Given the following standard free energies at \(25^{\circ} \mathrm{C}\), $$ \begin{array}{ll} \mathrm{SO}_{2}(g)+3 \mathrm{CO}(g) \longrightarrow \operatorname{COS}(g)+2 \mathrm{CO}_{2}(g) & \Delta G^{\circ}=-246.5 \mathrm{~kJ} \\ \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) & \Delta G^{\circ}=-28.5 \mathrm{~kJ} \end{array} $$ find \(\Delta G^{\circ}\) at \(25^{\circ} \mathrm{C}\) for the following reaction. $$ \mathrm{SO}_{2}(g)+\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{COS}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) $$

We are given two reactions with their standard free energies, \(\Delta G^{\circ}\), at \(25^{\circ} \mathrm{C}\): Reaction 1: $$\mathrm{SO}_{2}(g)+3 \mathrm{CO}(g) \longrightarrow \operatorname{COS}(g)+2 \mathrm{CO}_{2}(g), \quad \Delta G^{\circ}_1=-246.5 \mathrm{~kJ}$$ Reaction 2: $$\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g), \quad \Delta G^{\circ}_2=-28.5 \mathrm{~kJ}$$ Our goal is to find \(\Delta G^{\circ}\) for the following reaction: Desired Reaction: $$\mathrm{SO}_{2}(g)+\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{COS}(g)+2 \mathrm{H}_{2} \mathrm{O}(g), \quad \Delta G^{\circ}_\text{desired}?$$

First, we can notice that if we multiply Reaction 2 by 2 then we'll get exactly 2 moles of \(\mathrm{H}_{2}\mathrm{O}(g)\) produced in the desired reaction: $$2 \times (\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g)), \quad 2 \times \Delta G^{\circ}_2=-57.0 \mathrm{~kJ}$$ Now, we have: $$\mathrm{SO}_{2}(g)+2 \mathrm{CO}(g) \longrightarrow \mathrm{COS}(g)+\mathrm{CO}_{2}(g), \quad \Delta G^{\circ}_1=-246.5 \mathrm{~kJ}$$ $$2 \times (\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g)), \quad 2 \times \Delta G^{\circ}_2=-57.0 \mathrm{~kJ}$$ We can combine both reactions and cancel-out common terms to obtain the desired reaction: $$\mathrm{SO}_{2}(g) \cancel{+2 \mathrm{CO}(g)}+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{COS}(g) \cancel{-\mathrm{CO}_{2}(g)}+2 \mathrm{H}_{2} \mathrm{O}(g),$$

According to Hess's law, when we sum up the manipulated reactions to obtain the desired reaction, we can also sum up their corresponding standard free energy changes to find the \(\Delta G^{\circ}\) for the desired reaction: $$\Delta G^{\circ}_\text{desired} = \Delta G^{\circ}_1 + 2 \times \Delta G^{\circ}_2 = (-246.5) + 2 \times (-28.5) = -246.5 - 57.0 = -303.5 \text{ kJ}$$ So, the standard free energy change, \(\Delta G^{\circ}\), for the desired reaction at \(25^{\circ} \mathrm{C}\) is \(-303.5\,\mathrm{kJ}\).