Initial rate equations rapid equilibrium mechanism

Rapid Equilibrium Mechanism. If the rate-determining step is the catalytic step and all binding steps can be described by dissociation constants (e.g., K = [E][A]/ [EA]), then, in the absense of products i.e., [P] and [Q] 0), the initial rate equation for the rapid equihbrium Uni Bi mechanism is identical to that of the Uni Uni... 

The initial rate equation derived by steady-state analysis is of the second degree in A and B (SO). It simplifies to the form of Eq. (1) if the rates of dissociation of substrates and products from the complexes are assumed to be fast compared with the rates of interconversion of the ternary complexes k, k )] thus, the steady-state concentrations of the complexes approximate to their equilibrium concentrations, as was first shown by Haldane (14)- The kinetic coefficients for this rapid equilibrium random mechanism (Table I), together with the thermodynamic relations KeaKeab — KebKeba and KepKepq — KeqKeqp, suffice for the calculation of k, k and all the dissociation constants Kea = k-i/ki, Keab = k-i/ki, etc. 

The discussion thus far might lead one to say that the experimental evidence indicates that the initial collision process occurs between the ions. Actually, because of the rapidly established equilibrium [Eq. (7-34)], the ionic mechanism is no more probable on a kinetic basis than the collision process involving neutral molecules. To see why this is so we need only write down the rate equation for both mechanisms, including activity coefficients. For the ionic reactants. 

Figure 11.1 illustrates the behavior of Equation 11.6. By the assumption of rapid equilibrium the rate determining step is the unimolecular decomposition. At high substrate composition [S] KM and the rate becomes zero-order in substrate, v = Vmax = k3 [E0], the rate depends only on the initial enzyme concentration, and is at its maximum. We are dealing with saturation kinetics. The most convenient way to test mechanism is to invert Equation 11.6... 

Such a mechanism is a form of substrate-induced activation. If all of the binding steps are rapid relative to the ESA-to-EPA interconversion step, the initial-rate rapid-equilibrium equation for this scheme is... 

Equation (6) is identical in form with Eq. (4). In fact, if 3 2, k-2, Eq. (6) reduces to Eq. (4). Although Eq. (5) is a more realistic mechanism compared with Eq. (1), especially when the rapid-equilibrium treatment is applied to the reversible reaction, the information obtainable from initial-rate studies of such unireactant system remains nevertheless the same Vi and K. This serves to justify the simplification used by the kineticist that is, the elimination of certain intermediates to maintain brevity of the rate equation (provided the mathematical form is unaltered). Thus, the forward reaction of an ordered Bi Bi mechanism is generally written as diagrammed below. 

With chemical relaxation methods, the equilibrium of a reaction mixture is rapidly perturbed by some external factor such as pressure, temperature, or electric-field strength. Rate information can then be obtained by following the approach to a new equilibrium by measuring the relaxation time. The perturbation is small and thus the final equilibrium state is close to the initial equilibrium state. Because of this, all rate expressions are reduced to first-order equations regardless of reaction order or molecularity. Therefore, the rate equations are linearized, simplifying determination of complex reaction mechanisms (Bernasconi, 1986 Sparks, 1989),... 

An even more complicated situation is encountered in the alkaline hydrolysis ofp-substitutcd acetanilides (30) (Bender and Thomas, 1961a). The rate equation for the isotopic exchange of these compounds involves terms in [OH-] and [OH-]2. The original mechanism of hydrolysis proposed by Biechler and Taft (1957) in which there is a rapid pre-equilibrium addition and loss of hydroxide in the first step, is disproved by the fact that, although oxygen exchange is accompanied by hydrolysis, exchange is not as complete as it would be expected to be from that scheme. Bender considers the initial attack of hydroxide to be a rate process not an equilibrium. By use of low hydroxide ion concentrations... 

For mechanisms involving random addition of substrates, the King-Altman method gives squared terms in numerator and denominator of the rate equations, which are messy and difficult to work with. The method of Cha (10) treats each random segment as if it were in rapid equilibrium, and this simplifies the rate equation. The fact that data fit such a simplified equation does not prove that the mechanism is a rapid equilibrium one (see the rules in Section V,A,2 below) but does facilitate initial velocity analysis. 

Note that these rules correctly predict that in a two-substrate case the only sequential mechanism in which a term is missing from the denominator is an ordered one in which the first substrate addition is in rapid equilibrium. Rapid equilibrium binding in a random mechanism does not change the initial velocity rate equation (Rule 1), since both substrates can add in the second position. These rules can easily be generalized for cases with four or more substrates. 

---