Q$13.$ The catalytic isomerization of $n-$hexane to $2-$methylpentane and $3-$methylpentane was exhaustively discussed in Subsection $4\mathrm{.}7\mathrm{.}1.$ The effect of external mass transfer on selectivity was evaluated by assuming that the consecutive reactions were irreversible and first$-$order. In this problem, treat the realistic situation of reversible, first$-$order kinetics. Derive an expression for the selectivity of $2-$methylpentane with respect to $3-$methylpentane under isothermal conditions and for bulk$-$fluid concentrations of $c_{Ab},c_{Bb}$ and $c_{Rb}.$ Express the result in terms of the two first$-$order, forward$-$direction rate constants, $k_1$ and $k_2,$ the corresponding equilibrium constants, $K_1$ and $K_2^{**},$ and the necessary mass$-$transfer coefficients between bulk fluid and catalyst surface.