$($a$)$ Derive the equation to calculate the maximum concentration of the intermediate product $($i$.$e$.$ R$)$ of the series reaction $A{k}_{1}R{k}_{2}S$ that is taking place in a mixed flow reactor.

$($b$)$ What is the basic difference between 'Segregation Model' and 'Maximum Mixedness Model'? Derive an expression for mean conversion for the segregation model considering a second order liquid phase reaction. $)=(5$

$($c$)$ The first$-$order reaction $AB$ is carried out in a tubular reactor. The reaction rate constant is $0\mathrm{.}15mi{n}^{-1}.$ From the results of tracer test carried out on this reactor, the mean residence time and variance are found to be: $=5\mathrm{.}25$min,${}^2=6\mathrm{.}15mi{n}^2.$ Calculate the conversion using the tank$-$inseries model.

$($d$)$ What are the various mechanisms through which surface reaction takes place for heterogeneous catalytic reaction?