The electrode potential for the half cell reaction $Ag^{+}(aq)+e^{-}\rightleftharpoons Ag(s)$ is :

Question

Accepted Answer

The electrode potential for the half-cell reaction $Ag^{+}(aq) + e^{-} \rightleftharpoons Ag(s)$ can be determined using the Nernst equation. The Nernst equation for a general half-cell reaction is given by:

$$ E = E^{\circ} - \frac{RT}{nF} \ln Q $$

where:
- $E$ is the electrode potential.
- $E^{\circ}$ is the standard electrode potential.
- $R$ is the universal gas constant.
- $T$ is the temperature in Kelvin.
- $n$ is the number of moles of electrons transferred in the half-cell reaction.
- $F$ is Faraday's constant.
- $Q$ is the reaction quotient.

For the given half-cell reaction $Ag^{+}(aq) + e^{-} \rightleftharpoons Ag(s)$, the reaction quotient $Q$ is:

$$ Q = \frac{1}{[Ag^{+}]} $$

since the concentration of solid silver $Ag(s)$ is considered to be 1 (as it is a pure solid).

Substituting $Q$ into the Nernst equation, we have:

$$ E = E_{Ag^{+}|Ag}^{\circ} - \frac{RT}{F} \ln \left(\frac{1}{[Ag^{+]}}\right) $$

This simplifies to:

$$ E = E_{Ag^{+}|Ag}^{\circ} + \frac{RT}{F} \ln [Ag^{+}] $$

Thus, Option 2 is correct:

$$ E = E_{Ag^{+}|Ag}^{\circ} + \frac{RT}{F} \ln [Ag^{+}] $$

**Explanation for Incorrect Options:**

1. **Option 1:** $E = E_{Ag^{+}|Ag}^{\circ} - \frac{RT}{F} \ln [Ag^{+}]^{2}$

This option incorrectly uses $[Ag^{+}]^{2}$ in the logarithmic term. The reaction quotient $Q$ for this half-cell reaction is $\frac{1}{[Ag^{+}]}$, not $[Ag^{+}]^{2}$. Therefore, this option does not correctly represent the Nernst equation for the given reaction.

3. **Option 3:** $E = E_{Ag^{+}|Ag}^{\circ} - \frac{RT}{F} \ln [Ag^{+}]$

This option incorrectly has a negative sign in front of the logarithmic term. The correct form, derived from the Nernst equation, should have a positive sign when the reaction quotient is $\frac{1}{[Ag^{+}]}$, leading to a positive logarithmic term as shown in Option 2.

4. **Option 4:** $E = E_{Ag^{+}|Ag}^{\circ} + \frac{RT}{F} \ln \frac{1}{[Ag^{+}]}$

This option is incorrect because it does not simplify the logarithmic term correctly. The expression $\ln \frac{1}{[Ag^{+}]}$ should be simplified to $-\ln [Ag^{+}]$, which would then require a negative sign in front of the logarithmic term, contradicting the positive sign needed as shown in Option 2.

In conclusion, Option 2 is the correct representation of the electrode potential for the given half-cell reaction according to the Nernst equation.