Transition state reaction-rate expression

Eyring s transition-state reaction rate expression is... 

The major advance of the past decade is that, using quantum-chemical computations, activation energies (Eact) as well as activation entropies (AS ) can be predicted a priori for systems of catalytic interest. This implies much more reliable use of the transition-state reaction rate expression than before, since no assumption of the transition state-structure is necessary. This transition-state structure can now be predicted. However, the estimated absolute accuracy of computed transition states is approximately of the order of 20-30 kJ/mol. Here, we do not provide an extensive introduction to modern quantum-chemical theory that has led to this state of affairs excellent introductions can be found elsewhere [38,39]. Instead, we use the results of these techniques to provide structural and energetic information on catalytic intermediates and transition states. 

In order to appreciate the use of transition-state rate expressions, it is important to be reminded of the different time scales of the processes that imderpin the chemistry we wish to describe. The electronic processes that define the potential-energy surface on which atoms move have characteristic times that are of the order of femtoseconds, 10 sec, whereas the vibrational motion of the atoms is on the order of picoseconds, 10" sec. The overall time scale for bond activation and formation processes that control catalysis vary between 10 and 10 sec. This implies that on the time scale of the elementary reaction in a catalytic process, many vibrational motions occur. If energy transfer is efficient, then the assumption that all vibrational modes except the reaction coordinate of the chemical reaction are equilibrated is satisfied. Kramersl l defined this condition as Eb > 5kT. Under this condition the transition state reaction-rate expression applies ... 

To predict catalyst performance, one needs to predict the rates of the elementary reaction steps at the catalyst surface. This must ultimately be integrated into a kinetic simulation which treats the interactions between the many different adsorbates present on the catalyst surface. In this chapter, we presented rate expressions derived from transition state reaction rate theory as a bridge to connect ab initio quantum mechanical information to reaction rate predictions. In Chapter 3, we present a more extensive treatment of kinetic simulations including many-body interactions and their influence on the catalytic performance. 

In this book we intend to make a connection between molecular properties of species involved in catalytic reactions, their reactivity, and the expression for the reaction rate. Relations between the vibrational and rotational properties of molecules and their propensity to adsorb and react on a surface and to desorb into the gas phase, are derived from the field of statistical thermodynamics. The latter forms a substantial part of Chapter 4, where we also introduce the transition-state reaction-rate theory. 

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momenrnm along the reaction coordinate directed from reactant to product. Stated another way, is the equilibrium flux of reactant states across... 

Since the transition state formulation of a reaction rate expression treats the activated complex as being in equilibrium with the reactants, the resultant expression for the reaction rate constant depends similarly on the free energy difference between reactants and the activated complex. In this case equation 4.3.34 can be rewritten as... 

Following the determination of the geometry and the thermochemistry of transition states, the rate parameters for the two silane decomposition pathways can be obtained directly by the TST formulation presented earlier. These calculations have led to unimolecular rate constant expressions 10 exp(-91000/RT)s-S and 10 exp(-62000/l r)s" for Si-H bond scission and H2 elimination reactions, respectively. These results clearly... 

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... 

Thus, expression (59) enables us to describe the solid-state reaction rate constant dependence on the parameters of the potential barrier and medium properties in a wide temperature range, from liquid helium temperatures when the reaction runs by a tunneling mechanism to high temperatures (naturally, not exceeding the melting point) when the transition is of the activation type. 

With this choice of dividing surfaces, a generalized expression for the transition state theory rate constant for a bimolecular reaction is given by ... 

In classical transition-state theory, the expression for the rate constant of a bi-molecular reaction in solution is... 

Because the reaction rate can be expressed in terms of the concentration of the transition state, the rate can now be given by... 

It is not possible to cover all of the history or the theory of the chemical kinetics in the context of this chapter. However, the authors intention is to give the student an essential minimum in the theory of chemical kinetics to be able to follow the literature and to incorporate in the design of the chemical reaction units. This chapter is divided into two sections in the first part, the homogeneous kinetics will be covered in detail, covering the collision theory and the transition state theory for the determination of the rate constants and reaction rate expressions. Old but still valid approximations of pseudo-steady-state and pseudoequilibrium concepts will be given with examples. In the second part, the heterogeneous reaction kinetics will be discussed from a mechauis-tic point of view. 

In the limit of high pressure, collisions maintain the thermal distribution of reactant molecules over their internal energy states and consequently TST can be used to determine the thermal rate constants for dissociation and association. However, in the case where there is no maximum in the reaction path leading from reactants to products, it is necessary to take account of angular momentum (/) constraints as well as internal energy. The transition state is not found at a single separation but rather it depends on Eint and J. Then, in the language of the statistical adiabatic channel model (SACM), the partition function for the transition state can be expressed as ... 

For analysing equilibrium solvent effects on reaction rates it is connnon to use the thennodynamic fomuilation of TST and to relate observed solvent-mduced changes in the rate coefficient to variations in Gibbs free-energy differences between solvated reactant and transition states with respect to some reference state. Starting from the simple one-dimensional expression for the TST rate coefficient of a unimolecular reaction a— r... 

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... 

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 ... 

For gas-phase reactions, Eq. (5-40) offers a route to the calculation of rate constants from nonkinetic data (such as spectroscopic measurements). There is evidence, from such calculations, that in some reactions not every transition state species proceeds on to product some fraction of transition state molecules may return to the initial state. In such a case the calculated rate will be greater than the observed rate, and it is customaiy to insert a correction factor k, called the transmission coefficient, in the expression. We will not make use of the transmission coefficient. 

This form assumes that the effect of pressure on the molar volume of the solvent, which accelerates reactions of order > 1 by increasing the concentrations when they are expressed on the molar scale, has been allowed for. This effect is usually small, ignored but in the most precise work. Equation (7-41) shows that In k will vary linearly with pressure. We shall refer to this graph as the pressure profile. The value of A V is easily calculated from its slope. The values of A V may be nearly zero, positive, or negative. In the first case, the reaction rate shows little if any pressure dependence in the second and third, the applied hydrostatic pressure will cause k to decrease or increase, respectively. A positive value of the volume of activation means that the molar volume of the transition state is larger than the combined molar volume of the reactant(s), and vice versa. 

Clearly, neither rate expression yields to the ordinary interpretation. Transition states with a nonintegral number of atoms or a fractional ionic charge cannot exist (not that the one represented by Eq. (8-4) is fractional, but others we shall see would be). These reactions are believed to proceed by chain mechanisms. 

When F is equal to unity, the equation reduces to the rate expression of the well-known transition state theory. In most of the cases considered in this book, we will deal with reactions in condensed phases where F is not much different from unity and the relation between k and Ag follows the qualitative role given in Table 2.1. 

The reaction coordinate that describes the adsorption process is the vibration between the atom and the surface. Strictly speaking, the adsorbed atom has three vibrational modes, one perpendicular to the surface, corresponding to the reaction coordinate, and two parallel to the surface. Usually the latter two vibrations - also called frustrated translational modes - are very soft, meaning that k T hv. Associative (nondissociative) adsorption furthermore usually occurs without an energy barrier, and we will therefore assume that A = 0. Hence we can now write the transition state expression for the rate of direct adsorption of an atom via this transition state, applying the same method as used above for the indirect adsorption. 

The other extreme is direct adsorption, in which the molecule lands immediately at its final adsorption site without the possibility of moving over the surface. In this case the only degrees of freedom the molecule has in the transition state are vibrational, among which the vibration between the molecule and the surface represents the reaction coordinate. This leaves us with the following expression, which immediately indicates that the rate constant is small ... 

Note how the partition function for the transition state vanishes as a result of the equilibrium assumption and that the equilibrium constant is determined, as it should be, by the initial and final states only. This result will prove to be useful when we consider more complex reactions. If several steps are in equilibrium, and we express the overall rate in terms of partition functions, many terms cancel. However, if there is no equilibrium, we can use the above approach to estimate the rate, provided we have sufficient knowledge of the energy levels in the activated complex to determine the relevant partition functions. 

By applying the machinery of statistical thermodynamics we have derived expressions for the adsorption, reaction, and desorption of molecules on and from a surface. The rate constants can in each case be described as a ratio between partition functions of the transition state and the reactants. Below, we summarize the most important results for elementary surface reactions. In principle, all the important constants involved (prefactors and activation energies) can be calculated from the partitions functions. These are, however, not easily obtainable and, where possible, experimentally determined values are used. 

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