Page 268 - Instant notes

Physical chemistry 254

The standard approach is to consider N molecules in a system of total energy, E.
Within the system, each molecule may be in one of a number of states, each with energies
ε 0, ε 1, ε 2, ε 3, etc. ε 0 is the ground state and the subsequent terms have increasingly higher
energies. Each energy level, ε i, is occupied by n i molecules. The total number criterion
recognizes that the total number of molecules is the sum of the number of molecules in
each state:

The total energy of molecules in state ε i is equal to ε in i, and the total energy criterion
follows from this:

The huge number of possible ways in which the molecules can be distributed across the
available energy levels is known as the configuration of the system, and must comply
with the total energy and total number criteria. For each configuration there are a number
of ways, W, in which the molecules can be distributed amongst the available energy
levels, given by:

The macroscopic properties of the system will be the result of the total configuration of
the molecules in the system. The most probable configuration of the system will be that
with the largest value of W. The state with the maximum value of W, which complies
with the total energy and total number criteria, is obtained by a straightforward but
lengthy calculation. It is found that the maximum W is obtained when the population of
each state, n i, is given by the Boltzmann law:

If each energy level, ε j, has a degeneracy of g j, then the Boltzmann law may be rewritten
in terms of energy levels, rather than states:

The population ratio between two energy levels, ε I and ε j is then given by: