B. Chadhuri, A. A. Patwardham, and M. M. Sharma [Ind. Eng. Chem. Res. 29, XXXXXXXXXX)] studied the catalytic alkylation of m-cresol (M) and p-cresol (P) with a-methylstyrene (A). These reactions can...

B. Chadhuri, A. A. Patwardham, and M. M. Sharma [Ind. Eng. Chem. Res. 29, 1025 (1990)] studied the catalytic alkylation of m-cresol (M) and p-cresol (P) with a-methylstyrene (A). These reactions can be represented in schematic form as where B represents p-cumyl-m-cresol, C represents o-cumylp-cresol, and where each of the rate expressions is of mixed second-order form; that is, These reactions are of interest because of their involvement in a scheme proposed for separation of mixtures of o- and pcresols. Consider these reactions as they occur in the liquid phase at 60°C with cumene as the solvent in the presence of an Amberlyst 15 catalyst. For this system at this temperature, the ratio of rate constants (k1/k2) is 4.8. (a) If one starts with an equimolal mixture of species M and P (0.30 M in each species) and adds an amount of the olefin A to bring its concentration to 0.30 M, determine the concentrations of the species A, M, P, B, and C as functions of the fraction of the olefin (A) that has reacted. Express your results in the form of both equations and plots. (Hint: Start by deriving an equation relating the concentration of species M that will be present at a given time to the concentration of species P present at that time. Time itself should not appear as a variable in this equation. Then proceed to derive an equation relating the concentration of species A present at a given time to the concentration of species P at that same time.) (b) When 33% of the olefin (A) had reacted, the ratio of B to C in the product mixture was approximately 77 : 23, a result that supports the hypothesis that these reactions can constitute part of a process for separating the two isomers of cresol. Demonstrate whether or not a selectivity analysis based on the aforementioned reactivity ratio gives a result that is consistent with this value.