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 Incoming links: Ohmic polarization

Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 1 / Section B / 11 / ii / Solution Created 2026-10-20 Updated 2026-10-20

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On the stated

1 m^{2}

area, the total electric current is

I = 200 A

. The known ohmic polarization is

I R_{Ω} = 1.20 V

. Applying the supplied anodic Tafel equation gives

η_{a} = 0.295 lo g_{10} (200/1) = 0.6788 V .

(1)

Assume the quoted limiting current uses the same cathodic area convention. A steady diffusion-film model gives

c_{s} / c_{b} = 1 - j / j_{L} = 1/41

. The Nernst equation then makes the cathodic concentration overpotential magnitude

∣ η_{c, conc} ∣ = \frac{RT}{2 F} ln 41 = 0.04765 V .

(2)

Relative to the local surface equilibrium potential, the cathode also needs an activation overpotential. Neglecting anodic concentration polarization, the separated-loss accounting is

V_{app} ≃ 1.9265 V + ∣ η_{c, act} ∣ .

(3)

To calculate that activation term, adopt a single effective two-electron Butler-Volmer equation with constant transfer coefficients satisfying

α_{a} + α_{c} = 1

, no anodic concentration polarization and reduction proportional to the surface

Cu^{2 +}

concentration. The supplied anodic Tafel slope then determines the reverse slope through

α_{a} = \frac{RT l n 10}{2 F b} ≃ 0.10016, α_{c} = 1 - α_{a} ≃ 0.89984.

(4)

With

f = 2 F / (RT)

and cathodic overpotential

η_{c}

referenced to bulk equilibrium,

\frac{j _{c}}{j _{0}} = exp (α_{a} f η_{c}) - \frac{c _{s}}{c _{b}} exp (- α_{c} f η_{c}) .

(5)

Solving at

j_{c} = - 200 A m^{- 2}

gives

η_{c} ≃ - 0.12854 V

, including both concentration and activation effects. Thus

V_{app} ≃ 1.20 + 0.67880 + 0.12854 = 2.0073 V ≃ 2.01 V .

(6)

The

0.04765

V concentration loss must not be added again. The kinetic assumption matters: the supplied slope is for copper oxidation, and an exchange current density alone does not specify every cathodic current-potential relation. A multistep copper reaction need not obey the effective single-step relation used for this estimate.

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 1 / Section A(iii) / 7 / i / Solution 2026-10-20

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At fixed discharge current, delivered voltage can be represented as

V = E_{eq} (Q) - I R - η_{act} - η_{conc} - η_{phase},

(1)

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:

Ohmic polarization: electronic, ionic and contact resistance cause a rapid $I R$ drop on applying current. This need not wait for a concentration gradient to develop.
Activation overpotential: finite electrode-reaction kinetics require a departure from equilibrium to sustain charge transfer. It is important when the exchange current is small.
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.
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.

Figure 1.
An illustrative discharge voltage combines equilibrium changes and distinct losses
. 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. **Details**.

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