Earlier civilizations smelted iron from ore by heating it with charcoal from a wood fire: $$ 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 4 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g) $$ (a) Obtain an expression for \(\Delta G^{\circ}\) as a function of temperature. Prepare a table of \(\Delta G^{\circ}\) values at \(100-\mathrm{K}\) intervals between \(100 \mathrm{~K}\) and \(500 \mathrm{~K}\) (b) Calculate the lowest temperature at which the smelting could be carried out.

The smelting reaction is given by: $$ 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 4 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g) $$ For the given reaction, the expression for the Gibbs free energy change at standard conditions (\(\Delta G^{\circ}\)) can be written as: $$ \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} $$ where \(\Delta H^{\circ}\) is the enthalpy change at standard conditions and \(\Delta S^{\circ}\) is the entropy change at standard conditions.

As per the exercise, we assume that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are constant and we can use them to calculate \(\Delta G^{\circ}\) at different temperatures. However, the values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are not provided. So, we cannot provide the actual values of \(\Delta G^{\circ}\) at the mentioned temperature intervals. But, we can leave the expression as: $$ \Delta G^{\circ}(T) = \Delta H^{\circ} - T\Delta S^{\circ} $$ where \(T\) is the temperature at which \(\Delta G^{\circ}\) will be calculated.

Since we lack the values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\), we cannot calculate the numerical value of the lowest smelting temperature. However, we can provide the general approach to find it. At equilibrium, \(\Delta G = 0\), and the equilibrium constant \(K\) is given by: $$ K = e^{-\frac{\Delta G^{\circ}}{RT}} $$ where \(R\) is the universal gas constant and \(T\) is the temperature. As \(\Delta G\) approaches zero, the reaction approaches equilibrium. To find the lowest temperature for smelting, we can set \(\Delta G^{\circ} = 0\) and solve for the temperature \(T\): $$ 0 = \Delta H^{\circ} - T\Delta S^{\circ} $$ $$ T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} $$ The lowest smelting temperature can be calculated by finding the quotient of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\).