Combine the equations $$\Delta G^{\circ}=-n F \mathscr{E}^{\circ} \text { and } \Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}$$ to derive an expression for \(\mathscr{E}^{\circ}\) as a function of temperature. Describe how one can graphically determine \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) from measurements of \(\mathscr{E}^{\circ}\) at different temperatures, assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature. What property would you look for in designing a reference half-cell that would produce a potential relatively stable with respect to temperature?

To derive an expression for the standard cell potential, \(\mathscr{E}^{\circ}\), as a function of temperature, combine the given equations as follows: \( -n F \mathscr{E}^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \). Rearrange the terms to obtain, \( \mathscr{E}^{\circ} = \frac{\Delta H^{\circ}}{-n F} + \frac{T \Delta S^{\circ}}{-n F} \). To graphically determine the standard enthalpy change, \(\Delta H^{\circ}\), and standard entropy change, \(\Delta S^{\circ}\), plot \(\mathscr{E}^{\circ}\) against temperature, T. The plot will be linear, with \(\Delta H^{\circ} = -n F \times (T = 0 \text { intercept })\) and \(\Delta S^{\circ} = -n F \times (\text { slope })\). To design a reference half-cell with stable potential relative to temperature, look for a half-cell reaction with a low magnitude of the standard entropy change, \(\Delta S^{\circ}\). This minimizes the temperature dependence of the standard cell potential, \(\mathscr{E}^{\circ}\).