The cell potential for the given cell will be: $Mg|Mg^{2+}||Ag^{+}|Ag$

Question

Accepted Answer

To determine the correct expression for the cell potential of the given electrochemical cell, $Mg|Mg^{2+}||Ag^{+}|Ag$, we need to apply the Nernst equation. The Nernst equation for a general cell reaction is given by:

$$ E_{cell} = E_{cell}^{\circ} - \frac{RT}{nF} \ln Q $$

where:
- $ E_{cell} $ is the cell potential under non-standard conditions.
- $ E_{cell}^{\circ} $ is the standard cell potential.
- $ R $ is the universal gas constant.
- $ T $ is the temperature in Kelvin.
- $ n $ is the number of moles of electrons transferred in the balanced equation.
- $ F $ is Faraday's constant.
- $ Q $ is the reaction quotient.

For the given cell, the half-reactions are:

1. Oxidation at the anode: $ Mg \rightarrow Mg^{2+} + 2e^- $
2. Reduction at the cathode: $ Ag^+ + e^- \rightarrow Ag $

The overall cell reaction is:

$$ Mg + 2Ag^+ \rightarrow Mg^{2+} + 2Ag $$

From this reaction, we see that 2 moles of electrons are transferred ($ n = 2 $).

The reaction quotient $ Q $ is given by:

$$ Q = \frac{[Mg^{2+}]}{[Ag^{+}]^2} $$

Substituting these into the Nernst equation, we get:

$$ E_{cell} = E_{cell}^{\circ} - \frac{RT}{2F} \ln \frac{[Mg^{2+}]}{[Ag^{+}]^2} $$

This matches Option 4:

$$ E_{cell} = E_{cell}^{\circ} - \frac{RT}{2F} \ln \frac{[Mg^{2+}]}{[Ag^{+}]^2} $$

**Explanation for why Option 4 is correct:**

- The expression correctly accounts for the stoichiometry of the reaction, where 2 moles of $ Ag^+ $ are involved for every mole of $ Mg^{2+} $ produced. This is reflected in the $ [Ag^{+}]^2 $ term in the denominator of the reaction quotient $ Q $.

**Explanation for why the other options are incorrect:**

1. **Option 1:** $ E_{cell}=E_{cell}^{\circ}-\frac{RT}{2F} \ln \frac{[Mg^{2+}]}{[Ag^{+}]} $
   - This option incorrectly uses $ [Ag^{+}] $ instead of $ [Ag^{+}]^2 $ in the denominator, failing to account for the stoichiometry of the reaction.

2. **Option 2:** $ E_{cell}=E_{cell}^{\circ}-\frac{RT}{2F} \ln \frac{[Ag^{+}]^2}{[Mg^{2+}]} $
   - This option incorrectly inverts the reaction quotient, placing $ [Ag^{+}]^2 $ in the numerator and $ [Mg^{2+}] $ in the denominator, which is opposite to the correct expression.

3. **Option 3:** $ E_{cell}=E_{cell}^{\circ}+\frac{RT}{2F} \ln \frac{[Mg^{2+}]}{[Ag^{+}]^2} $
   - This option incorrectly uses a positive sign before the logarithmic term, which is not consistent with the Nernst equation for a spontaneous reaction under standard conditions.

Thus, Option 4 is the correct expression for the cell potential of the given electrochemical cell.