Transition-state theory limitations

Miller W H 1975 Semiclassical limit of quantum mechanical transition state theory for nonseparable systems J. Chem. Phys. 62 1899... 

As we expect, in the limit that 9 —> 1 the standard transition state theory result... 

Returning to the more general expression, in the low temperature limit we find that the transition state theory estimate of the rate is... 

For 5=1, the normal transition state theory rate constant is independent of temperature at high temperatures and varies exponentially with temperature in the limit of low temperatures kT small compared with the barrier height U ) as... 

Electrode kinetics lend themselves to treatment usiag the absolute reaction rate theory or the transition state theory (36,37). In these treatments, the path followed by the reaction proceeds by a route involving an activated complex where the element determining the reaction rate, ie, the rate limiting step, is the dissociation of the activated complex. The general electrode reaction may be described as ... 

We now carry the argument over to transition state theory. Suppose that in the transition state the bond has been completely broken then the foregoing argument applies. No real transition state will exist with the bond completely broken—this does not occur until the product state—so we are considering a limiting case. With this realization of the very approximate nature of the argument, we make estimates of the maximum kinetic isotope effect. We write the Arrhenius equation for the R-H and R-D reactions... 

Because the frequency of a weakly bonded vibrating system is relatively small, i.e. kBT hu we may approximate its partition function by the classical limit k T/hv, and arrive at the rate expression in transition state theory ... 

A pre-exponential factor and activation energy for each rate constant must be established. All forward rate constants involving alkyne adsorption (ki, k2, and ks) are assumed to have equal pre-exponential factors specified by the collision limit (assuming a sticking coefficient of one). All adsorption steps are assumed to be non-activated. Both desorption constants (k.i and k ) are assumed to have preexponential factors equal to 10 3 sec, as expected from transition-state theory [28]. Both desorption activation energies (26.1 kcal/mol for methyl acetylene and 25.3 kcal/mol for trimethylbenzene) were derived from TPD results [1]. 

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

Statistical rate theories have been used to calculate rate constants for gas-phase Sn2 reactions.1,7 For a SN2 reaction like Cl" + CH3Clb, which has a central barrier higher than the reactant asymptotic limit (see Figure 1), transition state theory (TST) assumes that the crossing of the central barrier is rate-limiting. Thus, the TST expression for the SN2 rate constant is simply,... 

The transition state theory provides a useful framework for correlating kinetic data and for codifying useful generalizations about the dynamic behavior of chemical systems. This theory is also known as the activated complex theory, the theory of absolute reaction rates, and Eyring s theory. This section introduces chemical engineers to the terminology, the basic aspects, and the limitations of the theory. 

The thermodynamic formulation of the transition state theory is useful in considerations of reactions in solution when one is examining a particular class of reactions and wants to extrapolate kinetic data obtained for one reactant system to a second system in which the same function groups are thought to participate (see Section 7.4). For further discussion of the predictive applications of this approach and its limitations, consult the books by Benson (59) and Laidler (60). Laidler s kinetics text (61) and the classic by Glasstone, Laidler, and Eyring (54) contain additional useful background material. 

Salt effects in kinetics are usually classified as primary or secondary, but there is much more to the subject than these special effects. The theoretical treatment of the primary salt effect leans heavily upon the transition state theory and the Debye-Hii ckel limiting law for activity coefficients. For a thermodynamic equilibrium constant one should strictly use activities a instead of concentrations (indicated by brackets). 

At high temperatures and low pressures, the unimolecular reactions of interest may not be at their high-pressure limits, and observed rates may become influenced by rates of energy transfer. Under these conditions, the rate constant for unimolecular decomposition becomes pressure- (density)-dependent, and the canonical transition state theory would no longer be applicable. We shall discuss energy transfer limitations in detail later. 

In the absence of transport limitations, the processes of adsorption, surface diffusion, surface reaction, and desorption can be treated via the transition state theory (Baetzold and Somorjai, 1976 Zhdanov et al, 1988). For example, the application of the TST to a single site adsorption process,... 

Phase space theory (PST) has been widely used for estimation of rates and energy partitioning for ion dissociations. It can be considered within the framework of transition-state theory as the limiting case of a loose transition state, in which all product degrees of freedom are statistically fully accessible at the transition state. As such, it is expected to give an upper limit for dissociation rates and to be best suited to barrierless dissociations involving reaction coordinates with simple bond cleavage character. 

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... 

Many of the 60 known reactions catalyzed by monoclonal antibodies involve kinetically favored reactions e.g., ester hydrolysis), but abzymes can also speed up kinetically disfavored reactions. Stewart and Benkovic apphed transition-state theory to analyze the scope and limitations of antibody catalysis quantitatively. They found the observed rate accelerations can be predicted from the ratio of equilibrium binding constants of the reaction substrate and the transition-state analogue used to raise the antibody. This approach permitted them to rationalize product selectivity displayed in antibody catalysis of disfavored reactions, to predict the degree of rate acceleration that catalytic antibodies may ultimately afford, and to highlight some differences between the way that they and enzymes catalyze reactions. 

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-... 

Transition state theory yields rate coefficients at the high-pressure limit (i.e., statistical equilibrium). For reactions that are pressure-dependent, more sophisticated methods such as RRKM rate calculations coupled with master equation calculations (to estimate collisional energy transfer) allow for estimation of low-pressure rates. Rate coefficients obtained over a range of temperatures can be used to obtain two- and three-parameter Arrhenius expressions ... 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do munerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

Fig. I. Illustration of the relationship between reactants (designated as 2-A-B), products (A-A and B-B), and the activated complex. According to transition state theory, reaction kinetics is limited by the irreversible decay of the activated complex minus the rate at which the activated complex reversibly breaks down to reactants.

Golden, D. M. 1979. Experimental and theoretical examples of the value and limitations of Transition State Theory. The Journal of Physical Chemistry, 83, 108-113. 

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