Up to now, we have assumed that the system in question (a battery) undergoes a change where the amount and composition do not change. It’s time to relax the latter assumption.
Chemical Potential
Gibbs energy may be interpreted as the energy released or absorbed that may be used for non-expansionary work. When the change in state that occurs is due to compositional changes, i.e. a reaction transforming one chemical species into another, we can define the change in Gibbs energy due to this process as the chemical potential,
$$\begin{equation} \mu_j := \frac{\partial G}{\partial n_j}\end{equation}$$
for a particular species $j$. If the only difference between two states is the amount of species (composition held constant), then we can write $G = G(n_1,n_2,\cdots,n_i)$, for the $i$ different species. Therefore,
$$\begin{equation} dG = \sum_j \frac{\partial G}{\partial n_j}dn_j \end{equation}$$
Integrating from zero to the $n_j$ of the amounts, we get,
$$\begin{equation} G = \sum_j \mu_j n_j\end{equation}$$
since $\mu_j$ is a constant at constant composition. Recall that we defined molar Gibbs energy as
$$\begin{equation} G_m(p) = G_m^{\theta} + RT\ln\left(\frac{\phi(p) p}{p^{\theta}}\right)\end{equation}$$
where now we can interpret this as the molar Gibbs energy as at a particular relative composition. Adding the molar Gibbs energy of each species and multiplying by their amounts, we get
$$\begin{equation} G = \sum_j n_jG_{m,j} = \sum_j n_j\left(G^{\theta}_j + RT\ln\left(\frac{\phi p_j}{p^{\theta}_j}\right) \right) \end{equation}$$
for the partial pressures, $p_j$. This implies that at each particular composition we may write
$$\begin{equation} \mu_j = G^{\theta}_j + RT\ln\left(\frac{\phi p_j}{p^{\theta}_j}\right) \end{equation} $$
Gibbs Energy Revisited
If instead we allow changes to all state variables, then the state function (whatever that may be) will imply that $V = V(p,T,n_1,n_2,\cdots,n_i)$. Hence, we can write, for the general $G = G(p,T,V,n_1,n_2,\cdots,n_i)$
$$\begin{equation} dG = \frac{\partial G}{\partial p} dp + \frac{\partial G}{\partial T} dT + \sum_j \frac{\partial G}{\partial n_j}dn_j \end{equation}$$
There exists Maxwell Relations (we don’t need to derive them) which state that those partial derivatives of G with respect to $p$ and $T$ are what we would imagine,
$$\begin{equation} dG = V dp -S dT + \sum_j \frac{\partial G}{\partial n_j}dn_j \end{equation}$$
or
$$\begin{equation} dG = V dp -S dT + \sum_j \mu_jdn_j \end{equation}$$
So, if we know the state function $f(p,V,n,T) = 0$, we can find out through the above equation how much energy should be expected to be released or absorbed by some chemical reaction. The larger the chemical potential, the larger the energy exchange.
Reaction Gibbs Energy
As a chemical reaction occurs, we can define a metric for how much of the reaction has occurred. When the extent of a reaction $\xi = 0$ the reaction has not occurred and all species are in their form as reactants. When the extent of a reaction $\xi = 1$, all species are products. We can define the change in Gibbs energy along the extent of the reaction by
$$\begin{equation} \Delta_rG := \frac{\partial G}{\partial \xi} \end{equation} $$
and we get that the amount of each species, $n_j$, changes with the extent of the reaction, $\xi$, proportionally to the stochiometric coefficients of the reaction, $\nu_j$, such that
$$\begin{equation} dn_j = \nu_j d\xi \end{equation} $$
The amount of any particular species, $n_j$, is a function of the extent of the reaction, $\xi$. Therefore,
$$\begin{equation} \Delta_rG_j = \frac{\partial G_j}{\partial \xi} =\frac{\partial G_j}{\partial n_j}\frac{d n_j}{d \xi}\end{equation} = \mu_j \nu_j$$
Summing up all the species, we get
$$\begin{eqnarray} \Delta_r G &=& \sum_j \nu_j \mu_j \nonumber \\ &=& \sum_j \nu_j \left( G^{\theta}_j + RT\ln\left(\frac{\phi p_j}{p^{\theta}_j}\right) \right) \nonumber \\ &=& \sum_j \nu_j \left( \mu^{\theta}_j + RT\ln(a_j) \right) \nonumber \\ &=& \Delta_r G^{\theta} + RT\ln\left(\Pi_j a_j^{\nu_j}\right) \end{eqnarray}$$
where $\Delta_r G^{\theta}$ is the standard Gibbs reaction energy (which ultimately is an empirically measurable term), and $a_j$ are activity functions. These activity functions replace the pressure and fugacity terms and are also empirical functions of the concentrations of the various species. The activity functions can be defined implicitly this way: they are the functions which satisfy the above relation.
Electrochemical Reactions
The final step is to equate the non-expansionary work (Gibbs energy) to electrical work. Over the course of a reaction, the non-expansionary work at any instant can be expressed as,
$$\begin{equation} dw = \Delta_r Gd\xi \end{equation}$$
The work of moving a charge, $Q$, across a potential difference, $d\phi$, is
$$\begin{equation} dw_{electric} = Qd\phi\end{equation}$$
During an electrochemical reaction, $\nu$ moles of charge are transferred. In terms of the extent of reaction, $\nu d\xi$ moles of electrons are transferred. The total charge transferred is then the total number of electrons transferred, $\nu N_Ad\xi$, for Avogadro’s number $N_A$, multiplied by the total charge, $-e$, yielding,
$$\begin{equation} Q = -\nu e N_A d\xi = -\nu Fd\xi\end{equation}$$
for Faraday’s constant, $F$. The charge is moved over a potential difference of $d\phi = E_{cell}$, the electrochemical cell’s voltage. Now, equating the non-expansionary work and the electrical work, we get,
$$\begin{equation} \Delta_r Gd\xi =-\nu FE_{cell}d\xi\end{equation}$$
Hence, the cell’s equilibrium voltage is,
$$\begin{equation} E_{cell} = \frac{-1}{\nu F} \Delta_r G = E_{cell}^{\theta} - \frac{RT}{\nu F}\ln\left(\Pi_j a_j^{\nu_j}\right) \end{equation}$$
for $E_{cell}^{\theta} = -\Delta_r G^{\theta}/(\nu F)$. This is the Nernst Equation!
Final Comments
Typically in the Li-S literature, the activity functions are taken to be the concentrations themselves; so $a_j = x_j$. This is an assumption based on dilute-solution theory, which essentially states that an ideal-dilute solution will follow Raoult’s Law as $x_j\to 0$ and Henry’s Law as $x_j\to 1$. Raoult’s Law is the formulation that $a_j = x_j$ and Henry’s Law is the formulation that $a_j = k_j x_j$, for some empirical constant $k_j$. The constant is therefore taken to be unity. The validity of these assumptions in real cells is questionable at best and constitutes an equilibrium condition only.
The Nernst Equation is an equilibrium equation. It assumes that the cell is in thermodynamic equilibrium. That is definitely not the case during charge and discharge. So, the validity of its predictions regarding the relation between measured voltages and concentrations of species will become evermore inaccurate as the battery deviates from thermodynamic equilibrium. More mathematical machinery is required to account for such deviations, such as over-potential and the Butler-Volmer equation. I will get into these topics in another series of posts.