For a practical spark$-$ignition engine cycle, determine the maximum cycle temperature $($no

round$-$off around the sharp corner$).$ The source of energy is the stoichiometric reaction of

isooctane $($C$8$H$18)$ in air but only a fraction of the heat is utilized to power the engine,

represented by a coefficient $.$ The temperature after compression is T$2.$ Assume constant

specific heat capacities cv and cp for the combustion product. Assume the enthalpies of

isooctance, air, CO$2,$ and H$2$O are known at T$2.$ Derive an expression for the maximum

cycle temperature.