Which graph correctly correlates `E_("cell")` as a function of concentration for the cell (for different values of M nad M' `Zn(s)+Cu^(2+)(M)rarrZn^(2+(M')+Cu(s), E_("cell")^(@)=1.10V`
X - axis : `log_(10).([Zn^(2+)])/([Cu^(2+)]),` Y - axis : `E_("cell")`

Options

A.
B.
C.
D.

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the Cell Reaction
The cell reaction is given as:
\[ \text{Zn(s)} + \text{Cu}^{2+}(M) \rightarrow \text{Zn}^{2+}(M') + \text{Cu(s)} \]

### Step 2: Identify Oxidation and Reduction
In this reaction:
- Zinc (Zn) is oxidized to Zinc ions (\( \text{Zn}^{2+} \)).
- Copper ions (\( \text{Cu}^{2+} \)) are reduced to Copper (Cu).

### Step 3: Determine the Number of Electrons Transferred
From the reaction, we can see that 2 electrons are involved in the process:
- Zn → \( \text{Zn}^{2+} + 2e^- \) (oxidation)
- \( \text{Cu}^{2+} + 2e^- \rightarrow \text{Cu} \) (reduction)

Thus, \( n = 2 \).

### Step 4: Write the Nernst Equation
The Nernst equation for this cell reaction is:
\[ E_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{RT}{nF} \ln Q \]
Where \( Q \) is the reaction quotient given by \( Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \).

### Step 5: Substitute Values into the Nernst Equation
Given \( E^{\circ}_{\text{cell}} = 1.10 \, \text{V} \), we can rewrite the Nernst equation as:
\[ E_{\text{cell}} = 1.10 - \frac{0.0591}{n} \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \]
Substituting \( n = 2 \):
\[ E_{\text{cell}} = 1.10 - 0.0591 \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \]

### Step 6: Analyze the Graph
In the equation, we see that:
- The y-intercept (when \( \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) = 0 \)) is \( E_{\text{cell}} = 1.10 \, \text{V} \).
- The slope is negative, indicating that as the log of the concentration ratio increases, \( E_{\text{cell}} \) decreases.

### Step 7: Conclusion
Based on the analysis, the graph of \( E_{\text{cell}} \) versus \( \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \) will have:
- An intercept at \( 1.10 \, \text{V} \).
- A downward slope.

Thus, the correct option is the one that shows a negative slope starting from the intercept of \( 1.10 \, \text{V} \).