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This doubt came to my mind while answering the following question:

Why does a voltaic cell not operate unless the two separate compartments are connected by an external circuit?

The answer seems to be that only on connecting the two halves of a Redox reaction via an external circuit, does one allow the electrons to be transferred from one species to another. And while the electrons move towards their goal (to a higher potential), they do work in the way (light a bulb or a motor or something).

Another way to view this process is that the free energy of the Redox reaction is used to do useful electrical work.

But then my brain started to wonder and there came up a question - what happens to all this energy when none of it is actually used to do electrical work, i.e, when the Redox reaction is let to occur spontaneously, where does all the free energy go? Does it get converted to heat, thus warming the solution? Or does it simply stay stored in the solution in some form?

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Free energy is a state function. No matter how you run the reaction, if you start with a certain state (set of concentration) and end with a certain state (set of concentrations at equilibrium), the change will be the same.

The free energy of reaction is the maximal amount of non-pV work (electrical work in this case) the reaction can do. If it does not do any work, that energy will be released as heat instead. For example, if you short the battery, the wire and the electrochemical cell will heat up while the reaction is going on.

If you want an argument from first principles, you should look at the first law of thermodynamics. The change in energy of a system is equal to the sum of the heat and the work exchanged with the surrounding. When making optimal use of the free energy, you maximize the work (even sometimes transferring heat from the surrounding to do even more work).

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All depends on the cell chemistry and geometry. It is usually a combination of keeping the energy and conversion of the free energy to thermal energy, when reagents diffuse and happen to meet each other. Or, some side reaction with solvent or auxiliary components may occur, like for $\ce{Li-ion}$ cells.

Some primary lithium cells last many years, while some early generation $\ce{NiMH}$ cells selfdischarge in few months.

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