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The explanation I have found is that primary cells have irreversible reactions, so passing current in the opposite direction cannot recharge the cell. But as far as I understood it, no reaction is really irreversible - what one calls irreversible simply has very high values of the equilibrium constant, so for all intents and purposes proceeds to completion. Nevertheless, some small - maybe undetectable practically - amount of reactant does remain and the reaction does reach equilibrium. So even in a primary cell, when it is used up, equilibrium has been achieved, so recharging should be possible - it might take long to build up the initial value of potential since the required concentration of the electrolytes will be very different from that at equilibrium, but it should be possible. Why then is it not?

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In some cases, a primary cell can be recharged, e.g. Mn-alkaline cells can take a partial charge. However, there are a number of reasons primary cells cannot be fully recharged, and that secondary cells have to be specifically designed to allow recharging.

  1. Gas evolution: once most of a cell is charged (though all material has not been converted to its original state), the electrolyte is electrolyzed (apologies for the pleonastic redundancy), usually evolving hydrogen and oxygen, causing the cell to rupture. Nickel-metal hydride cells, for example, get around that issue by incorporating a catalyst to recombine them, $\ce{2 H2 + O2 -> 2 H2O + heat}$.

  2. Physical change in the reactants: the standard "dry cell" (Leclanché cell) is made with an outer shell of zinc. As the cell discharges, parts of the shell are dissolved, eventually leaving holes. Reversing the current flow will cause zinc to redeposit, but not necessarily filling in the weakened areas. Even lead-acid cells designed for storage have limited lifetime because of the buildup of large, insoluble crystals of lead sulfate (sulfation, aka sulphation in GB), which may break off from the lead plates, preventing reversal of the chemical reaction during charging.

  3. Physical damage to the insulator: metal dendrite may grow during charge-discharge cycling, piercing the dielectric between cells, even in well-designed secondary cells.

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Irreversible in this context isn't a thermodynamic or kinetic thing, necessarily. It simply means that applying a current in reverse doesn't result in the original reagents. It may be that there is another possible reaction at a similar redox potential that leads to a different compound being formed or that the manner in which it is deposited is not suitable, e.g. the zinc in an alkaline battery being plated back onto the anode unevenly or such that it damages the ion conducting membrane.

It's also possible that recharging may make a primary cell inoperative by some other mechanism, such as gas formation in the potassium hydroxide electrolyte of an alkaline cell deforming the case or damaging the half-cell separator. Primary cells are also typically constructed cheaply as they don't have to withstand the stresses of secondary cells, e.g. the steel casing of alkaline batteries is susceptible to corrosion.

Things like the various lithium battery chemistries make better secondary cells as they use organic solvents which reduce the problem of gas evolution and lithium can form intercalated compounds where the lithium can be evenly spread instead of growing dendrites of solid lithium which can damage the cell when recharged. That's the main difference between disposable and rechargeable lithium cells: disposable cells use solid lithium electrodes because the anode only has to dissolve once and the difficulty of replating the anode is not an issue, whereas a rechargeable cell needs to return the anode to its original state many times and an intercalated lithium compound, though more expensive, is far easier to replenish with lithium.

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