Dependence of the population on the temperature in the Boltzmann distribution: a simple relation involving the average energy

The dependence on the temperature of the population of the ith state, Pi, in the Boltzmann distribution is analyzed by studying its derivative with respect to the temperature, T. A simple expression is found, involving Pi, the energy of the state, Ei, and the average energy, ?E?. This relation is completely general (it has the same form in all the thermodynamic ensembles), and it has a relevant didactic content, given that it predicts the qualitative variation of Pi with T even in complex systems. The derivation of this relation, the discussion of its properties, and its application to simple problems is appropriate for a statistical thermodynamics course in the chemistry curriculum.