Analyze the combustion reaction [begin{equation*}mathrm{CH}_{4}+2 mathrm{O}_{2} ightleftarrows mathrm{CO}_{2}+2 mathrm{H}_{2} mathrm{O}, tag{6.6.8}end{equation*}] assuming that at combustion temperatures

Analyze the combustion reaction

\[\begin{equation*}\mathrm{CH}_{4}+2 \mathrm{O}_{2} \rightleftarrows \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}, \tag{6.6.8}\end{equation*}\]

assuming that at combustion temperatures the equilibrium constant \(K(T) \gg 1\). Show that conducting combustion at the stoichiometric point or just a bit short of the stoichiometric point (so there is enough oxygen to oxidize all of the methane) will lead to low amounts of \(\mathrm{CH}_{4}\) in the exhaust. Determine the equilibrium amount of \(\mathrm{CH}_{4}\) in terms of the initial excess amount of \(\mathrm{O}_{2}\). Determine the equilibrium constant at \(T=1500 \mathrm{~K}\) from the data \(\beta \mu_{\mathrm{CO}_{2}}^{(0)}=-60.95\), \(\beta \mu_{\mathrm{O}_{2}}^{(0)}=-27.08, \beta \mu_{\mathrm{CH}_{4}}^{(0)}=-31.95\), and \(\beta \mu_{\mathrm{H}_{2} \mathrm{O}}^{(0)}=-44.62\).