Combine the equations $$ \Delta G^{\circ}=-n F \mathscr{C}^{\circ} \quad \text { and } \quad \Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ} $$ 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?

We are given two equations: 1. \( \Delta G^{\circ}=-n F \mathscr{E}^{\circ} \) 2. \( \Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ} \) To derive an expression for \( \mathscr{E}^{\circ} \), we will first equate the two expressions for \( \Delta G^{\circ} \): \( -n F \mathscr{E}^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \) Now, we will solve for \( \mathscr{E}^{\circ} \):

Dividing both sides by \( -nF \) and isolating \( \mathscr{E}^{\circ} \), we get: \( \mathscr{E}^{\circ} = -\frac{\Delta H^{\circ}}{nF} + \frac{T \Delta S^{\circ}}{nF} \)

To graphically determine \( \Delta H^{\circ} \) and \( \Delta S^{\circ} \) from the measurements of \( \mathscr{E}^{\circ} \) at different temperatures, we can rewrite the expression for \( \mathscr{E}^{\circ} \) in the form of a linear equation: \( \mathscr{E}^{\circ} = -\frac{\Delta H^{\circ}}{nF} + \Big(\frac{\Delta S^{\circ}}{nF}\Big) T \) Here, \( -\frac{\Delta H^{\circ}}{nF} \) is the y-intercept, and \( \frac{\Delta S^{\circ}}{nF} \) is the slope of the line. By plotting \( \mathscr{E}^{\circ} \) versus temperature (T) and fitting the data points to a linear regression, one can determine the slope and y-intercept of the line. With the values of slope and y-intercept, we can calculate \( \Delta H^{\circ} \) and \( \Delta S^{\circ} \) as follows: 1. \( \Delta H^{\circ} = -nF\times \text{y-intercept} \) 2. \( \Delta S^{\circ} = nF\times \text{slope} \)

In designing a reference half-cell with a potential relatively stable with respect to temperature, one would look for a property that minimizes the temperature dependence of the electromotive force, \( \mathscr{E}^{\circ} \). From the derived expression: \( \mathscr{E}^{\circ} = -\frac{\Delta H^{\circ}}{nF} + \frac{T \Delta S^{\circ}}{nF} \) A relatively stable potential with respect to temperature could be achieved by minimizing the \( \frac{\Delta S^{\circ}}{nF} \) term, as it directly depends on temperature. So, a reference half-cell with a small value of \( \Delta S^{\circ} \) would lead to a more stable potential with respect to temperature.