Equilibrium approximations

In addition to [A ] being qiiasi-stationary the quasi-equilibrium, approximation assumes a virtually unperturbed equilibrium between activation and deactivation (equation (A3.4.125)) ... 

Figure 3-9 shows plots of Eqs. (3-135) and (3-136) for some hypothetical systems. Obviously the equilibrium approximation is poor in the early stages of the reaction, but in the later stages the assumption can be quite good. The preequilibrium assumption, applied to Scheme XIV, amounts to the statement that 2 is negligible relative to ki and i. 

It, therefore, appears that the equilibrium approximation is a special case of the steady-state approximation, namely, the case i > 2- This may be, but it is possible for the equilibrium approximation to be valid when the steady-state approximation is not. Consider the extreme but real example of an acid-base preequilibrium, which on the time scale of the following slow step is practically instantaneous. Suppose some kind of forcing function were to be applied to c, causing it to undergo large and sudden variations then Cb would follow Ca almost immediately, according to Eq. (3-153). The equilibrium description would be veiy accurate, but the wide variations in Cb would vitiate the steady-state description. There appear to be three classes of practical behavior, as defined by these conditions ... 

This particular scheme has been the subject of many, many publications. Some deal with the rigorous solutions, but more treat various approximate solutions such as the steady-state and prior equilibrium approximations. Several assumptions, valid much of the time, convert the full expressions into more tractable forms. They are the subject of the next two sections. 

Prior-equilibrium approximation, 86-89 Product-catalyzed reactions, 36-37 Propagation cycle, 182... 

This is a further simplification of the quasi-equilibrium approximation, in which we simply neglect the reverse reaction of one or several steps. For instance, we may envisage a situation where the product concentration AB is kept so low that the reverse reaction in step (4) may be neglected. This greatly simplifies Eq. (161) since... 

The Most Abundant Reaction Intermediate (MARI) approximation is a further development of the quasi-equilibrium approximation. Often one of the intermediates adsorbs so strongly in comparison to the other participants that it completely dominates the surface. This intermediate is called the MARI. In this case Eq. (156) reduces to... 

The orders of reaction, U , ivith respect to A, B and AB are obtained from the rate expression by differentiation as in Eq. (11). In the rare case that we have a complete numerical solution of the kinetics, as explained in Section 2.10.3, we can find the reaction orders numerically. Here we assume that the quasi-equilibrium approximation is valid, ivhich enables us to derive an analytical expression for the rate as in Eq. (161) and to calculate the reaction orders as ... 

CO oxidation, an important step in automotive exhaust catalysis, is relatively simple and has been the subject of numerous fundamental studies. The reaction is catalyzed by noble metals such as platinum, palladium, rhodium, iridium, and even by gold, provided the gold particles are very small. We will assume that the oxidation on such catalysts proceeds through a mechanism in which adsorbed CO, O and CO2 are equilibrated with the gas phase, i.e. that we can use the quasi-equilibrium approximation. 

In solving the kinetics of a catalytic reaction, what is the difference between the complete solution, the steady-state approximation, and the quasi-equilibrium approximation What is the MARI (most abundant reaction intermediate species) approximation ... 

Why does the quasi-equilibrium approximation fail in the low-pressure... 

ILLUSTRATION 4.2 USE OF STEADY-STATE AND PSEUDO EQUILIBRIUM APPROXIMATIONS FOR INTERMEDIATE CONCENTRATIONS... 

We then use a Feautrier scheme [4] to perform spectral line formation calculations in local thermodynamic equilibrium approximation (LTE) for the species indicated in table 1. At this stage we consider only rays in the vertical direction and a single snapshot per 3D simulation. Abundance corrections are computed differentially by comparing the predictions from 3D models with the ones from ID MARCS model stellar atmospheres ([2]) generated for the same stellar parameters (a microturbulence = 2.0 km s-1 is applied to calculations with ID models). 

In almost every case differential equations for the quantitative description of the time dependence of particular species resulting from a catalytic cycle cannot be solved directly. This requires approximate solutions to be made, such as the equilibrium approximation [15], the Bodenstein principle [16], or the more generally valid steady-state approach [17]. A discussion of differences and similarities of different approximations can be found in [18]. 

A more detailed examination shows that, in case of equilibrium approximation, the value of fCM corresponds to the inverse stability constant of the catalyst-substrate complex, whereas in the case of the steady-state approach the rate constant of the (irreversible) product formation is additionally included. As one cannot at first decide whether or not the equilibrium approximation is reasonable for a concrete system, care should be taken in interpreting KM-values as inverse stability constants. At best, the reciprocal of KM represents a lower limit of a stability constant In other words, the stability constant quantifying the preequilibrium can never be smaller than the reciprocal of the Michaelis constant, but can well be significantly higher. 

From Merbach et al. (112). The solids in equilibrium approximate to LnCl3-4ROH for details consult reference (112). 

AEGL values for various levels of effect were derived using the following methods. The AEGL-1 was based on a controlled 1-h inhalation no-effect level of 8,000 ppm in humans. Because effects occurred in animal studies only at considerably higher concentrations, an intraspecies UF of 1 was applied. Because blood concentrations achieved equilibrium approximately 55 min into the exposure and circulating HFC-134a concentrations determine the level of effect, the 8,000 ppm concentration was applied across all time periods. 

Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158 160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by vrbb = k+ [EAB — k EPQ. Using the conservation of total enzyme... 

Thus we see that the pseudo-steady-state approximation gives orders of the reaction as the thermodynamic equilibrium approximation, the only difference being the definition of the rate constant... 

We are going to compare our hypergeometric approximation of the thermodynamic branch to the classic rate-limiting step and linear equilibrium approximations (see Sections 3.1.1 and 3.1.2). 

Figure 15 illustrates the case of step 2 limiting. Growth of the magnitude of kinetic parameters (in our case, 10-fold) unavoidably results in the degradation of the quality of rate-limiting type approximations (see Figure 15c). Whereas the equilibrium approximation works as expected in the vicinity of equilibrium, it does not produce good fit far from the equilibrium. The hypergeometric approximation produces uniformly good fit of the exact dependence (see Figure 15a-c).

Finally, Figure 17 compares all types of approximations and the exact reaction rate dependence. The equilibrium approximation works well at smaller values /2 (the equilibrium point is close to the origin). Limitation of the step 1 works at higher values of parameter /2 whereas limitation of step 2 fits the initial... 

Figure 15 Overall reaction rate and its approximations step 2 is rate-limiting. Dots represent the exact reaction rate dependence, solid line is the first-term hypergeometric approximation, dashed line corresponds to the reaction-rate equation that assumes the limitation of step 2 and dash-dots represent the equilibrium approximation. Hypergeometric approximation survives the 100-times increase in rate-limiting stage kinetic parameters and it works when there is no rate-limiting step at all. Parameters r, = 5, fj = 15, rj = 10 t2 = 0.2, fj = 0-1 (a) rj = 2,fj = - (b) t2 = 20, = 10, (c).

If all steps except one are fast, we can use the quasi-equilibrium approximation For the fast steps we use the corresponding equilibrium equations instead of the kinetic equations. 

This approximation will in most cases provide a very significant simplification in particular for large reaction mechanisms. In the quasi-equilibrium approximation the transient behavior is eliminated. Further, the description of changes in rate-limiting step has been lost. 

In the irreversible step approximation, we neglect the forward or backward rate for one of the steps. For small mechanisms the irreversible step approximation may be used alone, for larger mechanisms it is usually combined with the quasi equilibrium approximation... 

One of the reaction steps is assumed rate limiting, this step has a forward rate constant. This model maps onto the quasi equilibrium approximation. 

There are some common characteristics for gas-phase reaction systems that form the basis for understanding and describing the chemical behavior. In this section we will discuss some basic definitions and terms that are useful in kinetics, such as reaction order, molec-ularity, chain carriers, rate-limiting steps, steady-state and partial equilibrium approximations, and coupled/competitive reactions. 

For the hydrolysis reactions, five different components make up the reaction mixture the silane, water, acid, ethanol (to solubilize the water) and mesitylene (the internal standard). The water, acid, ethanol and mesitylene were weighed (to 0.001 g) into the vial, which was then sealed with the polypropylene cap. The vial was placed in a constant temperature bath and allowed to reach equilibrium (approximately 30 min). The silane to be used was kept in the constant temperature bath as well. The reaction was initiated by removing the reaction vial from the bath, quickly weighing in the desired amount of the silane, mixing the solution momentarily with a Vortex-Genie and then returning it to the bath. 

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