Consider the following reactions at \(25^{\circ} \mathrm{C}\) : $$ \begin{aligned} &\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O} \quad \Delta G^{\circ}=-2870 \mathrm{~kJ} \\ &\mathrm{ADP}(a q)+\mathrm{HPO}_{4}{ }^{2-}(a q)+2 \mathrm{H}^{+}(a q) \longrightarrow \mathrm{ATP}(a q)+\mathrm{H}_{2} \mathrm{O} \\ &\Delta G^{\circ}=31 \mathrm{~kJ} \end{aligned} $$ Write an equation for a coupled reaction between glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), and ADP in which \(\Delta G^{\circ}=-390 \mathrm{~kJ}\).

We are given two separate reactions: 1. The glucose reaction with oxygen with \(\Delta G^{\circ} = -2870 \, \mathrm{kJ}\) 2. The ADP reaction with \(\Delta G^{\circ} = 31 \, \mathrm{kJ}\) And we are asked to find a combined reaction between glucose and ADP with \(\Delta G^{\circ} = -390 \, \mathrm{k} \mathrm{J}\).

Let's say the number of glucose reactions required is \(x\), and the number of ADP reactions required is \(y\). We can write the following equation to find the ratio between these numbers: $$-2870x + 31y = -390$$

For simplicity, let's rearrange the equation from step 2 and solve for y in terms of x: $$y = \frac{-390 + 2870x}{31}$$ Now, we need to find an integer solution, as we cannot have reactions in fractions. By testing different values of \(x\), we can find an integer solution: For \(x = 1\), \(y = \frac{-390 + 2870}{31} = \frac{2480}{31}\), which is not an integer. For \(x = 2\), \(y = \frac{-390 + 5740}{31} = \frac{5350}{31} = 172.58...\), which is not an integer. For \(x = 3\), \(y = \frac{-390 + 8610}{31} = 260\), which is an integer. So, we find that for \(x = 3\) and \(y = 260\), we can achieve the desired Gibbs free energy change.

As we determined that 3 glucose reaction needs to be combined with 260 ADP reactions, we can write the final equation for a coupled reaction between glucose and ADP as: $$ \begin{aligned} 3(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a\mathrm{q})+6 \mathrm{O}_{2}(g) \longrightarrow 6\mathrm{CO}_{2}(g)+6\mathrm{H}_{2}\mathrm{O})+260(\mathrm{ADP}(a\mathrm{q})+\mathrm{HPO}_{4}{ }^{2-}(a\mathrm{q})+2 {\mathrm{H}^{+}(a\mathrm{q})} \longrightarrow \mathrm{ATP}(a\mathrm{q})+\mathrm{H}_{2} \mathrm{O}) \\\ \longrightarrow 3\mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}+260\mathrm{ADP}+260\mathrm{HPO}_{4}{ }^{2-}+18\mathrm{O}_{2}+520\mathrm{H}^{+} \longrightarrow 260\mathrm{ATP}+18\mathrm{CO}_{2}+525\mathrm{H}_{2}\mathrm{O} \end{aligned} $$