Physical Chemistry Third Edition

4.5 Multicomponent Systems 185

Exercise 4.15

a.Show thatμiAi+PVi

b.Show thatμiUi+PVi−TSi

There are some equations similar to the Maxwell relations that apply to multicom-

ponent open systems. We begin with the Gibbs equation, Eq. (4.5-3):

dG−SdT+VdP+

∑c

i 1

μidni (4.5-13)

Using the Euler reciprocity relation, Eq. (B-13) of Appendix B,

(

∂S

∂ni

)

T,P,n′

Si−

(

∂μi

∂T

)

P,n

(4.5-14)

A second use of the Euler reciprocity relation gives

(

∂V

∂ni

)

T,P,n′

Vi

(

∂μi

∂P

)

T,n

(4.5-15)

Various similar equations can be derived.

Exercise 4.16

Verify Eq. (4.5-14) and Eq. (4.5-15).

The Partial Molar Quantities in a One-Component System

The equilibrium thermodynamic state of a simple one-component open system can be

specified byT,P, andn, the amount of the single component. This gives the differential

relation for a general extensive quantity,Y, in a one-component system:

dY

(

∂Y

∂T

)

P,n

dT+

(

∂Y

∂P

)

T,n

dP+

(

∂Y

∂n

)

T,P

dn (4.5-16)

In a one-component system the molar quantityYmis given by

Ym

Y

n

(4.5-17)

The molar quantityYmis an intensive quantity. Because an intensive quantity cannot

depend on an extensive quantity,Ymdepends only onTandP. Therefore

Y

(

∂Y

∂n

)

T,P



(

∂(nYm)

∂n

)

T,P

Ym (4.5-18)