Rate law examples

Calculate a concentration, time, or rate constant by using an integrated rate law (Examples 13.3, 13.4, and 13.5). 

Example CD7-1 Deducing the Rate Law Example CD7- Find a Mechanism Living Example Problems... 

A large number of both homogeneous and heterogeneous reactions do not follow simple rate laws. Examples of reactions that don t follow simple elementary rate laws are discussed below. 

Sec. 7.1 Active Intermediates and Nonelementary Rate Laws Example 7-2 FSSH Applied to Thermal Cracking of Ethane... 

Determining the rate law from a mechanism Given a mechanism with an initial slow step, obtain the rate law (EXAMPLE 14.11). Given a mechanism with an initial fast, equilibrium step, obtain the rate law (EXAMPLE 14.12). 

Students should be aware that in real cases the numerical treatment does not work out as perfectly as it did in the above example. Some level of mathematical sophistication, and the application of logarithms, maybe necessary to determine the orders in the rate law. Example 20.6, later in this chapter, shows one such case where use of logarithms is necessary to determine the order of the reaction. 

Various specific rate laws and mechanisms are described in the next section the present discussion is in more general terms. For example, why is it that a contact catalyst is able to serve as such, that is, why is it able to provide a reac-... 

Sequences such as the above allow the formulation of rate laws but do not reveal molecular details such as the nature of the transition states involved. Molecular orbital analyses can help, as in Ref. 270 it is expected, for example, that increased strength of the metal—CO bond means decreased C=0 bond strength, which should facilitate process XVIII-55. The complexity of the situation is indicated in Fig. XVIII-24, however, which shows catalytic activity to go through a maximum with increasing heat of chemisorption of CO. Temperature-programmed reaction studies show the presence of more than one kind of site [99,1(K),283], and ESDIAD data show both the location and the orientation of adsorbed CO (on Pt) to vary with coverage [284]. 

The definitions of the empirical rate laws given above do not exclude empirical rate laws of another fomi. Examples are reactions, where a reverse reaction is important, such as in the cis-trans isomerization of 1,2-dichloroethene ... 

The second-order rate law for bimolecular reactions is empirically well confinned. Figure A3.4.3 shows the example of methyl radical recombination (equation (A3.4.36)) in a graphical representation following equation (A3.4.38) [22, 23 and 24]. For this example the bimolecular rate constant is... 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

The Lindemaim mechanism for thennally activated imimolecular reactions is a simple example of a particular class of compound reaction mechanisms. They are mechanisms whose constituent reactions individually follow first-order rate laws [11, 20, 36, 48, 49, 50, 51, 52, 53, 54, 55 and 56] ... 

For example the hydrolysis of optically active 2 bromooctane in the absence of added base follows a first order rate law but the resulting 2 octanol is formed with 66% inversion of configuration... 

A second requirement is that the rate law for the chemical reaction must be known for the period in which measurements are made. In addition, the rate law should allow the kinetic parameters of interest, such as rate constants and concentrations, to be easily estimated. For example, the rate law for a reaction that is first order in the concentration of the analyte. A, is expressed as... 

Unfortunately, most reactions of analytical interest do not follow the simple rate laws shown in equations 13.1 and 13.2. Consider, for example, the following reaction between an analyte. A, and a reagent, R, to form a product, P... 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

Several important points about the rate law are shown in equation A5.4. First, the rate of a reaction may depend on the concentrations of both reactants and products, as well as the concentrations of species that do not appear in the reaction s overall stoichiometry. Species E in equation A5.4, for example, may represent a catalyst. Second, the reaction order for a given species is not necessarily the same as its stoichiometry in the chemical reaction. Reaction orders may be positive, negative, or zero and may take integer or noninteger values. Finally, the overall reaction order is the sum of the individual reaction orders. Thus, the overall reaction order for equation A5.4 isa-l-[3-l-y-l-5-l-8. 

Demonstrating that a reaction obeys the rate law in equation A5.11 is complicated by the lack of a simple integrated form of the rate law. The kinetics can be simplified, however, by carrying out the analysis under conditions in which the concentrations of all species but one are so large that their concentrations are effectively constant during the reaction. For example, if the concentration of B is selected such that [B] [A], then equation A5.11 simplifies to... 

A variation on the use of pseudo-ordered reactions is the initial rate method. In this approach to determining a reaction s rate law, a series of experiments is conducted in which the concentration of those species expected to affect the reaction s rate are changed one at a time. The initial rate of the reaction is determined for each set of conditions. Comparing the reaction s initial rate for two experiments in which the concentration of only a single species has been changed allows the reaction order for that species to be determined. The application of this method is outlined in the following example. 

The order of the rate law with respect to the three reactants can be determined by comparing the rates of two experiments in which the concentration of only one of the reactants is changed. For example, in experiment 2 the [H+] and the rate are approximately twice as large as in experiment 1, indicating that the reaction is first-order in [H+]. Working in the same manner, experiments 6 and 7 show that the reaction is also first-order with respect to [CaHeO], and experiments 6 and 8 show that the rate of the reaction is independent of the [I2]. Thus, the rate law is... 

The value of the rate constant can be determined by substituting the rate, the [C3H5O], and the [H+] for an experiment into the rate law and solving for k. Using the data from experiment 1, for example, gives a rate constant of 3.31 X 10 s h The average rate constant for the eight experiments is 3.49 X 10-5 M-i s-i ... 

Figure 6.1 Volume of nitrogen evolved from the decomposition of AIBN at 77°C plotted according to the first-order rate law as discussed in Example 6.1. [Reprinted with permission from L. M. Arnett, /. Am. Chem. Soc. 14 2021 (1952), copyright 1952 by the American Chemical Society.]...

There are eight different rate laws and rate constants associated with these reactions. Equation (7.1), for example, is replaced by Eqs. (7.5) and (7.6). 

Flooding and Pseudo-First-Order Conditions For an example, consider a reaction that is independent of product concentrations and has three reagents. If a large excess of [BJ and [CJ are used, and the disappearance of a lesser amount of A is measured, such flooding of the system with all components butM permits the rate law to be integrated with the assumption that all concentrations are constant except A. Consequentiy, simple expressions are derived for the time variation of A. Under flooding conditions and using equation 8, if x happens to be 1, the time-dependent concentration... 

Much of the language used for empirical rate laws can also be appHed to the differential equations associated with each step of a mechanism. Equation 23b is first order in each of I and C and second order overall. Equation 23a implies that one must consider both the forward reaction and the reverse reaction. The forward reaction is second order overall the reverse reaction is first order in [I. Additional language is used for mechanisms that should never be apphed to empirical rate laws. The second equation is said to describe a bimolecular mechanism. A bimolecular mechanism implies a second-order differential equation however, a second-order empirical rate law does not guarantee a bimolecular mechanism. A mechanism may be bimolecular in one component, for example 2A I. 

The normal course of a kinetic investigation involves postulating likely mechanisms and comparing the observed rate law with those expected for the various mechanisms. Those mechanisms that are incompatible with the observed kinetics can be eliminated as possibilities. Let us consider aromatic nitration by nitric acid in an inert solvent as a typical example. We will restrict the mechanisms being considered to the three shown below. In an actual case, such arbitrary restriction would not be imposed, but instead all mechanisms compatible with existing information would be considered. 

These examples illustrate the relationship between kinetic results and the determination of reaction mechanism. Kinetic results can exclude from consideration all mechanisms that require a rate law different from the observed one. It is often true, however, that related mechanisms give rise to identical predicted rate expressions. In this case, the mechanisms are kinetically equivalent, and a choice between them is not possible on the basis of kinetic data. A further limitation on the information that kinetic studies provide should also be recognized. Although the data can give the composition of the activated complex for the rate-determining step and preceding steps, it provides no information about the structure of the intermediate. Sometimes the structure can be inferred from related chemical experience, but it is never established by kinetic data alone. 

Consider the data of Hull and von Ronsenberg in Example 8-3 for mixing in a fluidized bed. Suppose the solids in the fluidized bed were not aeting as a eatalyst, but were aetually reaeting aeeording to a first order rate law (-r) = kC, k = 1.2 min Compare the aetual eonversion with that of an ideal plug flow. 

Section 1.9 showed that as long as an oxide layer remains adherent and continuous it can be expected to increase in thickness in conformity with one of a number of possible rate laws. This qualification of continuity is most important the direct access of oxidant to the metal by way of pores and cracks inevitably means an increase in oxidation rate, and often in a manner in which the lower rate is not regained. In common with other phase change reactions the volume of the solid phase alters during the course of oxidation it is the manner in which this change is accommodated which frequently determines whether the oxide will develop discontinuities. It is found, for example, that oxidation behaviour depends not only on time and temperature but also on specimen geometry, oxide strength and plasticity or even on specific environmental interactions such as volatilisation or dissolution. 

The models derived for continuous oxide layers remain valuable when porous oxides are formed they provide a frame of reference against which deviations may be examined and give a basis for understanding the factors governing the location of new oxide. In many cases, however, the experimentally derived rate laws no longer have a unique interpretation. For example, the linear rate law relating the thickness of oxide, x, to the time, t... 

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