Activated complex, rate calculation

D24.3 The Eyring equation (eqn 24.53) results from activated complex theory, which is an attempt to account for the rate constants of bimolecular reactions of the form A + B iC -vPin terms of the formation of an activated complex. In the formulation of the theory, it is assumed that the activated complex and the reactants are in equilibrium, and the concentration of activated complex is calculated in terms of an equilibrium constant, which in turn is calculated from the partition functions of the reactants and a postulated form of the activated complex. It is further supposed that one normal mode of the activated complex, the one corresponding to displaconent along the reaction coordinate, has a very low force constant and displacement along this normal mode leads to products provided that the complex enters a certain configuration of its atoms, which is known as the transition stale. The derivation of the equilibrium constant from the partition functions leads to eqn 24.51 and in turn to eqn 24.53, the Eyring equation. See Section 24.4 for a more complete discussion of a complicated subject. 

The natiue of the rate constants k, can be discussed in terms of transition-state theory. This is a general theory for analyzing the energetic and entropic components of a reaction process. In transition-state theory, a reaction is assumed to involve the formation of an activated complex that goes on to product at an extremely rapid rate. The rate of deconposition of the activated con lex has been calculated from the assumptions of the theory to be 6 x 10 s at room temperature and is given by the expression ... 

In the context of Scheme 11-1 we are also interested to know whether the variation of K observed with 18-, 21-, and 24-membered crown ethers is due to changes in the complexation rate (k ), the decomplexation rate (k- ), or both. Krane and Skjetne (1980) carried out dynamic 13C NMR studies of complexes of the 4-toluenediazo-nium ion with 18-crown-6, 21-crown-7, and 24-crown-8 in dichlorofluoromethane. They determined the decomplexation rate (k- ) and the free energy of activation for decomplexation (AG i). From the values of k i obtained by Krane and Skjetne and the equilibrium constants K of Nakazumi et al. (1983), k can be calculated. The results show that the complexation rate (kx) does not change much with the size of the macrocycle, that it is most likely diffusion-controlled, and that the large equilibrium constant K of 21-crown-7 is due to the decomplexation rate constant k i being lower than those for the 18- and 24-membered crown ethers. Izatt et al. (1991) published a comprehensive review of K, k, and k data for crown ethers and related hosts with metal cations, ammonium ions, diazonium ions, and related guest compounds. 

Observations of reactivity are concerned with rate determining processes and require the knowledge of the structure and energy of the activated complexes. Up to now, the Hammond principle has been employed (see part 3.2) and reactive intermediates (cationic chain ends) have been used as models for the activated complexes. This was not successful in every case, therefore models of activated complexes related to the matter at hand were constructed, calculated and compared. For example, such models were used to explain the high reactivity of the vinyl ethers19 80). These types of obser-... 

The rate law of a reaction is an experimentally determined fact. From this fact we attempt to learn the molecularity, which may be defined as the number of molecules that come together to form the activated complex. It is obvious that if we know how many (and which) molecules take part in the activated complex, we know a good deal about the mechanism. The experimentally determined rate order is not necessarily the same as the molecularity. Any reaction, no matter how many steps are involved, has only one rate law, but each step of the mechanism has its own molecularity. For reactions that take place in one step (reactions without an intermediate) the order is the same as the molecularity. A first-order, one-step reaction is always unimolecular a one-step reaction that is second order in A always involves two molecules of A if it is first order in A and in B, then a molecule of A reacts with one of B, and so on. For reactions that take place in more than one step, the order/or each step is the same as the molecularity for that step. This fact enables us to predict the rate law for any proposed mechanism, though the calculations may get lengthy at times." If any one step of a mechanism is considerably slower than all the others (this is usually the case), the rate of the overall reaction is essentially the same as that of the slow step, which is consequently called the rate-determining step. ... 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

In equation (13.11), the first term corresponds to the catalysed part of the reaction and the remaining terms, which make a relatively small contribution, apply to the uncatalysed part. Kinetic data at constant acidity were in good agreement with the integrated form of the calculated rate expression. The rate coefficients k2,k, k, and the ratio k. jk were evaluated. Almost linear plots of log 2 versus log [ ] were obtained at four temperatures with slopes close to —1.8. This result suggests that the dominant activated complex is that formed by loss of two ions, viz. 

While the collision theory of reactions is intuitive, and the calculation of encounter rates is relatively straightforward, the calculation of the cross-sections, especially the steric requirements, from such a dynamic model is difficult. A very different and less detailed approach was begun in the 1930s that sidesteps some of the difficulties. Variously known as absolute rate theory, activated complex theory, and transition state theory (TST), this class of model ignores the rates at which molecules encounter each other, and instead lets thermodynamic/statistical considerations predict how many combinations of reactants are in the transition-state configuration under reaction conditions. 

The theory of equilibrium is treated on the basis of thermodynamics considering only the initial and final states. Time or intermediate states have no concern. However, there is a close relationship between the theory of rates and the theory of equilibria, in spite of there being no general relation between equilibrium and rate of reaction. A good approximation of equilibrium can be regarded between the reactants and activated state and the concentration of activated complex can, therefore, be calculated by ordinary equilibrium theory and probability of decomposition of activated complex and hence the rate of reaction can be known. 

The reason for this difference is very simple. It is that the density functional approach calculates the rate for the adiabatic channel. For AG x> this will proceed via an activated complex with successor-state formal electron density parameter A = 1 for AGq <-x, it will proceed via an activated... 

Figure 12. Comparison of simple RRKM rate-energy curves, using three different loose activated complexes giving the same rates at the energy corresponding to about 10 s" . Calculations are shown for Eq values of 1.86 and 3.10 eV. The three transition states are (a) uniform frequency multiplier of 0.9 (—) (b) four low-frequency vibrations (—) (c) low-frequency vibration to internal rotor ( -). The corresponding values are as follows 1.86 eV, (a) = 8.2 eu, (b) = 4.9 eu, (c) = 1.6 eu 3.10 eV (a) = 6.9 eu, (b) = 4.9 eu, (c) = 2.9 eu. Also shown is the semiclassical RRK functional form...

Theoretical descriptions of absolute reaction rates in terms of the rate-limiting formation of an activated complex during the course of a reaction. Transition-state theory (pioneered by Eyring "", Pelzer and Wigner, and Evans and Polanyi ) has been enormously valuable, and beyond its application to chemical reactions, the theory applies to a wider spectrum of rate processes (eg., diffusion, flow of liquids, internal friction in large polymers, eta). Transition state theory assumes (1) that classical mechanics can be used to calculate trajectories over po-... 

In transition-state theory, the absolute rate of a reaction is directly proportional to the concentration of the activated complex at a given temperature and pressure. The rate of the reaction is equal to the concentration of the activated complex times the average frequency with which a complex moves across the potential energy surface to the product side. If one assumes that the activated complex is in equilibrium with the unactivated reactants, the calculation of the concentration of this complex is greatly simplified. Except in the cases of extremely fast reactions, this equilibrium can be treated with standard thermodynamics or statistical mechanics . The case of... 

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. 

The first mechanism implies k19 = k5AK11A 11A the second leads to ki9 = k2iKlm- . Independent evidence suggests the existence of both intermediate species in the nitric oxide-oxygen system, and both mechanisms involve entirely reasonable collision complexes. In both, the equilibrium step is rapid, and the overall kinetics are third order. Theoretical calculations based on the activated complex theory were made by assuming a true termolecular reaction the predicted rates agree well with experiment.161 The experimental rate constants are summarized in Tables 4-3 and 4-4. 

The first studies of the kinetics of the NO-F2 reaction were reported by Johnston and Herschbach229 at the 1954 American Chemical Society (ACS) meeting. Rapp and Johnston355 examined the reaction by Polanyi s dilute diffusion flame technique. They found the free-radical mechanism, reactions (4)-(7), predominated assuming reaction (4) to be rate determining, they found logfc4 = 8.78 — 1.5/0. From semi-quantitative estimates of the emission intensity, they estimated 6//t7[M] to be 10-5 with [M] = [N2] = 10 4M. Using the method of Herschbach, Johnston, and Rapp,200 they calculated the preexponential factors for the bimolecular and termolecular reactions with activated complexes... 

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