Principle of Detailed Balance in Kinetics

1206 Journal of Chemical Education • Vol. 81 No. 8 August 2004 • www.JCE.DivCHED.org In 1975 Mahan (1) wrote a very nice article entitled “Microscopic Reversibility and Detailed Balance” that emphasized molecular collisions and the use of partition functions in chemical kinetics. One of the things that has happened since then is the development of computer programs for personal computers that make it possible to solve rather complicated sets of simultaneous differential equations (2). The concentrations of reactants as a function of time can be calculated for reaction systems that show the effects of detailed balance on chemical kinetics. In this article calculations are used to show what would happen in the chemical monomolecular triangle reaction for cases where the principle of detailed balance is violated. A discussion of the principle of detailed balance has to start with the principle of microscopic reversibility in mechanics. It was recognized in classical mechanics that when time t occurs in an equation, changing t to –t does not invalidate the equation. A simple example of this time-reversal invariance is a movie of a collision between two billiard balls. This movie appears to be in accord with classical mechanics whether the film is run forwards or backwards. When quantum mechanics was developed, it was found to be time-reversal invariant as well. This means that for every type of interaction between particles the exact reverse is also possible. When a system is at equilibrium this means that the forward rate of a molecular process has to be equal to the reverse rate of that process. Applying this to a chemical reaction observed on a macroscopic scale means that at equilibrium, the forward rate of each step is equal to the reverse rate of that step; this is the principle of detailed balance. This jump from the molecular scale to the macroscopic scale requires a leap of faith, but it is based on research that led Onsager (3) to his treatment of reciprocal relations, which is discussed in more advanced textbooks (4). Here we examine three simple cases that help clarify consequences of the principle of detailed balance in chemical kinetics and its relation to thermodynamic equilibrium, steady states, and oscillating reactions. The first question that should be discussed is “What do we mean by equilibrium?” We know that when a thermodynamic system is at equilibrium, it is not changing with time and the temperature and concentration are uniform throughout the system. But that is not enough. Callen (5) writes, “Operationally, a system is in an equilibrium state if its properties are consistently described by thermodynamic theory.” He recognized that this criterion is circular, but gives a good defense. In addition the principle of detailed balance has to be obeyed. Here we consider a system at specified temperature and pressure, and so the thermodynamic equilibrium state is the one that minimizes the Gibbs energy, G. When we study a reaction system, we know a lot about how changes in the temperature and pressure affect equilibrium constants. In some cases it is not easy to decide whether a system is at equilibrium or in a steady state, and so we need to ask “What is a steady state or stationary state?” Actually the state of the system is not literally steady or stationary, but the concentrations of one or more species are approximately constant while the concentrations of other species are changing much more rapidly. The rate of change of the concentration of one or more intermediates may be so close to zero that the steady-state approximation is useful in treating the kinetics of the reaction system. Usually chemical reactions approach equilibrium without oscillations, but sometimes oscillations are observed. However, these oscillations always appear far from equilibrium, and reactions never oscillate around the equilibrium composition. Thermodynamics does not allow a reaction system to go through an equilibrium state to a state of higher Gibbs energy. Equilibrium at constant temperature and pressure is reached at the lowest Gibbs energy. Reactions that show oscillations usually involve autocatalysis, and they eventually reach equilibrium if the system is closed. To clarify the application of the principle of detailed balance, calculations are presented here on the interconversion of three isomers A, B, and C. The general case is represented by: