Consider the reaction $$\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{H}_{2} \mathrm{O}(g)$$ a. Use \(\Delta G_{\mathrm{f}}^{\circ}\) values in Appendix 4 to calculate \(\Delta G^{\circ}\) for this reaction. b. Is this reaction spontaneous under standard conditions at \(298 \mathrm{K} ?\) c. The value of \(\Delta H^{\circ}\) for this reaction is \(100 .\) kJ. At what temperatures is this reaction spontaneous at standard conditions? Assume that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature.

We first calculate the standard Gibbs free energy change (\(\Delta G^{\circ}\)) for the reaction using the given standard Gibbs free energy of formation values \(\Delta Gf^{\circ}\) and the formula: \(\Delta G^{\circ} = \sum n \Delta G_{f}^{\circ}(\text{products}) - \sum n \Delta G_{f}^{\circ}(\text{reactants})\) Then, we determine the spontaneity of the reaction at 298 K by comparing the calculated \(\Delta G^{\circ}\) value with zero. If \(\Delta G^{\circ} < 0\), the reaction is spontaneous under standard conditions at 298 K. Lastly, to find the temperature range in which the reaction is spontaneous under standard conditions, we use the relationship between Gibbs free energy, enthalpy, and entropy: \(\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\) We determine \(\Delta S^{\circ}\) using the given \(\Delta H^{\circ}\) value, and then find the temperature range in which \(\Delta G^{\circ} < 0\).