Detailed equilibrium

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) ... 

A solution of the isolated platinum blue compound usually contains several chemical species described in the previous section. Such complicated behaviors had long been unexplored, but were gradually unveiled as a result of the detailed equilibrium and kinetic studies in recent years. The basic reactions can be classified into four categories (l)HH-HT isomerization (2) redox disproportionation reactions (3) ligand substitution reactions, especially at the axial coordination sites of both Pt(3.0+)2 and Pt(2.5+)4 and (4) redox reactions with coexisting solvents and atmosphere, such as water and 02. In this chapter, reactions 1-4 are summarized. 

Metal ion catalysis of salicyl phosphate hydrolysis is much more complicated than that of Sarin, since the former substrate can combine with metal ions to give stable complexes, and some of the complexes formed do not constitute pathways for the reaction. In addition the substrate undergoes intramolecular acid-base-catalyzed hydrolysis which is dependent on pH because of its conversion to a succession of ionic species having different reaction rates. Therefore a careful and detailed equilibrium study of proton and metal ion interactions of salicyl phosphate would be required before any mechanistic considerations of the kinetic behavior in the absence and presence of metal ions can be undertaken. 

The cyclization of 7-(2-hydroxyethoxy)-4-nitrobenzofurazan and 7-(2-hydroxyethoxy)-4-nitrobenzofuroxan to give the spiro Meisenheimer adducts 169 and 170, respectively, was investigated by Terrier et al.212 in aqueous solution by a detailed equilibrium and rate analysis similar to the one described for the reaction of 160. 

Spontaneous transitions are not the only possible transitions. Electronic transitions may be also induced by, for example, an external radiation field. According to the detailed equilibrium principle, the rate of transitions from all states of the lower level ct J into all states of the upper level aJ, caused by the absorption of photons from the radiation field, must be equal to the rates of spontaneous and induced transitions from the level a J into a J, i.e. 

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. 

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... 

Detailed equilibrium must occur in closed systems, whereas in open systems, particularly in those that are far from being in equilibrium due to their exchange with the environment, the situation is much more complicated. Primarily, steady-state solutions can be multiple, i.e. the rates of substance formation and consumption can be balanced on many points. 

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... 

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]. 

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]. 

It is of importance to understand that the limitation on rate constants (to be more precise, on their ratios, i.e. equilibrium constants) resulting from the detailed equilibrium principle, are fulfilled irrespective of the system under which the reaction takes place (either closed or open) since the rate constants are the same. The difference is that the right-hand sides in the equations for open systems contain additional factors accounting for the substance exchange with the environment. When choosing kinetic parameters, one must remember that not all of them are independent. It will reduce laborious difficulties and preclude probable mistakes. 

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. 

Let the equilibrium constants satisfy the conditions (72) (73) and (75). This suggests that there exists at least one positive PDE, c. Let us show that in this case any steady-state point is that of detailed equilibrium when the law of mass action (active surfaces) is valid. 

Note that the positive point for the G minimum in the reaction polyhedron is that of detailed equilibrium dG/dt < 0, hence at the point of minimum we have dG/d< = 0 (a decrease is possible "no-where ) and according to eqns. (83)-(85), dG/dt = 0 only at the PDEs. 

There are no damped oscillations near the point of detailed equilibrium. 

According to a given position of the detailed equilibrium point and a given initial composition, we can construct, using the above procedure, a region of compositions that can not be formed during the reaction. 

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). 

Any elementary reactions foUow the rules of detailed reversibility. Their detailed equilibrium, or Vy = 0 occurs at Pi = Pj (or that is the same as hi = hj). Thus, it is necessary that Ey = Eji. As a consequence. 

Fig. 6. IR spectra in the N-O stretching region of increasing dosed at about 150K onto Cu -ZSM-5 (see text for experimental details). Equilibrium pressures increasing from 6.7 to 267 Pa. (Adapted with permission from Prestipino et al. (S7).)...

From these studies, two motional processes have been postulated, a low-energy pseudorotational motion and a higher-energy arsenic atom inversion process. Detailed equilibrium studies have been conducted involving ring, chain, and oligomeric forms of these species. ... 

Effect of shaking. Shaking the micro titerplate during the first incubation (with antigen) results in a much faster establisment of equilibrium (see Fig 2 for details). Equilibrium was reached after shaking the plate during 30 minutes, and this time period was adopted for all incubation steps. 

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