List three different ways to calculate the standard free energy change, \(\Delta G^{\circ},\) for a reaction at \(25^{\circ} \mathrm{C}\) . How is $\Delta G^{\circ}\( estimated at temperatures other than \)25^{\circ} \mathrm{C} ?$ What assumptions are made?

Three different ways to calculate the standard free energy change, \(\Delta G^{\circ}\), at 25°C are: 1. Using the Gibbs-Helmholtz equation: \(\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\) 2. Using the equilibrium constant: \(\Delta G^{\circ} = -RT \ln(K)\) 3. Using data tables and the equation: \(\Delta G^{\circ} = \sum \Delta G_f^{\circ}(\text{products}) - \sum \Delta G_f^{\circ}(\text{reactants})\) To estimate \(\Delta G^{\circ}\) at other temperatures, we can use the van't Hoff equation: \[\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\] Assumptions made while estimating \(\Delta G^{\circ}\) at a different temperature include: 1. The reaction is assumed to be at equilibrium. 2. The standard free energy change, enthalpy change, and entropy change are assumed to be constant over the temperature range being considered. 3. The van't Hoff equation assumes that the change in standard free energy and enthalpy with temperature is negligible.