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Choose the ground-state energy as zero and write $x=\epsilon /(k_{B}T)$. The molecular partition function must sum individual states, so the triply degenerate excited level contributes three Boltzmann factors:The probabilities of occupying the two levels areEach of the three individual excited states has probability $e^{-x}/q$. At high temperature the four states become equiprobable, so the two levels have probabilities $1/4$ and $3/4$, not $1/2$ each. This is the distinction between a level and its state degeneracy.

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.

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.

Treat each species as a dilute classical ideal gas: molecules of the same species are indistinguishable, while different species are distinguishable. Let $N_i$ be the number of molecules of species $i$, $q_i(T,V)$ its molecular partition function with energies measured from a chosen internal reference, and ${\epsilon }^{0,i}$ that reference energy per molecule relative to a common chemical energy convention. Then the mixture's canonical partition function factorizes:The $N_i!$ corrects the overcounting of indistinguishable particles. The exponential restores the species-dependent reference energies that matter for reactions. Thus the Helmholtz free energy isUsing Stirling's approximation, $\ln (N_i!)\simeq N_i\ln N_i-N_i$, and differentiating at fixed $T,V,N_{j\ne i}$ yieldsThe ideal-gas $q_i$ does not depend on the number of molecules, so no additional $\partial q_i/\partial N_i$ term appears. These are chemical potentials per molecule; multiply by the Avogadro constant for a molar chemical potential. The result assumes the thermodynamic, nondegenerate-gas limit.