The method of initial rates

In the method of initial rates, which is often used in conjunction with the isolation method, the instantaneous rate is measured at the beginning of the reaction for several different initial concentrations of reactants. If the initial rate doubles when the initial concentration of the isolated reactant is doubled, then the reaction is first order in that reactant, and so on. 

To use the data more fully we suppose that the rate law for a reaction with A isolated is y = k[[kY, then the initial rate of the reaction, Vq, is given by the initial concentration of A  

This equation has the form of the equation for a straight line y = intercept -I- slope x x 

An important point to note is that the method of initial rates might not reveal the entire rate law, for in a complex reaction we may not be able to specify an order with respect to a reactant (see eqn 6.7a) or the products themselves might affect the rate. 

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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. 

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. 

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. 

Relative reactivities were determined by the method of initial rates and normalized to 35 C. "FOr pairs of compounds as indicated for the last eight entries, relative to PhC CPh = I. 

As wi the method of initial rates, various numerical and graphical technique. can be used to determine the appropriate algebraic equation for the rate law. 

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 cases, 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 q, and the initial rate of reaction, is determined for each run. The initial rate, can be found by differentiating the data and extrapolating to zero time. For example, in the tfi-tert-butyl peroxide decomposition shown in Example 5-1, the initial rate was found to be... 

One common experimental method of evaluating reaction order is called the method of initial rates. The method of initial rates determines reaction order by comparing the initial rates of a reaction carried out with varying reactant concentrations. To understand how this method works, let s use the general reaction aA + bB products. Suppose that this reaction is carried out with varying concentrations of A and B and yields the initial reaction rates shown in Table 17-3. 

Given the following experimental data, use the method of initial rates to determine the rate law for the reaction aA + t>B —> products. Hint Any number to the zero power equals one. For example,... 

Nitrogen monoxide gas and chlorine gas react according to the equation 2NO + CI2 — 2NOC1. Use the following data to determine the rate law for the reaction by the method of initial rates. Also, calculate the value of the specific rate constant. 

Formulating Models Create a table of concentrations, starting with O.IOOM concentrations of all reactants, that you would propose in order to establish the rate law for the reaction aA + fcB + cD products using the method of initial rates. 

In what is called the method of initial rates, analysis is simpler at / = 0 because products that may affect reaction rates are not present. 

The method of initial rates is often used to determine the rate law of a reaction. 

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