Using data from Appendix \(\mathrm{C}\) , calculate \(\Delta G^{\circ}\) for the following reactions. Indicate whether each reaction is spontaneous at 298 \(\mathrm{K}\) under standard conditions. (a) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (b) \(\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g)\) (c) \(6 \mathrm{Cl}_{2}(g)+2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \rightarrow 4 \mathrm{FeCl}_{3}(s)+3 \mathrm{O}_{2}(g)\) (d) \(\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

Use Appendix C to find the Gibbs free energy values for each species in the given reactions.

Use the equation \(\Delta G^\circ(products) - \Delta G^\circ(reactants)\) to calculate the Gibbs free energy change for each reaction. (a) $2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)$ \(\Delta G^\circ(2 SO_3) - \Delta G^\circ(2 SO_2 + O_2) = (-740.4) - (2(-300.2) + 0) = -140.0 \, kJ/mol \) (b) $\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g)$ \(\Delta G^\circ(3 NO) - \Delta G^\circ(NO_2 + N_2O) = (3(-86.6)) - (-51.3 - 103.6) = 46.9 \, kJ/mol \) (c) $6 \mathrm{Cl}_{2}(g)+2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \rightarrow 4 \mathrm{FeCl}_{3}(s)+3 \mathrm{O}_{2}(g)$ \(\Delta G^\circ(4 FeCl_3 + 3 O_2) - \Delta G^\circ(6 Cl_2 + 2 Fe_2O_3) = (4(-399.5) + 3(0)) - (6(0) + 2(-824.2)) = -784.0 \, kJ/mol \) (d) $\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)$ \(\Delta G^\circ(S + 2 H_2O) - \Delta G^\circ(SO_2 + 2 H_2) = (0 + 2(-228.6)) - (-300.2 + 2(0)) = -157.0 \, kJ/mol \)

If \(\Delta G^\circ < 0\), then the reaction is spontaneous. If \(\Delta G^\circ > 0\), the reaction is non-spontaneous. If \(\Delta G^\circ = 0\), the system is in equilibrium. (a) \(\Delta G^\circ = -140.0 \, kJ/mol < 0\): The reaction is spontaneous. (b) \(\Delta G^\circ = 46.9 \, kJ/mol > 0\): The reaction is non-spontaneous. (c) \(\Delta G^\circ = -784.0 \, kJ/mol < 0\): The reaction is spontaneous. (d) \(\Delta G^\circ = -157.0 \, kJ/mol < 0\): The reaction is spontaneous.