Collision rate model

Let us now deduce the factors that control the rate of conversion of B and C to D and E by imagining the transformation process is portrayed well by what is known as a collision rate model. (Strictly speaking, the collision rate model applies to gas phase reactions here we use it to describe interactions in solution where we are not specifying the roles played by the solvent molecules.) First, in order to be able to react, the molecules B and C have to encounter each other and collide. Hence, the rate of reaction depends on the frequency of encounters of B and C, which is proportional to the product of their concentrations. The rate is also related to how fast B and C move in the aqueous solution. Next, the rate is proportional to the probability that B and C meet with the right orientation to be able to react, which we may refer to as the orientation probability . Third, only a fraction of collisions have a sufficient amount of energy (greater then or equal to Ea) to break the relevant bonds in B and C... 

It should be noted that gravity can play an important role in collision rates for systems with large differences in density between the dispersed and continuous phases. Researchers (Al, El, S3) have also developed collision rate models taking into account gravity effects. In turbulent liquid-liquid dispersions with small fluid density differences, gravity is not an important factor in the collision rate model. 

Background and Useful Models The Marcus equation is an extension of earlier models from collision rate theory. As such, compliance with collision rate models is a prerequisite to defensible use of the Marcus equation. This is particularly important for reactions of charged species, and therefore, for reactions of many inorganic complexes. In these cases, the key question is whether electron transfer rate constants vary with ionic strength as dictated by electrolyte theory, on which the collision rate models are based. When they do not, differences between calculated and experimental values can differ by many orders of magnitude. 

The reaction mechanism is deduced from quantitative studies of the dependence of the rate upon the concentrations or pressures of the various reactants. To interpret such studies, we need to develop our collision theory model. 

Fig. 5.15. Theoretical dependences of HWHM on the rate of rotational energy relaxation perturbation theory asymptotics (1), classical weak-collision. /-diffusion model (2), quantum theory without (3) and with (4) adiabatic correction.

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

In 1990, Schroder and Schwarz reported that gas-phase FeO" " directly converts methane to methanol under thermal conditions [21]. The reaction is efficient, occuring at 20% of the collision rate, and is quite selective, producing methanol 40% of the time (FeOH+ + CH3 is the other major product). More recent experiments have shown that NiO and PtO also convert methane to methanol with good efficiency and selectivity [134]. Reactions of gas-phase transition metal oxides with methane thus provide a simple model system for the direct conversion of methane to methanol. These systems capture the essential chemistry, but do not have complicating contributions from solvent molecules, ligands, or multiple metal sites that are present in condensed-phase systems. 

Equation 21 has an exponential temperature dependence, which is much stronger than the weak temperature dependence of the collision rate itself. Let s now confirm that our model is consistent with the Arrhenius equation. When we take logarithms of both sides, we obtain... 

To understand why reaction rates depend on temperature, we need a picture of how reactions take place. According to the collision theory model, a bimolecu-lar reaction occurs when two properly oriented reactant molecules come together in a sufficiently energetic collision. To be specific, let s consider one of the simplest possible reactions, the reaction of an atom A with a diatomic molecule BC to give a diatomic molecule AB and an atom C ... 

Studies of proton transfers involving small ions with localized charge have shown that these reactions may proceed indeed with rate constants close to or even slightly larger than the collision rate constants predicted by the ADO theory (Mackay et al., 1976). However, rate-constant measurements of proton-transfer reactions between delocalized anions (Farneth and Brau-man, 1976) and sterically hindered pyridine bases (Jasinski and Brauman, 1980) and of SN2 displacement reactions (Olmstead and Brauman, 1977 Pellerite and Brauman, 1980 Pellerite and Brauman, 1983 Caldwell et al., 1984 for a review see Riveros et al., 1985) have shown that the rate constants can span the range from almost collision controlled values down to ones too slow to be observed. For these reactions the wide variation in rate constants has been explained on the basis of a double potential-well model which for a hypothetical SN2 substitution is schematically shown in Fig. 4. 

There is essentially a single modeling approach that has been developed, referred to here as the von Smoluchowski approach, and this method will be presented first. The von Smoluchowski approach requires analytical expressions to represent particle collision rates, to calculate collision efficiencies, and to dictate aggregate structure formation. These individual components are discussed in the subsequent sections, followed by analytical and numerical techniques of solving the von Smoluchowski equation. 

The collision theory model of chemical reactions can be used to explain the observed rate laws for both one-step and multi-step... 

The collision theory model suggests that the rate of any step in a reaction is proportional to the concentrations of the reagents consumed in that step. The rate law for a one-step reaction should, therefore, agree with the stoichiometry of the reaction. 

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