For a $$Mg|Mg^{2+}(aq)||Ag^{+}(aq)|Ag$$ the correct Nernst Equation is :

Find the correct Nernst equation for the cell $$Mg|Mg^{2+}(aq)||Ag^+(aq)|Ag$$. An anode (oxidation) half-reaction is $$Mg \rightarrow Mg^{2+} + 2e^-$$ and the cathode (reduction) half-reaction is $$2Ag^+ + 2e^- \rightarrow 2Ag$$. Combining these gives the net cell reaction:

$$ Mg + 2Ag^+ \rightarrow Mg^{2+} + 2Ag $$. Here $$n = 2$$ electrons are transferred in the balanced reaction. The reaction quotient $$Q$$ is given by

$$ Q = \frac{[Mg^{2+}]}{[Ag^+]^2} $$. (Pure solids Mg and Ag have activity = 1.)

Hence the Nernst equation is

$$ E_{\text{cell}} = E°_{\text{cell}} - \frac{RT}{nF}\ln Q = E°_{\text{cell}} - \frac{RT}{2F}\ln\frac{[Mg^{2+}]}{[Ag^+]^2} $$. This can be rewritten as

$$ E_{\text{cell}} = E°_{\text{cell}} + \frac{RT}{2F}\ln\frac{[Ag^+]^2}{[Mg^{2+}]} $$ (using $$-\ln(A/B) = \ln(B/A)$$).

The correct answer is Option B: $$E_{\text{cell}} = E°_{\text{cell}} + \frac{RT}{2F}\ln\frac{[Ag^+]^2}{[Mg^{2+}]}$$.