Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 47 / a / Solution

At fixed particle number and pressure, the joint isothermal-isobaric ensemble weight for a microstate of energy $E$ and volume $V$ is $e^{-\beta (E+PV)}$. Write its enthalpy as $H=E+PV$. Differentiating the isothermal-isobaric partition function with a temperature-independent volume measure givesAfter division by $Q_P$ this is the normalized ensemble average, soThe differentiation assumes that the microscopic Hamiltonian and the integration measure have no additional explicit temperature dependence. An extra temperature-dependent normalization factor must be differentiated separately.