Physical Chemistry Third Edition

184 4 The Thermodynamics of Real Systems

so that the natural independent variables forAareT,V,n 1 ,n 2 ,...,nc. By inspection

μi

(

∂U

∂ni

)

S,V,n′

(4.5-10)

and

μi

(

∂A

∂ni

)

T,V,n′

(4.5-11)

Exercise 4.14

Derive Eqs. (4.5-8) and (4.5-9).

The chemical potential is equal to four different partial derivatives with different

variables held fixed. The partial derivative in Eq. (4.5-4) identifies the chemical poten-

tial as thepartial molar Gibbs energy. A generalpartial molar quantityis a partial

derivative of an extensive quantity with respect to the amount of one component, keep-

ingT,P, and the amounts of all other components fixed. If the letterYstands for

any extensive quantity (U,H,A,G,S,V, and so on),the partial molar quantity for

substance numberi is denoted byYiand defined by

Yi

(

∂Y

∂ni

)

T,P,n′

(4.5-12)

The chemical potentialμi is equal toGi. The partial derivatives in Eqs. (4.5-7),

(4.5-10), and (4.5-11) to whichμiis equal are not partial molar quantities, because

PandTare not both held fixed in the differentiations.

EXAMPLE4.18

Find a relationship between the chemical potential and the partial molar enthalpy.

Solution

We begin with the relationship betweenGandH:

GH−TS

Differentiation of both sides at constantT,P, andn′gives

(

∂G

∂ni

)

T,P,n′



(

∂H

∂ni

)

T,P,n′

−T

(

∂S

∂ni

)

T,P,n′

or

μiHi−TSi