Bimolecular reactions, transition-state theory

In Chapter 5, the direct evaluation of k(T) via the reactive flux through a dividing surface on the potential energy surface was described. As a continuation of that approach, we consider in this chapter an—approximate—approach, the so-called transition-state theory (TST).1 We have already briefly touched upon this approximation, based on an evaluation of a stationary one-way flux, which implies that the rate constant can be obtained without any explicit consideration of the reaction dynamics. In this chapter, we elaborate on this important approach, in a form that takes some quantum effects into account. 

The intermediate nuclear configurations between reactants and products are all referred to as transition states for the reaction. The collection of atoms at the saddle point form a supermolecule , referred to as an activated complex, and their state is equivalent to one particular transition state for the reaction. This transition state obviously has a special status among all the transition states, and when one just refers to the transition state of a chemical reaction, it is tacitly assumed that one refers to the activated complex. The symbol f is used to represent activated complexes.2 

The saddle point is a stationary point on a multidimensional potential energy surface. It is a stable point in all dimensions except one, where the second-order derivative of the potential is negative see Fig. 6.0.1. 

This degree of freedom is the reaction coordinate (note that this definition coincides with the definition in Chapter 3). In Appendix E, we show that a multidimensional system close to a stationary point can be described as a set of uncoupled harmonic oscillators, expressed in terms of so-called normal-mode coordinates. The oscillator associated with the reaction coordinate has an imaginary frequency, which implies that the motion in the reaction coordinate is unbound. 

Example 6.1 Normal-mode frequencies at a saddle point 

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Bimolecular processes are very common in biological systems. The binding of a hormone to a receptor is a bimolecular reaction, as is substrate and inhibitor binding to an enzyme. The term bimolecular mechanism applies to those reactions having a rate-limiting step that is bimolecular. See Chemical Kinetics Molecularity Reaction Order Elementary Reaction Transition-State Theory... 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

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... 

Transition state theory. If the TST equation is applied to a bimolecular reaction, there appears to be a discrepancy in the units the left-hand side has dimensions of concentration-1 time-1, whereas the right-hand side has time-1. Comment. 

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. 

If hu0 is small compared with kT, the partition function becomes kT/hv0. The function kT/h which pre-multiplies the collision number in the transition state theory of the bimolecular collision reaction can therefore be described as resulting from vibration of frequency vq along the transition bond between the A and B atoms, and measures the time between each potential transition from reactants to product which will only occur provided that the activation energy, AE°0 is available. 

The transition-state theory (TST) provides the framework to derive accurate relationships between kinetic and thermochemical parameters. Consider the common case of the gas-phase bimolecular reaction 3.1, where the transient activated complex C is considered to be in equilibrium with the reactants and the products ... 

For a temperature of 1000 K, calculate the pre-exponential factor in the specific reaction rate constant for (a) any simple bimolecular reaction and (b) any simple unimolecular decomposition reaction following transition state theory. 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

Both unimolecular and bimolecular reactions are common throughout chemistry and biochemistry. Binding of a hormone to a reactor is a bimolecular process as is a substrate binding to an enzyme. Radioactive decay is often used as an example of a unimolecular reaction. However, this is a nuclear reaction rather than a chemical reaction. Examples of chemical unimolecular reactions would include isomerizations, decompositions, and dis-associations. See also Chemical Kinetics Elementary Reaction Unimolecular Bimolecular Transition-State Theory Elementary Reaction... 

ELECTROSTATIO BOND ELECTROSTATIO SUREAOE POTENTIAL ELECTROSTRIOTION ELECTROTAXIS ELECTROVALENT BOND ELEMENTARY OHARGE ELEMENTARY REACTION Elementary reaction stoichiometry, MOLECULARITY CHEMICAL KINETICS UNIMOLECULAR BIMOLECULAR TRANSITION-STATE THEORY ELEMENTARY REACTION Element effect,... 

STOICHIOMETRIC NUMBER Stoichiometry of elementary reactions, CHEMICAL KINETICS MOLECULARITY UNIMOLECULAR BIMOLECULAR TRANSITION-STATE THEORY ELEMENTARY REACTION STOKE S SHIFT... 

As seen in Tables 22—25, the Arrhenius preexponential factors Aa for the initiation step are very low, 10 in 7, 10 in 20, 10 " in 41 and 1in 44. These are very low values for bimolecular reactions for which values of about 10 ° are observed and also predicted by the Transition State Theory Thus step (a) belongs to a class of slow reactions , some of which might have ionic transition states . The activation entropies AS obtained from the Transition State Theory rate constant expression... 

Because a is a parameter that cannot be calculated from first principles. Equation 1-95 cannot be used to calculate reaction rate constant k from first principles. Furthermore, the collision theory applies best to bimolecular reactions. For monomolecular reactions, the collision theory does not apply. Tr3dng to calculate reaction rates from first principles for all kinds of reactions, chemists developed the transition state theory. 

Transition state theory treats a reacting system thermodynamically. Let us again take a bimolecular reaction between A and B. Transition state theory assumes that as A and B collide and start to react, they form a species called the activated complex, which corresponds to the A-B adduct at the peak of the energy hill lying between reactants and products. This activated complex is thus in a transition state and can either fall back to reactants or go on to form products. The activated complex is normally indicated with a double dagger symbol, AB. The reaction can thus be given as... 

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