Physical Chemistry Third Edition

190 4 The Thermodynamics of Real Systems

The Gibbs–Duhem Relation

From Euler’s theorem, Eq. (4.6-3), we can write an expression fordY, the differential

of an extensive quantity denoted byY:

dY

∑c

i 1

nidYi+

∑c

i 1

Yidni (4.6-7)

This equation represents the effect onYof any infinitesimal change in the state of the

system such as changing its temperature or pressure or adding one of the substances.

ConsideringYto be a function ofT,P, and then’s, we write another expression

fordY:

dY

(

∂Y

∂T

)

P,n

dT+

(

∂Y

∂P

)

T,n

dP+

∑c

i 1

Yidni (4.6-8)

We equate the right-hand sides of the two equations fordYand cancel equal sums:

∑c

i 1

nidYi

(

∂Y

∂T

)

P,n

dT+

(

∂Y

∂P

)

T,n

dP (4.6-9)

Equation (4.6-9) is called thegeneralized Gibbs–Duhem relation.

Theoriginal Gibbs–Duhem relationis a special case that applies to the Gibbs energy

at constantTandP:

∑c

i 1

nidμi 0

(the original Gibbs–Duhem

equation, valid at constantTandP)

(4.6-10)

In a two-component mixture, this equation specifies how much the chemical potential

of one component must decrease if the chemical potential of the other component

increases at constant temperature and pressure:

dμ 1 −

x 2

x 1

dμ 2 (two components at constantTandP) (4.6-11)

EXAMPLE4.22

A two-component ideal gas mixture at constant temperature and pressure has the partial

pressure of gas number 1 changed bydP 1. Show that the expression for the chemical potential

of a component of an ideal gas mixture, Eq. (4.5-26), is compatible with Eq. (4.6-11).

Solution

We need to manipulate−

x 2

x 1

dμ 2 into an expression fordμ 1. From Eq. (4.5-23)

μiμ◦i+RTln

(

Pi

P◦

)

(i1, 2)

At constantTandP,

dμ 2 

(

∂μ 2

∂P 2

)

dP 2 

RT

P 2

dP 2 −

RT

P 2

dP 1