Detailed balance principle, reversible energy

If the probability for the system to jump to the upper PES is small, the reaction is an adiabatic one. The advantage of the adiabatic approach consists in the fact that its application does not lead to difficulties of fundamental character, e.g., to those related to the detailed balance principle. The activation factor is determined here by the energy (or, to be more precise, by the free energy) corresponding to the top of the potential barrier, and the transmission coefficient, k, characterizing the probability of the rearrangement of the electron state is determined by the minimum separation AE of the lower and upper PES. The quantity AE is the same for the forward and reverse transitions. 

Equation (47) shows that in the Condon approximation the probabilities of forward and reverse transitions satisfy the detailed balance principle since the point q corresponds to the intersection of the potential energy surfaces (and free energy surfaces) where Haa = Hbb. Therefore, at the point q we have... 

The figure below (from Rimstidt and Barnes, 1980) shows the energy profiles in the dissolution (forward direction) and precipitation (reverse direction) reactions of amorphous Si02 and quartz, Q -Si02. Use the concept of detailed balancing (microscopic reversibility principle) to explain why the dissolution rate constant is directly proportional to the solubility of the Si02 solid phase. 

Since the microscopic rate constants of forward and reverse processes are connected by the detailed balance principle, the distribution function over the states of products of an exoergic reaction can be used to calculate the microscopic rates of an endoergic reaction. This can provide valuable information on the dependence of the endoergic reaction cross section on the energy of different degrees of freedom. 

For multichannel scattering where there are two or more open channels, the S matrix is a true matrix with elements Sy and the cross section for the transition from channel i to channel j is proportional to 5y - Sy 2. The symmetry of collision processes with respect to the time reversal leads to the symmetric property of the S matrix, ST = S, which, in turn, leads to the principle of detailed balance between mutually reverse processes. The conservation of the flux of probability density for a real potential and a real energy requires that SSf = SfS = I, i.e., S is unitary. For a complex energy or for a complex potential, in general, the flux is not conserved and S is non-unitary. 

We note, in passing, that this equation is consistent with the well-known equation for the temperature dependence of an equilibrium constant K = kf/kr, i.e., the van t Hoff equation. From the general principle of detailed balance, one can obtain a microscopic interpretation of the difference in activation energies between the forward and the reverse direction of an elementary reaction. Detailed balance, Eq. (2.34), implies... 

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. ... 

Based on the principle of microscopic reversibility ("detailed balance of fluxes"), the angular and kinetic energy distribution directed towards the surface, Eq. (27), and that of the molecules leaving the surface are the same in thermal equilibrimu The angular distribution of molecules leaving the surface are thus slightly peaked in the direction of the surface normal and the velocity distribution is Maxwellian. 

Consider the decay (C B + b) in which the compound nucleus C has initially an excitation energy Eq, and decays to form the nucleus B with energy of excitation E, and the particle 6, with kinetic energy e (see Fig. 4). Then E = Eq— — e, where is the binding energy of the particle b to the compound nucleus, C. The maximum kinetic energy of the particle b corresponds to E = 0, i.e. = E —Q b- The probability of decay of the compound nucleus may be obtained by application of the principle of detailed balancing. Suppose the residual nucleus B (at excitation E) and the particle b are contained in a box of volume V. Then the probability per unit time for the reverse reaction, B-f is a vjV... 

Some models assume that a system reaches a steady state rather than equilibrium. Equilibrium is defined by the principle of detailed balance, which requires that the forward and reverse rates are equal and that each step along the reaction path is reversible. The forward and reverse rates of steady-state processes are equal but the process steps that produce the forward rate are different from those that produce the reverse rate. At steady state, the state variables of an open system remain constant even though there is mass and/or energy flow through the system. The steady-state assumption is especially useful for processes that occur in a series, because the concentrations of intermediates that are formed and subsequently destroyed are constant. Perturbation of a steady-state system produces a transient state where the state variables evolve over time and approach a new steady state asymptotically. 

The application of microscopic reversibility to each molecular reactive collision in a chemical reaction system consisting of a statistically large assembly of molecules with a distribution of momenta and internal energy states is called the principle of detailed balance. Detailed balance requires one to write all elementary reactions as reversible, and it permits one to rule out some types of mechanisms, such as the cyclic sequence of the following equation ... 

Figure 9.1 The principle of detailed balance, (a) The equilibrium between three interconverting compounds A, B and C is a result of detailed balance between each pair of compounds, (b) Although a conversion from one compoimd to another can also produce concentrations that remain constant in time, this is not the equilibrium state, (c) The principle of detailed balance has a more general validity. The exchange of matter (or energy) between any two regions of a system is balanced in detail the amount of matter going from X to Y is balanced by exactly the reverse process, etc.

Detailed balance relates the rates of a particular activation and deactivation energy transfer process. Detailed balance thus provides a quantitative exact relation between rate constants that correspond to the same gap. This is unlike the principle of exponential gap tiiat provides an estimate of how the rate constants vary when the gap changes. The quahtative implication of detailed balance is that on a quantum state-to-quantum state basis, the rate constant for the activation process is always smaller than the rate constant for the reverse deactivation process. Take as an example the V—T process that we started this section with, A -I- BC(v = 0) A -I- BC(v = 1) and the reverse deactivation process, A -I- BC(v = 1) A -I- BC(v = 0). Detailed balance states that at equilibrium the rates of these two detailed ways of transferring populations between BC(v = 1) and BC(v = 0) must be equal. This is to be so even though there may be other processes that can transfer populations, such as transitions in the IR. Therefore, using the subscript eq to designate concentrations at equilibrium,... 

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