Rate Expressions bimolecular

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... 

Mechanism I is a three-step process in which the first step is rate-determining. When the first step of a mechanism is rate-determining, the predicted rate law is the same as the rate expression for that first step. Here, the rate-determining step is a bimolecular collision. The rate expression for a bimolecular collision is first order in each collision partner Rate = j i[03 ][N0 j Mechanism I is consistent with the experimental rate law. If we add the elementary reactions, we find that it also gives the correct overall stoichiometry, so this mechanism meets all the requirements for a satisfactory one. 

Mechanism II begins with fast reversible ozone decomposition followed by a rate-determining bimolecular collision of an oxygen atom with a molecule of NO. The rate of the slow step is as follows Rate = 2[N0][0 This rate expression contains the concentration of an intermediate, atomic oxygen. To convert the rate expression into a form that can be compared with the experimental rate law, assume that the rate of the first step is equal to the rate of its reverse process. Then solve the equality for the concentration of the intermediate ... 

These should be simple unlmolecular or bimolecular reactions yielding a single or at most two reaction paths. Rates may therefore be fit using Langmuir-Hinshelwood (LH) rate expressions. For A —> products this should be... 

Since 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 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... 

This rate expression is consistent with the observed kinetics, so this combination of a slow termolecular step with a rapid bimolecular step is a plausible mechanism based on the information we have been given. 

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. 

This equation shows that the reaction rate is neither first-order nor second-order with respect to species A. However, there are two limiting cases. At high pressures where [A] is lar e, the bimolecular deactivation process is much more rapid than the unimolecular decomposition (i.e., /c2[A][A ] /c3[A ]). Under these conditions the second term in the denominator of equation 4.3.20 may be neglected to yield a first-order rate expression. 

For the case of bimolecular reaction between two adsorbed A molecules, similar arguments lead to the following rate expression. 

In initial rate studies no products need be present in the feed, and the terms in the rate expression involving the partial pressures of these species may be omitted under appropriate experimental conditions. The use of stoichiometric ratios of reactants may also cause a simplification of the rate expression. If one considers a reversible bimolecular surface reaction between species A and ,... 

Rate Expressions for Enzyme Catalyzed Single-Substrate Reactions. The vast majority of the reactions catalyzed by enzymes are believed to involve a series of bimolecular or unimolecular steps. The simplest type of enzymatic reaction involves only a single reactant or substrate. The substrate forms an unstable complex with the enzyme, which subsequently undergoes decomposition to release the product species or to regenerate the substrate. 

The k2 step in equation (63) is interesting because it is bimolecular. The resulting rate expression is equation (64). Substituting equation (65) into this and taking logs gives equation (66) ... 

An interesting question then arises as to why the dynamics of proton transfer for the benzophenone-i V, /V-dimethylaniline contact radical IP falls within the nonadiabatic regime while that for the napthol photoacids-carboxylic base pairs in water falls in the adiabatic regime given that both systems are intermolecular. For the benzophenone-A, A-dimethylaniline contact radical IP, the presumed structure of the complex is that of a 7t-stacked system that constrains the distance between the two heavy atoms involved in the proton transfer, C and O, to a distance of 3.3A (Scheme 2.10) [20]. Conversely, for the napthol photoacids-carboxylic base pairs no such constraints are imposed so that there can be close approach of the two heavy atoms. The distance associated with the crossover between nonadiabatic and adiabatic proton transfer has yet to be clearly defined and will be system specific. However, from model calculations, distances in excess of 2.5 A appear to lead to the realm of nonadiabatic proton transfer. Thus, a factor determining whether a bimolecular proton-transfer process falls within the adiabatic or nonadiabatic regimes lies in the rate expression Eq. (6) where 4>(R), the distribution function for molecular species with distance, and k(R), the rate constant as a function of distance, determine the mode of transfer. 

As mentioned earlier, practically all reactions are initiated by bimolecular collisions however, certain bimolecular reactions exhibit first-order kinetics. Whether a reaction is first- or second-order is particularly important in combustion because of the presence of large radicals that decompose into a stable species and a smaller radical (primarily the hydrogen atom). A prominent combustion example is the decay of a paraffinic radical to an olefin and an H atom. The order of such reactions, and hence the appropriate rate constant expression, can change with the pressure. Thus, the rate expression developed from one pressure and temperature range may not be applicable to another range. This question of order was first addressed by Lindemann [4], who proposed that first-order processes occur as a result of a two-step reaction sequence in which the reacting molecule is activated by collisional processes, after which the activated species decomposes to products. Similarly, the activated molecule could be deactivated by another collision before it decomposes. If A is considered the reactant molecule and M its nonreacting collision partner, the Lindemann scheme can be represented as follows ... 

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. 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

In first-order reactions, the rate expression depends upon the concentration of only one species, whereas second-order reactions show dependence upon two species, which may be the same or different. The molecularity, or number of reactant molecules involved in the rate-determining step, is usually equivalent to the kinetic reaction order, though there can be exceptions. For instance, a bimolecular reaction can appear to be first order if there is no apparent dependence on the concentration of one of the... 

Despite occasional apparent anomalies such as this, the rate expression gives us valuable information about the likely reaction mechanism. If the reaction is unimolecular, the rate-determining step involves just one species, whereas the rate-determining step involves two species if it is bimolecular. As indicated in Table 5.1, we can then deduce the probable reaction, and our proposed mechanism must reflect this information. The kinetic rate expressions will be considered further as we meet specific types of reaction. 

Again, the molecularity of a reaction is always an integer and only applies to elementary reactions. Such is not always the case for the order of a reaction. The distinction between molecularity and order can also be stated as follows molecularity is the theoretical description of an elementary process reaction order refers to the entire empirically derived rate expression (which is a set of elementary reactions) for the complete reaction. Usually a bimolecular reaction is second order however, the converse need not always be true. Thus, unimolecular, bimolecular, and termolecular reactions refer to elementary reactions involving one, two, or three entities that combine to form an activated complex. 

With the assumption that the reaction is irreversible, bimolecular, and of constant density, the rate expressions are given by... 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

Recently, Praharso et al also developed a Langmuir-Hinshelwood type of kinetic model for the SR kinetics of i-Cg over a Ni-based catalyst. In their model, it was assumed that both the hydrocarbon and steam dissociatively chemisorb on two different dual sites on the catalyst surface. The bimolecular surface reaction between dissociated adsorbed species was proposed as the ratedetermining step. The following generalized rate expression was proposed ... 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

It has been known for some time that UV photopolymerization of multifunctional monomers does not obey the classical rate expression, Rp proportional to I0 5, but follows an approximately first-order relationship [196,197]. These results have been explained by postulating that, in these viscous monomers, radical occlusion competes with bimolecular termination. 

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... 

For bimolecular reactions involving two molecular reactants, e.g., 03 + alkenes, concentration-time profiles of both reactants can often be recorded simultaneously, and the data can be treated using the integrated second-order rate expressions to derive the corresponding rate constants ... 

A generalized rate expression for the reaction of allylic chloride (RC1) with metal stabilizers (MX2) includes terms for the unimolecular elimination of HC1 from RC1 and for the bimolecular reaction of RC1 with all metal containing species. 

We shall return to this important mechanism in a moment after a brief mention of first-order kinetics. The reaction between the acid chloride and the neutral alcohol to give an ester may not have the bimolecular rate expression expected for this mechanism rate = /cfR COCIH OH]. 

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