A chemical reaction below took place in a batch reactor. The conversion of A was $x_A.$ Temperature, pressure, and

volume at initial stage were $T_0,p_0,$ and $V_0,$ respectively. Initially, there were $A,B,$ inert I, and no products. The initial

compositions of $A,B,$ and I $($inert$)$ were same. Before reaction, the mixture was found to be an ideal gas mixture. During

the reaction, you could denote temperature, pressure and volume as $T,P,$ and $V,$ respectively. During the reaction, the

gaseous mixture did not seem to be an ideal gas mixture due to relatively high pressure.

Reaction: $A(g)+B(g)R(g)+S(g)$

$($a$)[20]$ Derive the relationship between the volume $V$ at the end of reaction and the initial volume $V_0.$

$($b$)[5]$ Express $C_A$ in gaseous phase. Do it by using stoichiometric coefficients and conversion of $A.$

$($c$)[20]$ Experimentalists carried out additional reaction at a lower reaction temperature than the original reaction

temperature. In this experiment the product $S$ totally condensed due to its boiling point. Experimentalists noticed that

the liquid fraction of the product was so small that they could disregard the liquid fraction in the volume of reaction

mixture. Derive the relationship between the gaseous volume $V$ and the end of reaction and the initial volume $V_0.$

$($d$)[5]$ In the $($c$)$ experiment, derive the $C_A$ expression in terms of stoichiometric coefficients and conversion of $A.$