A system initially containing \(3 \mathrm{~mol} \mathrm{CO}_{2}, 5 \mathrm{~mol} \mathrm{H}_{2}\) and \(1 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) is undergoing the following reactions:

\[ \begin{aligned} \mathrm{CO}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) & \rightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\ \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) & \rightarrow \mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \end{aligned} \]

Derive the expression for the mole fractions of various components in the reaction mixture in terms of extent of reaction.