Diffusion Stokes-Einstein relation

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

Since thermal agitation is the common origin of transport properties, it gives rise to several relationships among them, for example, the Nemst-Einstein relation between diffusion and conductivity, or the Stokes-Einstein relation between diffusion and viscosity. Although transport... 

Baxendale and Wardman (1973) note that the reaction of es with neutrals, such as acetone and CC14, in n-propanol is diffusion-controlled over the entire liquid phase. The values calculated from the Stokes-Einstein relation, k = 8jtRT/3jj, where 7] is the viscosity, agree well with measurement. Similarly, Fowles (1971) finds that the reaction of es with acid in alcohols is diffusion-controlled, given adequately by the Debye equation, which is not true in water. The activation energy of this reaction should be equal to that of the equivalent conductivity of es + ROH2+, which agrees well with the observation of Fowles (1971). 

Measurement of the translational diffusion coefficient, D0, provides another measure of the hydrodynamic radius. According to the Stokes-Einstein relation... 

With the help of the Stokes-Einstein relation, the translational diffusion coefficient may be calculated according to... 

Einstein s work on the diffusion of particles (1906) led to the well known Stokes-Einstein relation giving the diffusion coefficient D of a sphere ... 

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... 

Changes in fluidity of a medium can thus be monitored via the variations of Jo/J — 1 for quenching, and Ie/Im for excimer formation, because these two quantities are proportional to the diffusional rate constant kj, i.e. proportional to the diffusion coefficient D. Once again, we should not calculate the viscosity value from D by means of the Stokes-Einstein relation (see Section 8.1). 

The translational diffusion coefficient of micelles loaded with a fluorophore can be determined from the autocorrelation function by means of Eqs (11.8) or (11.9). The hydrodynamic radius can then be calculated using the Stokes-Einstein relation (see Chapter 8, Section 8.1) ... 

A unified understanding of the viscosity behavior is lacking at present and subject of detailed discussions [17, 18]. The same statement holds for the diffusion that is important in our context, since the diffusion of oxygen into the molecular films is harmful for many photophysical and photochemical processes. However, it has been shown that in the viscous regime, the typical Stokes-Einstein relation between diffusion constant and viscosity is not valid and has to be replaced by an expression like... 

The self-translational diffusion coefficient D is related to f, by the Stokes-Einstein relation and is given by... 

The z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

It should be noted that this derivation contains no assumptions about the shape of the particles. However, when the particles are assumed to be spherical, we can substitute Equation (8) for/, and the resulting equation for the diffusion coefficient is the well-known Stokes-Einstein relation. 

The use of the Stokes-Einstein relation with the above value of the average diffusion coefficient leads to a hydrodynamic radius of roughly 30 nm, which is consistent with the specification of the manufacturer. ... 

By comparing this result with Fick s first law (Eq. 18-6), we get the Stokes-Einstein relation between the diffusivity in aqueous solutions and the solution viscosity q ... 

As for Illustrative Example 18.2a (diffusivity of CFC-12 in air), these values agree fairly well with each other, except for the Stokes-Einstein relation, which was not meant to be a quantitative approximation but an expression to show qualitatively the relationship between diffusivity and other properties of both molecule and fluid. 

D = (1.8 0.6) x 10 12m2s 1. The encounter distance was estimated to be 1.32 nm, which, when used in the Stokes—Einstein relation for the mutual diffusion coefficient, eqn. (28), givesD as (1.1 0.03) x 10 I2m2 s-1, in reasonable agreement with the estimate from fitting experimental and theoretical decay curves (mentioned above). The germinate pair recombination probability at long times was measured and its increase correlates moderately well with T1/2/rj as noted for other systems (see Sect. 3.1) but was concave upwards (see ref. 22). 

In Fig. 3, the orientational diffusion time constants ror of the first solvation shell of the halogenie anions CD. Br, and D are presented as a function of temperature. From the observation that ror is shorter than rc, it follows that the orientational dynamics of the HDO molecules in the first solvation shell of the Cl ion must result from motions that do not contribute to the spectral diffusion, i.e. that do not affect the length of the O-H- -Cl hydrogen bond. Hence, the observed reorientation represents the orientational diffusion of the complete solvation structure. Also shown in Fig. 3 are fits to the data using the relation between ror and the temperature T that follows from the Stokes-Einstein relation for orientational diffusion ... 

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 ... 

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 (

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... 

The motion caused by thermal agitation and the random striking of particles in a liquid by the molecules of that liquid is called Brownian motion. This molecular striking results in a vibratory movement that causes suspended particles to diffuse throughout a liquid. If the colloidal particles can be assumed to be approximately spherical, then for a liquid of given viscosity (q), at a constant temperature (T), the rate of diffusion, or diffusion coefficient (D) is inversely related to the particle size according to the Stokes-Einstein relation (ref. 126) ... 

One aspect of the dynamics of micellar systems that has received a renewed interest during recent years is the translational motion of the micelles themselves. In the simplest approximation, the translational diffusion coefficient, D, of a spherical micelle is related to the hydrodynamic radius rM through the Stokes-Einstein relation... 

The diffusion constant Di of a particle in a solvent is related to the viscosity of the solvent by the Stokes-Einstein relation known from hydrodynamics ... 

Diffusionally mediated collisions between two floccules of equal size can be described by a second-order rate coefficient KD = 8irRD, where R is the radius and D is the diffusion coefficient of a floccule. Upon invoking the Stokes-Einstein relation, D = kBT/67ri7R, one derives Eq. 6.2. For an introductory discussion of the second-order rate law for particle collisions, see, for example, Chap. 11 in P. C. Hiemenz, Principles of Colloid and Surface Chemistry, Marcel Dekker, New York, 1986. 

This is known as the Stokes-Einstein relation and is independent of the charge of the species. Using this expression, diffusion coefficients can be estimated from viscosity measurements, so long as Stokes Law is applicable. It is used particularly for macromolecules. 

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

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