Initial rate assumption

Initial Rate Assumption. The entire reaction progress curve, or at least a substantial portion of it, is typically required to accurately determine the rate constant for a first-order or second-order reaction. Nonetheless, one can frequently estimate the rate constant by measuring the velocity over a brief period (known as the initial rate phase) where only a small amount of reactant is consumed. This leads to a straight-line reaction progress curve see Fig. 6) which is drawn as a tangent to the initial reaction velocity. 

The initial rate assumption is one of the most powerful and widely used assumptions in the kinetic characterization of enzyme action. The proper choice of reaction conditions that satisfy the initial rate assumption is itself a challenge, but once conditions are established for initial rate measurements, the kinetic treatment of an enzyme s rate behavior becomes much more tractablek In reporting initial rate data, investigators would be well advised to provide the following information ... 

Figure 1. Plot of the change in product concentration as a function of time of reaction. The initial rate phase corresponds to the early linear region, and a tangent to this early region has a slope corresponding to the initial reaction velocity. (For a detailed description of how one obtains rate constants using the initial rate assumption. See Chemical Kinetics.)...

There is, of course, an enzyme-product complex, EP, through which the reverse reaction proceeds. We assume in these analyses that the dissociation of EP is fast and so can be ignored in the forward reaction. The initial-rate assumption allows us to ignore the accumulation of the EP complex and the reverse reaction, since [P] is always very low. 

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

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 ... 

One of the basic assumptions in kinetic studies of an enzyme-catalyzed reaction is that true initial rates are being measured. In such cases, a plot of the product concentration versus time must yield a straight line. (This behavior is only observed when the substrate is at or near its initial (or, r = 0) concentration. As time increases, product accumulation and substrate depletion will result in a curvature of this progress curve hence, the reaction velocity at these later times would be correspondingly lower.)... 

A mathematical simplification of rate behavior of a multistep chemical process assuming that over a period of time a system displays little or no change in the con-centration(s) of intermediate species (i.e., d[intermedi-ate]/df 0). In enzyme kinetics, the steady-state assumption allows one to write and solve the differential equations defining fhe rafes of inferconversion of various enzyme species. This is especially useful in initial rate studies. 

The rate expression Eq. 3-32 requires a first-order dependence of the polymerization rate on the monomer concentration and is observed for many polymerizations [Kamachi et al., 1978], Figure 3-2 shows the first-order relationship for the polymerization of methyl methacrylate [Sugimura and Minoura, 1966], However, there are many polymerizations where Rp shows a higher than first-order dependence on [M], Thus the rate of polymerization depends on the -power of the monomer concentration in the polymerization of styrene in chlorobenzene solution at 120°C initiated by t-butyl peresters [Misra and Mathiu, 1967]. The benzoyl peroxide initiated polymerization of styrene in toluene at 80°C shows an increasing order of dependence of Rp on [M] as [M] decreases [Horikx and Hermans, 1953], The dependence is 1.18-order at [M] = 1.8 and increases to 1.36-order at [M] = 0.4. These effects may be caused by a dependence of the initiation rate on the monomer concentration. Equation 3-28 was derived on the assumption that Rt is independent of [M], The initiation rate can be monomer-dependent in several ways. The initiator efficiency / may vary directly with the monomer concentration... 

The various rate expressions were derived on the assumption that the rate-determining step in the initiation process is Reaction 5-4. If this is not the situation, the forward reaction in Eq. 5-3 is rate-determining. The initiation rate becomes independent of monomer concentration and is expressed by... 

Step 2 We now make an important assumption that the initial rate of reaction reflects a steady state in which [ES] is constant—that is, the rate of formation of ES is equal to the rate of its breakdown. This is called the steady-state assumption. The expressions in Equations 6-12 and 6-13 can be equated for the steady state, giving... 

Kinetic Considerations. The reaction kinetics are masked by a desorption process as shown below and are further complicated by rate deactivation. The independence of the 400-sec rate on reactant mole ratio is not indicative of zero-order kinetics but results because of the nature of the particular kinetic, desorption, and rate decay relationships under these conditions. It would not be expected to be more generally observed under widely varying conditions. The initial rate behavior is considered more indicative of the intrinsic kinetics of the system and is consistent with a model involving competitive adsorption between the two reactants with the olefin being more strongly adsorbed. Such kinetic behavior is consistent with that reported by Venuto (16). Kinetic analysis depends on the assumption that quasi-steady state behavior holds for the rate during rate decay and that the exponential decay extrapolation is valid as time approaches zero. Detailed quantification of the intrinsic kinetics was not attempted in this work. 

Butyllithium initiation of methylmethacrylate has been studied by Korotkov (55) and by Wiles and Bywater (118). Korotkov s scheme involves four reactions 1) attack of butyllithium on the vinyl double bond to produce an active centre, 2) attack of butyllithium at the ester group of the monomer to give inactive products, 3) chain propagation, and 4) chain termination by attack of the polymer anion on the monomer ester function. On the basis of this reaction scheme an expression could be derived for the rate of monomer consumption which is unfortunately too complex for use directly and requires drastic simplification. The final expression derived is therefore only valid for low conversions and slow termination, and if propagation is rapid compared to initiation. The mechanism does not explain the initial rapid uptake of monomer observed, nor the period of anomalous propagation often observed with this initiator. The assumption that kv > kt is hardly likely to be true even after allowance is made for the fact that the concentration of active species is much smaller than that of the added initiator. Butyllithium disappears almost instantaneously but propagation proceeds over periods from tens to hundreds of minutes. The rate constants finally derived therefore cannot be taken seriously (the estimated A is 2 x 105 that of k ) nor can the mechanism be regarded as confirmed. 

The thermal and kinetic models discussed above are the basis for determining the processing conditions for reactive processing by ionic polymerization,29 addition polymerization, vulcanization of rubbers and radical polymerization, although in the latter case additional assumptions of a constant initiation rate and a quasi-stationary concentration of radicals are made.89 These models can also be used to solve optimization problems to improve the performance and properties of end-products. 

Although not very commonly used (with the exception of the initial rate procedure for slow reactions), the differential method has the advantage that it makes no assumption about what the reaction order might be (note the contrast with the method of integration, Section 3.3.2), and it allows a clear distinction between the order with respect to concentration and order with respect to time. However, the rate constant is obtained from an intercept by this method and will, therefore, have a relatively high associated error. The initial rates method also has the drawback that it may miss the effect of products on the global kinetics of the process. 

The question regarding the statistical nature of the equilibrium configuration arises only in the case of low barriers. For chemical reactions, such barriers would correspond to very high rate constants so they would not satisfy the initial TST assumption that the barrier height... 

Initial rates represent a direct kinetic measurement devoid of any assumptions or interpretations regarding the rate law or stoichiometry. These qualities make the initial rates an indispensable tool in some complex situations, such as enzyme kinetics. Also, the initial rates are helpful in establishing the kinetic stoichiometry that can sometimes be difficult to obtain by other means. An example is provided by... 

In the present study, the value of y- which designates the distribution of initiation rates in respective regions is assumed to remain constant in the course of reactions. Evidently, this assumption is crude i.e., in the initial stages of reactions, most graft chains might be formed in the vicinities of the sur-... 

For example, in the acylation of veratrole with benzoic anydride,[14] following this mechanism, we assume that the veratrole chemisorption reduces the number of acid sites available for benzoic anydride, but that the reaction does not proceed between the two adsorbed species. Such an assumption leads to the corresponding initial rate equation as follows ... 

If the plug flow assumption holds and the reactor truly behaves in a differential manner, a plot of Xgg Vs. W/Fgg should be linear with the slope equal to the reaction rate. However, as is evident from Figure 1, slight curvature persists in each plot. Typical calculations revealed that intra and interparticle heat and mass transfer problems should not exist at the operating conditions. The reaction rates, therefore, were obtained by evaluating the slope of each curve at the origin and as such can be called initial rates of reaction, Rq. 

Nabi and Lu [13] have cast doubt on this general assumption that chemical reaction is first order with respect to the reactant and have studied the reduction Fe203 ->-Fe304 by H2 in the temperature range 650—800°C. They measured the initial rate of reaction of a perfectly... 

All the data discussed above are only an approximation or analogy of the real initiation of radical polymerization. The detailed investigation of the kinetics and mechanism of actual initiation processes is often our future task. Only rarely it is possible to obtain quantitative data on initiation, for example the value of the initiation rate constant, without simplifying assumptions. [133]. Some further information on this problem will be presented in Sect. 8.1 and in Chap. 5, Sect. 4.2 and in Chap. 8, Sect. 1.1. 

The most direct method of measuring the initiation rate is the determination of the incorporation rate of initiator fragments into the polymer. Experimentally this is a complicated method. The evaluation is usually based on the assumption that, in the stationary state at slow initiator consumption, i>p, uinil and v (mean kinetic chain length) remain almost constant. 

The kinetic model of styrene auto-initiation proposed by Hui and Hameilec [27] was used as a starting point for this work. The Mayo initiation mechanism was assumed (Figure 7.2) but the acid reaction was of course omitted. After invoking the quasi-steady-state assumption (QSSA) to approximate the reactive dimer concentration, Hui and Hameilec used different simplifying assumptions to derive initiation rate equations that are second and third order in monomer concentration. 

In the third article of the series the authors set out to determine the kinetic initiation parameters and the lifetime of the ionic chain carriers. The values of kj were cmnputed frmn the measured rates of carbenium icm formation assuming bimolecular initiation. This assumption is unacceptable a priori since the interaction between Brjinsted acids and olefins in solvents like the one used in this work has been shown to involve kinetic patterns whidi are almost always more complicated than a simple first order in each reactant (see Sect. III-B). As for the calculation of the mean lifetime of the active species based on the expression... 

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