A mixture of $50\%$ A and $50\%$ B is charged to a constant$-$volume batch reactor in which equilibrium is rapidly achieved. The initial total concentration is $3\mathrm{.}0$ mol$/$dm$3.$

$($a$)$ Calculate the equilibrium concentrations and conversion of A at $330$ K for the reaction sequence:

Reaction $1$: A $$ B $$ C $$ D Kc$1(330$ K$)=4\mathrm{.}0,$ Kc$1(350$ K$)=2\mathrm{.}63$

Reaction $2$: C $$ B $$ X $$ Y Kc$2(330$ K$)=1\mathrm{.}0,$ Kc$2(350$ K$)=1\mathrm{.}51$

$($b$)$ Suppose now the temperature is increased to $350$ K$.$ As a result, a third reaction must now be considered in addition to the reactions above:

Reaction $3$: A $$ X $$ Z Kc$3(350$ K$)=5\mathrm{.}0$ dm$3/$mol

Calculate the equilibrium concentrations, conversion of A and overall selectivities SCX$,$ SDZ$,$ and SYZ$.$

$($c$)$ In table, vary the temperature over the range $300$ to $500$ K to learn the effect of selectivity SCX$,$ SDZ$,$ and SYZ on temperature.

Additional information:

$$HR$1=-20,000$ J$/$mol A $$HR$2=+20,000$ J$/$mol B $$HR$3=-40,000$ J$/$mol A