Rate constant from activated complex theory

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. 

Transition-state theory is one of the earliest attempts to explain chemical reaction rates from first principles. It was initially developed by Eyring [124] and Evans and Polayni [122,123], The conventional transition-state theory (CTST) discussed here provides a relatively straightforward method to estimate reaction rate constants, particularly the preexponential factor in an Arrhenius expression. This theory is sometimes also known as activated complex theory. More advanced versions of transition-state theory have also been developed over the years [401],... 

A) In activated complex theory the rate constant can be expressed in terms of the free energy F of a system hypothetically constrained to exist on a certain hypersurface, the activated complex,99 cf. Marcus, R. A., J. Chem. Phys. 41, 2624 (1964). F can be expressed in terms of the free energy F of a system centered on that hypersurface (5). The difference between F and F contributes a factor to p. A second factor in p arises from the fluctuations in the separation distance of the reactants in the activated complex (5). 

First, we want to derive an expression for the microcanonical rate constant k(E) when the total internal energy of the reactant is in the range E to E + dE. From Eq. (7.43), the rate of reaction is given by the rate of disappearance of A or, equivalently, by the rate at which activated complexes A pass over the barrier, i.e., the flow through the saddle-point region. The essential assumptions of RRKM theory are equivalent to the assumptions underlying transition-state theory. 

On the other hand, following the activated complex theory, one can derive the value of the pre-exponential factor from the thermodynamic expression of the rate constant... 

Expression (67.Ill) can be considered as a "statistical formulation of the rate constant in that it represents a formal generalization of activated complex theory which is the usual form of the statistical theory of reaction rates. Actually, this expression is an exact collision theory rate equation, since it was derived from the basic equations (32.Ill) and (41. HI) without any approximations. Indeed, the notion of the activated complex has been introduced here only in a quite formal way, using equations (60.Ill) and (61.Ill) as a definition, which has permitted a change of variables only in order to make a pure mathematical transformation. Therefore, in all cases in which the activated complex could be defined as a virtual transition state in terms of a potential energy surface, the formula (67.HI) may be used as a rate equation equivalent to the collision theory expression (51.III). 

We will consider first the accurate collisional equations (51. HI) and (67 HI) for the rate constant. The factorization of these formally similar expressions allows us a separate evaluation of the three essential factors, including the activation energy, the partition functionf , and the corrections to the simple collision theory and the activated complex theory, respectively. A complete evaluation of the rate constant in this manner is, in principle, possible, if the potential energy surface is known from accurate or approximate calculations. If there is a col, both formulations are equivalent.Both involve the reaction dynamics through the transition probabilities in the corresponding correction factors and, which can be com-... 

Melander and Saunders (1980) have given a comprehensive description of the development of methods of computer calculations of isotope effects on the kinetics of chemical reactions. Such techniques, originally proposed by Wolfsberg and Stem (1964), Shiner (1975), Buddenbaum and Shiner (1977), and Schowen (1977), marry the methods ofEyring s absolute rate (activated complex) theory with detailed modeling of molecular vibrational properties. Input parameters are a mix of spectroscopically determined or quantum mechanically calculated force constants and/or force constant shifts. The method has resulted in informative and detailed molecular description of the molecular changes that occur as the system proceeds from reactant to product along the reaction coordinate. As a result, kinetic isotope effect studies now constitute one of the most important methods employed in the development of detailed... 

A reaction mechanism (see Table V) is a set of p reversible elementary processes involving c components. The net rate of the i reaction is written in the "mass action law" form. The rate constants, deduced from the activated complex theory, are characterized by an activation entropy and an activation enthalpy. 

Calculations of electron transfer rate constants from the energies of intervalence transfer (IT) bands in mixed-valence complexes have provided some interest (Table 1.1). The ion pair formed from paraquat (l,r-dimethyl-4,4 -bipyridine " ), and ferrocyanide, [PQ ", Fe(CN)6 ], shows an IT band from which the activation energy for thermal electron transfer within the ion pair can be derived (Figure 1.1) using Hush s theory to compare spectroscopic and kinetic data [equation (5)]. 

Eyring s theory of absolute reaction rates enables us to determine a concentration of activated complexes from rate measurements. The activated complexes proceed on to products in the manner of a first-order reachon, and the rate constant (specific rate) is k T/h for any reaction, where is Boltzmann s constant, T is the absolute temperature, and h is Planck s constant (Eq. 4.41). This is a very fast process its lifetime is comparable to a bond vibrahon hme. At 300 K, k T/h is 6.3 x 10 s . If the molar concentration of activated complexes is 0.16 x 10 , the rate of formation of products is 1 mol/l/s. (The activated complexes are also replenished at the same rate since they are in equihbrium with starting materials.)... 

Let us give the thermodynamic interpretation of the rate constant in the framework of the activated complex theory. The direct bimolecular reaction occurs. If the C and D products are rapidly removed from the system to prevent reverse reaction involving them, then reaction 1 can be presented in the form... 

For direct reactions, the reaction path profile has one potential barrier. In the frameworic of the theory of activated complex, the deviations of the temperature dependence of the rate constant from the Arrhenius equation for direct reactions can be explained by the temperature dependence of statistical sums of the reactants and activated complex. After inserting rotation, vibration, and translational statistical sums into expression (4.76), the temperature dependence of the rate constant is presented by the expression... 

Information concerning unimolecular potential energy surfaces can be acquired from several sources. Thermochemical measurements provide bond dissociation energies Dq and heats of reaction AHq. Analyses of the vibrational spectra of a unimolecular reaction s reactants and products yields their quadratic force constants, and if the data is sufficiently complete, also, their cubic and quartic force constants. From kinetic measurements of the unimolecular rate constant at high pressure the phenomenological Arrhenius A factor Aee and activation energy can be derived. If one can show that that the activated complex theory is valid for a specific unimolecular reaction its threshold energy Eq and the entropy difference between the activated complex and molecule can be found from... 

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 ... 

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. 

It can be difficult to estimate theoretically the bond lengths and vibrational frequencies for the activated complex and the energy barrier for its formation. It is of interest to assess how the uncertainty in these parameters affect the rate constant predicted from transition state theory (TST). For the exchange reaction... 

In classical kinetic theory the activity of a catalyst is explained by the reduction in the energy barrier of the intermediate, formed on the surface of the catalyst. The rate constant of the formation of that complex is written as k = k0 cxp(-AG/RT). Photocatalysts can also be used in order to selectively promote one of many possible parallel reactions. One example of photocatalysis is the photochemical synthesis in which a semiconductor surface mediates the photoinduced electron transfer. The surface of the semiconductor is restored to the initial state, provided it resists decomposition. Nanoparticles have been successfully used as photocatalysts, and the selectivity of these reactions can be further influenced by the applied electrical potential. Absorption chemistry and the current flow play an important role as well. The kinetics of photocatalysis are dominated by the Langmuir-Hinshelwood adsorption curve [4], where the surface coverage PHY = KC/( 1 + PC) (K is the adsorption coefficient and C the initial reactant concentration). Diffusion and mass transfer to and from the photocatalyst are important and are influenced by the substrate surface preparation. 

The standard entropy difference between the reactant(s) of a reaction and the activated complex of the transition state, at the same temperature and pressure. Entropy of activation is symbolized by either A5 or and is equal to (A// - AG )IT where A// is the enthalpy of activation, AG is the Gibbs free energy of activation, and T is the absolute temperature (provided that all rate constants other than first-order are expressed in temperature-independent concentration units such as molarity). Technically, this quantity is the entropy of activation at constant pressure, and from this value, the entropy of activation at constant volume can be deduced. See Transition-State Theory (Thermodynamics) Gibbs Free Energy of Activation Enthalpy of Activation Volume of Activation Entropy and Enthalpy of Activation (Enzymatic)... 

An equation for the bimolecular rate constant, k, obtained from transition-state theory. This constant is directly proportional to the equihbrium constant between reactants and the activated complex as well as the absolute temperature k = (RTlLh)K. See Transition-State Theory... 

Recall from transition state theory that the rate of a reaction depends on kg (the catalytic rate constant at infinite dilution in the given solvent), the activity of the reactants, and the activity of the activated complex. If one or more of the reactants is a charged species, then the activity coefficient of any ion can be expressed in terms of the Debye-Htickel theory. The latter treats the behavior of dilute solutions of ions in terms of electrical charge, the distance of closest approach of another ion, ionic strength, absolute temperature, as well as other constants that are characteristic of each solvent. If any other factor alters the effect of ionic strength on reaction rates, then one must look beyond Debye-Hiickel theory for an appropriate treatment. 

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