Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 2 / A8 / a / i / Solution

For an ideal mixture, the Helmholtz free energy is a sum over species. At fixed $T,V$ and all other populations, the molecular partition function $q_i$ does not depend on $N_i$. Differentiating the contribution written using each species' own energy reference givesThe desired additive energy term requires a consistent energy reference in molecular partition functions. Let ${\epsilon }^{0,i}$ be the energy per molecule of species $i$ in its reference state, measured relative to a common chemical energy zero, and let $q_i$ sum excitations above that reference. ThenThereforeOne convenient choice is each species' rovibrational ground state, so the molecular reference includes its zero-point energy. A different consistent reference is possible, but the corresponding partition function must change. The offset cannot arise from differentiation of a formula that has already omitted it, and it must not be counted twice. These chemical potentials are per particle; molar expressions instead use the molar gas constant.