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What is the exact definition of current efficiency of an electrolytic cell? I've looked online and there is no suitable answer to it.

I thought current efficiency would be the amount of electrons that actually participate in a desired reaction. For example, if we are electrolyzing a an aqueous solution of $\ce{FeCl3}$ and we have a $\pu{10A}$ current running for, say, $\pu{100 s}$, at $60\%$ current efficiency, two possible products are $\ce{H2}$ gas and reduced $\ce{Fe}$ metal. Current efficiency of $60\%$ means that half of the electrons that pass through this circuit would reduce the $\ce{Fe}$ instead of reducing $\ce{H2}$.

However, there is another way to think about it, in that it means 60% of the theoretical mass yield of $\ce{Fe}$ comes out of the reaction instead of my original definition, which is different because the reduction of iron (III) requires 3 electrons rather than only 2 electrons for the reduction of $\ce{H2O}$/$\ce{H+}$ ions. So what is the actual definition?

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The theoretical definition is the ratio of electrons used to make the useful product out of the total electrons that were supplied. The problem in the chemistry literature with efficiency definitions, however, is that you will more than likely find that different people will use different definitions for one reason or another (easier to calculate, some type of comparison in their work, or simply because it sounds or looks better). The right way to deal with efficiencies, then, would be to define how the efficiency is calculated in order for the reader to know what you are talking about. If this is an academic setting, I would stick to the theoretical definition, but ask your professor as to what their favorite definition would be.

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