To determine the Gibbs free energy change, $\Delta G^\circ_{\text{cell}}$, for the given Ni-Cad battery reaction, we can use the following formula:

$\Delta G^\circ_{\text{cell}} = -n F E^\circ_{\text{cell}}$

where:

• $n$ is the number of moles of electrons transferred,
• $F$ is Faraday's constant ($96485 \, \text{C/mol e}^-$),
• $E^\circ_{\text{cell}}$ is the standard cell potential.

Step 1: Calculate $E^\circ_{\text{cell}}$

The cell potential $E^\circ_{\text{cell}}$ can be calculated by finding the difference between the reduction potentials of the cathode and the anode.

The reduction half-reactions and their standard reduction potentials are:

1. Cathode (Reduction): $\text{NiO(OH)}(s) + 2\text{H}_2\text{O}(l) + 2e^- \rightarrow \text{Ni(OH)}_2(s) + 2\text{OH}^-(aq)$ with $E^\circ_{\text{red}} = 1.32 \, V$
2. Anode (Oxidation): $\text{Cd(OH)}_2(s) + 2e^- \rightarrow \text{Cd}(s) + 2\text{OH}^-(aq)$ with $E^\circ_{\text{red}} = -0.403 \, V$

To find $E^\circ_{\text{cell}}$: $E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} = 1.32 \, V - (-0.403 \, V) = 1.32 \, V + 0.403 \, V = 1.723 \, V$

Step 2: Determine $n$, the number of electrons

From the balanced half-reactions, 2 moles of electrons are transferred in the reaction.

Step 3: Calculate $\Delta G^\circ_{\text{cell}}$

Now, substitute the values into the Gibbs free energy formula: $\Delta G^\circ_{\text{cell}} = -n F E^\circ_{\text{cell}}$ $\Delta G^\circ_{\text{cell}} = - (2) (96485 \, \text{C/mol e}^-) (1.723 \, V)$

Calculating this: $\Delta G^\circ_{\text{cell}} = -2 \times 96485 \times 1.723 = -332,728.61 \, \text{J} = -332.73 \, \text{kJ}$

Final Answer:

$\Delta G^\circ_{\text{cell}} = -332.73 \, \text{kJ}$

Would you like further clarification on any step, or do you have additional questions?

Here are 5 related questions to explore:

1. How does the standard reduction potential relate to the spontaneity of a reaction?
2. What happens to $\Delta G^\circ$ if the cell potential is reversed?
3. Why is Faraday's constant important in electrochemical calculations?
4. How would the cell potential change if the concentration of $OH^-$ ions varied?
5. What is the impact of temperature on the cell potential?

Tip: Always verify the number of electrons transferred when calculating Gibbs free energy changes in electrochemical reactions, as it can vary between reactions.