Nernst equation calculations

Question

I am stuck on 2 problems and would be grateful if anyone decided to give me a proper explanation as to how to solve them. I am not sure how to use the given information to correctly solve what is being asked. I have included the answers so its not a matter of academic integrity, I am genuinely lost.

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1. Given the equation: $$\ce{2X^3+ + 3 Y -> 3 Y^2+ + 2X}$$ The standard reduction potential for the X cation is $\pu{0.498 V}$.
   The standard reduction potential for the Y cation is $\pu{0.187 V}$.
   Using the Nernst equation, calculate the $\log$ of the equilibrium constant $K$ for the above reaction.

The answer is $\log K = 31.52$.

Using the answer, I attempted to work backwards:

$$0.685 = \frac{0.0592}{n} \times 31.52 \implies n = 2.724 ,$$

which I don't understand since $n$ is supposed to be the number of electrons.

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2. Given the equation:
   $$\ce{2 X+ + Y -> Y^2+ + 2 X}$$
   The equilibrium constant for the reaction is $5.912 \cdot 10^{13}$.
   You measure the concentration of $\ce{Y^2+}$ to be $\pu{0.0674 M}$.
   Calculate molar concentration of $\ce{X^+}$.

The answer is $\pu{33.8 nM}$. This one I'm not even sure.

Answer 1

For (1), I think you made the mistake of using a different value for overall cell potential when you worked backwards. Because if you use the correct overall cell potential, you get $K = 31.52$, your $\log K$ would be different if you used $0.685$ (which would make $\log K = 69.43$). Using the correct $E_\mathrm{cell}$ will get you the correct value of $n$ that you're looking for.

To solve for (2), think of what $K$ stands for. It's the ratio of something. Rearrange the ratio to solve for the unknown. It might seem confusing since you might think of an electrochemical solution, but that isn't always the case.

Answer 2

Your cell potential is wrong. You've added up both the reduction potentials instead of adding up one reduction and one oxidation potential. The oxidation potential of $\ce{Y}$ would be the negative of its reduction potential, so on adding you get 0.311, which, when substituted in the Nernst equation, would give you the right answer. Here $n = 6$ because there are 2 $\ce{X}$ atoms losing 3 electrons each.