Kinetic studies rate expression

Techniques. Experimentally, several techniques are important tools for the study of photoimaging parameters. Though solution techniques common to photomechanistic studies can be applied in some instances, they must be used with care in photopolymer systems. The kinetic and rate expressions just described are only valid in model systems in which homopolymerization processes are the only ones which occur. Kinetic complications can result if crosslinking processes are important. Network formation is common, and represents a further complication. In practice, conversion must be kept to a low level in order to prevent depletion of initiator or monomer below acceptable levels. 

The goal of a kinetic study is to establish the quantitative relationship between the concentration of reactants and catalysts and the rate of the reaction. Typically, such a study involves rate measurements at enough different concentrations of each reactant so that the kinetic order with respect to each reactant can be assessed. A complete investigation allows the reaction to be described by a rate law, which is an algebraic expression containing one or more rate constants as well as the concentrations of all reactants that are involved in the rate-determining step and steps prior to the rate-determining step. Each concentration has an exponent, which is the order of the reaction with respect to that component. The overall kinetic order of the reaction is the sum of all the exponents in the... 

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. 

A detailed study of the solvolysis of L has suggested the following mechanism, with the reactivity of the intermediate M being comparable to that of L. Evidence for the existence of steps ki and k 2 was obtained fiom isotopic scrambling in the sulfonate M when it was separately solvolyzed and by detailed kinetic analysis. Derive a rate expression which correctly describes the non-first-order kinetics for the solvolysis of L. 

The order of reactivity of the hydrogen halides is HI > HBr > HCl, and reactions of simple alkenes with HCl are quite slow. The studies that have been applied to determining mechanistic details of hydrogen halide addition to alkenes have focused on the kinetics and stereochemistry of the reaction and on the effect of added nucleophiles. The kinetic studies often reveal complex rate expressions which demonstrate that more than one process contributes to the overall reaction rate. For addition of hydrogen bromide or Itydrogen... 

Structure-reactivity relationships can be probed by measurements of rates and equiUbria, as was diseussed in Chapter 4. Direct comparison of reaction rates is used relatively less often in the study of radical reactions than for heterolytic reactions. Instead, competition methods have frequently been used. The basis of competition methods lies in the rate expression for a reaction, and the results can be just as valid a comparison of relative reactivity as directly measured rates, provided the two competing processes are of the same kinetic order. Suppose that it is desired to compare the reactivity of two related compounds, B—X and B—Y, in a hypothetical sequence ... 

The slope of a concentration-time curve to define the rate expression can be determined. However, experimental studies have shown the reaction cannot be described by simple kinetics, but by the relationship ... 

This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using matliematical expressions show how physical... 

From a kinetic study [1281] of the reaction NH2COSNH4 = 2 NH3 + COS the specific rate expression 2k... 

The oxidation of CO on Pt is one of the best studied catalytic systems. It proceeds via the reaction of chemisorbed CO and O. Despite its complexities, which include island formation, surface reconstruction and self-sustained oscillations, the reaction is a textbook example of a Langmuir-Hinshelwood mechanism the kinetics of which can be described qualitatively by a LHHW rate expression. This is shown in Figure 2.39 for the unpromoted Pt( 111) surface.112 For low Pco/po2 ratios the rate is first order in CO and negative order in 02, for high pco/po2 ratios the rate becomes negative order in CO and positive order in 02. Thus for low Pcc/po2 ratios the Pt(l 11) surface is covered predominantly by O, at high pco/po2 ratios the Pt surface is predominantly covered by CO. 

The kinetics and mechanism of this reaction have been studied for years on Pt films deposited on doped Zr02.15 It has been found that at temperatures above 280°C the open-circuit catalytic kinetics can be described quantitatively by the rate expression... 

Consecutive Reactions. The prototypical reaction is A B C, although reactions like Equation (6.2) can be treated in the same fashion. It may be that the first reaction is independent of the second. This is the normal case when the first reaction is irreversible and homogeneous (so that component B does not occupy an active site). A kinetic study can then measure the starting and final concentrations of component A (or of A and A2 as per Equation (6.2)), and these data can be used to fit the rate expression. The kinetics of the second reaction can be measured independently by reacting pure B. Thus, it may be possible to perform completely separate kinetic studies of the reactions in a consecutive sequence. The data are fit using two separate versions of Equation (7.8), one for each reaction. The data will be the experimental values of for one sum-of-squares and b ut for another. 

The reduction of iodine by Fe(II) is, of course, the reverse of the ferric ion-iodide ion reaction (p. 408) and it influences the kinetics of the latter. However, the direct reaction has been studied, the rate expression being ... 

In its application to specific kinetics studies this general procedure may take on a variety of forms that are minor modifications of that outlined above. One modification does not require an explicit assumption of the form of 0(Q) including numerical values of the orders of the reaction with respect to the various species, but merely an assumption that the rate expression is of the following form. 

This chapter treats the descriptions of the molecular events that lead to the kinetic phenomena that one observes in the laboratory. These events are referred to as the mechanism of the reaction. The chapter begins with definitions of the various terms that are basic to the concept of reaction mechanisms, indicates how elementary events may be combined to yield a description that is consistent with observed macroscopic phenomena, and discusses some of the techniques that may be used to elucidate the mechanism of a reaction. Finally, two basic molecular theories of chemical kinetics are discussed—the kinetic theory of gases and the transition state theory. The determination of a reaction mechanism is a much more complex problem than that of obtaining an accurate rate expression, and the well-educated chemical engineer should have a knowledge of and an appreciation for some of the techniques used in such studies. 

Thus mechanism B, which consists solely of bimolecular and unimolecular steps, is also consistent with the information that we have been given. This mechanism is somewhat simpler than the first in that it does not requite a ter-molecular step. This illustration points out that the fact that a mechanism gives rise to the experimentally observed rate expression is by no means an indication that the mechanism is a unique solution to the problem being studied. We may disqualify a mechanism from further consideration on the grounds that it is inconsistent with the observed kinetics, but consistency merely implies that we continue our search for other mechanisms that are consistent and attempt to use some of the techniques discussed in Section 4.1.5 to discriminate between the consistent mechanisms. It is also entirely possible that more than one mechanism may be applicable to a single overall reaction and that parallel paths for the reaction exist. Indeed, many catalysts are believed to function by opening up alternative routes for a reaction. In the case of parallel reaction paths each mechanism proceeds independently, but the vast majority of the reaction will occur via the fastest path. 

If equations 4.2.25 and 4.2.26 are substituted for equations 4.2.11 and 4.2.15, respectively, in the mechanism described above, the effect is to replace kx by k [M] and k5 by /c 5[M] everywhere that they appear. Since these quantities appear as a ratio in the final rate expression, the third body concentration will drop out and kjks) becomes identical with k /k 5 The necessity for the use of the third body concentration thus is not obvious in kinetic studies of the thermal reaction. However, from studies of photochemical reaction between hydrogen and bromine, there is strong evidence that the termination reaction is termolecular. This fact and... 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

In this subsection we have treated a variety of higher-order simple parallel reactions. Only by the proper choice of initial conditions is it possible to obtain closed form solutions for some of the types of reaction rate expressions one is likely to encounter in engineering practice. Consequently, in efforts to determine the kinetic parameters characteristic of such systems, one should carefully choose the experimental conditions so as to ensure that potential simplifications will actually occur. These simplifications may arise from the use of stoichiometric ratios of reactants or from the degeneration of reaction orders arising from the use of a vast excess of one reactant. Such planning is particularly important in the early stages of the research when one has minimum knowledge of the system under study. 

Since most kinetic studies of this type involve initial rate experiments, it is usually necessary to rederive this expression in terms of the initial concentrations and the initial rate. From material balance considerations... 

A comparison of equations 7.3.43 and 7.3.38 shows that they are of the same mathematical form. Both can be written in terms of four measurable kinetic constants in the manner of equation 7.3.40. Only the relationship between the kinetic constants and the individual rate constants differs. Thus, no distinction can be made between the two mechanisms using steady-state rate studies. In general, the introduction of unimolecular steps involving only isomerization between unstable intermediate complexes does not change the form of the rate expression. 

In the design of an industrial scale reactor for a new process, or an old one that employs a new catalyst, it is common practice to carry out both bench and pilot plant studies before finalizing the design of the commercial scale reactor. The bench scale studies yield the best information about the intrinsic chemical kinetics and the associated rate expression. However, when taken alone, they force the chemical engineer to rely on standard empirical correlations and prediction methods in order to determine the possible influence of heat and mass transfer processes on the rates that will be observed in industrial scale equipment. The pilot scale studies can provide a test of the applicability of the correlations and an indication of potential limitations that physical processes may place on conversion rates. These pilot plant studies can provide extremely useful information on the temperature distribution in the reactor and on contacting patterns when... 

Equation 8.3.4 may also be used in the analysis of kinetic data taken in laboratory scale stirred tank reactors. One may directly determine the reaction rate from a knowledge of the reactor volume, flow rate through the reactor, and stream compositions. The fact that one may determine the rate directly and without integration makes stirred tank reactors particularly attractive for use in studies of reactions with complex rate expressions (e.g., enzymatic or heterogeneous catalytic reactions) or of systems in which multiple reactions take place. 

As noted in Chapter 16, transition state theory does not require that kinetic rate laws take a linear form, although most kinetic studies have assumed that they do. The rate law for reaction of a mineral A can be expressed in the general nonlinear form,... 

MnO, 2%MgO, and 5%A1203 with surface areas between 18.9 and 35.9 m2/g. Although in some kinetic experiments C02 was added, for practical purposes, the kinetics were studied in essentially the region of irreversibility. Therefore, as again the constant m was obtained to be unity, the rate expression reduced further to ... 

A general difficulty encountered in kinetic studies of outer-sphere electron-transfer processes concerns the separation of the precursor formation constant (K) and the electron-transfer rate constant (kKT) in the reactions outlined above. In the majority of cases, precursor formation is a diffusion controlled step, followed by rate-determining electron transfer. In the presence of an excess of Red, the rate expression is given by... 

Ashby and Yu have studied the kinetics of reduction of benzophenone with TIBA in ether and showed that the overall kinetic rate expression is second order, first order in TIBA and first order in ketone (151). The observed activation parameters were AG - 18.8 kcal/mol AH = 15.8 kcal/mol and AS = - 10.1 e.u. The negative entropy of activation is consistent with a cyclic transition state for the rate-determining hydride-transfer step. A Hammett study gave a value of p = 0.362, supporting nucleophilic attack by the aluminum alkyl on the carbonyl group in the rate-determining step. 

In biological systems, electron transfer kinetics are determined by many factors of different physical origin. This is especially true in the case of a bimolecular reaction, since the rate expression then involves the formation constant Kf of the transient bimolecular complex as well as the rate of the intracomplex transfer [4]. The elucidation of the factors that influence the value of Kf in redox reactions between two proteins, or between a protein and organic or inorganic complexes, has been the subject of many experimental studies, and some of them are presented in this volume. The complexation step is essential in ensuring specific recognition between physiological partners. However, it is not considered in the present chapter, which deals with the intramolecular or intracomplex steps which are the direct concern of electron transfer theories. 

First-order reactions are extremely common and form the bulk of reported kinetic studies. The rate of loss of the reactant A decreases as the concentration of A decreases. The differential form (1.18) leads to a number of equivalent integrated expressions, in addition to (1.19) and (1.19 a) ... 

---