= Solution
At fixed discharge current, delivered voltage can be represented as
$$
V=E_{\rm eq}(Q)-IR-\eta_{\rm act}-\eta_{\rm conc}-\eta_{\rm phase},
$$
where $Q$ is discharged charge and the terms denote distinct physical losses. A typical trace has an initial drop from open circuit, an approximately flat useful region and a final decline toward cutoff. Its exact shape depends on the cell chemistry and current; not every battery has an equally flat plateau.
Four distinguishable contributions are:
* \b[<Ohmic polarization>:] electronic, ionic and contact resistance cause a rapid $IR$ drop on applying current. This need not wait for a concentration gradient to develop.
* \b[<Activation overpotential>:] finite electrode-reaction kinetics require a departure from equilibrium to sustain charge transfer. It is important when the exchange current is small.
* \b[<Concentration overpotential>:] electrolyte transport and diffusion within active particles can deplete reactants or accumulate products near reaction sites. These losses often grow near the end of discharge or at high current.
* \b[<Phase-transformation polarization>:] nucleation and growth of a new phase can require extra driving force, produce an initial voltage dip or cause hysteresis. It is especially relevant to conversion and two-phase electrodes, rather than a universal independent loss in every chemistry.
Terminology sometimes groups solid-state transport or phase-change effects differently. The important distinction is the mechanism, not counting the same diffusion loss under two names. A falling equilibrium voltage as composition changes is also not, by itself, polarization.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-1-additional-discharge.png]
{title=An illustrative discharge voltage combines equilibrium changes and distinct losses}
{description=Original stipulated model, not battery measurements. The loss components show an instantaneous resistive contribution, activation loss, a transient nucleation-related contribution and increasing late-discharge transport loss. The equilibrium curve is plotted separately.}
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