A jet engine burns a weak mixture of octane (left(mathrm{C}_{8} mathrm{H}_{18} ight)) and air, with an equivalence

A jet engine burns a weak mixture of octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) and air, with an equivalence ratio, \(\phi=2\). The products of combustion, in which dissociation may be neglected, enter the nozzle with negligible velocity at a temperature of \(1000 \mathrm{~K}\). The gases, which may be considered to be ideal, leave the nozzle at atmospheric pressure of 1.013 bar with an exit velocity of \(500 \mathrm{~m} / \mathrm{s}\).

The nozzle may be considered to be adiabatic and frictionless.

Determine:

(a) the specific heat at constant pressure \(c_{p}\), of the products as a function of temperature;

(b) the molecular weight, \(m_{w}\), of the products;

(c) the temperature of the products at the nozzle exit;

(d) the pressure of the products at the nozzle inlet.

Specific heat at constant pressure \(c_{p, m}\) in \(\mathrm{J} / \mathrm{kmol} \mathrm{K}\), with \(T\) in \(\mathrm{K}\)
\[\begin{array}{cc}
\mathrm{CO}_{2} & c_{p, m}=21 \times 10^{3}+34.0 T \\
\mathrm{H}_{2} \mathrm{O} & c_{p, m}=33 \times 10^{3}+8.3 T \\
\mathrm{O}_{2} & c_{p, m}=28 \times 10^{3}+6.4 T \\
\mathrm{~N}_{2} & c_{p, m}=29 \times 10^{3}+3.4 T \end{array}\]
\(\left[0.9986 \times 10^{3}+0.2105 T ; 28.71 ; 895.7 \mathrm{~K} ; 1.591 \mathrm{bar}\right]\)