Chemical rate law

The rate of a chemical reaction is a function of the temperature, the composition of the reacting mixture, and, if present, the catalyst. The relationship between the reaction rate and these parameters is commonly called the rate expression or, sometimes, the rate law. Chemical kinetics is the branch of chemistry that deals with reaction mechanisms and provides a theoretical basis for the rate expression. When such information is available, we use it to obtain the rate expression. In many instances, the reaction rate expression is not available and should be determined experimentally. 

It is not always possible to solve kinetic problems using simple approaches. Even if the mechanism only involves a few steps the data may not readily suggest the coupled rate laws. Chemical intuition is required to propose a plausible mechanism which then must be tested. The natural procedure is to integrate the mechanistic rate equations and to compare the predicted concentrations with the observed ones. If a set of kinetic parameters can be found that satisfactorily accounts for the data, the mechanism is a possible one. Such calculations always involve extensive numerical work. Thus reasonable starting points are needed, even if modern high-speed computers are used. 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

For a closed chemical system witli a mass action rate law satisfying detailed balance tliese kinetic equations have a unique stable (tliennodynamic) equilibrium, In general, however, we shall be concerned witli... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Because of tire underlying dissipative nature of tire chemical systems tliat tire ODEs (C3.6.2) represent, tliey have anotlier important property any volume in F will shrink as it evolves. For a given set of initial chemical concentrations tire time evolution under tire chemical rate law will approach arbitrarily close to some final set of points in... 

For tliis model tire parameter set p consists of tire rate constants and tire constant pool chemical concentrations l A, 1 (Most chemical rate laws are constmcted phenomenologically and often have cubic or otlier nonlinearities and irreversible steps. Such rate laws are reductions of tire full underlying reaction mechanism.)... 

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... 

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. 

In this section we review the application of kinetics to several simple chemical reactions, focusing on how the integrated form of the rate law can be used to determine reaction orders. In addition, we consider how rate laws for more complex systems can be determined. 

Since the rate of a chemical reaction only depends on the slowest, or rate-determining step, and any preceding steps, species B will not show up in the rate law. 

Since this is the first occasion we have had to examine the rates at which chemical reactions occur, a few remarks about mechanistic steps and rate laws seem appropriate. The reader who feels the need for additional information on this topic should consult the discussions which will be found in any physical chemistry text. 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

A mechanism is a series of simple reaction steps which, when added together, account for the overall reaction. The rate law for the individual steps of the mechanism may be written by inspection of the mechanistic steps. The coefficients of the reactants in the chemical equation describing the step become the exponents of these concentrations in the rate law for... 

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.]...

Intrinsic Kinetics. Chemisorption may be regarded as a chemical reaction between the sorbate and the soHd surface, and, as such, it is an activated process for which the rate constant (/ ) follows the familiar Arrhenius rate law ... 

The Rate Law The goal of chemical kinetic measurements for weU-stirred mixtures is to vaUdate a particular functional form of the rate law and determine numerical values for one or more rate constants that appear in the rate law. Frequendy, reactant concentrations appear raised to some power. Equation 5 is a rate law, or rate equation, in differential form. 

Results may be reported for any component. The functional form of the rate law and the exponents x,j, w,... are not affected by such an arbitrary choice. The rate constants, however, may change in numerical value. Similarly, the stoichiometric chemical equation may be written in alternative but equivalent forms. This also affects, at most, the numerical value of rate constants. Consequentiy, one must know the chemical equation assumed before using any rate constant. 

Mechanisms. Mechanism is a technical term, referring to a detailed, microscopic description of a chemical transformation. Although it falls far short of a complete dynamical description of a reaction at the atomic level, a mechanism has been the most information available. In particular, a mechanism for a reaction is sufficient to predict the macroscopic rate law of the reaction. This deductive process is vaUd only in one direction, ie, an unlimited number of mechanisms are consistent with any measured rate law. A successful kinetic study, therefore, postulates a mechanism, derives the rate law, and demonstrates that the rate law is sufficient to explain experimental data over some range of conditions. New data may be discovered later that prove inconsistent with the assumed rate law and require that a new mechanism be postulated. Mechanisms state, in particular, what molecules actually react in an elementary step and what products these produce. An overall chemical equation may involve a variety of intermediates, and the mechanism specifies those intermediates. For the overall equation... 

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. 

The rate of a chemical reaction is always taken as a positive quantity, and the rate constant k is always positive as well. A negative rate constant is thus without meaning. An equation such as Eq. (1-4), which gives the reaction rate as a function of concentration, usually at constant temperature, is referred to as a rate law. The determination of the form in which the different concentrations enter into the rate law is one of the initial goals of a kinetic study, since it allows one to infer certain features of the mechanism. 

Presto, a third-order rate law This multiplication should not be taken as representing a chemical event or as carrying such implications it is only a valid mathematical manipulation. Other similar transformations can be given,2 as when one multiplies by another factor of unity derived from the acid ionization equilibrium of HOC1. (The reader may show that this gives a second-order rate law.) These considerations illustrate that it is the rate law and not the reaction itself that has associated with it a unique order. 

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. 

Rather than the use of instantaneous or initial rates, the more usual procedure in chemical kinetics is to measure one or more concentrations over the timed course of the reaction. It is the analysis of the concentrations themselves, and not the rates, that provides the customary treatments. The concentration-time data are fitted to an integrated form of the rate law. These methods are the subjects of Chapters 2, 3, and 4. 

Each of these variables will be considered in this book. We start with concentrations, because they determine the form of the rate law when other variables are held constant. The concentration dependences reveal possibilities for the reaction scheme the sequence of elementary reactions showing the progression of steps and intermediates. Some authors, particularly biochemists, term this a kinetic mechanism, as distinct from the chemical mechanism. The latter describes the stereochemistry, electron flow (commonly represented by curved arrows on the Lewis structure), etc. 

To this point we have focused on reactions with rates that depend upon one concentration only. They may or may not be elementary reactions indeed, we have seen reactions that have a simple rate law but a complex mechanism. The form of the rate law, not the complexity of the mechanism, is the key issue for the analysis of the concentration-time curves. We turn now to the consideration of rate laws with additional complications. Most of them describe more complicated reactions and we can anticipate the finding that most real chemical reactions are composites, composed of two or more elementary reactions. Three classifications of composite reactions can be recognized (1) reversible or opposing reactions that attain an equilibrium (2) parallel reactions that produce either the same or different products from one or several reactants and (3) consecutive, multistep processes that involve intermediates. In this chapter we shall consider the first two. Chapter 4 treats the third. 

The route from kinetic data to reaction mechanism entails several steps. The first step is to convert the concentration-time measurements to a differential rate equation that gives the rate as a function of one or more concentrations. Chapters 2 through 4 have dealt with this aspect of the problem. Once the concentration dependences are defined, one interprets the rate law to reveal the family of reactions that constitute the reaction scheme. This is the subject of this chapter. Finally, one seeks a chemical interpretation of the steps in the scheme, to understand each contributing step in as much detail as possible. The effects of the solvent and other constituents (Chapter 9) the effects of substituents, isotopic substitution, and others (Chapter 10) and the effects of pressure and temperature (Chapter 7) all aid in the resolution. 

The transition state composition is, moreover, uncertain as regards the number of solvent molecules it contains. Thus, we might write for Eq. (6-1), carried out in water, a composition of [Fe(CN)5N2H4C>33 nH20]t. The number of water molecules is of more than academic interest here, in that the net reaction extrudes one molecule of water. It has been suggested (from chemical evidence,1 not as a deduction from the rate law) that water is eliminated first, and that the reaction may occur by way of the following transition state (or intermediate) ... 

Base catalysis. Develop chemical equations and rate laws that illustrate specific and general base catalysis. Equations (10-37)—(10-44) can be used as a model. 

Chain reactions, 181 branching, 189 initiation step, 182 propagation steps, 182 rate laws for, 188 termination step, 182 well-behaved, 187 Chemical mechanism, 9 Chemical relaxation, 255-260 Coalescence temperature, 262 Col, 170... 

The order of a reaction cannot in general be predicted from the chemical equation a rate law is an empirical law ... 

The rate law for a reaction is experimentally determined and cannot in general be inferred from the chemical equation for the reaction. 

We stressed in Section 13.3 that we cannot in general write a rate law from a chemical equation. The reason is that all but the simplest reactions are the outcome of several, and sometimes many, steps called elementary reactions. Each elementary reaction describes a distinct event, often a collision of particles. To understand how a reaction takes place, we have to propose a reaction mechanism, a sequence of elementary reactions describing the changes that we believe take place as reactants are transformed into products. 

To construct an overall rate law from a mechanism, write the rate law for each of the elementary reactions that have been proposed then combine them into an overall rate law. First, it is important to realize that the chemical equation for an elementary reaction is different from the balanced chemical equation for the overall reaction. The overall chemical equation gives the overall stoichiometry of the reaction, but tells us nothing about how the reaction occurs and so we must find the rate law experimentally. In contrast, an elementary step shows explicitly which particles and how many of each we propose come together in that step of the reaction. Because the elementary reaction shows how the reaction occurs, the rate of that step depends on the concentrations of those particles. Therefore, we can write the rate law for an elementary reaction (but not for the overall reaction) from its chemical equation, with each exponent in the rate law being the same as the number of particles of a given type participating in the reaction, as summarized in Table 13.3. 

Rate laws and rate constants are windows on to the molecular processes of chemical change. We have seen how rate laws reveal details of the mechanisms of reactions here, we build models of the molecular processes that account for the values of the rate constants that appear in the rate laws. 

A rate law summarizes the dependence of the rate on concentrations. However, rates also depend on temperature. The qualitative observation is that most reactions go faster as the temperature is raised (Fig. 13.22). An increase of 10°C from room temperature typically doubles the rate of reaction of organic species in solution. That is one reason why we cook foods heating accelerates reactions that lead to the breakdown of cell walls and the decomposition of proteins. We refrigerate foods to slow down the natural chemical reactions that lead to their decomposition. 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

As in a unimolecular chemical reaction, the rate law for nuclear decay is first order. That is, the relation between the rate of decay and the number N of radioactive nuclei present is given by the law of radioactive decay ... 

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