4.5 Multicomponent Systems
185
Exercise 4.15
a. Show that µ i A i + PV i
b. Show that µ i U i + PV i − T S i
There are some equations similar to the Maxwell relations that apply to multicomponent open systems. We begin with the Gibbs equation, Eq. (4.5-3):
dG −SdT + VdP +
c
i1
µ i dn i
(4.5-13)
Using the Euler reciprocity relation, Eq. (B-13) of Appendix B,
∂S
∂n i
T ,P,n
S i −
∂µ i
∂T
P,n
(4.5-14)
A second use of the Euler reciprocity relation gives
∂V
∂n i
T ,P,n
V i
∂µ i
∂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 by T , P, and n, 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 quantity Y m is given by
Y m
Y
n
(4.5-17)
The molar quantity Y m is an intensive quantity. Because an intensive quantity cannot
depend on an extensive quantity, Y m depends only on T and P. Therefore
Y
∂Y
∂n
T ,P
∂ (nY m )
∂n
T ,P
Y m
(4.5-18)