Problem $2$: A shock$-$tube experiment begins with a mixture composed of $99\backslash \%($by volume$)$ of Argon with $\backslash (0\mathrm{.}2\backslash \%\backslash )$ propane $\backslash (\backslash$left$(\backslash$mathrm$\{$C$\}\_\{3\}\backslash$mathrm$\{$H$\}\_\{8\}\backslash$right$)\backslash )$ and $\backslash (0\mathrm{.}8\backslash \%\backslash$mathrm$\{$O$\}\_\{2\}\backslash ).$ Arrival of a shock wave instantly brings this mixture to a temperature and pressure of $1600$ K and $10$ atm $,$ respectively; reaction begins shortly thereafter.

a$)$ What is the equivalence ratio of the mixture?

b$)$ Determine $($using STANJAN$)$ the equilibrium temperature, pressure, and composition of the mixture after complete combustion, making suitable assumptions for the product species for the following two cases: $1)$ assume the combustion process is constant $\backslash (\backslash$mathrm$\{$H$\},\backslash$mathrm$\{$P$\}$ ; $2\backslash ))$ constant $\backslash (\backslash$mathrm$\{$U$\},\backslash$mathrm$\{$V$\}\backslash ).$

c$)$ Repeat the $\backslash (\backslash$mathrm$\{$H$\},\backslash$mathrm$\{$P$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$U$\},\backslash$mathrm$\{$V$\}\backslash )$ calculations for a shock$-$tube mixture that contains initially a mixture of propane and air $\backslash (\backslash$left$(\backslash$mathrm$\{$O$\}\_\{2\}+3\mathrm{.}76\backslash$mathrm$\{$~N$\}\_\{2\}\backslash$right$)\backslash )$ at the same equivalence ratio as part a$.$