For the galvanic cell `Zn(s)abs(Zn^(2+) ) abs(Cd^(2+)) Cd(s),E_(cell ) = 0.30 V and E_(cell)^(o) = 0.36 V,` then value of `[Cd^(2+)]//[Zn^(2+)]` will be:

Options

A. 10
B. 1
C. `0.1`
D. 100

Answer

To solve the problem, we will use the Nernst equation, which relates the cell potential to the standard cell potential and the concentrations of the reactants and products.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( E_{\text{cell}} = 0.30 \, \text{V} \)
- \( E_{\text{cell}}^{\circ} = 0.36 \, \text{V} \)
- The half-reactions are:
- \( \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \) (oxidation)
- \( \text{Cd}^{2+} + 2e^- \rightarrow \text{Cd} \) (reduction)

2. **Write the Nernst Equation:**
The Nernst equation is given by:
\[ E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{0.0591}{n} \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) \]
where \( n \) is the number of moles of electrons transferred in the reaction. Here, \( n = 2 \).

3. **Substitute the Known Values:**
Substitute the values into the Nernst equation:
\[ 0.30 = 0.36 - \frac{0.0591}{2} \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) \]

4. **Rearrange the Equation:**
Rearranging gives:
\[ 0.30 - 0.36 = - \frac{0.0591}{2} \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) \]
\[ -0.06 = - \frac{0.0591}{2} \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) \]

5. **Solve for the Logarithm:**
Multiply both sides by -1:
\[ 0.06 = \frac{0.0591}{2} \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) \]
Now, multiply both sides by \( \frac{2}{0.0591} \):
\[ \frac{0.06 \times 2}{0.0591} = \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) \]
\[ \log \left( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \right) = \frac{0.12}{0.0591} \approx 2.03 \]

6. **Convert Logarithm to Concentration Ratio:**
To find the ratio, we take the antilogarithm:
\[ \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} = 10^{2.03} \approx 107.2 \]

7. **Final Answer:**
The concentration ratio \( \frac{[\text{Cd}^{2+}]}{[\text{Zn}^{2+}]} \) is approximately 107.2.