First, we discuss the chemical potential of “Li”, or Li ions in the positive electrode. The “reaction” in Li-ion batteries consists of inserting Li atoms into a host lattice, in this case $\text{CoO}_2$. Therefore we want to investigate the Gibbs free energy per lattice site (specific Gibbs free energy, $\bar{G}$). This will depend on the composition $X$ of Li in $\text{Li}_X\text{CoO}_2$, where $X$ is the fraction of lithium sites that are occupied. The $X$ is related, but not exactly equal, to the state of charge of the battery. For positive electrodes, an $X$ of 1 usually indicates a fully discharged battery. That being said, many battery electrodes cannot be cycled stably from $X=0$ to $X=1$. Thus in practice, a state of $X=0.5$ may indicate that a battery is fully charged.

At constant temperature and pressure the chemical potential is defined as

where $N$ is the number of Li atoms. Writing the composition as $X = N/J$, with $J$ the number of “lattice sites” within a $\text{CoO}_2$ host lattice, we have

Then, in integral form,

For a completely filled site ($X=1$) the specific Gibbs free energy per filled lattice site should equal the chemical potential of a purely lithiated (LCO) lattice, $\mu^0_{\text{LCO}}$, while for an empty site ($X=0$) it equals the chemical potential of the empty host, $\mu^0_{\text{CO}}$. A simple weighted-average model gives

or, rearranged,

However, because $G = H - TS$ and there is a configurational entropy associated with mixing Li in the host lattice, the dependence on $X$ is modified. In fact, one writes

with

For a lattice gas the entropy of mixing is given by

Thus, the full expression for the specific Gibbs free energy in the positive electrode becomes

Graphically (see Figure 4.8), the instantaneous chemical potential of Li in the positive electrode is the derivative of $\bar{G}_{\text{“Li”}}$ with respect to $X$, yielding

where