Debye rate constant

To treat more realistic models for the kinetics of biomolecular encounters, a simulation approach has been developed and applied to proteins.370,371,3718 This approach merges stochastic dynamics methodology (Chapt. IV.D) with the analytic result for the Debye rate constant for a pair of particles, moving diffusively through solvent with a centrosymmetric interaction potential.365 The analytical expression for the Debye rate constant, ko(,R), to first achieve a separation, R, is given by... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

It then also follows that the rate constant for a first-order reaction, whether or not the solvent is involved, is also independent of ionic strength. This statement is true at ionic strengths low enough for the Debye-Huckel equation to hold. At higher ionic strengths, predictions cannot be made about reactions of any order because all of the kinetic effects can be expected to show chemical specificity. 

This composite rate constant is predicted to have a different ionic strength dependence for the two schemes. According to the Br0nsted-Debye-Hiickel equation, the composite rate constant for Eq. (9-76) will be independent of ionic strength if Scheme ... 

Of the ethers, rate constants for es reactions are available for tetrahydrofuran (THF). Since the neutralization reaction, THF+ + es, is very fast, only fast reactions with specific rates 10u-1012 M s"1 can be studied (see Matheson, 1975, Table XXXII). Bockrath and Dorfman (1973) compared the observed rate of the reaction es + Na+ in THF, 8 x 1011 M 1s 1, with that calculated from the Debye equation, <3 x 1011 M-1s-1. Although the reaction radius is not well known, the authors note on a spectroscopic basis that Na+ and es are strongly coupled in THF Thus, the reaction of a solute with (Na+, es) in THF is much slower, sometimes by an order of magnitude, than the corresponding reaction with es only. Reaction with pyrene is an example. 

Rate constant for homogeneous self exchange, corrected for electrostatic work terms using Debye-Huckel-Bronsted model. Data taken from sources quoted in ref. 15 unless otherwise stated. 

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. 

Some reactants have rate constants higher than 3xl0 M sec examples are nitrobenzene and o-dinitrobenzene. These two compounds have large dipole moments of 4.1 and 6.1 Debye, respectively, and it has been shown [110] that the rate constants in cyclohexane increase with dipole moment because the reaction radius increases. That dependence is given by [117] ... 

The simulation results of the electron ion recombination rate constant obtained in Ref. 39 are plotted in Fig. 5. The figure shows that the rate constant becomes lower than the Debye-Smoluchowski value when the electron mean free path exceeds —O.Olrc. At higher values of X, the ratio kjk further decreases with increasing mean free path. The simulation results are found to be in good agreement with the experimental data on the electron ion recombination rate constant in liquid methane, which are also plotted in Fig. 5. 

Figure 5 The rate constant of bulk electron-ion recombination, relative to the Debye-Smoluchowski value [Eq. (36)], as a function of the electron mean free path X. The solid line represents the simulation results, and the circles show the experimental data for liquid methane [49]. (From Ref. 39.)...

In the liquid phase, observed electron-ion recombination rate constants kr in a variety of nonpolar media were, as shown in Fig. 15, in good agreement with the values of k, calculated from the reduced Debye equation,... 

On the other hand, if the rate constant for the quenching step exceeds that expected for a diffusion-controlled process, a modification of the parameters in the Debye equation is indicated. Either the diffusion coefficient D as given by the Stokes-Einstein equation is not applicable because the bulk viscosity is different from the microviscosity experienced, by the quencher (e.g. quenching of aromatic hydrocarbons by O, in paraffin solvents) or the encounter radius RAb is much greater than the gas-kinetic collision radius. In the latter case a long-range quenching... 

The first step is described by back-reaction boundary conditions with intrinsic rate constants Aj and k.d. This is followed by a diffusion second step in which the hydrated proton is removed from the parent molecule. TTiis latter step is described by the Debye-Smoluchowski equation (DSE). 

The rate constants of reactions of hydrated electrons with some accep-tors-anions substantially exceed the diffusion rate constants calculated with the help of the Debye equation [Chap. 2, eqn. (45)l(see Chap. 2, Sect. 4). This excess is usually attributed to the capture of electrons by acceptors via tunneling at distances exceeding the sum of the reagents [28,89,111,1201- In this case, the tunneling distance can be estimated from experimental rate constants for reactions of eaq with acceptors [109] by means of the expression... 

The rates for many of the e aq reactions in Table II are very fast, exceeding 1010M-1 sec.-1, and therefore, may be limited by the rates of diffusion-controlled encounters. The equation from which the diffusion-limited rate constants may be calculated for ionic species is due to Debye... 

The values of th2o s,h2o in Table II indicate a value for th2o ranging from 10 9 to 10 10 sec., based on a value for s.h,o of 6 X 10 M 1 sec."1 from Debye s equation (11). The constants in Table II are also a measure of the relative reactivity of solute with H20 for any particular form of radiation for which th2o can be considered a constant. The relative reactivities of solute with H20 and e aq for Co60 7-radiation differ slightly but significantly as shown in Table III. Relative reactivities with e aq are based on measurements of absolute rate constants by pulsed-radiolysis techniques (4). 

The diffusional rate constant kD is calculated on the basis of the Debye-Hiickel theory (Equation 6.107), where the distance tr is the sum of A and B radii in the hard-sphere approximation. 

The quite another temperature dependence of the rate constant at helium temperatures is resulted in the case when the principal contribution to dispersion a in formula (25a) gives the acoustic phonons. Their frequencies lie in the interval [0, lud], where tuD is Debye s frequency. Even if hin0 kT, it exists always in the range of such low frequencies that haxkT. It is these phonons that give the contribution depending on the temperature in the dispersion a [15], One assumes that the displacements of the equilibrium positions of phonon modes Sqs do not depend on frequency. Then, the calculation of the rate constant gives at low temperatures, hcou>kT,... 

The more recent theories of chemical conversions [59-61] take into account the fact that the process of overcoming the activation barrier involves a cooperative change of more than one degree of freedom for the starting reagents subsystem. For the surface processes this is expected to lead to a need for considering the dynamics of the solid atom motion and, at least, the model should include information on Debye frequencies for its atoms (see, e.g., Ref. [62]). An additional inconvenience of the models for the elementary surface processes is associated with the fact that the frequencies of the surface atom oscillations differ from those inside the solid. Consideration of the multiphonon contributions to the probabilities that the elementary process can take place results in a significant modification of its rate constant up to the complete disappearance of the activation form of the temperature dependence [63,64]. 

The initial step of chemical reactions is an encounter of reactants by diffusion, and the subsequent reactions proceed to give products from the activated complex. The diffusion energy in solution is 15 kJ/mol, while many chemical reactions need an activation energy of 40 kJ-100 kJ/ mol. If the activation energy of the reaction is low enough compared to the diffusion energy, then the diffusion determines the overall reaction, which has been referred to as a diffusion-controlled or -limited reaction. From Debye s equation on the diffusion-limited bimolecule reaction, the maximum value for the second-order reaction rate constant is estimated to be 109-1010 M 1 s l (25 °C). The fastest reaction in aqueous solution is that of oxonium and OH- ions at a rate constant of 1.4 X 10nM 1 s 1 (25°C) ... 

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