Integrated rate equations linear form

A major limitation of the linearized forms of the Michaelis-Menten equation is that none provides accurate estimates of both Km and Vmax. Furthermore, it is impossible to obtain meaningful error estimates for the parameters, since linear regression is not strictly appropriate. With the advent of more sophisticated computer tools, there is an increasing trend toward using the integrated rate equation and nonlinear regression analysis to estimate Km and While this type of analysis is more complex than the linear approaches, it has several benefits. First, accurate nonbiased estimates of Km and Vmax can be obtained. Second, nonlinear regression may allow the errors (or confidence intervals) of the parameter estimates to be determined. 

Figure 8.7 shows the way the concentration of hydrogen peroxide decreases with time. The trace is clearly curved, and Figure 8.8 shows a graph constructed with the linear form of the first-order integrated rate equation, Equation (8.26). This latter graph is clearly linear. 

An alternative form of the integrated rate equation is the so-called linear form... 

In theory, Equ.(2) can be rearranged into Equ.(5) as a linear function of Vm and Km- In Equ.(5), the instantaneous reaction time at the moment for Si is preset as zero so that there is no treatment of flag. When the signal for Si is not treated as a nonlinear parameter, kinetic analysis of reaction curve by fitting with Equ.(5) can be finished within 1 s with a pocket calculator. However, parameters estimated with Equ.(5) always have so large errors that Equ.(5) is scarcely practiced in biomedical analyses. Hence, the proper form of an integrated rate equation after validating should be selected carefully. 

An easier way consists of presenting the integrated rate equation in such a form that a linear plot can be obtained. For example, in the case of a first-order reaction, a plot of log (along y-axis) against t (along... 

A value of q is assumed and values of k are calculated for each data point. The correct value of q has been chosen when the values of/c are nearly constant or show no drift. This procedure is applicable for a rate equation of any complexity if it can be integrated. Eqs. (7-28) and (7-29) can also be put into linear form ... 

The linear form of the integrated first-order rate equation... 

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...

Figure 8.11 Graph plotted with data from Figure 8.10, plotted with the axes of the linear form of the integrated first-order rate equation, with ln[A] as y against time t as V...

The integrated form of this equation yields the familiar linear first-order rate equation in which In stands for the logarithm to the base e, and the subscripts o and t refer to the initial value of DP and to the value at any time, t, respectively ... 

Ideally, integration of equation 5.9 should reproduce the TG curve of a given sample. Integration can be performed easily if a hyperbolic temperature programme of the form IT=r-st is used, where r and s are constants. For a linear heating programme with constant heating rate, (j), equation 5.9 becomes... 

At this point, we can see that (1) if a plot of [A] versus time is Unear, the reaction is zero order and (2) if a plot of ln[A] versus time is linear, the reaction is first order. From the form of the integrated rate law equations for these cases, you should also be able to see that the slope of the linear plot must be equal to -k. So we can find both the rate constant and the reaction order from our graph. Next we will examine the correct model for second-order kinetics. 

When the complexity of the mechanism is increased to a two step reaction, then the solutions of the Laplace forms of the equations involve the two roots of a quadratic as indicated above for second order differential equations. Recently Zhang, Strand White (1989) have suggested how a general matrix solution of rate equations in the Laplace form can be used to model kinetic mechanisms. Zhang et al. (1989) suggest this method as an alternative to numerical integration, but its use is, of course, restricted to linear equations like that of the more elegant matrix method described in section 4.2. 

Rate equations, and the integrated forms of rate equations, can be tested visually against experimental data. The rate equation is first linearized, and the data are then plotted in a straight-line form, as suggested by the linearized rate equation. 

In equation (13.11), the first term corresponds to the catalysed part of the reaction and the remaining terms, which make a relatively small contribution, apply to the uncatalysed part. Kinetic data at constant acidity were in good agreement with the integrated form of the calculated rate expression. The rate coefficients k2,k, k, and the ratio k. jk were evaluated. Almost linear plots of log 2 versus log [ ] were obtained at four temperatures with slopes close to —1.8. This result suggests that the dominant activated complex is that formed by loss of two ions, viz. 

If the rate law (for ( rA)) is such that the integral can be evaluated analytically, then it is only necessary to make measurements <°f cA or /A) at the inlet and outlet, Sin and Sout, respectively, of the reactor. Thus, if the rate law is given by equation 3.4-1, integration of the right side of equation 2.4-4b results in an expression of the form g(JA)lkA, where g(/A) is in terms of the order n, values of which can be assumed by trial, and kA is unknown. The left side of equation 2.4-4b for a given reactor (V) can be varied by changing Fao> and g(fA) is a linear function of V/FAo with slope kA, if the correct value of n is used. 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

First-Order Kinetics, K[A] Unimolecular processes, such as ligand dissociation from a metal center or a simple homolytic or heterolytic cleavage of a single bond, provide a straightforward example of a first-order reaction. The kinetics of this simple scheme, Equation 8.5, is described by a first-order rate law, Equation 8.6, where A stands for reactants, P for products, [A]0 for initial concentration of A, and t for time. The integrated form is shown in Equation 8.7 and a linearized version in Equation 8.8. 

Integrated forms of the simple mass-action rate functions produce linear equations that are easily tested by graphical methods. The advantage of piu ameter optimization methods is that the computer programs can be writ-Icn lo generate statistics for a more quantitative estimation of goodness-of-I ii rather than the visual estimation that graphical methods provide. 

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