Stokes-Einstein theory

The model was also checked by evaluating the center-of-mass friction. It was shown that hydrodynamic interactions are important for solvent-separated atoms, 8 A, but not for the diatomic with 2.66 A. The mass dep>endences of the isolated iodine and argon frictions were not consistent with hydrodynamics estimates of the Stokes-Einstein theory (Section III E). Rather, they are in agreement with the Enskog theory corrected for caging by the Herman-Adler results for hard spheres. Further studies are required which avoid the use of Eq. (5.8). 

In the Stokes-Einstein theory of brownian motion, the particle motion at very low concentrations depends on the viscosity of tne suspending liquid, the temperature, and the size of the particle. If viscosity and temperature are known, the particle size can be evaluated from a measurement of the particle motion. At low concentrations, this is the hydrodynamic diameter. 

This rule has the obvious advantage of drastically reducing the number of useful recombination constants simply related to the self-recombinations. The kcoii evaluation deserves some further consideration. First of all, the Stokes-Einstein theory suggests that when molecules are not too long (the exclusion regards the case of polymers or that one of solvents with extended linear backbones), the diffusion coefficient is... 

The dependence of reaction rate constants on the viscosity of the solvent predicted by this model can be derived from Equation (2.8) along with the assumption that D oc T/ij as in the Stokes-Einstein theory. For simplicity we take as an example the case of a reaction between equal-sized molecules, for which ko = 4RT/rf (Equation (2.2)). Inserting this in Equation (2.8), we obtain ... 

Motional factors determining the linewidths can be envisaged in terms of a frictional model (Debye-Stokes-Einstein theory) according to which... 

Intcrmolecular dipole-dipole relaxation depends on the correlation time for translational motion rather than rotational motion. Intermolecular dipole-dipole interactions arise from the fluctuations which are caused by the random translational motions of neighboring nuclei. The equations describing the relaxation processes are similar to those used to describe the intramolecular motions, except is replaced by t, the translation correlation time. The correlation times are expressed in terms of diffusional coefficients (D), and t, the rotational correlation time and the translational correlation time for Brownian motion, are given by the Debye-Stokes-Einstein theory ... 

From slopes of such figures, experimental activation volumes AV obs were obtained. Except for TEMPO in [emim][BF4], all other measurements in [bmim][BF4], [omim][BF4], [omimJpFe], [emim][BF4] and [bmim][PF6] showed experimental AF obs Values which corresponded, within less than 10%, to the calculated AV, -values obtained from the Debye-Stokes-Einstein theory, see Fig. 7(b). 

In the derivation of both Eqs. (9.4) and (9.9), the disturbance of the flow streamlines is assumed to be produced by a single particle. This is the origin of the limitation to dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated as the sum of the individual nonoverlapping disturbances. When more than one sphere is involved, the same limitation applies to Stokes law also. In both cases contributions from the walls of the container are also assumed to be absent. 

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

For this case, the need for a molecular theory is cleverly avoided. The Stokes-Einstein equation is (Bird et al.)... 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

The validity of Eq. (12) for correlations of limiting-current measurements was first questioned by Arvia et al. (A5), and later by Wragg and Ross (W13a). The latter found that limiting currents in an annular flow cell could be correlated in better agreement with the Leveque mass-transfer theory if a lower mobility (Stokes-Einstein) product were employed, such as... 

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 gas A must transfer from the gas phase to the liquid phase. Equation (1) describes the specific (per m2) molar flow (JA) of A through the gas-liquid interface. Considering only limitations in the liquid phase, this molar flow notably depends on the liquid molecular diffusion coefficient DAL (m2 s ). Based on the liquid state theories, DA L can be calculated using the Stokes-Einstein expression, and many correlations have been developed in order to estimate the liquid diffusion coefficients. The best-known example is the Wilke and Chang (W-C) relationship, but many others have been established and compared (Table 45.4) [28-33]. 

Various modifications of the Stokes-Einstein relation have been proposed to take into account the microscopic effects (shape, free volume, solvent-probe interactions, etc.). In particular, the diffusion of molecular probes being more rapid than predicted by the theory, the slip boundary condition can be introduced, and sometimes a mixture of stick and slip boundary conditions is assumed. Equation (8.3) can then be rewritten as... 

Another microscopic approach to the viscosity problem was developed by Gierer and Wirtz (1953) and it is worthwhile describing the main aspects of this theory, which is of interest because it takes account of the finite thickness of the solvent layers and the existence of holes in the solvent (free volume). The Stokes-Einstein law can be modified using a microscopic friction coefficient ci micro... 

Hasinoff noted that the rate coefficient of formation of encounters pairs, fcD, was smaller than predicted from the Smoluchowski—Stokes—Einstein rate coefficient [eqn. (29)]. In aqueous glycerol, this reduction was by 0.14 times, in aqueous polyethylene glycol by 0.30 times, and in aqueous ethylene glycol by 0.11 times. Hasinoff compared these reductions in rate of diffusive rate of formation of encounter pairs with three theories of anisotropic reactivity due to Weller [262], Schmitz and Schurr [257] and... 

It is interesting to compare conductance behavior with that of the shear viscosity, because conventional hydrodynamic conductance theories relate A to the frictional resistance of the surrounding medium. At first glance, one would expect from the Stokes-Einstein equation a critical anomaly of the... 

Other than dynamical correlations, transport properties have also been derived using hydrodynamic theory. In hydrodynamics the diffusion of a tagged particle is defined by the Stoke-Einstein relation that is given by the following well-known expression ... 

The above-mentioned computer simulation and experimental studies have addressed various aspects of mass dependence, but they all show that the selfdiffusion coefficient of a tagged molecule exhibits a weak mass dependence, especially for solutes with size comparable to or larger than the size of the solvent molecules. Sometimes this mass dependence can be fitted to a power law, with a small exponent less than 0.1 [99]. This weak mass dependence has often been considered as supportive of the hydrodynamic picture. In hydrodynamics the diffusion of a solute is conventionally described by the well-known Stokes-Einstein (SE) relation, which predicts that the diffusion is totally independent of the mass of the solute. Kinetic theory, on the other... 

Figure 8. The ratio of the self-diffusion coefficient of the solute (Di) to that of the solvent molecules (D ) plotted as a function of the solvent-solute size ratio (

The water proton NMRD profile of Cu(II) aqua ion at 298 K [108] (Fig. 5.36) is in excellent accordance with what expected from the dipole-dipole relaxation theory, as described by the Solomon equation (Eq. (3.16)). The best fitting procedure applied to a configuration of 12 water protons bound to the metal ion provides a distance between water protons and the paramagnetic center equal to 2.7 A, and a correlation time equal to 2.6 x 10 11 s, which defines the position of the cos dispersion. The correlation time is determined by rotation as expected from the Stokes-Einstein equation (Eq. (3.8)). The electron relaxation time is in fact expected to be one order of magnitude longer (see Table 5.6). This also ensures... 

Figure 3.8. (a) The linear viscosity dependence of the inverse ionization rate in the reaction studied in Ref. 98. Bullets—experimental points solid line—fit performed with the generalized Collins—Kimball model, (b) The effective quenching radius for the same reaction in the larger range of the viscosity variation. Bullets—experimental points solid fine—fit performed with the encounter theory for the exponential transfer rate. The diffusion coefficient D given in A2/ns was calculated from the Stokes—Einstein relationship corrected by Spemol and Wirtz [100]. 

The ability to correctly reproduce the viscosity dependence of the dephasing is a major accomplishment for the viscoelastic theory. Its significance can be judged by comparison to the viscosity predictions of other theories. As already pointed out (Section II.C 22), existing theories invoking repulsive interactions severely misrepresent the viscosity dependence at high viscosity. In Schweizer-Chandler theory, there is an implicit viscosity dependence that is not unreasonable on first impression. The frequency correlation time is determined by the diffusion constant D, which can be estimated from the viscosity and molecular diameter a by the Stokes-Einstein relation ... 

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