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 i_th state, P_i, in the Boltzmann distribution is analyzed by studying its derivative with respect to the temperature, T. A simple expression is found, involving P_i, the energy of the state, E_i, 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 P_i 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