3.2 The Canonical Ensemble: Closed Systems
143
at temperature T , then the relative probability for the states of these systems to
be occupied will be determined by the Boltzmann distribution via the canonical
partition function
z(T ) =
levels
level e
−βE level ,
as
p i (T ) =
i e −βE i
z(T )
,
with β = (k B T ) −1 as usual. Thus, for systems A and B, respectively, we have
z A (T ) = e
−ββ 0 + e
−ββ 1
= e
−ββ 0 (1 + e
−359.7/T )
and
z B (T ) = e
−ββ 0 (1 + 3e
−359.7/T ) .
Note that we have replaced 250 cm −1 by 250 cm −1 /(0.69503 cm −1 K −1 ) = 359.7 K
for calculational convenience: i.e., we have employed the Boltzmann constant in
cm −1 units as k B = 0.69504 cm −1 K −1 , thereby effectively expressing energy in
kelvin temperature units.
We shall examine the difference in behaviour between systems A and B due to
the presence of the triple degeneracy in 1 for system B. If we examine the fractional
populations of the ground and excited states, given by
p
A
0 (T ) =
e −ββ 0
z A (T )
=
e −ββ 0
e −ββ 0 (1 + e −359.7/T )
=
1
1 + e −359.7/T ,
p
A
1 (T ) =
e −ββ 1
z A (T )
=
e −ββ 0 e −359.7/T
e −ββ 0 (1 + e −359.7/T )
=
e −359.7/T
1 + e −359.7/T ,
we see that they are independent of the value for 0 . We may thus deduce from this
result that the value of 0 is irrelevant for our purposes, and hence we are free to
choose 0 to be zero. The fractional populations for system B can now simply be
written down as