Rate constants from transition-state theory

The transmission coefficient kgh is, as previously, a measure of the departure of the rate constant from transition-state theory, kqh is given as the ratio between the frequencies with and without friction, that is,... 

F. Jousse and S. M. Auerbach, /. Chem. Phys., 107, 9629 (1997). Activated Diffusion of Benzene in NaY Zeolite Rate Constants from Transition State Theory with Dynamical Corrections. 

In the present work, we show that the production of ethane from the reaction of methane with methyl is grossly overestimated by the above rate constant. From transition-state theory (TST) and highly correlated ab initio calculations, we find rates that are between three (at 1500 K) and ten (at 300 K) orders of magnitudes slower than thc e used by Norinaga and Deutschmann [63]. This leads us to conclude that the role of this reaction in the mechanism must be reexamined. Furthermore, we show that the barrier height... 

The models developed here account for unmeasurable intermediates such as adsorbed ions or free radicals. Microkinetic analysis, pioneered by Dumesic and cowokers"", is an example of this approach. It quantifies catalytic reactions in terms of the kinetics of elementary surface reactions. This is done by estimating the gas-phase rate constants from transition state theory and adjusting these constants for surface reactions. For instance, isobutane cracking over zeolite Y-based FCC catalysts has 21 reversible steps defined by 60 kinetic parameters." The rate constants are estimated from transition state theory... 

Brennecke and coworkers (90,320) investigated the uncatalyzed esterification of phthalic anhydride with methanol in SCCO2 as a probe reaction to show that augmented local densities and cosolvent compositions in the near-critical region represent enhanced reactant concentrations that result in increased reaction rates. The authors report kinetic data for the esterification reaction at both 40 C and 50 C and pressures ranging from 97.5 to 166.5 bar. Based on bulk fluid compositions, a dramatic pressure effect was exhibited for the measured bimolecular rate constants. For example, at 50°C the value of the rate constant decreased 25-fold from 0.0348 L/mol-min at 97.5 bar to 0.00138 L/mol-min at 166.5 bar, which represents one of the largest pressure effects ever reported for a reaction in an SCF. Based on calculations of the thermodynamic pressure effect on the rate constant from transition state theory and estimates of the local concentrations from literature data, the authors conclude that the dramatic pressure... 

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 observed solvent effect can be expressed quantitatively with the aid of the Leffler-Grunwald operator 5m introduced in Chapter 7. For rate constant k measured in medium M we have, from transition state theory, k = (kT//i)exp ( —AGm// T) and similarly for rate constant ko measured in a reference solvent. Combining these two expressions gives... 

The equation from transition state theory (Chapter 7) provides the basis for correlating the rate constants as a function of temperature. It may be written in either exponential or logarithmic form ... 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

The activation parameters from transition state theory are thermodynamic functions of state. To emphasize that, they are sometimes designated A H (or AH%) and A. 3 4 These values are the standard changes in enthalpy or entropy accompanying the transformation of one mole of the reactants, each at a concentration of 1 M, to one mole of the transition state, also at 1 M. A reference state of 1 mole per liter pertains because the rate constants are expressed with concentrations on the molar scale. Were some other unit of concentration used, say the millimolar scale, values of AS would be different for other than a first-order rate constant. 

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... 

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]. 

To calculate the mixed solvent isotope effect on rate constants one applies simple ideas from transition state theory to evaluate the isotope effect on the... 

From transition-state theory, the bimolecular rate constant ki2 can be written as ... 

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. 

From transition-state theory we can derive an expression that relates the magnitude of a rate constant to the activation energy ... 

In summary, the measured rate constants (based on bulk concentrations) increase as the pressure is decreased near the critical point. This cannot be explained solely on the basis of the pressure effect on the rate constant predicted from transition state theory or cage effects. As a result, we believe that local composition increases near the critical point play an important role in the rate increase. 

These expressions may also be derived from transition state theory a rate constant k is given by... 

For example, we now use transition state theory to estimate the rate constant for an adsorption process. From transition state theory, the adsorption of species A is expressed by the reaction... 

The Arrhenius form of the rate constant specifies that both A and E are independent of T. Note that when the formulation derived from transition-state theory is compared to the Arrhenius formulation [Equations (2.3.10) and (2.3.11)], both A and E do have some dependence on T. However, A//J is very weakly dependent on T and the temperature dependence of ... 

As this chapter focuses on hydrogen transfers in liquids and solids, it will be assumed that the transfer constitutes a rate process which can be described in terms of rate constants, for which the usual rate theories can be applied, in particular those derived from transition state theory. 

The relationship between the rate constants kei for an electrode reaction and fee for the corresponding self-exchange electron transfer reaction is not obvious because kgi can be strongly influenced by the nature and history of the electrode surface and by solvent dynamic effects if present. Electrode properties, however, are not expected to be sensitive to pressures in the 0-200 MPa range. Moreover, the signature of solvent dynamical effects is a dependence of reaction rate on solvent viscosity, but the viscosity of water is effectively independent of such pressures at near-ambient temperatures. Consequently, for typical aqueous electrode reactions, Ai/, = O.SAV, regardless of any involvement of solvent dynamics, and so AVg can be predicted from transition state theory (TST) according to Eqs (5.5)-... 

Equations (1.3-14) and (1.3-15) thus give the prediction from transition-state theory for the rate of a reaction in terms appropriate for an SCF. The rate is seen to depend on (i) the pressure, the temperature and some universal constants (ii) the equilibrium constant for the activated-complex formation in an ideal gas and (iii) a ratio of fugacity coefficients, which express the effect of the supercritical medium. Equation (1.3-15) can therefore be used to calcu-late the rate coefficient, if Kp is known from the gas-phase reaction or calculated from statistical mechanics, and the ratio (0a 0b/0cO estimated from an equation of state. Such calculations are rare an early example is the modeling of the dimerization of pure chlorotrifluoroethene = 105.8 °C) to 1,2-dichlor-ohexafluorocyclobutane (Scheme 1.3-2) and comparison with experimental results at 120 °C, 135 °C and 150 °C and at pressures up to 100 bar [15]. 

There are many ways to express reaction rates, and keeping track of notation for different kinds of rates along with the units of their accompanying rate equations is challenging. For a simple rate equation such as (3.1), the rate and the rate constant have units of mol/sec, which are the units expected from transition-state theory (Chapter 5). A reaction rate can also be expressed in terms of the time rate of change of concentration of a species (R, mol/kg sec = molal/sec), by dividing both sides of Eq. (3.1) by the mass of water (M) in the system. 

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