The Gibbs energy for the decomposition of $Al_2O_3$ at $500^{\circ}C$ is as follows :<br>$\large\frac{2}{3}$$Al_2O_3 \to \large\frac{4}{3}$$Al+O_2, \Delta_rG=+966\;kJ\;mol^{-1}$ <br> The potential difference needed for electrolytic reduction of $Al_2O_3$ at $500^{\circ}C$ is at least