Physical Chemistry Third Edition

4.5 Multicomponent Systems 183

However, the partial derivatives in this equation are not equal to any simple thermody-

namic variables as are the partial derivatives in Eq. (4.5-2). We therefore say that the

natural independent variablesfor the Gibbs energy areT,P,n 1 ,n 2 ,...,nc.

EXAMPLE4.17

Use an analogue of Eq. (B-7) of Appendix B to write a relation between (∂G/∂ni)T,V,n′

andμi.

Solution

(

∂G

∂ni

)

T,V,n′



(

∂G

∂ni

)

T,P,n′

+

(

∂G

∂P

)

T,n

(

∂P

∂ni

)

T,V,n′

μi+V

(

∂P

∂ni

)

T,V,n′

The internal energy, the enthalpy, and the Helmholtz energy have their own sets

of natural independent variables. From Eq. (4.5-3), Eq. (4.5-8), and the relation

GH−TS,

dHdG+TdS+SdT

dH−SdT+VdP+

∑c

i 1

μidni+TdS+SdT

dHTdS+VdP+

∑c

i 1

μidni (4.5-6)

The natural independent variables forHareS,P,n 1 ,n 2 ,...,nc. We can see from

Eq. (4.5-6) that

μi

(

∂H

∂ni

)

S,P,n′

(4.5-7)

Similarly, sinceUH−PV,

dUTdS−PdV+

∑c

i 1

μidni (4.5-8)

so that the natural independent variables forUareS,V,n 1 ,n 2 ,...,nc. Also

dA−SdT−VdP+

∑c

i 1

μidni (4.5-9)