$(15$ pt$)$ In a continuous stirred tank reactor the reversible reaction $RP$ takes place under constant

temperature. The reaction rates are first order in both directions, with forward reaction rate constant $k_f$

and backward reaction rate constant $k_b.$ Assume that the reactor volume $V$ and the volumetric flowrate

$F$ of the feed are both constant.

$($a$)$ Derive the transfer function between the reactant concentration in the feed $C_{R0}$ and the product

concentration in the reactor $C_P.$ Calculate $,$ and $K$ for this system.

$($b$)$ Could the response of the product concentration $C_P$ exhibit oscillations in response to a step

change in the concentration of the reactant in the feed?