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Consider the following electrochemical cell: $$\ce{Zn|Zn^{2+}}(0.1~\mathrm{M})||\ce{Zn^{2+}}(0.01~\mathrm{M})\ce{|Zn}$$ How should I write the overall cell reaction for this cell? I can identify the oxidation and reduction reactions but when they are added, all the species cancel out, leaving my with no overall equation.

Anode: $\ce{Zn -> Zn^{2+} + 2e}$

Cathode: $\ce{Zn^{2+} + 2e -> Zn}$

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The species do cancel out. However, the concentrations of the species do not. It is a bit of a matter of preference as to how you express this. The way I would write it is:

$$\ce{Zn^{2+}_{(aq)} ($c = 0.01\text{ M}$) + Zn_{(s)} <=> Zn^{2+}_{(aq)} ($c = 0.1\text{ M}$) + Zn_{(s)}}$$

for which $E = E^0 - \frac{RT}{2F}\,\ln\!\left(\frac{0.1}{0.01}\right) = -0.030 \text{ V}$ at 298 K, since $E^0 = 0 \text{ V}$. A negative value of $E$ agrees with the reverse reaction above being spontaneous, which is of course what you would expect since $0.1 > 0.01$.

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