19
Statistical thermodynamics:
the concepts
Solutions to exercises
Discussion questions
E19.1(b)
Consider the value of the partition function at the extremes of temperature. The limit of
q
as
T
approaches zero, is simply
g
0
, the degeneracy of the ground state. As
T
approaches inFnity, each
term in the sum is simply the degeneracy of the energy level. If the number of levels is inFnite, the
partition function is inFnite as well. In some special cases where we can effectively limit the number
of states, the upper limit of the partition function is just the number of states. In general, we see
that the molecular partition function gives an indication of the average number of states thermally
accessible to a molecule at the temperature of the system.
E19.2(b)
The statistical entropy may be deFned in terms of the Boltzmann formula,
S
=
k
ln
W
, where
W
is the
statistical weight of the most probable conFguration of the system. The relation between the entropy
and the partition function is developed in two stages. In the Frst stage, we justify Boltzmann’s formula,
in the second, we express
W
in terms of the partition function. The justiFcation for Boltzmann’s
formula is presented in
Justifcation
19.6. Without repeating the details of this justiFcation, we can
see that the entropy deFned through the formula has the properties we expect of the entropy.
W
can be thought of as a measure of disorder, hence the greater
W
, the greater the entropy; and the
logarithmic form is consistent with the additive properties of the entropy. We expect the total disorder
of a combined system to be the product of the individual disorders and
S
=
k
ln
W
=
k
ln
W
1
W
2
=
k
ln
W
1
+
k
ln
W
2
=
S
1
+
S
2
.
In the second stage the formula relating entropy and the partition function is derived. This derivation
is presented in
Justifcation
19.7. The expression for
W
, eqn 19.1, is recast in terms of probabilities,
which in turn are expressed in terms of the partition function through eqn 10. The Fnal expression
which is eqn 19.34 then follows immediately.
E19.3(b)
Since
β
and temperature are inversely related, strictly speaking one can never replace the other. The
concept of temperature is useful in indicating the direction of the spontaneous transfer of energy in
the form of heat. It seems natural to us to think of the spontaneous direction for this transfer to be
from a body at high
T
to one at low
T
. In terms of
β
, the spontaneous direction would be from low
to high and this has an unnatural feel.
On the other hand,
β
has a direct connection to the energy level pattern of systems of atoms and
molecules. It arises in a natural, purely mathematical, manner from our knowledge of how energy
is distributed amongst the particles of our atomic/molecular system. We would not have to invoke
the abstract laws of thermodynamics, namely the zeroth and second laws in order to deFne our
concept of temperature if we used
β
as the property to indicate the natural direction of heat ±ow.