PX284 - E1a -internal energy

\begin{aligned} U & = \sum_{i} E_{i} P_{i} \\ = \sum_{i} \frac{E_{i} \exp (- β E_{i})}{Z} \end{aligned}

Z = \sum_{i} \exp (- β E_{i}) ⟹ \frac{d Z}{d β} = - E_{i} \sum_{i} \exp (- β E_{i})

∴ U = - \frac{d \ln Z}{d β}