The gas-phase reaction between methanol and acetic acid to form methyl acetate and water CH_3 OH + CH_3 COOH CH_3 COOCH_3 + H_2 O takes place in a batch reactor. When the reaction mixture comes to equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant K_y = Y_C Y_D/Y_A Y_B = 4.87 (a) Suppose the feed to the reactor consists of n_A0, n_B0, n_c0, n_D0, and n_I0 gram-moles of A, B, C, D, and an inert gas, I, respectively. Let xi be the extent of reaction. Write expressions for the gram-moles of each reactive species in the final product, n_A (xi), n_B(xi), n_C(xi), and n_D(xi). Then use these expressions and the given equilibrium constant to derive an equation for xi_e, the equilibrium extent of reaction, in terms of n_A0, ... , n_I0. (see Example 4.6-2.) (b) If the feed to the reactor contains equimolar quantities of methanol and acetic acid and no other species, calculate the equilibrium fractional conversion. (c) It is desired to produce 70 mol of methyl acetate starting with 75 mol of methanol. If the reaction proceeds to equilibrium, how much acetic acid must be fed? What is the composition of the final product? (d) Suppose it is important to reduce the concentration of methanol by making its conversion at 99%. A feed to the reactor contains only gain assuming the equilibrium as high as possible, say methanol and acetic acid and that it is desired to produce 70 mol of methyl acetate, determine the extent of reaction and quantities of methanol and acetic acid that must be fed to the reactor. (e) If you wanted to carry out the process of Part (b) or (c) commercially, what would you need to know besides the equilibrium composition to determine whether the process would be profitable? (List several things.)