Rate constants practical equations

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Although equation (3) provides a rigorous quantum definition of the cumulative reaction probability - and thus the canonical and microcanonical rate constants via equations (2) and (4) - it is not helpful in a practical sense because a complete state-to-state reactive scattering calculation is required to obtain the 5-matrix. We seek a more direct (and thus presumably more efficient) route to N(E), but without approximation, to which approximations can be incorporated later as needed in specific applications. 

VER in liquid O 2 is far too slow to be studied directly by nonequilibrium simulations. The force-correlation function, equation (C3.5.2), was computed from an equilibrium simulation of rigid O2. The VER rate constant given in equation (C3.5.3) is proportional to the Fourier transfonn of the force-correlation function at the Oj frequency. Fiowever, there are two significant practical difficulties. First, the Fourier transfonn, denoted [Pg.3041]

As with the case of energy input, detergency generally reaches a plateau after a certain wash time as would be expected from a kinetic analysis. In a practical system, each of its numerous components has a different rate constant, hence its rate behavior generally does not exhibit any simple pattern. Many attempts have been made to fit soil removal (50) rates in practical systems to the usual rate equations of physical chemistry. The rate of soil removal in the Launder-Ometer could be reasonably well described by the equation of a first-order chemical reaction, ie, the rate was proportional to the amount of removable soil remaining on the fabric (51,52). In a study of soil removal rates from artificially soiled fabrics in the Terg-O-Tometer, the percent soil removal increased linearly with the log of cumulative wash time. 

In practice it is found that the concentration of radicals rapidly reaches a constant value and the reaction takes place in the steady state. Thus the rate of radical formation Ej becomes equal to the rate of radical disappearance V. It is thus possible to combine equations (2.1) and (2.3) to obtain an expression for [M—] in terms of the rate constants... 

In practice n changes only slowly with changes in y and it is possible to postulate that over a range of shear rates it is constant. If equation (8.7) is therefore integrated we obtain... 

We can make two different uses of the activation parameters AH and A5 (or, equivalently, E and A). One of these uses is a very practical one, namely, the use of the Arrhenius equation as a guide for interpolation or extrapolation of rate constants. For this purpose, rate data are sometimes stored in the form of the Arrhenius equation. For example, the data of Table 6-1 may be represented (see Table 6-2) as... 

Miller226 applied the Hammett equation to the rate constants for the reaction of 4-substituted l-chloro-2-nitrobenzenes with OMe in methanol at 50°C. a values (denoted ct in accordance with the practice briefly in vogue at that time, 1956) were used for + R substituents, and S02Me conformed well at a a value of 1.04952. Act value of 1.117 for S02Ph was derived from the Hammett plot, intermediate between the values based on phenol and anilinium ionizations by Szmant and Suld88 at about the same time. 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

The complexity of the integrated form of the second-order rate equation makes it difficult to apply in many practical applications. Nevertheless, one can combine this equation with modem computer-based curve-fitting programs to yield good estimates of reaction rate constants. Under some laboratory conditions, the form of Equation (A1.25) can be simplified in useful ways (Gutfreund, 1995). For example, this equation can be simplified considerably if the concentration of one of the reactants is held constant, as we will see below. 

The non-linear equations 21 and equations 35-38 can be solved iteratively to give directly, the instantaneous efficiencies and the the ratios of rate constants (kx/kp), (ktc/kp2) and (ktc /kp2). The values obtained for the rate constants have been summarized in Table III and in Figures 3 and 6. The results from the calculations show a small difference (ie less than 1 ) betwen ktc and ktc. Therefore, for all practical purposes they can be considered equal... 

The RC1 reactor system temperature control can be operated in three different modes isothermal (temperature of the reactor contents is constant), isoperibolic (temperature of the jacket is constant), or adiabatic (reactor contents temperature equals the jacket temperature). Critical operational parameters can then be evaluated under conditions comparable to those used in practice on a large scale, and relationships can be made relative to enthalpies of reaction, reaction rate constants, product purity, and physical properties. Such information is meaningful provided effective heat transfer exists. The heat generation rate, qr, resulting from the chemical reactions and/or physical characteristic changes of the reactor contents, is obtained from the transferred and accumulated heats as represented by Equation (3-17) ... 

In a typical experiment, the sample is a solution (e.g., in benzene) of both the ferf-butoxyl radical precursor (di-tert-butylperoxide) and the substrate (phenol). The phenol concentration is defined by the time constraint referred to before. The net reaction must be complete much faster than the intrinsic response of the microphone. Because reaction 13.23 is, in practical terms, instantaneous, that requirement will fall only on reaction 13.24. The time scale of this reaction can be quantified by its lifetime rr, which is related to its pseudo-first-order rate constant k [PhOH] and can be set by choosing an adequate concentration of phenol, according to equation 13.25 ... 

As will now be discussed, the exchange current is proportional to the standard rate constant, thus resulting in the common practice of using i0 instead of k° in kinetic equations. 

Hence the dimension ("the order") of the reaction is different, even in the simplest case, and hence a comparison of the two rate constants has little meaning. Comparisons of rates are meaningful only if the catalysts follow the same mechanism and if the product formation can be expressed by the same rate equation. In this instance we can talk about rate enhancements of catalysts relative to another. If an uncatalysed reaction and a catalysed one occur simultaneously in a system we may determine what part of the product is made via the catalytic route and what part isn t. In enzyme catalysis and enzyme mimics one often compares the k, of the uncatalysed reaction with k2 of the catalysed reaction if the mechanisms of the two reactions are the same this may be a useful comparison. A practical yardstick of catalyst performance in industry is the space-time-yield mentioned above, that is to say the yield of kg of product per reactor volume per unit of time (e.g. kg product/m3.h), assuming that other factors such as catalyst costs, including recycling, and work-up costs remain the same. 

The exact steady-state solution for this mechanism is too complicated to be of any interest in this context. If, however, the rate constant ks for disappearance of the ternary complex EAB is so small that there is practically equilibrium as far as EA, EB, and EAB are concerned, the equation reduces to a simple form. In this case, the scheme (76) leads to the kinetic constants W = s[E]o, KA = k-2/k2, Kb = h -2/k2, and K = k-i/ki. 

A plot of In k against the reciprocal of the absolute temperature (an Arrhenius plot) will produce a straight line having a slope of —EJR. The frequency factor can be obtained from the vertical intercept. In A. The Arrhenius relationship has been demonstrated to be valid in a large number of cases (for example, colchicine-induced GTPase activity of tubulin or the binding of A-acetyl-phenylalanyl-tRNA to ribosomes ). In practice, the Arrhenius equation is only a good approximation of the temperature dependence of the rate constant, a point which will be addressed below. 

Radioactive decay follows the same rate equation as first-order chemical kinetics (Section 2.5) the half-life t j2, the time required for one-half of the sample to decay, is given by (In 2)/(rate constant). Decay of a sample is considered arbitrarily to be practically complete after 10ti/2, which is 240,000 years for 239Pu. 

It is easy to see from Equation 8 why —NH3+ ion does not catalyze the mutarotation The positively charged ion cannot extract the proton from the hydroxyl group on carbon 1. When no NaOH is added, in the presence of 0.0114M Cd(N03)2, the rate constant of mutarotation is 0.0122, practically the same as in the absence of the metal. This is as expected, since no glucosamine complex is present. 

Non-Newtonian fluids are generally those for which the viscosity is not constant even at constant temperature and pressure. The viscosity depends on the shear rate or, more accurately, on the previous kinematic history of the fluid. The linear relationship between the shear stress and the shear rate, noted in Equation (1), is no longer sufficient. Strictly speaking, the coefficient of viscosity is meaningful only for Newtonian fluids, in which case it is the slope of a plot of stress versus rate of shear, as shown in Figure 4.2. For non-Newtonian fluids, such a plot is generally nonlinear, so the slope varies from point to point. In actual practice, the data... 

When Cg (i.e., concentration of B which reacts with A) is much larger than C, Cg can be considered approximately constant, and k Cg) can be regarded as the pseudo first-order reaction rate constant (T ). The dimensionless group y, as defined by Equation 6.23, is often designated as the Hatta number (Ha). According to Equation 6.22, if y > 5, it becomes practically equal to E, which is sometimes also called the Hatta number. For this range. 

Two practical points should be noted. The kinetic mechanisms in equations 4.71 and 4.74 may be distinguished by the concentration dependence of l/r2. For 4.71 this increases with increasing [S] for 4.74 it decreases. But there are situations that are difficult to resolve. For example, in equation 4.74, if [E ] > [E] there will be a burst of formation of ES with relaxation time r(, followed by a small increase at relaxation time r2 as E converts to E. The concentration dependence of r2 will be small, since kl > k-y for [E ] > [E], This can be mistaken for the scheme in equation 4.71, where only a little ES is formed. In this case also, the concentration dependence of 1/t2 is small, because k-2 > k2. In both cases the amplitudes of the changes will often be small and the rate constants difficult to measure accurately. 

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