Question

\(Al_2O_3\) can be electrolyzed with an inert anode or a carbon anode.
Overall reactions are as follows:
Reaction I: For inert anode,
\[ \frac{2}{3}Al_2O_3(s) \to \frac{4}{3}Al(l) + O_2(g); \quad \Delta G^\circ \text{ (in Joules)} = 1124800 - 218T \]
Reaction II: For carbon anode,
\[ \frac{2}{3}Al_2O_3(s) + C(s) \to \frac{4}{3}Al(l) + CO_2(g); \quad \Delta G^\circ \text{ (in Joules)} = 730700 - 218T \]
T denotes temperature in Kelvin.
Given: Faraday constant = 96500 Coulomb.
If the inert anode is replaced by a carbon anode during electrolysis of \(Al_2O_3\) at temperature 1300 K, the decrease in the magnitude of decomposition potential between the two reactions (rounded off to two decimal places) is _________________ Volts.

Hint:

Find n=4 electrons from the 4/3 mol of Al produced, convert each ΔG° into E = -ΔG°/(nF) at T=1300 K, then take the difference in magnitude.

Answer 1

Correct Answer: 1

Step 1: Find the number of electrons transferred in each reaction.
Aluminum forms as \(Al^{3+}\) is reduced to \(Al\), so each mole of Al needs 3 electrons. Both reactions produce \(\frac{4}{3}\) mol of Al, so the number of electrons transferred per reaction as written is
\[ n = \frac{4}{3}\times 3 = 4 \]

Step 2: Compute \(\Delta G^\circ\) for each reaction at T = 1300 K.
For Reaction I (inert anode):
\[ \Delta G_1^\circ = 1124800 - 218(1300) = 1124800-283400 = 841400 \text{ J} \]
For Reaction II (carbon anode):
\[ \Delta G_2^\circ = 730700 - 218(1300) = 730700-283400 = 447300 \text{ J} \]

Step 3: Convert each Gibbs energy to a decomposition potential.
The decomposition potential of an electrolytic cell is related to the Gibbs free energy of the reaction it drives through
\[ \Delta G^\circ = -nFE \]
so
\[ E = -\frac{\Delta G^\circ}{nF} \]
For the inert anode,
\[ E_1 = -\frac{841400}{4\times96500} = -\frac{841400}{386000} = -2.1798 \text{ V} \]
For the carbon anode,
\[ E_2 = -\frac{447300}{4\times96500} = -\frac{447300}{386000} = -1.1588 \text{ V} \]

Step 4: Find the decrease in the magnitude of the decomposition potential.
Switching from the inert anode to the carbon anode lowers the magnitude of the decomposition potential from \(2.1798\) V to \(1.1588\) V. The decrease is
\[ |E_1| - |E_2| = 2.1798 - 1.1588 = 1.0210 \text{ V} \]

Final Answer:
The magnitude of the decomposition potential decreases by about 1.02 V, which lies inside the accepted band of 1.00 to 1.05 V.
\[ \boxed{\Delta E \approx 1.02 \text{ V}} \]