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

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.

For an ideal gas, translation makes the molecular partition function proportional to volume. Define the molecular partition function density and number concentration byThenBoth $f_i$ and $c_i$ have units of inverse volume, so their ratio is dimensionless. If concentrations are specified in moles per volume instead, replace $c_i$ in this microscopic expression by $N_A$ times that molar concentration, with consistent volume units.