Question

For a \( \text{Mg} | \text{Mg}^{2+} (aq) || \text{Ag}^+ (aq) | \text{Ag} \), the correct Nernst Equation is:

• A. \( E_{\text{cell}} = E^\circ_{\text{cell}} + \frac{RT}{2F} \ln \left( \frac{[\text{Ag}^+]}{[\text{Mg}^{2+}]} \right) \)
• B. \( E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Ag}^+]}{[\text{Mg}^{2+}]} \right) \)
• C. \( E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]} \right) \)
• D. \( E_{\text{cell}} = E^\circ_{\text{cell}} + \frac{RT}{2F} \ln [\text{Ag}^+]^2 \)

Hint:

When writing the Nernst equation, ensure that the ion concentrations are correctly placed in the logarithmic expression.

Answer 1

The Correct Option is C

To determine the correct Nernst equation for the given electrochemical cell, Mg | Mg2+ (aq) || Ag+ (aq) | Ag, we must first understand the Nernst equation's general form:

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

where:

• \( E_{\text{cell}} \) is the cell potential under non-standard conditions.
• \( E^\circ_{\text{cell}} \) is the standard cell potential.
• \( R \) 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 reaction.
• \( F \) is Faraday’s constant (96485 C/mol).
• \( Q \) is the reaction quotient.

In this cell, magnesium is oxidized and silver ions are reduced. The half-reactions are:

• Oxidation: \( \text{Mg} \rightarrow \text{Mg}^{2+} + 2e^- \)
• Reduction: \( \text{Ag}^+ + e^- \rightarrow \text{Ag} \) (this occurs twice in the balanced equation for cell reaction)

The balanced overall reaction is:

\( \text{Mg} + 2\text{Ag}^+ \rightarrow \text{Mg}^{2+} + 2\text{Ag} \)

The reaction quotient \( Q \) is given by:

\( Q = \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]^2} \)

Here, \( n = 2 \) based on the number of electrons transferred. Substituting these into the Nernst equation, we have:

\( E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]^2} \right) \)

Since only the concentration ratio's argument in the logarithmic function is squared with respect to silver ions, this simplifies the expression further when compared to the given options. Thus, the correct form aligns with:

\( E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]} \right) \)

Answer 2

Step 1: Understand the electrochemical cell.
The given cell is:
\[ \text{Mg} | \text{Mg}^{2+}(aq) || \text{Ag}^+(aq) | \text{Ag} \] This represents a galvanic cell where magnesium acts as the anode and silver acts as the cathode.

Step 2: Identify the half-cell reactions.
At the anode (oxidation occurs):
\[ \text{Mg (s)} \rightarrow \text{Mg}^{2+}(aq) + 2e^- \] At the cathode (reduction occurs):
\[ 2\text{Ag}^+(aq) + 2e^- \rightarrow 2\text{Ag (s)} \] Overall cell reaction:
\[ \text{Mg (s)} + 2\text{Ag}^+(aq) \rightarrow \text{Mg}^{2+}(aq) + 2\text{Ag (s)} \]

Step 3: Write the general Nernst equation.
The Nernst equation for a cell is:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q \] where \( Q \) is the reaction quotient, \( R \) is the gas constant, \( T \) is the temperature (in K), \( F \) is Faraday’s constant, and \( n \) is the number of electrons transferred.

Step 4: Write the expression for \( Q \).
From the cell reaction:
\[ Q = \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]^2} \] Since \( n = 2 \), the Nernst equation becomes:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]^2} \right) \] Simplify the logarithmic expression:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]} \right) \]

Final Answer:
\[ \boxed{E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{2F} \ln \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^+]} \right)} \]