Microscopic reversibility equilibrium

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... 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

In any equilibrium process the sequence of intermediates and transition states encountered as reactants proceed to products m one direction must also be encountered and m precisely the reverse order m the opposite direction This is called the principle of microscopic reversibility Just as the reaction... 

A catalyst is a substance that increases the rate of a reaction without affecting the position of equilibrium. It follows that the rate in the reverse direction must be increased by the same factor as that in the forward direction. This is a consequence of the principle of microscopic reversibility (Section 3.3), which applies at equilibrium, and rates are often studied far from equilibrium. 

Lack of termination in a polymerization process has another important consequence. Propagation is represented by the reaction Pn+M -> Pn+1 and the principle of microscopic reversibility demands that the reverse reaction should also proceed, i.e., Pn+1 -> Pn+M. Since there is no termination, the system must eventually attain an equilibrium state in which the equilibrium concentration of the monomer is given by the equation Pn- -M Pn+1 Hence the equilibrium constant, and all other thermodynamic functions characterizing the system monomer-polymer, are determined by simple measurements of the equilibrium concentration of monomer at various temperatures. 

Without further refinement, the value of the equilibrium constant for reaction (7-73) seemingly would depend on [H+]. Clearly, that supposition violates a very fundamental precept of thermodynamics. With the help of microscopic reversibility, the dilemma is resolved. It specifies that the equilibrium constant is separately realized for each pathway ... 

The WGS reaction is a reversible reaction, that is, it attains equilibrium with reverse WGS reaction. Thus the fact that the WGS reaction is promoted by H20(a reactant), in turn, implies that the reverse WGS reaction may also be promoted by a reactant, H2 or CO2. In fact the decomposition of the surface formates produced from H2+CO2 is promoted 8-10 times by gas-phase hydrogen. The WGS and reverse WGS reactions can conceivably proceed on different formate sites of the ZnO surface unlike usual catalytic reaction kinetics, while the occurrence of the reactant-promoted reactions does not violate the principle of microscopic reversibility[63]. 

An expression for the equilibrium occupancy of pARt can again be obtained using the methods outlined in Chapter 1. A potential complication is that this mechanism contains a cycle, so the product of the reaction rates in both clockwise and counterclockwise directions should be equal in order to ensure the principle of microscopic reversibility is maintained. In this case, microscopic reversibility is maintained. Thus,... 

Thus isomerization of the trans to cis ketone is stereospecific while isomerization of cis to trans results in about 28% racemization. Postulating different pathways for trans to cis and cis to trans isomerizations would appear at first hand to violate the principle of microscopic reversibility. However, what we are dealing with here are the excited state intermediates as reactants and these are not necessarily in equilibrium for the two isomers. For example, taking the analogy... 

For reversible reactions the principle of microscopic reversibility (Section 4.1.5.4) indicates that a material that accelerates the forward reaction will also catalyze the reverse reaction. In several cases where the catalytic reaction has been studied from both sides of the equilibrium position, the observed rate expressions are consistent with this statement. 

The determinations of absolute rate constants with values up to ks = 1010 s-1 for the reaction of carbocations with water and other nucleophilic solvents using either the direct method of laser flash photolysis1 or the indirect azide ion clock method.8 These values of ks (s ) have been combined with rate constants for carbocation formation in the microscopic reverse direction to give values of KR (m) for the equilibrium addition of water to a wide range of benzylic carbocations.9 13... 

Table 2 gives rate and equilibrium constants for the deprotonation of and nucleophilic addition of water to X-[6+]. These data are plotted as logarithmic rate-equilibrium correlations in Fig. 5, which shows (a) correlations of log ftp for deprotonation of X-[6+] and log Hoh for addition of water to X-[6+] with logXaik and log KR, respectively (b) correlations of log(/cH)soiv for specific-acid-catalyzed cleavage of X-[6]-OH (the microscopic reverse of nucleophilic addition of water to X-[6+]) and log( H)aik for protonation of X-[7] (the microscopic reverse of deprotonation of X-[6+]) with log Xafc and log XR, respectively. 

Fig. 5 Logarithmic plots of rate-equilibrium data for the formation and reaction of ring-substituted 1-phenylethyl carbocations X-[6+] in 50/50 (v/v) trifluoroethanol/water at 25°C (data from Table 2). Correlation of first-order rate constants hoh for the addition of water to X-[6+] (Y) and second-order rate constants ( h)so1v for the microscopic reverse specific-acid-catalyzed cleavage of X-[6]-OH to form X-[6+] ( ) with the equilibrium constants KR for nucleophilic addition of water to X-[6+]. Correlation of first-order rate constants kp for deprotonation of X-[6+] ( ) and second-order rate constants ( hW for the microscopic reverse protonation of X-[7] by hydronium ion ( ) with the equilibrium constants Xaik for deprotonation of X-[6+]. The points at which equal rate constants are observed for reaction in the forward and reverse directions (log ATeq = 0) are indicated by arrows.

We recognize the right-hand side of the equation as the equilibrium constant K. We give the term microscopic reversibility to the idea that the ratio of rate constants equals the equilibrium constant K. 

The principle of microscopic reversibility demonstrates how the ratio of rate constants (forward to back) for a reversible reaction equals the reaction s equilibrium constant K. 

SAQ 8,22 A simple first-order reaction has a forward rate constant of 120 s 1 while the rate constant for the back reaction is 0.1 s F Calculate the equilibrium constant K of this reversible reaction by invoking the principle of microscopic reversibility. 

The existence of the limit (3) guarantees that, after a large enough number of steps, the different configurations are generated following a probability density II. Then it is said that a distribution of stationary probability or situation of static equilibrium has been reached. If II has been previously chosen, the method consists of selecting pij so that the conditions (2) and (4) are fulfilled. We must stress the fact that the condition of microscopic reversibility ... 

The WGS reaction is a reversible reaction that is, the WGS reaction attains equilibrium with the reverse WGS reaction. Thus, the fact that the WGS reaction is promoted by H20 (a reactant), in turn implies that the reverse WGS reaction may also be promoted by a reactant, H2 or C02. In fact, the decomposition of the surface formates produced from H2+C02 was promoted 8-10 times by gas-phase hydrogen. The WGS and reverse WGS reactions conceivably proceed on different formate sites of the ZnO surface unlike usual catalytic reaction kinetics, while the occurrence of the reactant-promoted reactions does not violate the principle of microscopic reversibility. The activation energy for the decomposition of the formates (produced from H20+CO) in vacuum is 155 kJ/mol, and the activation energy for the decomposition of the formates (produced from H2+C02) in vacuum is 171 kJ/mol. The selectivity for the decomposition of the formates produced from H20+ CO at 533 K is 74% for H20 + CO and 26% for H2+C02, while the selectivity for the decomposition of the formates produced from H2+C02 at 533 K is 71% for H2+C02 and 29% for H20+C0 as shown in Scheme 8.3. The drastic difference in selectivity is not presently understood. It is clear, however, that this should not be ascribed to the difference of the bonding feature in the zinc formate species because v(CH), vav(OCO), and v/OCO) for both bidentate formates produced from H20+C0 and H2+C02 show nearly the same frequencies. Note that the origin (HzO+CO or H2+C02) from which the formate is produced is remembered as a main decomposition path under vacuum, while the origin is forgotten by coadsorbed H20. 

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). 

In the application of the principle of microscopic reversibility we have to be careful. We cannot apply this concept to overall reactions. Even Eqs. (4.43) - (4.45) cannot be applied unless we know that other reaction steps (e.g., diffusional transport) are not rate controlling. In a given chemical system there are many elementary reactions going on simultaneously. Rate constants are path-dependent (which is not the case for equilibrium constants)and may be changed by catalysts. For equilibrium to be reached, all elementary processes must have equal forward and reverse rates... 

An accurate knowledge of the thermochemical properties of species, i.e., AHf(To), S Tq), and c T), is essential for the development of detailed chemical kinetic models. For example, the determination of heat release and removal rates by chemical reaction and the resulting changes in temperature in the mixture requires an accurate knowledge of AH and Cp for each species. In addition, reverse rates of elementary reactions are frequently determined by the application of the principle of microscopic reversibility, i.e., through the use of equilibrium constants, Clearly, to determine the knowledge of AH[ and S for all the species appearing in the reaction mechanism would be necessary. 

This principle has only limited application to reactions that are not at equilibrium. Furthermore, the Principle of Microscopic Reversibility does not apply to reactions commencing with photochemical excitation. See also... 

The latter condition is commonly known as microscopic reversibility or local detailed balance. This property is equivalent to time reversal invariance in deterministic (e.g., thermostatted) dynamics. Although it can be relaxed by requiring just global (rather than detailed) balance, it is physically natural to think of equilibrium as a local property. Microscopic reversibility, a common assumption in nonequilibrium statistical mechanics, is the crucial ingredient in the present derivation. 

Developments in FTs. FTs are simple results that provide a new view to better understanding issues related to irreversibility and the second law of thermodynamics. The main assumption of FTs is microscopic reversibility or local equilibrium, an assumption that has received some criticism [192-194]. Establishing limitations on the validity of FTs is the next task for the future. At present, no experimental result contradicts any of the FTs, mainly because the underlying assumptions are respected in the experiments or because current techniques are not accurate enough to detect systematic discrepancies. Under some experimental conditions, we... 

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. 

The model satisfied microscopic reversibility with the equilibrium constraints determined a priori from free energy data. 

It follows that the equilibrium constant K is given by kf/kr. The reverse reaction is inverse second order in iodide, and inverse first-order in H+. This means that the transition state for the reverse reaction contains the elements of arsenious acid and triiodide ion less two iodides and one hydrogen ion, namely, H2As03I. This is the same as that for the forward reaction, except for the elements of one molecule of water, the solvent, the participation of which cannot be determined experimentally. The concept of a common transition state for the forward and reverse reactions is called the principle of microscopic reversibility. 

In contrast. Figure 2 shows that the percentage of protein in solution for soy isolates remains constant as the amount of added protein is increased (1 ). In other words, the amount of protein in solution increases linearly with increasing amounts of added protein. This behavior is observed for all the isolates we have studied up to the highest concentration of 18 percent. Thus, soy isolates behave as if they are composed of a completely soluble fraction (A) and a completely insoluble fraction (B). Upon the addition of solvent, the soluble fraction (A) dissolves completely while the insoluble fraction (B) remains unchanged. There is no equilibrium established between A and B such that, if B is separated from A and reslurried in additional amounts of solvent, no additional protein will go into solution. (More precisely, no evidence of microscopic reversibility was found on the time scale of the experiment,... 

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