A liquid-phase chemical reaction with stoichiometry A → B takes place in a semi batch reactor. The rate of consumption of A per unit volume of the reactor contents is given by the first-order rate expression (see Problem 11.14) rA [mol/(L∙s)] kCA where CA (mol A/L) is the reactant concentration. The tank is initially empty. Beginning at a time = 0, a solution containing A at a concentration CA0 (mol A/L) is fed to the tank at a steady rate V (L/s).

(a) Write a differential balance on the total mass of the reactor contents. Assuming that the density of the contents always equals that of the feed stream, convert the balance into an equation for dV/dt, where V is the total volume of the contents, and provide an initial condition. Then write a differential mole balance on the reactant, A, letting NA (t) equal the total moles of A in the vessel, and provide an initial condition. Your equations should contain only the variables NA, V, and t and the constants v and CA0. (You should be able to eliminate CA as a variable.)

(b) Without attempting to integrate the equations, derive a formula for the steady-state value of NA.

(c) Integrate the two equations to derive expressions for V (t) and NA (t), and then derive an expression for CA (t). Determine the asymptotic value of NA as t → ∞ and verify that the steady-state value obtained in part (b) is correct. Briefly explain how it is possible for NA to reach a steady value when you keep adding A to the reactor and then give two reasons why this value would never be reached in a real reactor.

(d) Determine the limiting value of CA as t → ∞ from your expressions for NA (t) and V (t). Then explain why your result makes sense in light of the results of part (c).