Examples of Rate Expressions

The normal course of a kinetic investigation involves the posmlation of likely mechanisms and comparison of the observed rate expression with those expected for the various mechanisms. Those mechanisms that are incompatible with the observed kinetics can be eliminated as possibilities. One common kind of reaction involves proton transfer occurring as a rapid equilibrium preceding the rate-determining step, for example, in the reaction of an alcohol with hydrobromic acid to give an alkyl bromide  

The overall rate being measured is that of Step 2, but there may be no means of directly measuring [ROH2]. The concentration of the protonated intermediate ROH2 can be expressed in terms of the concentration of the starting material by taking into consideration the equilibrium constant, which relates [ROH], [Br ], and [H+]  

To illustrate the development of a kinetic expression from a postulated reaction mechanism, let us consider the base-catalyzed reaction of benzaldehyde and acetophenone. Based on general knowledge of base-catalyzed reactions of carbonyl compounds, a reasonable sequence of steps can be written, but the relative rates of the steps is an open question. Furthermore, it is known that reactions of this type 

PhCHCH2CPh + OC2H5 PhCH=CHCPh + OH + C2H5OH dehydration 

Because proton transfer reactions between oxygen atoms are usually very fast. Step 3 can be assumed to be a rapid equilibrium. With the above mechanism assumed, let us examine the rate expression that would result, depending upon which of the steps is rate determining. If Step 1 is rate controlling the rate expression would be 

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The dependence of the rate of reaction on the concentrations can often be expressed as a direct proportionality in which the concentrations may appear to the zero, first, or second power. The power to which the concentration of a substance appears in the rate expression (formula) is the order of the reaction with respect to that substance. Table 20-1 provides examples of rate expressions. 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

In the case of the hexacarbonyls, the rate-expression contains not only the same type of first-order term but in addition one second-order overall. For good entering groups (but not CO, for example) the rate expression contains a term strictly first-order in both the complex and the entering nucleophile. The first-order rates of CO exchange are practically identical with the rates of substitution in hydrocarbon solvents, but there is nevertheless some acceleration in ether (THF, dioxan) solutions. This solvent-dependence is not so well-marked ° as in the case of nickel tetracarbonyl. The second-order rate of substitution very strongly depends upon the basicity of the entering nucleophile... 

The random error arises from the measurement of y the true value of which is not known. The measurements are assumed to be free of systematic errors. The modeling equations contain adjustable parameters to account for the fact that the models are phenomenological. For example, kinetic rate expressions contain rate constants (parameters) the value of which is unknown and not possible to be obtained from fundamental principles. 

Examples of derivation of rate expressions Rj — R,( T, ck, 6m x) are presented in the following sections, depending on the amount of knowledge on the catalytic chemistry, either empirical or mechanistically consistent forms can be adopted. 

Quite often it is found that one of the above-mentioned steps is much slower than the others and therefore controls the rate of the catalytic reaction. As an example, the rate expression for the bimolecular reaction... 

In some cases it may be desirable to use a series of stirred-.tank reactors, with the exit stream from the first serving as the feed to the second, and so on. For constant density the exit concentration or conversion can be solved by consecutive application of Eq. (4-6) to each reactor. MacDonald and Piret have derived solutions for a number of rate expressions and for systems of reversible, consecutive, and simultaneous reactions. Graphical procedures have also been developed. The kinds of calculations involved are illustrated for the simple case of a first-order reaction in Example 4-9. 

Reversible exothermic reactions have an ultimate decrease of rate with temperature, and so the heat generation curve turns down (as in Example 10.4.a-l) however, the qualitative features remain the same. The heat generation curve for complex reactions can have more than one hump, and thus more than three steady states are possible for a given operating condition. The humps also tend to be smaller, leading to more readily obtained transitions between steady states, and so on—Westerterp [30]. Also, other types of multiple steady states and instabilities can occur. For example, with certain forms of rate expressions highly nonlinear in concentration, just the mass balance Eq. 10.4.a-i may have more than one solution. This is summarized in Perlmutter [31] (as well as many other techniques). 

Because an elementary reaction occurs on a molecular level exactly as it is written, its rate expression can be determined by inspection. A unimolecular reaction is a first-order process, bimolecular reactions are second-order, and termolecular processes are third-order. However, the converse statement is not true. Second-order rate expressions are not necessarily the result of an elementary bimolecular reaction. While a stoichiometric chemical equation remains valid when multiplied by an arbitrary factor, a mechanistic eqnation loses its meaning when multiplied by an arbitrary factor. Whereas stoichiometric coefficients and reaction orders may be integers or nonintegers, the molecularity of a reaction is always an integer. The following examples indicate the types of rate expressions associated with various molecularities. Unimolecular ... 

THE CHANGE OF CONCENTRATION WITH TIME We learn that rate equations can be written to express how concentrations change with time and look at several examples of rate equations zero-order, first-order, and second-order reactions. 

This example demonstrates how multiple CSTR steady states may arise in nonisothermal systems, even when the associated kinetics are simple, and temperature is assumed to be linear in terms of concentration. The inherent nonlinear nature of rate expressions in general thus often leads to complex behavior even when the energy balance is of a simple form. Multiple steady states must be included in the AR in order to understand the true bounds of achievability. Omission of these states may have important implications on subsequent optimizations, such as if we wish to maximize the concentration of component B. 

This formula can be transformed to the fatality rate. An example of this expression is fatalaties exposure, accidents, fatalaties... 

The goal of a kinetic study is to establish the quantitative relationship between the concentration of reactants and catalysts and the rate of reaction. Typically, such a study involves rate measurement 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 reactant species that are involved in the rate-determining step or a step prior to the rate-determining step. Each concentration term has an exponent that is equal to the order of the reaction with respect to that component. The overall kinetic order of the reaction is equal to the sum of all the exponents in the rate expression. Several examples of rate laws that illustrate the variety observed are presented in Scheme 4.1. Some are relatively simple others are more complex. 

Most reported PrOx kinetic studies have focused on the CO oxidation [25-27] and neglected the influence of r-WGS reaction [28]. However, the incorporation of rate expressions of the coupled H2 oxidation and r-WGS reactions is necessary for accurate representation of PrOx reaction behavior Despite the importance of the evaluation of all three expressions, only a few groups have addressed kinetic expressions for all three PrOx reactions [29]. As an example, based on the observation of characteristic PrOx behavior and other PrOx kinetic studies in the literature [29], the kinetic expressions for PrOx on Pt/Al203 are as shown in Equations 27.4-27.9. 

When the electrode acts as a source or sink for electrons with the reactant sitting in the outer Helmholtz plane (see Section 1.4.4), reaction rate is not usually dependent on the nature of the electrode material. For some reactions to proceed, however, the reactant or an intermediate must be adsorbed on the electrode surface. Then the nature of the electrode material often has a significant effect on the reaction rate by virtue of its ability or lack of ability to provide the necessary sites. We shall first discuss briefly the phenomenon of adsorption and then derive an example of kinetic expressions when adsorption is taken into account. 

See, for example, Poliak E 1986 Theory of activated rate processes a new derivation of Kramers expression J. Chem. Phys. 85 865... 

Fast transient studies are largely focused on elementary kinetic processes in atoms and molecules, i.e., on unimolecular and bimolecular reactions with first and second order kinetics, respectively (although confonnational heterogeneity in macromolecules may lead to the observation of more complicated unimolecular kinetics). Examples of fast thennally activated unimolecular processes include dissociation reactions in molecules as simple as diatomics, and isomerization and tautomerization reactions in polyatomic molecules. A very rough estimate of the minimum time scale required for an elementary unimolecular reaction may be obtained from the Arrhenius expression for the reaction rate constant, k = A. The quantity /cg T//i from transition state theory provides... 

The algebraic form of the expression (9.24) for the enhancement factor is specific to the particular reaction rate expression we have considered, and corresponding results can easily be obtained for other reactions in binary mixtures, for example the irreversible cracking A—2B. ... 

The rate of a reaction r is dependent on the reactant concentrations. For example, a bimolecular reaction between the reactants B and C could have a rate expression, such as... 

The selectivity relationship merely expresses the proportionality between intermolecular and intramolecular selectivities in electrophilic substitution, and it is not surprising that these quantities should be related. There are examples of related reactions in which connections between selectivity and reactivity have been demonstrated. For example, the ratio of the rates of reaction with the azide anion and water of the triphenylmethyl, diphenylmethyl and tert-butyl carbonium ions were 2-8x10 , 2-4x10 and 3-9 respectively the selectivities of the ions decrease as the reactivities increase. The existence, under very restricted and closely related conditions, of a relationship between reactivity and selectivity in the reactions mentioned above, does not permit the assumption that a similar relationship holds over the wide range of different electrophilic aromatic substitutions. In these substitution reactions a difficulty arises in defining the concept of reactivity it is not sufficient to assume that the reactivity of an electrophile is related... 

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