A weak mixture of propane (left(mathrm{C}_{3} mathrm{H}_{8} ight)) and (50 %) excess air is ignited in a

A weak mixture of propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) and \(50 \%\) excess air is ignited in a constant volume combustion chamber. The initial conditions were \(1 \mathrm{bar}\) and \(300 \mathrm{~K}\) and the final composition was \(7.87 \% \mathrm{CO}_{2}, 0.081 \% \mathrm{CO}, 10.58 \% \mathrm{H}_{2} \mathrm{O}, 0.02 \% \mathrm{H}_{2}, 6.67 \% \mathrm{O}_{2}\) and \(74.76 \% \mathrm{~N}_{2}\) by volume. Evaluate the final temperature and pressure of the products. Prove that the temperature obeys the conservation of energy, if the calorific value of propane is \(46440 \mathrm{~kJ} /\) \(\mathrm{kg}\) at \(300 \mathrm{~K}\).

The equilibrium constant for the carbon dioxide reaction is given by

\(K_{p}=\frac{\left(p_{\mathrm{CO}}\right)^{2} p_{\mathrm{O}_{2}}}{\left(p_{\mathrm{CO}_{2}}\right)^{2}}\) where \(\log _{10} K_{p}=8.46-\frac{28600}{T}\), and \(T=\) temperature in K.

The specific heats at constant volume \(\left(c_{v, m}\right)\) of the constituents, in \(\mathrm{kJ} / \mathrm{kmol} \mathrm{K}\), may be taken as

\begin{tabular}{cccc}

\(\mathrm{CO}_{2}\) & \(12.7+22 \times 10^{-3} T\) & \(\mathrm{H}_{2} \mathrm{O}\) & \(22.7+8.3 \times 10^{-3} T\) \\

\(\mathrm{CO}\) & \(19.7+2.5 \times 10^{-3} T\) & \(\mathrm{~N}_{2}\) & \(17.7+6.4 \times 10^{-3} T\) \\

\(\mathrm{O}_{2}\) & \(20.7+4.4 \times 10^{-3} T\) & \(\mathrm{H}_{2}\) & \(17.0+4.0 \times 10^{-3} T\)\end{tabular}

\(\mathrm{C}_{3} \mathrm{H}_{8}\) neglect the internal energy in the reactants The internal energies of formation of the constituents at \(300 \mathrm{~K}\) are

\[\mathrm{CO}_{2} \quad-393 \times 10^{3} ; \mathrm{H}_{2} \mathrm{O} \quad-241 \times 10^{3} ; \quad \mathrm{CO}-112 \times 10^{3} \mathrm{~kJ} / \mathrm{kmol}\]