`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

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the given electrochemical cell and apply the Nernst equation to determine how the cell potential (E_cell) varies with the logarithm of the concentration ratio of magnesium ions to zinc ions.

### Step-by-Step Solution:

1. **Identify the Cell Reaction**:
The cell is represented as:
\[ \text{Mg(s)} | \text{Mg}^{2+}(aq) || \text{Zn}^{2+}(aq) | \text{Zn(s)} \]
The half-reactions are:
- Oxidation: \(\text{Mg} \rightarrow \text{Mg}^{2+} + 2e^-\)
- Reduction: \(\text{Zn}^{2+} + 2e^- \rightarrow \text{Zn}\)

2. **Write the Overall Cell Reaction**:
The overall cell reaction can be written as:
\[ \text{Mg} + \text{Zn}^{2+} \rightarrow \text{Mg}^{2+} + \text{Zn} \]

3. **Apply the Nernst Equation**:
The Nernst equation for the cell potential is given by:
\[ E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]}\right) \]
Where:
- \(E^\circ_{cell} = +3.13 \, \text{V}\) (given)
- \(n = 2\) (number of electrons transferred)

4. **Substitute Values into the Nernst Equation**:
Substituting the values into the Nernst equation:
\[ E_{cell} = 3.13 - \frac{0.0591}{2} \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]}\right) \]
Simplifying gives:
\[ E_{cell} = 3.13 - 0.02955 \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]}\right) \]

5. **Identify the Plot Characteristics**:
- On the y-axis, we have \(E_{cell}\).
- On the x-axis, we have \(\log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]}\right)\).
- The slope of the line is negative (\(-0.02955\)), indicating that as the log ratio increases, the cell potential decreases.

6. **Determine the Correct Plot**:
Since the slope is negative, 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.

### Conclusion:
The correct option for the plot of \(E_{cell}\) versus \(\log \left( \frac{[\text{Mg}^{2+}]}{[\text{Zn}^{2+}]}\right)\) is the one that shows a negative slope, which corresponds to option 2.