Derivation of rate equations for

The systematic method is equally convenient for the derivation of rate equations for simple mechanisms. Scheme 1, for example, can be redrawn as an enclosed figure after deleting the pathways between unlabeled enzyme forms. 

Ishikawa, H., Maeda, T., Hikita, H and Miyatake, K. (1988) The computerised derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption. Biochem. J. 251, 175-181. 

It is clear that the material given in this chapter is quite classical and has been known in the literature since the 1930s and 1940s in the field of surface chemistry and catalysis. In fact this is the extent of knowledge used to date in the derivation of rate equations for gas solid catalytic reactions. To be more specific most of the studies on the development of gas-solid catalytic reactions do not even use the information and knowledge related to the rates of chemisorption (activated or non-activated) and desorption. Even the most detailed kinetic studies, usually rely on the assumption of equilibrium adsorption-desorption and use one of the well known equilibrium isotherms (usually the Langmuir isotherm) in order to relate the surface concentration to the concentration of the gas just above the surface of the catalyst. 

The derivation of rate equations for simultaneous hydrogenation of TMP-aldol and formaldehyde was based on a plausible surface reaction mechanism. According to the mechanism, the aldol, formaldehyde and hydrogen undergo competitive adsorption on the nickel-alumina surface. Adsorbed hydrogen atoms add to the carbonyl bonds of the aldol and formaldehyde. These irreversible reaction steps are presumed to be rate-determining, whereas the product desorption is regarded as rapid. Consequently, the reaction mechanism is written as follows ... 

The same strategy was applied in the derivation of rate equations for w-step nucleation according to a power law (cf. Eq. (21)) [133, 134], the combination of nucleation laws with anisotropic growth regimes [153], as well as truncated nucleation due to time-dependent concentration gradients of monomers [136]. MC simulations verified that the Avrami theorem is valid for instantaneous [184], progressive [185], and n-step nucleation according to a power law [184-187]. 

The derivation of rate equations for simple monosubstrate reactions was described in Chapter 3. For bisubstrate reactions, the derivation is usually much more complex, and requires the application of a special mathematical apparatus and special mathematical procedures. 

The interaction factors a, p, and y represent, respectively, the effect of A on the binding of B, the effect of I on the binding of B, and the effect of I on the binding of A. The factor d represents the effect of I on the catalytic activity of the EABI complex. The derivation of rate equation for reaction (6.14) would be extremely laborious from the steady-state assumptions. Therefore, the general velocity equation is derived from Ae rapid equilibrium assumptions ... 

In this section, we shall reviewthe rate equations forthe majortypes of trisubstrate mechanisms, written in the absence of products (Cleland, 1963 Plowman, 1972 Fromm, 1975,1979). All trisubstrate mechanisms in the rapid equilibrium category are relatively rare and the steady-state mechanisms are more common. However, the derivation of rate equations for rapid equilibrium mechanisms, in the absence of products, is less demanding, as it requires only the rapid equilibrium assumptions and, therefore, the resulting rate equations are relatively simple. 

The derivation of rate equations for isotope exchange away from equilibrium may be understood in terms of Eq. (16.7) for the Ordered Bi Bi mechanism... 

Let us start the derivation of rate equations for kinetic isotope effects with analysis of a simple monosubstrate enzyme reaction ... 

Stationary state approximation is widely used for the derivation of rate equations for non-elementary reactions. We will see a few more applications of stationary state approximations in the forthcoming sections. 

It is usually assumed in the derivation of isothermal rate equations based on geometric reaction models, that interface advance proceeds at constant rate (Chap. 3 Sects. 2 and 3). Much of the early experimental support for this important and widely accepted premise derives from measurements for dehydration reactions in which easily recognizable, large and well-defined nuclei permitted accurate measurement. This simple representation of constant rate of interface advance is, however, not universally applicable and may require modifications for use in the formulation of rate equations for quantitative kinetic analyses. Such modifications include due allowance for the following factors, (i) The rate of initial growth of small nuclei is often less than that ultimately achieved, (ii) Rates of interface advance may vary with crystallographic direction and reactant surface, (iii) The impedance to water vapour escape offered by... 

Derive the rate equation for the formation of the stable double helix and express the rate constant of the overall reaction in terms of the rate constants of the individual steps. 

The catalytic process is a sequence of elementary steps that form a cycle from which the catalyst emerges unaltered. Identifying which steps and intermediates have to be taken into account may be difficult, requiring spectroscopic tools and computational approaches, as described elsewhere (see Chapter 7). Here we will assume that the elementary steps are known, and will describe in detail how one derives the rate equation for such processes. 

Ha and Choun [51] confirmed these findings from the investigation of cyclic oligomer formation at 270 °C. They derived a rate equation for cyclic oligomer formation taking thermal degradation of the polymer into account. 

Derivation of rate equations is an integral part of the effective usage of kinetics as a tool. Novel mechanisms must be described by new equations, and famihar ones often need to be modified to account for minor deviations from the expected pattern. The mathematical manipulations involved in deriving initial velocity or isotope exchange-rate laws are in general quite straightforward, but can be tedious. It is the purpose of this entry, therefore, to present the currently available methods with emphasis on the more convenient ones. 

This use of the determinant method for complex enzyme mechanisms is time-consuming because of the stepwise expansion and the large number of positive and negative terms that must be canceled. It is quite useful, however, in computer-assisted derivation of rate equations. ... 

Derive a rate equation for formation of C in the following mechanism, assuming the stationary state for B ... 

Derive the rate equation for rate of formation of E in terms of concentrations of reactants A and B in the following mechanism, assuming that the rates of steps kx and... 

Derive the rate equation for formation of F in terms of concentrations of A, B, and D in the following mechanisms, assuming that A, B, and C are in equilibrium and E is a highly reactive intermediate. 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

For catalytic cycles with more than three or four members, the long-hand derivation of rate equations gets out of hand. However, a general formula comparable to that given in Section 6.3 for noncatalytic simple pathways was established as early as 1931 by Christiansen [36-38]. ... 

In copolymerization, several different combinations of initiation and termination mechanisms are possible, giving rise to a variety of different polymerization rate equations. Only two cases will be singled out here free-radical copolymerization with termination by coupling, and ionic polymerization with termination by chain transfer to a deactivating agent or impurity. For other combinations, the derivation of rate equations follows along the same lines. 

Fi0. 3. Transitions between states of radical occupancy requited for derivation of OToole equation for rate of transition of loci across notional barrier between states i and i +1. 

This is the central equation to start the derivation of rate equations, and we will see below how this equation can be used for various quantities X. 

The desorption of an adatom that does not feel neighboring adatoms is the simplest case to derive macroscopic rate equations for. The derivation in this section will be exact. This is due to the fact that there is no interaction between adatoms. 

The general idea of the three previous cases (i.e., Figures 10-31 through 10-33)is to arrange the data in such a fashion as to arrive at functional groupings of measured variables that will be linear with time. The particular functional groups will vary with (1) type of reactor used to collect the data, (2) reaction order of the main reaction, and (3) the decay reaction order. For the three main types of reactors, three main reaction rate laws, and three decay rates, 27 different types of plots could result. We leave derivation of the equation for each of these plots to the reader and point out that only one additional step is needed in our solution algorithm. That step is the decay rate law ... 

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