3.2 The Canonical Ensemble: Closed Systems
153
To see how the canonical partition function may be evaluated, we begin with its
definition as a ‘sum over states’ of e −ββ ij for a system of interest that consists of two
distinguishable atoms/molecules, one of chemical species A, the other of chemical
species B, in thermal equilibrium with a bath system. The partition function Z AB
for this two-particle system is defined by
Z AB ≡
ij
e
−ββ ij =
ij
e
−ββ Ai e
−ββ Bj
=
i
e
−ββ Ai
j
e
−ββ Bj .
If we now recall the definition of the canonical partition function for an individual
particle such as A (or B), namely z A =
i e −ββ Ai , we may then rewrite our
expression for the total partition function z AB as
Z AB = z A z B .
(3.2.25)
More generally, we can apply this argument to any set of statistically independent
and distinguishable single-particle subsystems to obtain
Z = z A z B z C . . . .
(3.2.26)
A concrete example of such a splitting of a larger system into smaller independent
subsystems is to consider air as a mixture of two ideal gases, oxygen and nitrogen
(we shall for the sake of simplicity ignore all the other minor constituents of the air
around us). Thus, if we choose to treat air as a mixture of O 2 and N 2 only, nitrogen
and oxygen are certainly distinguishable (chemically, if not otherwise) and, in the
ideal gas limit, they are statistically independent, so that
Z air = Z O 2 Z N 2 .
We can already see from this example chosen to illustrate the concept of distinguishability that were we to extend this argument to Z N 2 , we should immediately
encounter a problem, as we cannot possibly distinguish one nitrogen molecule from
another, no matter how hard we try to do so. We may, however, still consider one
nitrogen molecule to be statistically independent of all other nitrogen molecules,
as it is equivalent to considering nitrogen to be an ideal gas. Thus, to proceed
properly beyond this point, we obviously must now consider the case of statistically
independent but indistinguishable subsystems.
We shall retain the approximation in which we ignore interactions. Let us
examine, as before, two subsystems labelled A and A , and let us compare the
result obtained by treating the subsystems as distinguishable with what we obtain
by working directly with the definition of the canonical partition function for the
total system. Were we simply to ignore the indistinguishability of the A and A
subsystems, we would obtain the result
153
To see how the canonical partition function may be evaluated, we begin with its
definition as a ‘sum over states’ of e −ββ ij for a system of interest that consists of two
distinguishable atoms/molecules, one of chemical species A, the other of chemical
species B, in thermal equilibrium with a bath system. The partition function Z AB
for this two-particle system is defined by
Z AB ≡
ij
e
−ββ ij =
ij
e
−ββ Ai e
−ββ Bj
=
i
e
−ββ Ai
j
e
−ββ Bj .
If we now recall the definition of the canonical partition function for an individual
particle such as A (or B), namely z A =
i e −ββ Ai , we may then rewrite our
expression for the total partition function z AB as
Z AB = z A z B .
(3.2.25)
More generally, we can apply this argument to any set of statistically independent
and distinguishable single-particle subsystems to obtain
Z = z A z B z C . . . .
(3.2.26)
A concrete example of such a splitting of a larger system into smaller independent
subsystems is to consider air as a mixture of two ideal gases, oxygen and nitrogen
(we shall for the sake of simplicity ignore all the other minor constituents of the air
around us). Thus, if we choose to treat air as a mixture of O 2 and N 2 only, nitrogen
and oxygen are certainly distinguishable (chemically, if not otherwise) and, in the
ideal gas limit, they are statistically independent, so that
Z air = Z O 2 Z N 2 .
We can already see from this example chosen to illustrate the concept of distinguishability that were we to extend this argument to Z N 2 , we should immediately
encounter a problem, as we cannot possibly distinguish one nitrogen molecule from
another, no matter how hard we try to do so. We may, however, still consider one
nitrogen molecule to be statistically independent of all other nitrogen molecules,
as it is equivalent to considering nitrogen to be an ideal gas. Thus, to proceed
properly beyond this point, we obviously must now consider the case of statistically
independent but indistinguishable subsystems.
We shall retain the approximation in which we ignore interactions. Let us
examine, as before, two subsystems labelled A and A , and let us compare the
result obtained by treating the subsystems as distinguishable with what we obtain
by working directly with the definition of the canonical partition function for the
total system. Were we simply to ignore the indistinguishability of the A and A
subsystems, we would obtain the result