Method of initial rates

We note that the reaction order is the same as that in Examples 5-1 and 5-2 however. the value of k is about 8% larger. 

Model Discrimination. One can also determine which model or equation best fits the experimental data by comparing the sums of the squares for each model and then choosing the equation with a smaller sum of squares and/or carrying out an F-test. Alternatively, we can compare the residual plots for each model. The.se plots show the error associated with each data point, and one looks to see if the error is randomly distributed or if there is a trend in the error. When the error is randomly distributed, this is an additional indication that the correct rate law has been chosen. An example of model discrimination using nonlinear regression is given on the CD-ROM in Chapter 10 of the Sum-marv Notes. 

The use of the differential method of data analysis to determine reaction orders and specific reaction rates is clearly one of the easiest, since it requires only one experiment. However, other effects, such as the presence of a significant reverse reaction, could render the differential method ineffective. In these ca.ses, the method of initial rates could be used to determine the reaction order and the specific rate constant. Here, a series of experiments is carried out at different initial concentrations, C o. and the initial rate of reaction, is determined for each run. The initial rate. Tao h found by differentiating the data and extrapolating to zero time. For example, in the Trityl-Methanol reaction shown in Example 5-1, the initial rate was found to be 0.00028 mol/dm min. By various plotting or numerical analysis techniques relating —to Cao, we can obtain the appropriate rate law. If the rate law is in the form 

An importanl leaciion for enhancement of oil flow in carbonate reservoirs 

Example 5— Method of Initial Rates in Soiid Liquid Dissolution Kinetics 

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Pseudo-Order Reactions and the Method of Initial Rates Unfortunately, most reactions of importance in analytical chemistry do not follow these simple first-order and second-order rate laws. We are more likely to encounter the second-order rate law given in equation A5.11 than that in equation A5.10. 

Once v, is determined under one set of conditions, the procedure is then repeated, varying the concentrations of reactant, catalyst, buffer, etc. The resulting family of v, values can be used to formulate the rate law. This desirable method is probably deserving of wider use in general chemical reactions, just as it is used in biochemical reactions. The method of initial rates is, however, not without its problems. For one thing, the accurate determination of product in the presence of so much substrate is not always feasible. For another, this approach may conceal important effects that come into play only later in the course of the reaction. If the method of initial rates is used, separate experiments must be performed to check these points. 

Since X + In X is a transcendental function, Eq. (2-67) cannot be solved for [A], Two methods are usually used. The method of initial rates is the more common one, since it converts the differential equation into an algebraic one. Values of v(, determined as a function of [A]o, are fit to the equation given for v. This application to enzyme-catalyzed reactions will be taken up in Chapter 4. The other method regularly used relies on numerical integration these techniques are given in Chapter 5. 

This peculiar form applies when a dimeric molecule dissociates to a reactive monomer that then undergoes a first-order or pseudo-first-order reaction. This scheme is considered in Section 4.3. Unless one can work at either of the limits, the form is such that a numerical solution or the method of initial rates will be needed, since the integrated equation has no solution for [A]r. 

Initial rate method. Consider a first-order reaction studied by the method of initial rates. If AY/At represents the slope of the initial linear portion of the recording of Y against time, show that the rate constant is given by... 

The study of reactions with rates that He outside the time frame of ordinary laboratory operations requires specialized instrumentation and techniques. This chapter presents the wide range of methods currently in use for very fast reactions. Extraordinarily slow reactions, on the other hand, have received very little attention. For them, one may resort to measuring a tiny concentration of product over normal times, as in the method of initial rates. 

Method of initial rates (see Initial rates, method of)... 

Equations and describe how concentration changes with time when only a single reactant is involved. However, most reactions involve concentration changes for more than one species. Although it is possible to develop equations relating concentration and time for such reactions, such equations are more complicated and more difficult to interpret than the equations that involve just one reactant. Fortunately, it is often possible to simplify the experimental behavior of a reaction. We describe two experimental methods that accomplish this, the isolation method and the method of initial rates. 

A second way to simplify the behavior of a reaction is the method of initial rates. In this method, we measure the rate at the very beginning of the reaction for different concentrations. A set of experiments is done, changing only one initial concentration each time. Instead of measuring the concentration at many different times during the reaction, we make just one measurement for each set of concentrations. The reaction orders can be evaluated from the relationships between the changes in concentration and the changes in initial rates. We illustrate how this works using a gas-phase reaction of H2 with NO 2H2(g) -b 2NO(g) N2(g) + 2H2 0(g)... 

Example provides practice in applying the method of initial rates. 

The kinetics results of the batch reactor runs lead to the following qualitative observations At low CO pressures (less than about 1 atm) the catalysis appears to be first order in ruthenium over the range 0.018 M to 0.072 M and also in Pco as illustrated by the log Pco vs time plots of Fig. 2 and also shown by the method of initial rates. Changes in the sulfuric acid and water concentrations over the respective ranges 0.25 M to 2.0 M and 4 M to 12 M have relatively small effects on the catalysis rates, although the functionalities are complicated and show concave rate vs concentration curves with maximum rates... 

This graphical method of determining a rate constant, illustrated in Worked Example 12.6, is an alternative to the method of initial rates used in Worked Example 12.3. It should be emphasized, however, that a plot of ln [A] versus time will give a straight line only if the reaction is first order in A. Indeed, a good way of testing whether a reaction is first order is to examine the appearance of such a plot. 

Note that the slope is negative, k is positive, and the value of k agrees with the value obtained earlier in Worked Example 12.3 by the method of initial rates. 

The simplest way to determine a rate law is the method of initial rates. If a reaction is slow enough, it can be allowed to proceed for a short time, At, and the change in a reactant or product concentration measured. Repeating the experiment for different concentrations, the concentration-dependence of the rate can be deduced. 

FIGURE 5.9 Method of initial rate measurement for determining the order of reaction. 

A kinetic study has been reported recently for the nitrosation of many symmetrical tertiary amines in aqueous acid-acetate buffers (Gowenlock et al., 1979). One experimental difficulty in tertiary amine nitrosations generally is that the reactions are much slower than for the analogous secondary amines, and are more conveniently studied at a higher temperature, typically 75°C, where the decomposition of nitrous acid is quite rapid. In this study, rate constants were obtained from the less accurate method of initial rate measurements. Nevertheless, for acidities less than pH 3.1, rate eqn (16) was established. The acid dependence is complicated by the protonation of the... 

The most common method for directly determining the form of the differential rate law for a reaction is the method of initial rates. The initial rate of a reaction is the instantaneous rate determined just after the reaction begins (just after t = 0). The idea is to determine the instantaneous rate before the initial concentrations of reactants have changed significantly. Several experiments are carried out using different initial concentrations, and the initial rate is determined for each run. The results are then compared to see how the initial rate depends on the initial concentrations. This procedure allows the form of the rate law to be determined. We will illustrate the method of initial rates by using the following reaction ... 

The most common method for experimentally determining the differential rate law is the method of initial rates. In this method several experiments are run at different initial concentrations, and the instantaneous rates are determined for each at the same value of t as close to t = 0 as possible. The point is to evaluate the rate before the concentrations change significantly from the initial values. From a comparison of the initial rates and the initial concentrations, the dependence of the rate on the concentrations of various reactants can be obtained—that is, the order in each reactant can be determined. 

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