In which of the following $\Delta G^{\circ}$ is correctly related to E°cell?

Q: In which of the following $\Delta G^{\circ}$ is correctly related to E°cell?

In which of the following $\Delta G^{\circ}$ is correctly related to E°cell?

Question

Accepted Answer

The relationship between the standard Gibbs free energy change ($\Delta G^{\circ}$) and the standard cell potential ($E^{\circ}_{\text{cell}}$) is a fundamental concept in electrochemistry. The correct relationship is given by the equation:

$$
\Delta G^{\circ} = -nF E^{\circ}_{\text{cell}}
$$

where:
- $\Delta G^{\circ}$ is the standard Gibbs free energy change.
- $n$ is the number of moles of electrons transferred in the balanced redox reaction.
- $F$ is the Faraday constant, approximately $96485 \, \text{C/mol}$.
- $E^{\circ}_{\text{cell}}$ is the standard cell potential.

**Explanation for Option 4:**

Option 4 correctly states that $\Delta G^{\circ} = -nF E^{\circ}_{\text{cell}}$. This equation is derived from the relationship between Gibbs free energy and the work done by an electrochemical cell. The negative sign indicates that a positive cell potential ($E^{\circ}_{\text{cell}}$) corresponds to a spontaneous reaction, which results in a negative $\Delta G^{\circ}$. This is consistent with the thermodynamic principle that spontaneous processes have a negative Gibbs free energy change.

**Why the Other Options are Incorrect:**

1. **Option 1: $\Delta G^{\circ} = RT \ln E^{\circ}_{\text{cell}}$**

This option is incorrect because it incorrectly relates $\Delta G^{\circ}$ to the natural logarithm of the cell potential. The correct relationship involves the number of moles of electrons and the Faraday constant, not the gas constant $R$ and the natural logarithm.

2. **Option 2: $\Delta G^{\circ} = -RT \ln E^{\circ}_{\text{cell}}$**

Similar to Option 1, this option incorrectly uses the natural logarithm of the cell potential. The correct formula does not involve the gas constant $R$ or the natural logarithm of $E^{\circ}_{\text{cell}}$.

3. **Option 3: $\Delta G^{\circ} = nF E^{\circ}_{\text{cell}}$**

This option is incorrect because it lacks the negative sign. The negative sign in the correct equation ($\Delta G^{\circ} = -nF E^{\circ}_{\text{cell}}$) is crucial as it indicates the direction of spontaneity. Without the negative sign, the relationship would suggest that a positive cell potential results in a positive $\Delta G^{\circ}$, which contradicts the principle of spontaneity.

In conclusion, Option 4 is the only option that correctly represents the relationship between $\Delta G^{\circ}$ and $E^{\circ}_{\text{cell}}$ in accordance with electrochemical principles.