Combine the equations $$ \Delta G^{\circ}=-n F \mathscr{E}^{\circ} \quad \text { and } \quad \Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ} $$ 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 \(\mathscr{E}^{\circ}\) in terms of temperature, solve for \(\mathscr{E}^{\circ}\): \(\mathscr{E}^{\circ} = \frac{\Delta H^{\circ}}{-nF} + \frac{T \Delta S^{\circ}}{-nF}\) Now we have an expression for \(\mathscr{E}^{\circ}\) as a function of temperature: \(\mathscr{E}^{\circ} = \frac{-\Delta H^{\circ}}{nF} + \frac{T \Delta S^{\circ}}{nF}\) #Step 3: Graphically determine \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\)#

Since we want the reference half-cell to have a potential relatively stable with respect to temperature, we should look for a reference half-cell with a small or zero value for the temperature-dependent term in the expression we derived: \(\frac{T \Delta S^{\circ}}{nF}\) Ideally, we would want a reference half-cell with a nearly zero value for \(\Delta S^{\circ}\), as this would minimize the effect of temperature on the potential. In summary, we have derived an expression for \(\mathscr{E}^{\circ}\) as a function of temperature, and discussed how to graphically determine \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) from measurements of \(\mathscr{E}^{\circ}\) at different temperatures. To design a reference half-cell with a potential relatively stable with respect to temperature, we should look for a reference half-cell with a small or zero value for \(\Delta S^{\circ}\).