A gas engine with a volumetric compression ratio of 10:1 is run on a weak mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and air, with an equivalence ratio, \(\phi=0.9\). If the initial temperature and pressure at the commencement of compression are \(60{ }^{\circ} \mathrm{C}\) and 1 bar respectively, calculate the maximum temperature and pressure reached during combustion at constant volume if compression is isentropic, and \(10 \%\) of the heat released during the combustion period is lost by heat transfer.

Assume the ratio of specific heats, \(\kappa\), during the compression stroke is 1.4 , and the heat of reaction at constant volume for methane at \(25^{\circ} \mathrm{C}\) is \(-8.023 \times 10^{5} \mathrm{~kJ} / \mathrm{kmol} \mathrm{CH}_{4}\) [2817 K; 84.59 bar].