Finite-Time Optimization of Chemical Reactions: nA ⇄ mB

We investigate the optimization of chemical reactions of the type n A ⇄ m B in a closed container. The goal is to produce a maximal amount of A or B within a given finite time. The controls are taken to be volume and temperature, restricted to allowed regions in ( V,β)-space with β = 1/ kT . We show that the optimal path is achieved by choosing V and β such that, for the current amounts of reaction products N A and N B, the reaction rate f ( N B, N A, V,β) = d N B /d t is at all times maximal or minimal. For different combinations of endo/exothermicity and activation energies switches between extremal values of V and/or β may be included in the optimal path. The resulting paths are described qualitatively for general values of n and m, while the reactions 2NH 3 ⇄ N 2 + 3H 2, N 2 O 4 ⇄ 2NO 2, and 2NO 2 ⇄ N 2 + 2O 2 are used as concrete examples. It is observed that the optimal and equilibrium paths differ by a constant Δβ; a possible connection with a constant thermodynamic speed path is discussed.