Chapter 10, Problem 10.25P

A liquid-phase chemical reaction with stoichiometry A ? B takes place in a semibatch reactor. The rate of consumption of A per unit volume of the reactor contents is given by the first-order rate expression (see Problem 10.19)

   $r_A\left[mol/\left(L\cdot s\right)\right]=k{C}_A$

where C_A(mol A/L) is the reactant concentration. The tank is initially empty. Beginning at a time t = 0, a solution containing A at a concentration C_A0(mol A/L) is fed to the tank at a constant rate $\stackrel{\dot{}}{V}\left(L/s\right)$.

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 N_A(t) equal the total moles of A in the vessel, and provide an initial condition. Your equations should contain only the variables (V_A, V, and t and the constants $\stackrel{\dot{}}{V}$

1. and C_A0. (You should be able to eliminate C_Aas a variable.)
2. Without attempting to integrate the equations, derive a formula for the steady-state value of N_A.

Integrate the two equations to derive expressions for V(t) and N_A(t), and then derive an expression for C_A(t). Determine the asymptotic value of N_aas t ? 8 and verify that the steady-state value obtained in Part (b) is correct. Briefly explain how it is possible for N_Ato 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.

Determine the limiting value of C_Aas t ? 8 from your expressions for N_A(t) and V(t). Then explain why your result makes sense in light of the results of Part (c).