The Potential to add a Chemical Species

A very important quantity is identified when the Gibbs free energy is introduced as in Equation 19-8.

$$\input{equations/13-5A}$$ (19-11)

so that

$$\input{equations/13-5B}$$ (19-12)

Note that the chemical potential is an intensive variable as a potential should be.

$\mu^\alpha_i(P,T)$ is the chemical potential of species $i$ in phase $\alpha$: it is rate at which reversible work that must be done on a the subsystem $\alpha$ to add a species $i$ at constant $P$ and $T$.

Question: How would Equation 19-8 be modified if the chemical species that were being added were charged?

Question: How would the equilibrium condition Equation 19-8 (or more generally Equation 19-1) be modified if the chemical species that were being added were charged?

Work out in the space provided below, the general conditions on equilibrium for a closed system (i.e.,one that has a fixed numbers of independent chemical species) at fixed pressure and temperature. Let the system have $f$ different phases.