Smoluchowski rate constant

Here ko = 471 gD is the rate constant for a diffusion-controlled reaction (Smoluchowski rate constant) for a perfectly absorbing sphere. For long times, kf t) approaches its asymptotic constant value kf = kofkD/(kqf + kD) as... 

In this case, 4tt r D is the Smoluchowski rate constant, which is useful for describing radial diffusion to a cell surface totally covered with transport sites. The rate corresponding with fractional surface coverages of carriers has been derived both analytically [35,237,348] and numerically [349] by assuming that the radius of the cell is large compared to the carrier radii, a, and that the NR carriers are sufficiently dispersed so as to be independent and noncompeting. The result ... 

Diffusion-limited aggregation (DLA) is often referred to as a fast process because the above model predicts the maximum aggregation rate. Reaction-limited aggregation (RLA) rates are slower than predicted by the DLA model. The effect of the slow attachment reaction rate on the aggregation rate is expressed as a stability ratio (W) and is defined as the ratio of the Smoluchowski rate constant (ks) to the observed rate constant (ko). 

Table I. Diameters of the polystyrene microspheres and Smoluchowski rate constants for the monodisperse suspensions in 1 mol dm KCl...

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

The temperature dependence of a diffusion-controlled rate constant is very small. Actually, it is just the temperature coefficient of the diffusion coefficient, as we see from the von Smoluchowski equation. Typically, Ea for diffusion is about 8-14 kJ mol"1 (2-4 kcal mol-1) in solvents of ordinary viscosity. 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

In liquids, collisional energy transfer takes place by multistep diffusion (the rate determining step) followed by an exchange interaction when the pair is very close. The bimolecular-diffusion-controlled rate constant is obtained from Smoluchowski s theory the result, including the time-dependent part, may be written as... 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

If the bimolecular process is diffusion-limited kq is identical to the diffusional rate constant ki, which can be written in the following simplified form (proposed for the first time by Smoluchowski) ... 

In reality, the diffusional rate constant is time-dependent, as explained at the end of Section 4.2.1, and should be written as ki(t). Several models have been developed to express the time-dependent rate constant (see Box 4.1). For instance, in Smoluchowski s theory, ki (t) is given by... 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

In contrast to Smoluchowski s theory, the rate constant for the intrinsic reaction has a finite value kR when A and B are at the distance of closest approach... 

By comparing time-resolved and steady-state fluorescence parameters, Ross et alm> have shown that in oxytocin, a lactation and uterine contraction hormone in mammals, the internal disulfide bridge quenches the fluorescence of the single tyrosine by a static mechanism. The quenching complex was attributed to an interaction between one C — tyrosine rotamer and the disulfide bond. Swadesh et al.(()<>> have studied the dithiothreitol quenching of the six tyrosine residues in ribonuclease A. They carefully examined the steady-state criteria that are useful for distinguishing pure static from pure dynamic quenching by consideration of the Smoluchowski equation(70) for the diffusion-controlled bimolecular rate constant k0,... 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

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.)...

The kinetic rate constant for the association process (7cjN) has an upper limit set by diffusion. In other words, the rate of the fastest association processes cannot exceed the rate by which the host and the guest diffuse to encounter in solution. The maximum value of kD can then be estimated using the well-known Smoluchowski equation8 ... 

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). 

To estimate the rate constant for a reaction that is controlled strictly by the frequency of collisions of particles, we must ask how many times per second one of a number n of particles will be hit by another of the particles as a result of Brownian movement. The problem was analyzed in 1917 by Smoluchowski,30/31 who considered the rate at which a particle B diffuses toward a second particle A and disappears when the two codide. Using Fick s law of diffusion, he concluded that the number of encounters per milliliter per second was given by Eq. 9-26. 

The rate constant kp is given in terms of physical parameters (Boltzmann Constant KB, the absolute temperature T, and the absolute viscosity rj) that characterize these transport conditions. In the case of not completely destabilized colloids, when according to v. Smoluchowski so-called slow coagulation is observed, the rate constant contains in addition the collision efficiency factor, p, the fraction of collisions leading to permanent attachment under perikinetic conditions ... 

Figure 9 shows the temperature dependence of the recovered kinetic rate coefficients for the formation (k bimolecular) and dissociation (k unimolecular) of pyrene excimers in supercritical CO2 at a reduced density of 1.17. Also, shown is the bimolecular rate coefficient expected based on a simple diffusion-controlled argument (11). The value for the theoretical rate constant was obtained through use of the Smoluchowski equation (26). As previously mentioned, the viscosities utilized in the equation were calculated using the Lucas and Reichenberg formulations (16). From these experiments we obtain two key results. First, the reverse rate, k, is very temperature sensitive and increases with temperature. Second, the forward rate, kDM, 1S diffusion controlled. Further discussion will be deferred until further experiments are performed nearer the critical point where we will investigate the rate parameters as a function of density. 

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