\begin{equation} E=\sum_{i}N_{i}\epsilon_{i}=\sum_{i}\frac{N}{q}g_{i}e^{-\beta{\epsilon_{i}}}\epsilon_{i}=\frac{N}{q}\sum_{i}g_{i}\epsilon_{i}e^{-{\epsilon_{i}}/kT} \label{ec31} \end{equation} Derivando $q=\sum_{i}g_{i}e^{-\epsilon_{i}/kT}$ respecto de $T$ a volumen: \begin{equation} \left(\frac{\partial{q}}{\partial{T}}\right)_{V}=\frac{1}{kT^2}\sum_{i}g_{i}\epsilon_{i}e^{-\epsilon_{i}/kT} \label{ec32} \end{equation} Despejando el sumatorio de la ecuación (~\ref{ec32}): \begin{equation} kT^{2}\left(\frac{\partial{q}}{\partial{T}}\right)_{V}=\sum_{i}g_{i}\epsilon_{i}e^{-\epsilon_{i}/kT} \label{ec33} \end{equation} Sustituyendo (~\ref{ec33}) en (~\ref{ec31}) \begin{equation} E=NkT^{2}\left(\frac{\partial{lnq}}{\partial{T}}\right)_{V} \label{ec34} \end{equation}