Batteries

Three families, one question. Every named battery is sorted by asking whether its cell reaction is reversible on recharging.

• Primary cells — the reaction runs once. Reactants are consumed, side products build up around the electrodes, and the cell is discarded. Leclanché (dry) cell and mercury cell belong here.
• Secondary cells — an external current forces the reaction exactly backwards, regenerating the original electrodes and electrolyte. The lead storage accumulator is the syllabus example; it can be discharged and recharged hundreds of times.
• Fuel cells — fuel and oxidant are fed in continuously from an external reservoir rather than stored inside, so the cell keeps producing current as long as it is supplied. The $\text{H}_2$–$\text{O}_2$ fuel cell is the standard case.

Why voltage behaves differently across the three. A cell's EMF tracks the concentrations of the species at its electrodes (the same logic behind the Nernst equation you met with simple galvanic cells). In the Leclanché cell the electrolyte paste is consumed and side products accumulate, so the EMF drifts down with use. In the mercury cell the overall reaction $\text{Zn(Hg)} + \text{HgO} \rightarrow \text{ZnO} + \text{Hg}$ involves no dissolved ion whose concentration changes — $\text{OH}^-$ is regenerated at the cathode exactly as fast as it is used at the anode — so the EMF stays essentially flat until the reactants nearly run out.

Building a battery from cells. A single galvanic cell rarely supplies the voltage a real device needs, so practical batteries wire several identical cells in series, positive terminal to negative terminal, so their EMFs add:

where: $E_{\text{battery}}$ is the terminal voltage of the assembled battery, $E_{\text{cell}}$ is the EMF of one identical cell, and $n$ is the number of cells connected in series. This is exactly why a lead-acid car battery, built from cells each near $2\ \text{V}$, is packaged as six cells in series to give the familiar $12\ \text{V}$.

The lead storage battery is the one named example that is truly rechargeable, and the reason is baked into what it produces on discharge. Both the anode reaction ($\text{Pb} + \text{SO}_4^{2-} \rightarrow \text{PbSO}_4 + 2e^-$) and the cathode reaction ($\text{PbO}_2 + \text{SO}_4^{2-} + 4\text{H}^+ + 2e^- \rightarrow \text{PbSO}_4 + 2\text{H}_2\text{O}$) deposit the same insoluble solid, $\text{PbSO}_4$, directly onto the electrode surface it formed at — it does not dissolve away or drift off as a gas. Because the products stay put, physically attached exactly where the reverse reaction will need them, an external current can push electrons the other way and rebuild $\text{Pb}$ and $\text{PbO}_2$ in place. Contrast this with the Leclanché cell, where $\text{NH}_3$ gas and other soluble by-products disperse through the electrolyte paste; even if you tried to force current backwards, there is no clean solid deposit sitting in the right place to reassemble. Reversibility, in other words, is not a property of "how strong" a reaction is — it is a property of whether the products remain physically available, undisturbed, at the electrode.

Problem: A hearing aid needs a steady $4.05\ \text{V}$ supply. Mercury cells are chosen because their voltage does not drift with use. Given a single mercury cell's EMF is $1.35\ \text{V}$, (a) how many cells must be connected in series, and (b) why is the mercury cell — rather than the Leclanché cell — the right choice for this device?

Step 1. Apply equation (1) with the target battery voltage and single-cell EMF: $E_{\text{battery}} = n \times E_{\text{cell}} \Rightarrow 4.05 = n \times 1.35$.

Step 2. Solve for $n$: $n = 4.05 / 1.35 = 3$. Three mercury cells wired in series deliver the required $4.05\ \text{V}$.

Step 3. Justify the choice of cell type: the mercury cell's overall reaction $\text{Zn(Hg)} + \text{HgO} \rightarrow \text{ZnO} + \text{Hg}$ involves no net change in dissolved-ion concentration, so its EMF holds steady near $1.35\ \text{V}$ for almost its entire working life. A Leclanché cell of nominal $1.5\ \text{V}$ would seem to need fewer cells for a similar voltage, but its EMF sags as the electrolyte paste is consumed — unacceptable for a hearing aid, which needs a dependable, unchanging reference voltage rather than just an initial value. Answer: 3 mercury cells in series, chosen for their flat discharge voltage rather than raw cell count.

• Calling the lead storage battery a "primary cell" — it is secondary, precisely because $\text{PbSO}_4$ stays fixed at each electrode and can be driven back to $\text{Pb}$ and $\text{PbO}_2$ by recharging.
• Assuming equation (1) applies to any combination of cells — it only adds EMFs correctly when identical cells are connected in series (terminal to terminal); mixing cell types or connecting them in parallel does not simply multiply the voltage.
• Treating fuel cells as batteries that "store" fuel internally — they are supplied continuously from an external tank, which is what lets them run indefinitely rather than being an unusually large battery.
• Mixing up which primary cell has the steadier voltage — both Leclanché and mercury cells are primary with zinc anodes, but only the mercury cell's reaction leaves ion concentrations unchanged.

Ask the tutor: if you would like a side-by-side comparison table of all four named batteries, or want to work through why fuel cells are considered more environmentally friendly than conventional batteries.