Question

Given that: \[ E^\circ_{\text{Fe}^{3+}/\text{Fe}} = -0.036\, \text{V} \quad \text{and} \quad E^\circ_{\text{M}^{x+}/\text{M}} = 0.15\, \text{V} \] A Galvanic cell is formed using above electrodes, whose \( E_{\text{cell}} = 0.2047\, \text{V} \) when reaction quotient of cell reaction is \(10^{-2}\). Find the value of \(x\). [Nearest integer]

• A. \(1 \)
• B. \(2 \)
• C. \(3 \)
• D. \(4 \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The cell potential is calculated using the Nernst equation. Since \( E^\circ_{\text{M}^{x+}/\text{M}} > E^\circ_{\text{Fe}^{3+}/\text{Fe}} \), the metal \( M \) acts as the cathode and \( Fe \) acts as the anode. The value of \( n \) in the Nernst equation represents the total moles of electrons transferred in the balanced cell reaction.

Step 2: Key Formula or Approach:
\[ E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} \] \[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.059}{n} \log Q \] Reaction: \( x \text{Fe} + 3 \text{M}^{x+} \to x \text{Fe}^{3+} + 3 \text{M} \), where \( n = 3x \).

Step 3: Detailed Explanation:
First, calculate \( E^\circ_{\text{cell}} \):
\[ E^\circ_{\text{cell}} = 0.15 - (-0.036) = 0.186 \, \text{V} \] Apply the Nernst Equation:
\[ 0.2047 = 0.186 - \frac{0.059}{n} \log(10^{-2}) \] \[ 0.2047 - 0.186 = - \frac{0.059}{n} (-2) \] \[ 0.0187 = \frac{0.118}{n} \] \[ n = \frac{0.118}{0.0187} \approx 6.31 \] Since \( n = 3x \):
\[ 3x = 6 \implies x = 2 \]

Step 4: Final Answer:
The value of \( x \) is 2.