Quasi-equilibrium assumption, validity

Note that the transfer coefficient obtained here is not in any way related to the symmetry factor. It follows from the quasi-equilibrium assumption and should therefore be a true constant, independent of potential and temperature, as long as the assumptions leading to Eq. 43F are valid. 

The rate equations valid for one rapid step (Equation 2.30 or 2.32) can be obtained more easily by applying the quasi-equilibrium assumption directly. For instance, if step I is rapid,... 

The quasi-equilibrium assumption, that is the basis for the use of the Boltzmann distribution law, may lose its validity for rapid reactions. In such reactions, the most energetic reactant molecules may disappear very rapidly and the concentration of species at the transition state may be lesser than that for a trae equilibrium. In practice, even when EJRTk 3, as in the Cl+HBr—>HCH-Br hydrogen atom abstraction, internal-state nonequilibrium effects are very small [6]. 

The derivation of initial velocity equations invariably entails certain assumptions. In fact, these assumptions are often conditions that must be fulfilled for the equations to be valid. Initial velocity is defined as the reaction rate at the early phase of enzymic catalysis during which the formation of product is linear with respect to time. This linear phase is achieved when the enzyme and substrate intermediates reach a steady state or quasi-equilibrium. Other assumptions basic to the derivation of initial rate equations are as follows ... 

Quasi-equilibrium treatments, on the other hand, assume that all elementary steps prior to the rds are almost in equilibrium, i.e. they can occur sufficiently fast not to alter significantly their equilibrium conditions under net charge flow at the interface. For this assumption to be valid, the elementary step rate coefficients must be at least ten times larger than that of the rds. 

In spite of the above justification for the kinetic approach to the estimate of l, this has a number of drawbacks. First of all, there is no point in using a kinetic approach to determine a thermodynamic equilibrium quantity such as l. The justification of the validity ofEqs. (42) and (45) by the resulting equilibrium condition of Eq. (46) is far from rigorous, just as is the justification of the empirical Butler-Volmer equation by the thermodynamic Nernst equation. Moreover, the kinetic expressions of Eq. (41) involve a number of arbitrary assumptions. Thus, considering the adsorption step of Eq. (38a) in quasi-equilibrium under kinetic conditions cannot be taken for granted a heterogeneous chemical step, such as a deformation of the solvation shell of the... 

The reduced models in Table 11.1 rely on the validity of the Bodenstein approximation for all intermediates except the aldehyde in hydroformylation, but are otherwise free of assumptions. In every case, equations that are as simple or even simpler have long been derived, but only with much more restrictive assumptions, most commonly that of a single rate-controlling step and quasi-equilibrium everywhere else. Of course, such equations should be used in preference if their assumptions can be substantiated. 

In the following it is assummed that the diffusion in the pores is the rate determining process and that the other steps are that fast that they can be considered to be in quasi-equilibrium. The thicker the membrane, the more this assumption is valid. Barrer [67] gives criteria to check this assumption, and in the data shown above this assumption is almost always valid. 

In general, such nonlinear differential equations are difficult to solve exactly. Therefore, Michaelis and Menten (1913) made the simplifying assumption that the intermediate complex is in equilibrium with the free enzyme and substrate (the quasi-equilibrium postulate). While this is not strictly true, it is nearly so when k2 is much less than k i or ki, as was the case for Michaelis and Menten, who were concerned with the enzyme invertase. More often valid is the quasi-steady state postulate, which was first developed in detail by Briggs and Haldane (1925). 

If the potential energy surface has a col, then a separation of the reaction coordinate is possible also in the vicinity of the saddle-point. Supposing that the system remains there sufficiently long time for a stationary state to be established, we may assume the existence of a quasi-equilibrium energy distribution in that transition state, too. Then, we can apply the above statistical treatment of reactants also to the transition state of the reacting system. However, we will not introduce at this stage the above restrictive assumptions, since our aim is to derive a collision theory rate expression of a possibly general validity. 

Mass spectrometry measures the mass of fragment ions formed in a unimolecular dissociation of an energy-rich parent ion. To apply a transition state theory approach to the determination of flie branching into different products requires the assumption that after ionization there is a sufficient delay for the ion to distribute the excess energy over all the available modes. Mass spectrometrists refer to this assumption as the quasi-equilibrium hypothesis, and they are actively concerned about its validity. See Lifshitz (1989), Lorquet (1994, 2000). 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

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