = n H1 1 2 2 i i i i

i i T p n , , j ipartial molar enthalpy

U

ni

, , j i

= Ui U = nU + n U +"+ nU = i1 1 2 2 i i nUii T p n

partial molar energy

Lets compare of a pure ideal gas to in a mixture of ideal gases.

Chemical potential in a pure (1-component) ideal gas

o oFrom ( , ) = G ( ) + RTlnp

( , ) = T + pG T p T T p ( ) RTlnp0 p0

Chemical potential in a mixture of ideal gases

Consider the equilibrium

pA +pB =ptot

mixedA,B

pureApAp 'A , p 'B

Fixed Partition

\

At equilibrium mix T p, , = pureT pA( tot) A( , , A)

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5.60 Spring 2008 Lecture #14 page 5

Also pA(pure) =pA (mix) =ptotXA

Daltons Law

SoA(mix T p, , tot) = A(pure T ptotXA), ,

A( ) + RT p X

= o T ln tot A p0

= Ao ( )T + RTln

ptot + RTlnXAp0

(pure T ptot)A , ,

A( , , tot) = A(pure T ptot) + RTlnXAmix T p , ,Note XA < 1 A (mix T p , tot) < A ( , tot), ,pure T pThe chemical potential of A in the mixture is always less than the

chemical potential of A when pure, at the same total pressure. Thisis at heart a reflection about entropy, the chemical potential of A

in the mixture is less than if it were pure, under the same (T,p)

conditions, because of the underlying (but hidden in this case)

entropy change!