Optimal path of the consecutive chemical reactions xA ↔ yB ↔ zC

The optimal path of the consecutive chemical reactions xA double left right arrow yB double left right arrow zC (x, y and z are the orders of chemical reactions) is analyzed numerically by taking temperature as a control variable, using the optimal-control theory based on Bak et al's work (2002 J. Phys. Chem. A 106 10961-4). Starting with pure A and maximizing the yield of B at the end of the given process duration, the optimal path starts with a branch at infinite temperature. A curve in which switching from this temperature to a lower temperature is possible is derived. For given parameters, there is a unique 'maximal useful time' that results in the highest possible yield of B. If a duration longer than this is specified, all reactions should be shut off during that excess amount of time in the optimal path. A numerical example for the optimal path is provided by the Taylor method. Finally, the results obtained are compared with the results of A, B, C (the orders of chemical reactions are all equal to 1) (Bak et al 2002 J. Phys. Chem. A 106 10961-4). When the orders of chemical reactions are taken into account, the optimal time sequences of the concentrations change markedly, the maximum obtainable yield is smaller, the initial values of co-state variables are bigger and the optimal phase trajectory changes.