An isothermal reversible reaction A ⇄ B is carried out in an aqueous solution. The reaction is first-order in both directions. The forward rate constant is 0.4 h^–1 and the equilibrium constant is 4.0. The feed to the plant contains 100 kg/m³ of A and enters at the rate of 12 m³/h. Reactor effluents pass to a separator, where B is completely recovered. The reactor is a stirred tank of volume 60 m³. A fraction, f₁, of the unreacted effluent is recycled as a solution containing 100 kg/m³ of A and the remainder is discarded. Product B is worth $2 per kilogram and operating costs are $50 per cubic meter of solution entering the separator. What value of f maximizes the operational profit of the plant? What fraction A fed to the plant is converted at the optimum? Source: H. . Shankar, IIT Mumbai.