4.1 Canonical Ensemble: Closed Systems
181
We can see from this expression that the internal energy for a multicomponent
mixture is thus given as the sum of the internal energies of the component species.
As C V ,mix (T ) is simply the temperature derivative (at constant volume) of U mix , the
same property clearly holds for it.
A similar argument applies to the pressure P mix in a multicomponent mixture, as
it is defined through an extension of Eq. (4.1.13) as
P mix = k B T
∂Z mix
∂V mix
T ,N Nα
= k B T
M
α=1
∂ ln Z α
∂V mix
T ,N α
(4.1.33a)
or
P mix =
M
α=1
P α .
(4.1.33b)
For a mixture of M ideal gases, expression (4.1.33b) will be recognized as a
statement of Dalton’s law of partial pressures.
If we examine the Helmholtz energy A mix (T , V mix , N 1 , · · · , N M ) for a multicomponent mixture beginning with the generalization of Eq. (4.1.17) to
A mix (T , V mix , N 1 , · · · , N M ) = − k B T ln Z mix (T , V mix , N 1 , · · · , N M ) ,
(4.1.34a)
we see that the properties of the natural logarithm give directly
A mix (T , V mix , N 1 , · · · , N M ) =
M
α=1
A α (T , V mix , N α ) .
(4.1.34b)
Multicomponent mixture expressions for the other thermodynamic state functions
may be obtained in a similar fashion.
The chemical potential μ α (T , V mix , N α ) is defined by Eq. (4.1.29) for α = A, and
more generally as
μ α (T , V mix , N α ) = − k B T
∂ ln Z mix
∂N α
T ,V mix ,N β =N α
= − k B T
∂ ln Z α
∂N α
T ,V mix
.
(4.1.35)
It is clear from this expression for μ α (T , V mix , N α ) that it is not possible to define
a chemical potential μ mix for a mixture as a whole and that the individual chemical
potentials for the components of the mixture, unlike the chemical potential for a
pure substance, do not simply represent the Gibbs energies per component particle.