The kinetics of three coupled irreversible elementary reactions: two parallel mixed second order reactions followed by a first order reaction

A semi-analytical solution for the time dependence of the concentration of the intermediate is derived, in the case of two parallel mixed second order reactions followed by a first order reaction. The solution is restricted to equal initial concentrations for the reactants, and it is connected to the exponential integral. From the solution and through a proper choice of the dimensionless time (u) and concentration of the intermediate (y) one obtains a very simple relation between the maximum concentration of the intermediate (y_max) and the time associated with this concentration (u_max). This relation is governed by a parameter (β) which depends on the three rate constants and on the initial concentration. The smaller is β the larger are u_max and y_max, and the slower is the decay of y. An approximate expression connecting u_max and β, has also been obtained, and it yields maximum errors of ~ 8% and ~ 15% for u_max and y_max, respectively. The obtained expression can be very useful from the experimental point of view, as it allows an a priori selection of the most suitable experimental technique to detect the intermediate, simply comparing its time resolution with t_max (that is, u_max transformed to a time unit). An illustrative calculation is also discussed.