Statistical thermodynamics 1: the concepts - W.H. Freeman

PC8eC16 1/26/06 14:34 Page 561
16.1 CONFIGURATIONS AND WEIGHTS 561
The distribution of molecular states
We consider a closed system composed of N molecules. Although the total energy is
constant at E, it is not possible to be definite about how that energy is shared between
the molecules. Collisions result in the ceaseless redistribution of energy not only
between the molecules but also among their different modes of motion. The closest
we can come to a description of the distribution of energy is to report the population
of a state, the average number of molecules that occupy it, and to say that on average
there are n i molecules in a state of energy ε i . The populations of the states remain
almost constant, but the precise identities of the molecules in each state may change
at every collision.
The problem we address in this section is the calculation of the populations of states
for any type of molecule in any mode of motion at any temperature. The only restriction
is that the molecules should be independent, in the sense that the total energy
of the system is a sum of their individual energies. We are discounting (at this stage)
the possibility that in a real system a contribution to the total energy may arise from
interactions between molecules. We also adopt the principle of equal a priori probabilities,
the assumption that all possibilities for the distribution of energy are equally
probable. A priori means in this context loosely ‘as far as one knows’. We have no reason
to presume otherwise than that, for a collection of molecules at thermal equilibrium,
vibrational states of a certain energy, for instance, are as likely to be populated as
rotational states of the same energy.
One very important conclusion that will emerge from the following analysis is that
the populations of states depend on a single parameter, the ‘temperature’. That is, statistical
thermodynamics provides a molecular justification for the concept of temperature
and some insight into this crucially important quantity.
16.1 Configurations and weights
Any individual molecule may exist in states with energies ε 0 , ε 1 ,....We shall always
take ε 0 , the lowest state, as the zero of energy (ε 0 = 0), and measure all other energies
relative to that state. To obtain the actual internal energy, U, we may have to add a
constant to the calculated energy of the system. For example, if we are considering the
vibrational contribution to the internal energy, then we must add the total zero-point
energy of any oscillators in the sample.
(a) Instantaneous configurations
At any instant there will be n 0 molecules in the state with energy ε 0 , n 1 with ε 1 , and so
on. The specification of the set of populations n 0 , n 1 ,...in the form {n 0 , n 1 ,...} is a
statement of the instantaneous configuration of the system. The instantaneous
configuration fluctuates with time because the populations change. We can picture a
large number of different instantaneous configurations. One, for example, might be
{N,0,0, ...}, corresponding to every molecule being in its ground state. Another
might be {N − 2,2,0,0, ...}, in which two molecules are in the first excited state.
The latter configuration is intrinsically more likely to be found than the former
because it can be achieved in more ways: {N,0,0, ...} can be achieved in only one
way, but {N − 2,2,0, ...} can be achieved in – 1 2
N(N − 1) different ways (Fig. 16.1; see
Justification 16.1). At this stage in the argument, we are ignoring the requirement
that the total energy of the system should be constant (the second configuration has
a higher energy than the first). The constraint of total energy is imposed later in this
section.
Fig. 16.1 Whereas a configuration
{5,0,0, ...} can be achieved in only one
way, a configuration {3,2,0, ...} can be
achieved in the ten different ways shown
here, where the tinted blocks represent
different molecules.