 Solution

Chemical equilibrium for $\mathrm{M}\rightleftharpoons {\mathrm{M}}^{+}+{\mathrm{e}}^{-}$ requiresInsert the given chemical potential form and define the positive ionization energyAfter collecting logarithms,soDefine a dimensionless concentration equilibrium constant using a standard number density $c^{\circ }$:Each $f_i$ is a molecular partition function density, with units of inverse volume. Consequently $f_e{f}_{{\mathrm{M}}^{+}}/f_{\mathrm{M}}$ has density units and division by $c^{\circ }$ makes $K_c$ dimensionless. All ground-state energies must use the same energy reference.