Principle of detailed equilibrium

So far (Sect. 1) we have discussed only approaches to derive chemical kinetic equations for closed systems, i.e. those having no exchange with the environment. Now let us study their dynamic properties. For this purpose let us formulate the basic property of closed chemical systems expressed by the principle of detailed equilibrium a rest point for the closed system is a point of detailed equilibrium (PDE), i.e. at this point the rate of every step equals zero 

Hence if in the closed system c = N = 0, then for every step wB = 0. 

Fundamental results in substantiating and extending the principle of detailed equilibrium to a wide range of chemical processes were obtained in 1931 by Onsager, though chemists had also applied this principle (see Chap. 2). A derivation of this principle from that of microscopic reversibility was reported by Tolman [19] and Boyd [20], In the presence of an external magnetic field it is possible that equilibrium is not detailed. Respective modifications of this principle were reported by de Groot and Mazur [21]. 

Note that, when speaking about closed systems, one should remember not only the extent of the closed nature, i.e. the absence of in-flux and off-flux of the substance, but also about the equilibrium of the environment with which the system interacts. The ideal interaction with an equilibrium environment can be of several types, e.g. (a) according to the heat, they are isothermal (interacting with a thermostat) or heat insulating and (b) according to volume and pressure these interactions are isobaric or isochoric. 

The principle of detailed equilibrium accounts for the specific features of closed systems. For kinetic equations derived in terms of the law of mass/ surface action, it can be proved that (1) in such systems a positive equilibrium point is unique and stable [22-25] and (2) a non-steady-state behaviour of the closed system near this positive point of equilibrium is very simple. In this case even damped oscillations cannot take place, i.e. the positive point is a stable node [11, 26-28]. 

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A relation between the coefficients ar, ai, ctz, and a4 as well as between ai+, cc2+, a3+, and a4+ can be obtained from the conditions of equilibrium prior to illumination, which have the following form (the principle of detailed equilibrium) ... 

We easily find the dependence of the evaporation rate on superheating by applying the principle of detailed equilibrium. The accommodation coefficient, i.e., the probability that molecules of vapor falling onto the surface of the liquid will stick, we take equal to 1. The number of molecules which evaporate in unit time is equal to the number of molecules which fall in unit time onto the surface at equilibrium pressure which, in turn is equal to the product of half the number n of molecules in a unit volume of vapor and the average velocity cx of the molecules in the direction normal to the surface. 

Thus, assuming that one of the mechanisms (either the Langmuir -Hinshelwood or the Eley-Rideal) is irreversible, the second mechanism must also be assumed to be irreversible provided that K2 = 0. If the process is carried out at high temperatures and K2 is a minute value, the equality K4 = K2K3 can also be fulfilled in the case when the fourth step is reversible and the third is practically irreversible. It does not contradict the principle of detailed equilibrium. 

Limitations on the rate constants imposed by the principle of detailed equilibrium (see Sect. 2) have been fulfilled, since steps (3) and (4) are simultaneously taken to be irreversible. Stoichiometric step vectors are... 

It is evident that each PDE (wkj = wjk) is PCB. The opposite is incorrect. For example, though any steady-state point of the linear mechanism is a PCB (complexes are substances, Z, = y,), the principle of detailed equilibrium for it is not always valid (if the system is open). 

As mentioned in Sec.3.II., the initial conditions for integration of the classical equations of motion (14.11) may be chosen in such a way that they correspond to quantized values of the molecular vibration and rotation energies. The shortcoming of this semiclassical approach is that it does not satisfy the principle of detailed equilibrium, which states that the probability of forward transition (i— -f) equals the probability of reverse transition (f——i). This is easily seen if one introduces the action-angle variables /105/... 

Equations of a relation between the parameters of a system based on a phase Gibbs rule follows from the principle of detailed equilibrium in its different displays. Thus, for a multi-phase system the principle of a detailed equilibrium requires the equilibrium of any two phases with each other. This permits to separate them and to consider them separately from others. General conditions for an equilibrium in an isolated system are reduced to partial conditions of thermal (temperature of all phases is equal), mechanical (at plain... 

Simplification not only is a means for the easy and efficient analysis of complex chemical reactions and processes, but also is a necessary step in understanding their behavior. In many cases, to understand means to simplify. Now the main question is Which reaction or set of reactions is responsible for the observed kinetic characteristics The answer to this question very much depends on the details of the reaction mechanism and on the temporal domain that we are interested in. Frequently, simplification is defined as a reduction of the original set of system factors (processes, variables, and parameters) to the essential set for revealing the behavior of the system, observed through real or virtual (computer) experiments. Every simplification has to be correct. As a basis of simplification, many physicochemical and mathematical principles/methods/approaches, or their efficient combination, are used, such as fundamental laws of mass conservation and energy conservation, the dissipation principle, and the principle of detailed equilibrium. Based on these concepts, many advanced methods of simplification of complex chemical models have been developed (Marin and Yablonsky, 2011 Yablonskii et al., 1991). 

The relation between a and the thermodynamic functions may be rigorously established by means of the general partition function or by making use of Guggenheim s intuitive method, based on the principle of detailed equilibrium. According to this principle if a system is in equilibrium, each single process has to be balanced by its reverse. The processes that must be considered in these circumstances are evaporation of an r-mer molecule from solution followed by condensation of r monomer molecules in the r sites previously occupied and the reverse process, i.e. evaporation of r monomer molecules from these sites followed by condensation of one r-mer molecule on the sites considered. The rate of the first process is proportional to f(rp) and the vapour pressure of the monomer at an exponent r, p and that of the second is proportional to /(rS) and the vapour pressure of r-mer pg, which is negligible. The ratio may then be vritten as ... 

The plot of CE = Pout/Ps (from Eqs (5.10.33) and (5.10.37)) versus Ag for AM 1.2 is shown in Fig. 5.65 (curve 1). It has a maximum of 47 per cent at 1100 nm. Thermodynamic considerations, however, show that there are additional energy losses following from the fact that the system is in a thermal equilibrium with the surroundings and also with the radiation of a black body at the same temperature. This causes partial re-emission of the absorbed radiation (principle of detailed balance). If we take into account the equilibrium conditions and also the unavoidable entropy production, the maximum CE drops to 33 per cent at 840 nm (curve 2, Fig. 5.65). 

The principle of detailed balancing provides an automatic check on the self-consistency of postulated reaction mechanisms when equilibrium can be approached from both sides. 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

The principle we have applied here is called microscopic reversibility or principle of detailed balancing. It shows that there is a link between kinetic rate constants and thermodynamic equilibrium constants. Obviously, equilibrium is not characterized by the cessation of processes at equilibrium the rates of forward and reverse microscopic processes are equal for every elementary reaction step. The microscopic reversibility (which is routinely used in homogeneous solution kinetics) applies also to heterogeneous reactions (adsorption, desorption dissolution, precipitation). 

The backward velocity is obtained by Plummer et al. (1978) by application of the principle of detailed balancing. Denoting as Ki, K2, Kc, and respectively, the equilibrium constants of the processes... 

To determine n, we utilize the condition of electron equilibrium on the surface. In the absence of illumination, this is of the form (principle of detailed balance)... 

The pairwise rates are presumed to be the same in both the forward and backward directions. The fact that the lines in the spectrum are all equally intense places little restriction on the three pairwise rates. The principle of detailed balance shows that the symmetry of the individual processes (i.e., equal forward and reverse rates) is sufficient to ensure that all the lines have equal intensity at equilibrium. 

The principle of detailed balance is a result of the microscopic reversibility of electron kinetics. A prerequisite for the establishment of thermal equihbrium requires that the forward and reverse rates are identical. For isothermal reactions, the equilibrium constant remains unchanged. The principle of detailed balance is of fundamental importance to estabhsh helpful relations between reaction and equilibrium constants because both are at the initial thermal equilibrium in addition, at the new equihbrium after the relaxation of the perturbation, the net forward and reverse reaction rates are zero. 

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 ). 

In a system of connected reversible reactions at equilibrium, each reversible reaction is individually at equilibrium. This is the principle of microscopic reversibility or its corollary, the principle of detailed balance. 

Any dynamic system becomes stable eventually and comes to the rest point, i.e. attains its equilibrium or steady state. For closed systems, a detailed equilibrium is achieved at this point. This is not so simple as it would seem, as substantiated by a principle of the thermodynamics of irreversible processes. At a point of detailed equilibrium not only does the substance concentration remain unchanged (dcjdt = 0), but also the rate of each direct reaction is balanced by that of its associated reverse counterpart... 

We assume that thermal equilibrium, at the temperature T, has been established in the translational degrees of freedom. Then, according to the principle of detailed balance (Eq. (B.37)),... 

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... 

Both Newton s equation of motion for a classical system and Schrodinger s equation for a quantum system are unchanged by time reversal, i.e., when the sign of the time is changed. Due to this symmetry under time reversal, the transition probability for a forward and the reverse reaction is the same, and consequently a definite relationship exists between the cross-sections for forward and reverse reactions. This relationship, based on the reversibility of the equations of motion, is known as the principle of microscopic reversibility, sometimes also referred to as the reciprocity theorem. The statistical relationship between rate constants for forward and reverse reactions at equilibrium is known as the principle of detailed balance, and we will show that this principle is a consequence of microscopic reversibility. These relations are very useful for obtaining information about reverse reactions once the forward rate constants or cross-sections are known. Let us begin with a discussion of microscopic reversibility. 

Detailed balance provides a relation between the macroscopic rate constants kf and kr for the forward and reverse reactions, respectively. On a macroscopic level, the relation is derived by equating the rates of the forward and reverse reactions at equilibrium. Here it will be shown that the principle of detailed balance can be readily obtained as a direct consequence of the microscopic reversibility of the fundamental equations of motion. 

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