We began the discussion of thermodynamics by comparing microscopic and macroscopic views of matter. This connection is formalized by the field of statistical mechanics. In broad terms we want to consider averaging over occupied molecular energy levels to obtain macroscopic quantities

The Boltzmann relation

We would like to prove the Boltzmann hypothesis that the relative populations of levels are determined by an exponential function of temperature (e.g. P₁/P₀ = exp{ -E/k_BT }. In this section we offer an intuitive proof based on simple comparison of three energy levels. If energies are additive, but populations enter as ratios, then we show that the only possible conclusion is that the relationship involves an exponential function in terms of energy.

The concept of a partition function

The idea that we can treat a molecule as a collection of energy levels of different types, electronic, vibrational, rotational and translational, gives us a method to consider how populations of molecules will occupy those levels. By studying the occupation of the energy levels as a function of temperature we can understand quantities such as entropy and heat capacity from a more fundamental point of view. While the energy is dependent upon how much thermal energy is available to populate the levels, the entropy is dependent upon the number of levels that are available at a given temperature. This is subtle distinction. The entropy depends strongly on the degeneracy, but also on the total number of ways that a population can be distributed among "available" energy levls. The number of available levels is, of course, dependent on the amount of thermal energy, on k_BT (or RT on a molar basis).