184
4 The Thermodynamics of Real Systems
so that the natural independent variables for A are T , V , n 1 , n 2 , … , n c . By inspection
µ i
∂U
∂n i
S,V ,n
(4.5-10)
and
µ i
∂A
∂n i
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 potential as the partial molar Gibbs energy. A general partial molar quantity is a partial
derivative of an extensive quantity with respect to the amount of one component, keeping T , P, and the amounts of all other components fixed. If the letter Y stands for
any extensive quantity (U, H, A, G, S, V , and so on), the partial molar quantity for
substance number i is denoted by Y i and defined by
Y i
∂Y
∂n i
T ,P,n
(4.5-12)
The chemical potential µ i is equal to G i . The partial derivatives in Eqs. (4.5-7),
(4.5-10), and (4.5-11) to which µ i is equal are not partial molar quantities, because
P and T are not both held fixed in the differentiations.
E X A M P L E 4.18
Find a relationship between the chemical potential and the partial molar enthalpy.
Solution
We begin with the relationship between G and H:
G H − TS
Differentiation of both sides at constant T , P, and n gives
∂G
∂n i
T ,P,n
∂H
∂n i
T ,P,n
− T
∂S
∂n i
T ,P,n
or
µ i H i − T S i
4 The Thermodynamics of Real Systems
so that the natural independent variables for A are T , V , n 1 , n 2 , … , n c . By inspection
µ i
∂U
∂n i
S,V ,n
(4.5-10)
and
µ i
∂A
∂n i
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 potential as the partial molar Gibbs energy. A general partial molar quantity is a partial
derivative of an extensive quantity with respect to the amount of one component, keeping T , P, and the amounts of all other components fixed. If the letter Y stands for
any extensive quantity (U, H, A, G, S, V , and so on), the partial molar quantity for
substance number i is denoted by Y i and defined by
Y i
∂Y
∂n i
T ,P,n
(4.5-12)
The chemical potential µ i is equal to G i . The partial derivatives in Eqs. (4.5-7),
(4.5-10), and (4.5-11) to which µ i is equal are not partial molar quantities, because
P and T are not both held fixed in the differentiations.
E X A M P L E 4.18
Find a relationship between the chemical potential and the partial molar enthalpy.
Solution
We begin with the relationship between G and H:
G H − TS
Differentiation of both sides at constant T , P, and n gives
∂G
∂n i
T ,P,n
∂H
∂n i
T ,P,n
− T
∂S
∂n i
T ,P,n
or
µ i H i − T S i