Question

A cell is given as $M(s) | M^{n+}(aq) || M^{z+}(aq) | M(s)$. For which of the following condition, $E_{\text{cell}}$ is positive:

• A. $C_1<C_2$ (If $C_1$ is concentration at cathode)
• B. $C_2<C_1$ (If $C_1$ is concentration at anode)
• C. $C_1<C_2$ (If $C_2$ is concentration at anode)
• D. $C_1>C_2$ (If $C_1$ is concentration at cathode)

Hint:

In a concentration cell, the spontaneous direction of reaction moves ions from the higher concentration compartment (cathode) to the lower concentration compartment (anode), thereby increasing entropy.

Answer 1

The Correct Option is D

The given question involves understanding the conditions under which the electromotive force (\(E_{\text{cell}}\)) of a given electrochemical cell is positive. The cell is represented as \(M(s) | M^{n+}(aq) || M^{z+}(aq) | M(s)\), hinting at a concentration cell where the same electrode and electrolyte are used, but at different concentrations.

In this situation, the Nernst equation is used to calculate the cell potential, \(E_{\text{cell}}\): 

\(E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln \frac{C_2}{C_1}\)

Where:

• \(E^\circ_{\text{cell}}\) is the standard cell potential (which is zero for concentration cells since the electrodes are the same).
• \(R\) is the universal gas constant (8.314 J/mol·K).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the cell reaction.
• \(F\) is Faraday's constant (96485 C/mol).
• \(C_2\) and \(C_1\) are the concentrations of the ions in the solutions at the two electrodes.

For the cell potential to be positive, we need \(\ln \frac{C_2}{C_1}\) to be negative, which happens when \(C_1 > C_2\). This means the concentration of ions at the cathode (\(C_1\)) should be greater than that at the anode (\(C_2\)).

Thus, the correct condition for a positive \(E_{\text{cell}}\) is:

• \(C_1 > C_2\) (If \(C_1\) is the concentration at the cathode)

Therefore, the correct answer is $C_1>C_2$ (If $C_1$ is concentration at cathode).

This ensures the operation of the cell as a concentration cell where diffusion of ions from high to low concentration drives the generation of a potential difference, consistent with the principles of thermodynamics.