Calculations of rate constants

The transfer of electrons between a metal or a semi-conductor electrode and a dissolved or surface-bound reactant is no different in kind from the homogenous processes, previously described. In the model proposed by Marcus the electrochemical rate constant, is given by 

Several expressions have been proposed for the pre-exponential factor. In terms of a translational partition function, Z is 

Marcus has pointed out that the energy barrier for a heterogeneous process should be half of that for the homogeneous process in solution, AG q, described above, 

Curtiss et al. [44] have carried out measurements for reaction rates of the electrochemical oxidation of Fe(OH2)6 to Fe(OH2)g at a gold electrode in a 0.5 M HCIO4 aqueous solution. The experimental design of a pressurised flow-system allowed electrochemical studies over a vast range of temperatures, from 25 to 250 °C. A good Arrhenius behaviour was found with = 56.8 1.5 kJ mol , a frequency factor of Z = 6 X 10 cm sec and a Tafel coefficient t] = 0.425 0.01, independent of the temperature. The measured standard rate constant at 25 °C is = 6 x 10 cm sec T 

The activation energy of the electrochemical oxidation of FefOHj)/ can be calculated with the structural and electronic data in Appendix III for = 3.75 X 

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Dellago C, Boihuis P G, Csajka F S and Chandler D 1998 Transition path sampling and the calculation of rate constants J. Chem. Phys. 108 1964-77... 

C. Dellago, P.G. Bolhuis, F.S. Csajka, and D. Chandler, Transition path sampling and the calculation of rate constants , a preprint. 

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. 

If the intrinsic barrier AGq could be independently estimated, the Marcus equation (5-69) provides a route to the calculation of rate constants. An additivity property has frequently been invoked for this purpose.For the cross-reaction... 

A small square wave modulation of the current is applied in order to disturbe the non-equilibrium steady state of the discharge. The exponential decay of the concentrations leads to characteristic relaxation times, which allow the calculation of rate constants used in the modeling of the deposition mechanism. 

Dellago, C. Bolhuis, P. G. Chandler, D., On the calculation of rate constants in the transition path ensemble, J. Chem. Phys. 1999,110, 6617-6625... 

Rate constant temperature dependence Processing threshold Calculation of rate constants at different temperatures, including collision numbers and concentrations of species in steady state Calculation of the rate of photodissociation and cosmic ray-induced molecular processing from photon and particle fluxes... 

Turning to the calculation of rate constants, our results are summarized in Table II. We consider first the direct addition mechanism. This is expected to lead to second-order kinetics, and... 

The parameters used in the IPM are presented in Table 4.16 and Table 4.17. In these tables, the additional parameters for the reactions of hydroperoxides with molecules and free radicals are given. The reactions of hydroperoxide with free radicals are important for the chain processes of the decomposition of hydroperoxides (see later). The results of the calculation of rate constants of various hydroperoxide reactions are collected in Tables 4.18-4.21. The comparison of the calculated values with the experimental values helped to introduce a few corrections in the traditional view on the bimolecular reactions of hydroperoxides. 

Since these reaction products exhibit considerable absorbance at the wave lengths utilized in the rate measurements, the calculation of rate constants required a technique incorporating this factor. Two methods of calculation were employed successfully. In some cases, limiting absorbances (A00) were determined and the rates were obtained from the slopes of graphs of log (A0—A00)/(A—A0o) vs. time. These served to demonstrate the pseudo-first-order nature of the rate constant however, the more general calculation procedure was that due to Guggenheim (11). The first-order dependence of the rate on the concentration of alkyl halide was shown by varying initial concentrations. 

An apparent compensation effect can result from errors in the experimental data used for an Arrhenium plot. Besides trivial errors, there may also occur errors in the calculation of rate constants, for instance when a homogeneous and a heterogeneous reaction occur simultaneously or when a heterogeneous reaction undergoes a change from a certain reaction order to another order. A temperature dependence of the activation energy, and the variability of the effective surface of the catalyst with temperature, especially caused by diffusion processes, may also account for apparent compensation effects. 

The relation between pre-exponential factors and activation energies given in Figure 5 is regular and will probably permit calculation of rate constants for reactions of atoms with molecules. 

The calculation of rate constants from steady state kinetics and the determination of binding stoichiometries requires a knowledge of the concentration of active sites in the enzyme. It is not sufficient to calculate this specific concentration value from the relative molecular mass of the protein and its concentration, since isolated enzymes are not always 100% pure. This problem has been overcome by the introduction of the technique of active-site titration, a combination of steady state and pre-steady state kinetics whereby the concentration of active enzyme is related to an initial burst of product formation. This type of situation occurs when an enzyme-bound intermediate accumulates during the reaction. The first mole of substrate rapidly reacts with the enzyme to form stoichiometric amounts of the enzyme-bound intermediate and product, but then the subsequent reaction is slow since it depends on the slow breakdown of the intermediate to release free enzyme. 

The calculation of rate constants for OH radical addition to >C=C< and -C=C- bonds assumes that the rate constant for OH radical addition to these carbon-carbon unsaturated bonds depends on the number, identity, and position of substituent groups around the >C=C< or -C=C- bond(s). Conjugated double bond systems are dealt with by considering the entire conjugated >C=C-C=C< system as a single unit (Atkinson, 1986), rather than as conjugated >C=C< sub-units as Ohta (1983) did. 

C2H5NH2 decomposition - calculation of rate constant from total pressure with time, 23-24... 

The same arguments apply with even greater force to rates of reactions. The calculation of rate constants is a much more difficult problem than the calculation of equilibria because we cannot determine the structures of transition states experimentally. We should in principle calculate the whole potential energy... 

Activation entropies are useful because they can give information on the structure of a transition state (as stated above, a more confined transition state is signalled by a negative, unfavorable, activation entropy), but the ab initio calculation of rate constants [148] from activation free energies is not as straightforward as... 

Fig. 5.30 Reactions used to illustrate the calculation of rate constants and halflives with Eq. (5.198). Cf. Fig. 5.21...

In the following chapters, we will consider an approach to the calculation of rate constants— transition-state theory—that do not take into account such details of the reaction dynamics. The theory will be based on the basic axioms of statistical mechanics where all partitionings of the total energy are equally likely, and it is assumed that all these partitionings are equally effective in promoting reaction. 

The KMC method requires that a certain set of reactions be specified. This set includes transformations from reagents to products that correspond to local minima on the PES. Therefore, in the general case, to construct the table of reactions in KMC method, one should determine local minima on the PES and then determine the rate constants of transitions between them. Since PES is modified after each reaction, the process of searching for local minima should be dynamically performed in the course of the KMC run. This implementation of the KMC method is called dynamic KMC, since the set of all possible reactions at a given time is determined dynamically during the run rather than specified before calculations. Therefore, to implement dynamic KMC, it is required to specify an energy functional for the calculation of PES, methods of searching for local minima, and methods for the calculation of rate constants for transitions between local minima. 

Calculation of rate constants for cross reactions from known data on self-exchange reactions. 

Calculation of rate constants (k]2) for organic electron transfer processes, using the Marcus cross relations (62) and (63)... 

In the following, we shall confine ourselves to some indications and remarks based mainly on our own experience. These have mostly to be of a qualitative nature since systematic quantitative measurements that allow calculation of rate constants have been started only recently (Section 2). 

Thus, if a putative radical scavenger is incubated with hydrogen peroxide and the reaction mixture sampled for analysis of hydrogen peroxide at various times, the rates of loss of hydrogen peroxide can be measured to allow calculation of rate constants. 

Polzius R, DieBel E, Bier F, Bilitewski U (1997) Real-time observation of affinity reactions using grating couplers determination of the detection limit and calculation of rate constants. Anal Biochem 248 269-276... 

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