Stokes-Einstein volume

To = temperature of the solvent at which tan A goes through a maximum. These values are presented in Table II. VSE (the Stokes-Eeinstein volume) is calculated for a spherical molecule if the molecule is aspherical this calculation (VSE) is called Vapparent The Vapparent can be smaller or larger than the Stokes-Einstein volume and varies from the equivalent sphere volume obtained by solution of equations 3,4 and 5. 

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. 

Table II. Comparison of Stokes-Einstein (Vapparent) Volume to the Perrin Volume...

The Stokes-Einstein equation predicts that DfxITa is independent of the solvent however, for real solutions, it has long been known that the product of limiting interdiffusion coefficient for solutes and the solvent viscosity decreases with increasing solute molar volume [401]. Based upon a large number of experimental results, Wilke and Chang [437] proposed a semiempirical equation,... 

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. 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

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... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Flocculation kinetics can be described in different ways. Here we introduce a treatment first suggested by Smoluchowski [547], and described in Ref. [538], p. 417. The formalism can also be used to treat the aggregation of sols. A prerequisite for coalescence is that droplets encounter each other and collide. Smoluchowski calculated the rate of diffusional encounters between spherical droplets of radius R. The rate of diffusion-limited encounters is SttDRc2, where c is the concentration of droplets (number of droplets per unit volume). For the diffusion coefficient D we use the Stokes-Einstein relation D = kBT/finr/R. The rate of diffusion-limited encounters is, at the same time, the upper limit for the decrease in droplet concentration. Both rates are equal when each encounter leads to coalescence. Then the rate of encounters is given by... 

Here, R is the radius of the sphere, q is the coefficient of viscosity, kB is Boltzmann s constant, and T is temperature. Equation (4) implies that rR should vary linearly with volume, or mass, in the range where the Stokes-Einstein equation is valid. Figure 5 shows the roughly linear behavior of rR for these compounds, and illustrates why polymeric conjugates of Gd3+ chelates remain a very attractive method of modulating both rR and the intravenous retention time (t1/2) of BPCAs. 

Application of the Stokes-Einstein equation requires a value for the solute radius. A simple approach is to assume the molecule to be spherical and to calculate the solute radius from the molar volume of the chemical groups making up the molecule. Using values for the solute radius calculated this way along with measured and known diffusion coefficients of solutes in water, Edward [26]... 

S nuclear quadrupole coupling constants have been determined from line width values in some 3- and 4-substituted sodium benzenesulphonates33 63 and in 2-substituted sodium ethanesulphonates.35 Reasonably, in sulphonates R — SO3, (i) t] is near zero due to the tetrahedral symmetry of the electronic distribution at the 33S nucleus, and (ii) qzz is the component of the electric field gradient along the C-S axis. In the benzenesulphonate anion, the correlation time has been obtained from 13C spin-lattice relaxation time and NOE measurements. In substituted benzenesulphonates, it has been obtained by the Debye-Stokes-Einstein relationship, corrected by an empirically determined microviscosity factor. In 2-substituted ethanesulphonates, the molecular correlation time of the sphere having a volume equal to the molecular volume has been considered. 

Molecular weight and molecular volume Drag diffusion in simple liquids is expressed by Stokes-Einstein equation ... 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

The diffusion coefficients associated with translational motions when the radii of the diffusing radicals are not much larger than that of the solvent are expressed more accurately by D = kTI6nrr T (where r is the radius of the diffusing radical assuming a spherical shape and r (=yxr ) is the microviscosity. The value of /, the microfriction factor, can be calculated or taken equal to DsE/f exptb the ratio between the Stokes-Einstein diffusion coefficient (that considers van der Waals volumes, but not interstitial volumes) and the experimentally measured diffusion coefficient, Dexpti- As will be discussed later, these relationships appear to hold even in some polymer matrices. 

Assume one takes a 1 1 molten salt for which the increase of volume on melting is 20%, while the intemuclear distance shrinks by 5%. Calculate on the basis of simple statistics the fraction of paired vacancies. For simplicity, assume the radii of the cation and the anion are equal (as in KF) and use the Stokes-Einstein equation to calculate the diffusion coefficient of the ions and that of the paired vacancies. (The viscosity of KE at 1000 K is 1.41 centipoise the mean radius... 

Various models have been suggested to describe the interdependence of an ion s ionic mobility, its size, and its charge. Three of these models are described later (a) the Stokes-Einstein model, (b) the free-volume, or sieving. 

Initially, Roberts et al. [77] assumed a free-volume model. The results for different sets of data are analyzed by Yoshida and Roberts [68] and reproduced in Figure 9. Despite scatter, the slopes are similar. Yoshida and Roberts [62,68] have also compared three models that may describe the interdependence of solute ionic mobility molecular size and charge. These are the Stokes-Einstein, free-volume, and pore-restriction models. We now examine each of these models in more detail. 

Yoshida and Roberts [62,68] have expressed the size dependence of in terms of the Stokes-Einstein, free-volume, and pore-restriction models and... 

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