`Mg(s)|Mg^(2+)(aq)||Zn^(2+)(aq)|Zn(s),E^(@)=+3.13V`
The correct plot of `E_("cell")` versus `log.([Mg^(2+)])/([Zn^(2+)])` will be represented as

Options

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Answer

To solve the problem, we need to analyze the electrochemical cell given and apply the Nernst equation to plot the relationship between the cell potential (E_cell) and the logarithm of the concentration ratio of the ions involved.

### Step-by-Step Solution:

1. **Identify the Half-Reactions**:
- At the anode (oxidation):
\[ \text{Mg(s)} \rightarrow \text{Mg}^{2+}(aq) + 2e^- \]
- At the cathode (reduction):
\[ \text{Zn}^{2+}(aq) + 2e^- \rightarrow \text{Zn(s)} \]

2. **Write the Overall Cell Reaction**:
The overall cell reaction can be combined from the half-reactions:
\[ \text{Mg(s)} + \text{Zn}^{2+}(aq) \rightarrow \text{Mg}^{2+}(aq) + \text{Zn(s)} \]

3. **Apply the Nernst Equation**:
The Nernst equation relates the standard cell potential (E°) to the cell potential (E) under non-standard conditions:
\[ E = E° - \frac{RT}{nF} \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]} \right) \]
Where:
- \( E° = +3.13 \, \text{V} \)
- \( R \) is the universal gas constant (8.314 J/(mol·K))
- \( T \) is the temperature in Kelvin (assume standard temperature, 298 K)
- \( n = 2 \) (number of moles of electrons transferred)
- \( F \) is Faraday's constant (96485 C/mol)

4. **Rearranging the Nernst Equation**:
We can rearrange the equation in the form of \( y = mx + c \):
\[ E = E° - \frac{RT}{nF} \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]} \right) \]
Here, \( y = E \), \( x = \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]} \right) \), \( m = -\frac{RT}{nF} \), and \( c = E° \).

5. **Determine the Slope and Intercept**:
- The slope \( m \) is negative because \( -\frac{RT}{nF} < 0 \).
- The intercept \( c = E° \) is positive since \( E° = +3.13 \, \text{V} \).

6. **Plotting the Graph**:
- Since the slope is negative, the graph of \( E_{cell} \) versus \( \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]} \right) \) will be a straight line that slopes downwards from left to right.
- The y-intercept will be at \( +3.13 \, \text{V} \).

### Conclusion:
The correct plot of \( E_{cell} \) versus \( \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]} \right) \) will be a straight line with a negative slope and a positive intercept.