Amazing variational approach to chemical reactions

In this letter we analyse an amazing variational approach to chemical reactions. Our results clearly show that the variational expressions are unsuitable for the analysis of empirical data obtained from chemical reactions. Some time ago Liu and He [1] proposed the application of a variational approach to chemical reactions. Their method consisted of the transformation of the differential equation for the reaction rate into a kind of Newton equation that they easily converted into a variational problem for the mechanical energy. By means of an appropriate ansatz they derived approximate expressions for the extent of reaction and half–time. The purpose of this letter is to investigate if those variational expressions may be of any utility for a chemist. Liu and He [1] considered the chemical reaction nA → C +D (1) 1 e–mail: [email protected] Preprint submitted to Elsevier 4 June 2009 If NA(t), NB(t), and NC(t) are the number of molecules of the species A, B, and C, respectively, at time t then Liu and He [1] assumed that NA(0) = a, and NB(0) = NC(0) = 0. If we call x = NB(t) = NC(t), then we conclude that NA(t) = a − nx, where x is known as the extent of reaction [2]. The unique rate of reaction can be defined in terms of the extent of reaction as v = dx/dt. Liu and He [1] further assumed that the rate law is given by dx dt = k(a− x) (2) At this point we stress the fact that this expression is correct only if the chemical reaction (1) is elementary, otherwise the rate law may be more complicated. Most chemical reactions are not elementary and therefore the reaction order and molecularity do not necessarily agree, as discussed in any book on physical chemistry [2]or chemical kinetics [3]. What is more, the order of reaction may not even be a positive integer [2,3]. This fact is known by any undergraduate student of chemical kinetics but Liu and He seem to be ignorant of it [1]. For concreteness here we assume that the rate law (2) is correct. Liu and He [1] tried and solved the differential equation (2) approximately by means of a variational approach based on the so–called semi–inverse method. It consisted in finding the minimum of the integral expression