Detailed balancing principle example

These MC methods then are based on the Metropolis algorithm [251], by which one constructs a stochastic trajectory through the crmfiguration space (X) of the system, performing transitions W(X X ). The transition probability must be chosen such that it satisfies the detailed balance principle with the probability distribution that one wishes to study. For example, for classical statistical mechanics, the canonical ensemble distribution is given in terms of the total potential energy t/(X), where X = ( i, 2,..., iV) stands symbolically for a point in configuration space [17] ... 

One of the most significant recent insights in surface chemical dynamics is the idea that the principle of detailed balance may be used to infer the properties of a dissociative adsorption reaction from measurements on an associative desorption reaction.51,52 This means, for example, that the observation of vibrationally-excited desorption products is an indicator that the dissociative adsorption reaction must be vibrationally activated, or vice versa the observation of vibrationally-cold desorption products indicates little vibrational promotion of dissociative adsorption. In this spirit, it is... 

Capture and dissociation rates are related by the principle of detailed balance. Returning to our example of H+ + A- 5 AH, let us define rAH as the lifetime of an AH complex with respect to breakup into H+ and A-. 

In the example discussed above, the transition X- X sta s simply for a single spin flip at a randomly chosen lattice site, and W(X - ) = 1 if 5< 0 while W(X- X ) = Q — 3 lkgT) for >0 should be interpreted as transition probability per unit time. Note that other choices for W would also be possible provided they satisfy the principle of detailed balance ... 

Under equilibrium conditions the currents and i , and also icp and i, are equal to each other by the absolute value, in accordance with the principle of detailed balancing (see, for example, Landau and Lifshitz, 1977). These equilibrium values (i ) = (i )° = i° and (i )° = (i )° = i represent, by definition, exchange currents of an electrode reaction passing through the valence band (i°) and through the conduction band (i ). 

The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. 

SO that the approach to equilibrium is always an exponential decay and never a damped oscillation. This is a general characteristic of the approach to equilibrium in reaction systems. The principle of microscopic reversibility (also called the principle of detailed balancing) asserts that in a complex reacting system at equilibrium each individual reaction must be at equilibrium. This excludes the possibility of continuous cycles in which for example —> B —> C — the rates being such as to keep the concentrations... 

Inverse mass balance modeling here only employs the mass balance principle thermodynamics and equilibrium are not considered. Inverse models are usually nonunique. A number of combinations of mass transfer reactions can produce the same observed concentration changes along the flow path. Mass transfer reactions here refer to the reactions that result in the mass transfer between two or more phases, such as the dissolution of solid and gas or precipitation of solids. Chapter 9 describes the details of the models and shows a few examples. 

This relationship, then, is a direct consequence of the principle of detailed balance, and when it is used to simplify Eqs. (1-39), we find, for example, that... 

A condition closely related to detailed balance which also can be used to help understand the second example is that the equilibrium state is independent of the reaction path. This, of course, follows from the definition of thermodynamic equilibrium. A further important consequence of these equilibrium principles is that if the rate law for the reaction in the forward direction is known, the rate law for the reverse reaction can be derived from it and the equilibrium expression for the overall reaction. As an example. 

Thus far we have considered only perturbations of equilibrium states. This generally requires that the equilibrium constants be such that appreciable concentrations of both reactants and products are present. However, perturbations of steady states also can be realized. The mathematical analysis is quite similar to that already discussed for equilibrium systems except that steady-state concentrations are utilized rather than equilibrium concentrations and the principle of detailed balance cannot be used. For example, a rapid mixing apparatus might be used to establish a steady state which is then perturbed by a temperature jump. While steady-state perturbations have not yet been extensively used, they represent a potentially important application of relaxation methods. 

The relationship between the rate constants of this reversible reaction and the equilibrium constant (Equation 14.18) is an example of the principle of detailed balance, which states that, at equilibrium, the rates of forward and reverse processes are equal ... 

If we compare the schemes for the chemically activated part (R + R ) and the thermal dissociation parts (unimolecular reactions of A and B), we will notice that several apparent rate constants are defined twi e. For example, we obtain apparent rate constants not only for R -I- R A, but also for A R -I- R. The principle of detailed balancing requires that both rate constants must be thermodynamically consistent and this can be used as internal check of the kinetic analysis. 

Hint Apply the principle of detailed balancing. See, for example. Exercise 3.4,... 

In a closed system reactant concentration must decrease. Thus perpetual oscillations in the concentration of intermediates are not possible. However, if initially A is in great excess and conditions (ii) and (iii) are met, very slowly damped oscillations may be observed. In the remainder of this chapter we discuss some specific examples. Figure 7.5 illustrates allowed and forbidden chemical oscillation. Detailed balance forecloses oscillations about equilibrium, as shown in Fig. 7.5a. Oscillations about the steady state, as shown in Fig. 7.5b, violate no thermodynamic principle. 

The principle of detailed balance implies a /axa = K(T), regardless of the mechanism this can be seen through the following example. Assume the reaction really consists of two steps ... 

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,... 

This result, which also follows directly from the Hermitian property of the perturbation operator, is an example of microscopic reversibility which provides the quantum mechani cal basis for the principle of detailed balancing used in section 9.2. 

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