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4 Mean Values and Thermodynamics
4.1.2 Canonical Partition Function for a Mixture
We have seen in Sect. 3.2.3 that the canonical partition function Z AB (N, T , V ) for
a binary ideal gas mixture of atoms A and B is given via Eq. (3.2.25) in terms of the
canonical partition functions z A (N A , T , V ) and z B (N B , T , V ) for atoms A and B as
Z AB (N, T , V ) =
z
N A
A (T , V )z
N B
B (T , V )
N A !N B !
; N = N A + N B .
(4.1.28)
If we now utilize the Stirling approximation (see Appendix C) to write ln Z AB as
ln Z(N, T , V ) = N A ln z A + N B ln z B − N A ln N A + N A − N B ln N B + N B ,
we may obtain expressions for the chemical potential μ A (N A , T , V ), for example,
as
μ A ≡ −k B T
∂ ln Z
∂N A
V ,T ,N B
= −k B T ln
z A
N A
.
(4.1.29)
A completely analogous expression can also be obtained for the chemical potential
μ B (N B , T , V ) for constituent B.
More generally, the canonical partition function Z mix for a mixture of M
chemically distinct species is given by
Z mix (T , V mix , N 1 , · · · , N M ) =
M
α=1
Z α (T , V mix , N α ) ,
(4.1.30)
with V mix the volume of the mixture, and the individual canonical partition functions
Z α (T , V mix , N α ) given by Eq. (3.2.27) in terms of the partition function z α (T , V mix )
for chemical species α as
Z α (T , V mix , N α ) ≡
z
N α
α (T , V mix )
N α !
.
(4.1.31)
Upon utilizing Eq. (4.1.3b) to obtain the internal energy U mix in terms of the
temperature derivative of ln Z mix , we obtain the expression
U mix (T , V mix , N 1 , · · · , N M ) = k B T
2
M
α=1
∂ ln Z mix
∂T
T ,V mix ,N β =N α
=
M
α=1
U α (T , V mix , N α ) .
(4.1.32)