$A+\frac{1}{2}BC+\frac{3}{2}D$ reaction is carried out isothermally in a closed vessel at $T=500K.$ Initially, the vessel is loaded with $50\%$A and $50\%$ B at a pressure $P=12$atm.

a$)$ Find $.$

b$)$ Find the concentration of the initial components and all components after conversion of $70\%$ according to the limiting reactant. Find initial partial pressures and total pressure inside the reactor and partial pressures after $70\%$ conversion.

c$)$ Since it is known that the reaction rate constant is $k=3\mathrm{.}2Lmo{l}^{-1}mi{n}^{-1},$ calculate the reaction rate $(-r_A)$ for $70\%$ conversion if the reaction is of the $1$ st order for A and $B.$

d$)$ If the reaction was carried out isothermally and isobarically in a variable volume vessel, find the concentration of all components after $70\%$ conversion. Find the reaction rate $(-r_A)$ for the case under $c).$