Page 269 - Instant notes

Statistical thermodynamics 255

If the lowest energy level has a degeneracy of one, and is regarded as having an energy of
zero, then it follows that the population n j of an energy level, ε j relative to that of the
lowest level n 0 is given by:

at low temperature, , and the number of molecules with energies above that of
the ground state approaches zero.

The partition function

The denominator in the Boltzmann law is of considerable importance in statistical
thermodynamics, and is referred to as the partition function, q.

The significance of q is seen when the total number of molecules is summed over all
energy levels:

Hence, q=N/n 0.

q is a temperature-dependent dimensionless number. Because q is the reciprocal of the
fraction of molecules in the ground state, it provides a measure of the ability of molecules
to access energy levels above the ground state. The larger the value of q, the more
molecules access energy levels above ε 0 (Fig. 1). The value of q varies from 1 at absolute
zero (n 0=N) to an exceedingly large value where the energy levels are closely spaced and
at high temperature (n 0→0).