4.5 Multicomponent Systems
183
However, the partial derivatives in this equation are not equal to any simple thermodynamic variables as are the partial derivatives in Eq. (4.5-2). We therefore say that the
natural independent variables for the Gibbs energy are T , P, n 1 , n 2 , … , n c .
E X A M P L E 4.17
Use an analogue of Eq. (B-7) of Appendix B to write a relation between (∂G/∂n i ) T ,V ,n
and µ i .
Solution
∂G
∂n i
T ,V ,n
∂G
∂n i
T ,P,n
+
∂G
∂P
T ,n
∂P
∂n i
T , V , n
µ i + V
∂P
∂n i
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
G H − TS,
dH dG + TdS + SdT
dH −SdT + VdP +
c
i1
µ i dn i + TdS + SdT
dH TdS + VdP +
c
i1
µ i dn i
(4.5-6)
The natural independent variables for H are S, P, n 1 , n 2 , … , n c . We can see from
Eq. (4.5-6) that
µ i
∂H
∂n i
S,P,n
(4.5-7)
Similarly, since U H − PV ,
dU TdS − PdV +
c
i1
µ i dn i
(4.5-8)
so that the natural independent variables for U are S, V , n 1 , n 2 , … , n c . Also
dA −SdT − VdP +
c
i1
µ i dn i
(4.5-9)