308
6 Molecular Systems
with all volume dependence contained, and the factor N ! included, in the translational component, and U int expressed as a function only of the canonical ensemble
thermodynamic variable 2 T .
We have argued earlier that if the interaction energies ij between pairs of the
constituent atoms/molecules that make up a canonical ensemble can be neglected
relative to the energies i and j associated with the individual constituents,
then the partition function Z N (T , V ) for an ensemble of N indistinguishable
atoms/molecules can be represented [see Eq. (3.2.27)] as
Z N (T , V ) =
z N (T , V )
N!
,
in terms of the partition function z(T , V ) for an individual constituent of the
ensemble. Moreover, we have also argued in Sect. 4.1 that the thermodynamic
internal energy U for a canonical ensemble of N atoms/molecules will be given
by U(T , V ; N) = Nu(T , V ), with u(T , V ) the thermodynamic internal energy
per particle which is, in turn, given via Eq. (4.1.1) as the ensemble average energy,
≡
i p i i , for a single constituent atom/molecule. Due to the separability of
the translational and internal state contributions to the partition function, we may
apply the same reasoning to the internal state contribution to the thermodynamic
internal energy, U int (T ; N).
We may express U int (T ; N) either via Eq. (4.1.3b) in terms of the temperature
partial derivative of ln Z int (T ; N) or, alternatively, in terms of the ‘more natural’
canonical ensemble variable β ≡ (k B T ) −1 , the so-called dimensionless reciprocal
temperature. Thus, in terms of the derivative with respect to β, we write
U int (T ; N) = −
∂ ln Z int
∂β
N
.
(6.2.136a)
Upon recalling Eq. (6.1.4) relating Z int (β; N) to z int (β) by Z int (T ; N) = z N
int (β),
we obtain U int (T ; N) as
U int (T ; N) = −N
d ln z int
dβ
= −
N
z int (β)
dz int
dβ
,
(6.2.136b)
with z int (β) given explicitly by
2 Although we have written U int (T ; N), within the context of the canonical ensemble, the only truly
independent variable is the temperature T . The number of particles N has, strictly speaking, a fixed
value and should be considered as a parameter characterizing particular canonical ensembles: it
thus plays a slightly different role in our description of statistical thermodynamics based upon the
canonical ensemble than do the truly independent variables T and V .
6 Molecular Systems
with all volume dependence contained, and the factor N ! included, in the translational component, and U int expressed as a function only of the canonical ensemble
thermodynamic variable 2 T .
We have argued earlier that if the interaction energies ij between pairs of the
constituent atoms/molecules that make up a canonical ensemble can be neglected
relative to the energies i and j associated with the individual constituents,
then the partition function Z N (T , V ) for an ensemble of N indistinguishable
atoms/molecules can be represented [see Eq. (3.2.27)] as
Z N (T , V ) =
z N (T , V )
N!
,
in terms of the partition function z(T , V ) for an individual constituent of the
ensemble. Moreover, we have also argued in Sect. 4.1 that the thermodynamic
internal energy U for a canonical ensemble of N atoms/molecules will be given
by U(T , V ; N) = Nu(T , V ), with u(T , V ) the thermodynamic internal energy
per particle which is, in turn, given via Eq. (4.1.1) as the ensemble average energy,
≡
i p i i , for a single constituent atom/molecule. Due to the separability of
the translational and internal state contributions to the partition function, we may
apply the same reasoning to the internal state contribution to the thermodynamic
internal energy, U int (T ; N).
We may express U int (T ; N) either via Eq. (4.1.3b) in terms of the temperature
partial derivative of ln Z int (T ; N) or, alternatively, in terms of the ‘more natural’
canonical ensemble variable β ≡ (k B T ) −1 , the so-called dimensionless reciprocal
temperature. Thus, in terms of the derivative with respect to β, we write
U int (T ; N) = −
∂ ln Z int
∂β
N
.
(6.2.136a)
Upon recalling Eq. (6.1.4) relating Z int (β; N) to z int (β) by Z int (T ; N) = z N
int (β),
we obtain U int (T ; N) as
U int (T ; N) = −N
d ln z int
dβ
= −
N
z int (β)
dz int
dβ
,
(6.2.136b)
with z int (β) given explicitly by
2 Although we have written U int (T ; N), within the context of the canonical ensemble, the only truly
independent variable is the temperature T . The number of particles N has, strictly speaking, a fixed
value and should be considered as a parameter characterizing particular canonical ensembles: it
thus plays a slightly different role in our description of statistical thermodynamics based upon the
canonical ensemble than do the truly independent variables T and V .