Rate constant rapid equilibrium assumption

Equations (2.10) and (2.12) are identical except for the substitution of the equilibrium dissociation constant Ks in Equation (2.10) by the kinetic constant Ku in Equation (2.12). This substitution is necessary because in the steady state treatment, rapid equilibrium assumptions no longer holds. A detailed description of the meaning of Ku, in terms of specific rate constants can be found in the texts by Copeland (2000) and Fersht (1999) and elsewhere. For our purposes it suffices to say that while Ku is not a true equilibrium constant, it can nevertheless be viewed as a measure of the relative affinity of the ES encounter complex under steady state conditions. Thus in all of the equations presented in this chapter we must substitute Ku for Ks when dealing with steady state measurements of enzyme reactions. 

In order for an equilibrium to exist between E -E S and ES, the rate constant kp would have to be much smaller than k i However, for the majority of enzyme activities, this assumption is unlikely to hold true. Nevertheless, the rapid equilibrium approach remains a most useful tool since equations thereby derived often have the same form as those derived by more correct steady-state approaches (see later), and although steady-state analyses of very complex systems (such as those displaying cooperative behavior) are almost impossibly complicated, rapid equilibrium assumptions facilitate relatively straightforward derivations of equations in such cases. 

If the rate constant k2 is much smaller than the rate constant k., of the enzyme, the substrate and the enzyme-substrate complex are in equilibrium, which is not disturbed by the decomposition of ES into E and P ( rapid equilibrium-assumption ). Based on this assumption, Michaelis and Menten derived the following rate equation (Eq. (17)) ... 

Although the derivation of reaction rates using the steady state assumption is more exact, often the rapid equilibrium assumption is used because it allows a simple derivation of the rate equation from the relevant enzyme-substrate complexes (see below) and allows fitting of the kinetic data. The following explanations are based on the rapid equilibrium assumption, and therefore all following constants K are used as dissociation constants with the component dissociating from the enzyme as the subscript, e.g. Ka, Kb, and the component remaining at the enzyme as second subscript (e.g. Kf, see below). 

The rate equations for fully random and ordered mechanisms for three-substrate reactions are shown in Table II and can only be briefly discussed here. For the random mechanism, the rate equation derived by the rapid equilibrium assumption 43) contains all the terms of Eq. (2), and from experimental values for the eight kinetic coefficients for the reaction in each direction the dissociation constants for all the complexes may be calculated (c/. 43). 

This mechanism is based on the assumption that the enzyme (E) binds the substrate in a rapid and reversible step to give a noncovalent enzyme—substrate complex (ES) known as the Michaelis-Menten complex. The ES slowly turns over to the product with a first-order rate constant 2- The free enzyme can resume the catalytic cycle. When k i > k2, the rapid equilibrium assumption holds and the ES is in equilibrium with E and S. Under the rapid equilibrium assumption, the rate expression is given by Eq. (1.7)... 

It is interesting to note that, although the intrinsic rate of desorption is slower than that of adsorption, both rates were found to be sufficiently fast under our experimental conditions so that the adsorption-desorption process on the Pt surface can be assumed to rapidly equilibrate at all times that is, even a ten-fold increase in both the adsorption and desorption rate constants (while keeping their ratio constant) did not significantly change the predicted step responses. With the assumption of chemisorption equilibrium, Equations (1) and (4) can be combined into the form (35)... 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

For the types of comparisons reported here it has generally been convenient to use steady state assumptions, but these clearly do not apply to conditions after forest spraying. Monitoring studies typically report rapid penetration of pesticides to forest streams followed by rapid dissipation of residues by a number of processes. Most published bioconcentration equations do not contain a time term and so they cannot readily be applied to short intervals when only a small fraction of the time to reach equilibrium would apply. The rate constants and other descriptive equations offer the possibility of predicting bioconcentration under non-equilibrium conditions. 

The Sj-Sj annihilation rate constant was determined in the following manner. First, the concentration of the excimer was obtained by dividing the observed absorbance by its extinction coefficient and by the effective cell length where the Sj<-Sp absorbance at 266nm was 1. In addition, in pure liquid benzene as well as in solution, there exists rapid equilibrium between monomer and excimer, of which time constants of association and dissociation are in the order of a few ps °. Hence, the sum of the monomer and excimer concentrations obtained by the equilibrium constant at each temperature was used as the concentration of the excited singlet species for the analysis. Although this assumption may affect to some extent the accuracy of the obtained rate constant, the error of this estimation would not depend upon the temperature. 

The traditional theory for the rate of chemical reactions is the transition-state theory [21] (abbreviated as TST). In fact, all the rate constants given so far in previous sections were formulated, in general terms, within the framework of the TST. It is tacitly assumed in this theory that fluctuations in the reactant state are so rapid that all the substates comprising the reactant state are always thermally equilibrated in the course of reaction. According to this assumption, the reactant population in the transition state is always maintained in thermal equilibrium with the population in the reactant state since both states are located on the reactant-state adiabatic (or diabatic) potential. Therefore, calculation of the rate constant is greatly simplified... 

Finally, the validity of the chromatographic methods for the determination of isotherms is based on the assumption that phase equilibrium is reached rapidly during the experiment. The rate constant for phase equilibration must be large enough for the experimental results to be independent of the mobile phase velocity. When carrying out FA, FACP, or ECP measurements on proteins that tend to equilibrate slowly, it is advisable to check the influence of the flow velocity on the isotherms (Figures 3.15 and 3.44 [38]). 

In this case, v is the velocity of the reaction, [S] is the substrate concentration, Vmax (also known as V or Vj ) is the maximum velocity of the reaction, and is the Michaelis constant. From this equation quantitative descriptions of enzyme-catalyzed reactions, in terms of rate and concentration, can be made. As can be surmised by the form of the equation, data that is described by the Michaelis-Menten equation takes the shape of a hyperbola when plotted in two-dimensional fashion with velocity as the y-axis and substrate concentration as the x-axis (Fig. 4.1). Use of the Michaelis-Menten equation is based on the assumption that the enzyme reaction is operating under both steady state and rapid equilibrium conditions (i.e., that the concentration of all of the enzyme-substrate intermediates (see Scheme 4.1) become constant soon after initiation of the reaction). The assumption is also made that the active site of the enzyme contains only one binding site at which catalysis occurs and that only one substrate molecule at a time is interacting with the binding site. As will be discussed below, this latter assumption is not always valid when considering the kinetics of drug metabolizing enzymes. 

Although the assumptions of rapid adsorption and local equilibrium at the interface are justified in many situations of interest, sometimes the rates of adsorption and desorption must be considered. Equation 6.41 still applies, but the analysis must be modified. The terms adsorption barrier and desorption barrier are sometimes used when kinetic limitations exist for the respective processes. If a surface active solute diffuses between phases imder conditions where there is an appreciable desorption barrier, for example, interfacial concentration r will attain higher values than in the absence of the barrier, and interfacial tension will be lower. England and Berg (1971) and Rubin and Radke (1980) have studied such situations. Figure 6.11 shows an example of predicted interfacial tension as a fimction of time for various values of a dimensionless rate constant. The low transient interfacial tension is evident. 

The increase in phase angle is the more considerable, the more rapid the adsorption process [29]. These qualitative considerations are at present illustrated in a number of theoretical and experimental papers. Faradaic impedance with allowance for adsorption of electrochemically active substances at the equilibrium potential was frequently [30-35] investigated theoretically. Originally [30, 35] it was assumed that the adsorption of the reaction components can be described by the Langmuir isotherm and that the adsorption and desorption rate constants did not depend on the potential. A more general analysis avoiding these assumptions was made by Senda and Delahay [31] on the supposition that only adsorbed particles were electrolyzed and by Lorenz [32], who took account... 

To determine Katrp, the activator, Mt Lm, is reacted with an alkyl halide, and the deactivator concentration (experimentally accessible through ESR or electronic spectroscopy) is monitored as a function of time. Then, a plot of pCMt Lm] vx. t is constructed, and Katrp is determined from the slope provided that the termination rate constant kt is known. The equilibrium must be established rapidly and, due to the assumptions mentioned above, this strategy is only applicable for comparatively low conversions of activator and initiator. If these conditions are notmet, the dependence (8.12) is not observed. This is a severe limitation when the values of Katrp for active catalysts should be determined, where both the concentrations of activator and initiator decrease rapidly. 

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