Reversibility and Detailed Balance

One of the first things which any student of chemical kinetics learns is that chemical equilibrium arises not because reaction stops, but because conditions are reached where reaction proceeds equally fast in the forward and reverse directions. Chemical equilibrium is not static, but dynamic, and it is trivial to show that 

Equation (1.9) states the principle of detailed balance. It defines the relationship between rate constants which describe the kinetics of chemical change in thermally equilibrated systems. At the other, least averaged, end of the scale, the ultimate experiment referred to earlier, would measure differential state-to-state reaction cross sections, F, 0), connecting 

Most kineticists are familiar with the principles of detailed balance and microscopic reversibility, as stated in equations (1.9) and (1.10). The successive stages of averaging over initial state distributions and summing over final state distributions by which one can proceed rigorously from (1.10) to (1.9) are less well-known. This procedure is important, not only for its own sake, but also because it yields equations that relate the parameters describing the kinetics of forward and reverse processes at all levels of detail. It is now outlined for a reaction of the type 

For any reaction, the principle of microscopic reversibility implies that the ratio of detailed rate coefficients, given by Visin n and VrS n n Vr) 

Detailed rate constants for reactions between species in defined v, J states, but occurring in collisions with a thermal spread of energies (or relative translational velocities), are derived from detailed rate coefficients by carrying out an integration such as 

According to the law of mass action the differential rate equation is 

At equilibrium v = 0, the chemical fluxes in the forward and reverse directions are equal, and we write 

These are applications of the principle of detailed balancing, which can be stated  

In a system of connected reversible reactions at equilibrium, each reversible reaction is individually at equilibrium. 

Detailed balance is a chemical application of the more general principle of microscopic reversibility, which has its basis in the mathematical conclusion that the equations of motion are symmetric under time reversal. Thus, any particle trajectory in the time period t = 0 to / = ti undergoes a reversal in the time period t = —ti to t = 0, and the particle retraces its trajectoiy. In the field of chemical kinetics, this principle is sometimes stated in these equivalent forms  

---

Gray, B. F., Scott, S. K. and Gray, P., 1984, Multiplicity for isothermal autocatalytic reactions in open systems the influence of reversibility and detailed balance. J. Chem. Soc. Faraday Trans. 1 80, 3409. 

The system has to stay far from equilibrium, or better the turnover of the major constituents is not at equilibrium, where formation and decomposition would be microscopically reversible and detailed balance would hold. This requires a metabolism sustained by a supply of energy-rich monomeric building blocks of the polymeric constituents. (It should be stressed that at equilibrium the first two conditions are not sufficient to cause selective evolutionary self-organization.) As a consequence of microscopic reversibility, all eigenvalues will always turn out to be real and negative. 

To elaborate, in Eig. 4 is shown simulated [by Eq. (40)] Tafel plots demonstrating the effect of varying k2, the forward rate constant of the rds, with respect to k. All other quantities are held constant in this simulation. The effective concentrations Aa, and have been set equal to 0.5 mol dm , the rate constants fc i, k, and fc 3 have all been set to 1 x 10 cm s and k-2 (i-6.> -rds) to 1 x 10 cm s 2 and ki are the only rate constants that vary in this simulation where the ratio, k /k2, (i.e., prior step relative to the rds) changes from 10 to 10 . Note that ki does not actually change independently since the principle of microscopic reversibility and detailed balance (Ref. 15, p. 285) requires that the following equality must hold ... 

Walz, D., Caplan, S.R. (1988) Energy coupling and thermokinetic balancing in enzyme kinetics. Microscopic reversibility and detailed balance revisited. Cell Biophys, 12, 13-28. 

Microscopic reversibility and detailed balance are important kinetic concepts. The former provides a mechanical argument for the reversibility of chemical reactions, while the latter shows a connection between kinetics and thermodynamics, and provides an important principle that must be applied in writing mechanisms. 

The proper application of the principles of microscopic reversibility and detailed balancing can be helpful in mechanistic assessments, as illustrated by the CO exchange in Mn(CO)5X systems. Johnson et al. initially claimed that all the CO ligands were being exchanged at a similar rate and proposed the mechanism in Scheme 2.1. 

The principles of microscopic reversibility and detailed balancing imply a fundamental relationship (see Table V) between the equilibrium constant (in terms of concentrations) and the rate constants of the reversible processes. Of course, two relations are obtained, one between entropies, and one between enthalpies. 

It might be thought that since chemisorption equilibrium was discussed in Section XVIII-3 and chemisorption rates in Section XVIII-4B, the matter of desorption rates is determined by the principle of microscopic reversibility (or, detailed balancing) and, indeed, this principle is used (see Ref. 127 for... 

When microscopic reversibility is present in a complex system composed of many particles, every elementary process in a forward direction is balanced by one in the reverse direction. The balance of forward and backward rates is characteristic of the equilibrium state, and detailed balance exists throughout the system. Microscopic reversibility therefore requires that the forward and backward reaction fluxes in Fig. 2.1 be equal, so that... 

The principle of microscopic reversibility or detailed balance is used in thermodynamics to place limitations on the nature of transitions between different quantum or other states. It applies also to chemical and enzymatic reactions each chemical intermediate or conformation is considered as a state. The principle requires that the transitions between any two states take place with equal frequency in either direction at equilibrium.52 That is, the process A — B is exactly balanced by B — A, so equilibrium cannot be maintained by a cyclic process, with the reaction being A — B in one direction and B — > C — A in the opposite. A useful way of restating the principle for reaction kinetics is that the reaction pathway for the reverse of a reaction at equilibrium is the exact opposite of the pathway for the forward direction. In other words, the transition states for the forward and reverse reactions are identical. This also holds for (nonchain) reactions in the steady state, under a given set of reaction conditions.53... 

There is a useful application of the Principle of Microscopic Reversibility (or Detailed Balancing) in the study of surface processes. This is a principle that requires that, when carried out under identical conditions, the reverse of any process should proceed by exactly the same route as the forward process thus whatever energy input is needed for the chemisorption of a hydrogen molecule will be recovered and released when the two atoms recombine and desorb. Measurement of the relaxation of the vibrational and translational energy of the desorbing molecule therefore provides information on the needs in dissociation, and values of S can also be derived. ... 

Following the discussion of microscopic reversibility for cross-sections and detailed balancing for rate coefficients given in ref. 87, one can derive an essential property of the thermal averaged kj ... 

TIME-REVERSIBLE MARKOV CHAINS AND DETAILED BALANCE 117... 

The simple use of a chemically irreversible chemical reaction step representing a chemical process is physically unrealistic, because the law of microscopic reversibility or detailed balance [94] is violated. More realistic is the use of an reaction scheme (Eq. 11.1.22, Fig. 11.1.21b). Even for the relatively simple reaction scheme, interesting additional consequences arise when the possibility of reversibility of the chemical step is considered. In Fig. II. 1.2lb, cyclic voltammograms for the case of a reversible electron transfer process coupled to a chemical process with kf = 10 s and fcb = 10 s" are shown. At a scan rate of 10 mV s a well-defined electrochemically and chemically reversible voltammetric wave is found with a shift in the reversible half-wave potential E1/2 from Ef being evident due to the presence of the fast equilibrium step. The shift is AEi/2 = RT/F ln(X) = -177 mV at 298 K in the example considered. At faster scan rates the voltammetric response departs from chemical reversibility since equilibrium can no longer be maintained. The reason for this is associated with the back reaction rate of ky, = 10 s or, correspondingly, the reaction layer, Reaction = = 32 pm. At Sufficiently fast scan rates, the product B is irre-... 

Because of their prevalence in physical adsorption studies on high-energy, powdered solids, type II isotherms are of considerable practical importance. Bmnauer, Emmett, and Teller (BET) [39] showed how to extent Langmuir s approach to multilayer adsorption, and their equation has come to be known as the BET equation. The derivation that follows is the traditional one, based on a detailed balancing of forward and reverse rates. 

Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

The frequency (number per second) of /transitions from all g. degenerate initial internal states and from the p. d E. initial external translational states is equal to tire reverse frequency from the g degenerate final internal states and the pyd final external translational states. The detailed balance relation between the forward and reverse frequencies is therefore... 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

When the integrator used is reversible and symplectic (preserves the phase space volume) the acceptance probability will exactly satisfy detailed balance and the walk will sample the equilibrium distribution... 

---