Kinetic equations, linear forms

The Arrhenius equation is best viewed as an empirical relationship that describes kinetic data very well. It is commonly applied in the linearized form... 

We wish to apply weighted linear least-squares regression to Eq. (6-2), the linearized form of the Arrhenius equation. Let us suppose that our kinetic studies have provided us with data consisting of Tj, and for at least three temperatures, where o, is the experimental standard deviation of fc,. We will assume that the error in T is negligible relative to that in k. For convenience we write Eq. (6-2) as... 

The choice of the y-variable is also important. If one records a series of concentrations, or a quantity proportional to them, then this set is a valid quantity to be fitted by linear least squares. On the other hand, if the equation is rearranged to a form that can be displayed in a linear graph, then the new variable may not be so suitable. Consider the equations for second-order kinetics. The correct form for least-squares fitting is... 

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A... 

Matheson, I. B. C. The method of successive integration a general technique for recasting kinetic equations in a readily soluble form which is linear in the coefficients and sufficiently rapid for real time instrumental use. Anal Instrum. 1987, 16, 345-373. 

If there is more than one reactant, as in Examples 3-3 or 3-5, with a rate law given by (-rA) = kAcaAcl, the procedure to determine (-rA) is similar to that for one reactant, and the kinetics parameters are obtained by use of equation 3.4-4, the linearized form of the rate law. 

Figure 8.9 Kinetics of a second-order reaction the racemization of glucose in aqueous mineral acid at 17 °C (a) graph of concentration (as y ) against time (as V) (b) graph drawn according to the linear form of the integrated second-order rate equation, obtained by plotting 1 / A, (as V) against time (as V). The gradient of trace (b) equals the second-order rate constant k2, and has a value of 6.00 x 10-4 dm3mol 1s 1...

The behaviour patterns ensuing when bulk monomers are diluted by solvents are very varied. The most detailed information concerns the VE. My re-examination of the results shows that, contrary to current belief, no one kinetic scheme will fit all the systems over the whole range of m. My interpretations were facilitated considerably by the availability of the dependence of c on m, which for most systems can be expressed by a linear equation of the form c = Am + B, where in some systems A is positive, in others negative. By making this substitution in the kinetic equations it becomes obvious why for most systems the external kinetic order with respect to m is greater than unity, an effect noted, but hitherto not explained convincingly. 

Many kinetic equations can be suitably linearized to the form of Eq. (20). For example, Eq. (1) can be transformed logarithmically, or Eq. (2) can be transformed reciprocally. Two equations proposed for describing pentane-isomerization data (Cl, Jl) are the single site... 

Of particular value in kinetic studies are residual plots using the linearized form of the Hougen-Watson equation. For the model of Eq. (18), for example, we obtain... 

In the lumped kinetic model, various kinetic equations may describe the relationship between the mobile phase and stationary phase concentrations. The transport-dispersive model, for instance, is a linear film driving force model in which a first-order kinetics is assumed in the following form ... 

Formally, we call the set of reaction solvable, if there exists a linear transformation of coordinates a- a such that kinetic equation in new coordinates for all values of reaction constants has the triangle form ... 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

This substitution has only a small effect on the form of kinetic equations in particular, the linear character of the dependence of S0l on PC2h4o/Po2 is preserved. 

The most straightforward way is to plot r against Cs as shown in Figure 2.2. The asymptote for r will be rmax and KM is equal to Cs when r = 0.5 rmax. However, this is an unsatisfactory plot in estimating rmax and KM because it is difficult to estimate asymptotes accurately and also difficult to test the validity of the kinetic model. Therefore, the Michaelis-Menten equation is usually rearranged so that the results can be plotted as a straight line. Some of the better known methods are presented here. The Michaelis-Menten equation, Eq. (2.11), can be rearranged to be expressed in linear form. This can be achieved in three ways ... 

Since in irreversible linear polymerization the cyclic molecules once formed do not react further, the kinetics equations can be written thus (here Q is the concentration of cycles) ... 

Before adequate computer hardware and software were widely available, fitting experimental kinetic data to a curve to determine Km and Vmax was a significant challenge. Lineweaver and Burk rearranged the Michaelis-Menten equation to form a new linear relationship, the Lineweaver-Burk equation (Equation 4.13).6... 

For linear mechanisms we have obtained structurized forms of steady-state kinetic equations (Chap. 4). These forms make possible a rapid derivation of steady-state kinetic equations on the basis of a reaction scheme without laborious intermediate calculations. The advantage of these forms is, however, not so much in the simplicity of derivation as in the fact that, on their basis, various physico-chemical conclusions can be drawn, in particular those concerning the relation between the characteristics of detailed mechanisms and the observable kinetic parameters. An interesting and important property of the structurized forms is that they vividly show in what way a complex chemical reaction is assembled from simple ones. Thus, for a single-route linear mechanism, the numerator of a steady-state kinetic equation always corresponds to the kinetic law of the overall reaction as if it were simple and obeyed the law of mass action. This type of numerator is absolutely independent of the number of steps (a thousand, a million) involved in a single-route mechanism. The denominator, however, characterizes the "non-elementary character accounting for the retardation of the complex catalytic reaction by the initial substances and products. 

The theory of linear mechanisms is a sufficiently developed field of catalytic kinetics. Let us present its principal results. In accordance with the law of acting surfaces, kinetic equations for a linear mechanism are of the form... 

When the principal linear law of conservation is of the form LrriiZi = const., elementary reactions entering into the mechanism without interactions are (d/m A - (dlmj)AJ and the corresponding kinetic equations and Jacobian matrix will be... 

This equation is independent of the order in which the steps are numbered. Temkin suggested an algorithm on the basis of eqn. (30) to obtain an explicit form of the steady-state kinetic equations. For linear mechanisms in this algorithm it is essential to apply a complex reaction graph. In some cases the derivation of a steady-state equation for non-linear mechanisms on the basis of eqn. (30) is also less difficult. 

GENERAL FORM OF STEADY-STATE KINETIC EQUATION FOR COMPLEX CATALYTIC REACTIONS WITH MULTI-ROUTE LINEAR MECHANISMS... 

It can be concluded that, for linear multi-route mechanisms, a class has been specified for which the representation of a kinetic equation in the form of the Horiuti-Boreskov equation, eqn. (59), is valid. Note that Khomenko et al. [43] have analyzed a kinetic equation for the two-route reaction, one of which is in equilibrium. For the results of the analysis for a non-linear one-route mechanism, see ref. 44. 

Basic assumptions 1 to 3 were put forward as a result of analysis, firstly, of the reaction-diffusion mechanism described in detail by V.I. Arkha-rov1,46,47 and, secondly, of a linear-parabolic equation derived for the first time by U.R. Evans from somewhat different considerations. It should be noted that similar assumptions were used earlier by B.Ya. Pines,9,131 who in deriving differential forms of kinetic equations summed up the duration of external and internal diffusion. 

Simple, graphical methods for testing the fit of rate data to Equation 9 and for estimating the kinetic parameters k2 and KR involve using linearized forms of Equation 9. By far the most widely used linear form is that of Kitz and Wilson (14). Taking reciprocals of both sides of Equation 9, we obtain... 

Eisenthal and Cornish-Bowden (21) and Cornish-Bowden (20) have described a rather different type of enzyme kinetics plot that should be useful in analyzing affinity-labeling kinetics. The equation that forms the basis of the direct linear plot (21) is obtained from Equation 11 by rearrangement of terms to give Equation 12... 

The problem consists in seeking such a combination of the values of constants k which gives the minimum value of Q ( >mm). Before computers became commonly available, the kinetic equations had usually been transformed into a linear form and the linear regression ( least-squares method ) had been applied to find the best set of constants. This procedure is not statistically correct in most cases. Therefore, only the nonlinear regression method can be recommended to optimize constants in kinetic equations that have a nonlinear form [48-51]. 

Klier et al. investigated several cases of kinetics in which methanol is formed by a surface reaction between CO and hydrogen adsorbed on the Aox sites competitively or noncompetitively with CO on the Aox sites and hydrogen elsewhere on the surface in each case C02 effects sub (/) and (iii) above were taken into account. In addition, it was found empirically that a small amount of C02 is hydrogenated to methanol at a rate that linearly depended on partial pressure of C02. All kinetic equations that successfully described the C02 effects had the general form... 

Since the use of equilibrium (Freundlich) type with n > 1 is uncommon, we also attempted the kinetic reversible approach given by equation 12.2 to describe the effluent results from the Bs-I column. The use of equation 12.2 alone represents a fully reversible S04 sorption of the n-th order reaction where kj to k2 are the associated rates coefficients (Ir1). Again, a linear form of the kinetic equation is derived if m = 1. As shown in Figure 12.7, we obtained a good fit of the Bs-I effluent data for the linear kinetic curve with r2 = 0.967. The values of the reaction coefficients kj to k2, which provided the best fit of the effluent data, were 3.42 and 1.43 h with standard errors of 0.328 and 0.339 h 1, respectively (see Table 12.3). Efforts to achieve improved predictions using nonlinear (m different from 1) kinetics was not successful (figures not shown). We also attempted to incorporate irreversible (or slowly reversible) reaction as a sink term (see equation 12.5) concurrently with first-order kinetics. A value of kIIT = 0.0456 h 1 was our best estimate, which did not yield improved predictions of the effluent results as shown in Figure 12.7. 

Let us demonstrate that the presence of the thermodynamic conjugation with the undesired side transformation channels allows, in principle, the target process of the conversion of benzene to ethylbenzene to be achieved with the 100% selectivity provided that the undesired DEB product is added in a certain amount to the initial reaction mixture. Consider the first and second stepwise alkylation processes as thermodynamically conjugate (the third stepwise process is linearly dependent on these two processes). In doing so, the kinetic equations of the formation of ethylbenzene and diethylbenzene can be written in the Horiuti Boreskov Onsager form as... 

Assuming a constant surface area, dissolution at a solution-solid interface (Case I) results in linear kinetics in which the rate of mass transfer is constant with time (equation 1). Analytical solutions to the diffusion equation result in parabolic rates of mass transfer (, 16) (equation 2). This result is obtained whether the boundary conditions are defined so diffusion occurs across a progressively thickening, leached layer within the silicate phase (Case II), or across a growing precipitate layer forming on the silicate surface (Case III). Another case of linear kinetics (equation 1) may occur when the rate of formation of a metastable product or leached layer at the fresh silicate surface becomes equal to the rate at which this layer is destroyed at the aqueous... 

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