Animated Solution for Chemistry - Electrochemistry: The standard Gibbs energy for the given cell reaction in ${\text{kJ mol}}^{-1}$ at $298\text{ K}$ is $\text{Zn}(s)+{\text{Cu}}^{2+}(aq)⟶{\text{Zn}}^{2+}(aq)+\text{Cu}(s),E^{\circ }=2\text{ V}$ at $298\text{ K}$ (Faraday's constant, $F=96000{\text{ C mol}}^{-1}$)

Imagine a classic galvanic cell, specifically the Daniell cell. We have a solid zinc electrode immersed in a solution of zinc ions, and a solid copper electrode immersed in a solution of copper ions. The overall cell reaction is given by:

This is a spontaneous redox reaction. Zinc undergoes oxidation (losing electrons), and copper undergoes reduction (gaining electrons). This flow of electrons generates an electrical potential, which is given as $E_{\text{cell}}^{\circ }=2.0\text{ V}$.

To find the standard Gibbs free energy change ($\Delta G^{\circ }$), which tells us about the maximum useful non-expansion work the cell can perform, we use the fundamental thermodynamic relationship:

Here, $n$ is the number of moles of electrons transferred in the balanced equation, $F$ is Faraday's constant (the charge of one mole of electrons), and $E_{\text{cell}}^{\circ }$ is the standard cell potential.

Let's carefully identify our variables from the given data: 1. $n=2$: In the reaction, $\text{Zn}$ goes to ${\text{Zn}}^{2+}$ (losing 2 electrons) and ${\text{Cu}}^{2+}$ goes to $\text{Cu}$ (gaining 2 electrons). 2. $F=96000{\text{ C mol}}^{-1}$: This is provided in the question. 3. $E_{\text{cell}}^{\circ }=2.0\text{ V}$: Also provided.

Now, we substitute these values into our master equation. It is crucial not to forget the negative sign!

Multiplying these together:

Since the options are given in ${\text{kJ mol}}^{-1}$, we must divide our result by $1000$:

The negative sign is a beautiful confirmation of thermodynamics: a positive cell potential ($E^{\circ }>0$) always corresponds to a negative Gibbs free energy change ($\Delta G^{\circ }<0$), indicating that the reaction is spontaneous.