Law Stokes-Einstein

The random walk model discussed in the previous section assumes the diffusion of molecules to evolve in a free environment. It assumes the collisions between molecules and helps in understanding the diffusion of one gas molecule to another. However, when the small particles or molecules are immersed in a dense phase, that is, liquid of viscosity fi, the role of dense phase must be explicitly brought out. The Stokes-Einstein law describes the diffusion coefficient D as 

In the previous section, we described the diffusion process using the motion of individual particles/molecules. Let us consider now a group of diffusers or particles instead of isolated walkers. The mass concentration (p) is defined as 

8m is the mass of collection of diffusing particles contained in volume 5V. The mass fiux through surface area 8A is defined as 

Fick s law describes the relation between the material fiux and concentration as 

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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... 

As seen in the preceding section, the counterions play a crucial role in the mobility of the polyelectrolyte molecules. Even in the absence of an external electric field, the counterions exert an induced electric field in the immediate environment of a charged segment which in turn significantly modifies the collective diffusion coefficient of the polymer. This additional contribution is absent for uncharged polymers, where the cooperative diffusion coefficient Dc is given by the Stokes-Einstein law in dilute solutions. 

Chromium(III) has a ground state in pseudo-octahedral symmetry. The absence of low-lying excited states excludes fast electron relaxation, which is in fact of the order of 10 -10 ° s. The main electron relaxation mechanism is ascribed to the modulation of transient ZFS. Figure 18 shows the NMRD profiles of hexaaqua chromium(III) at different temperatures (62). The position of the first dispersion, in the 333 K profile, indicates a correlation time of 5 X 10 ° s. Since it is too long to be the reorientational time and too fast to be the water proton lifetime, it must correspond to the electron relaxation time, and such a dispersion must be due to contact relaxation. The high field dispersion is the oos dispersion due to dipolar relaxation, modulated by the reorientational correlation time = 3 x 10 s. According to the Stokes-Einstein law, increases with decreasing temperature, and... 

We can find corresponding experimental results in relation to the limitations of the uniform continuum model. The halide ions have negative B coefficients in water while the Einstein laws predicts only positive B value. However, the electrical conductivities for the same ions are at most 20% larger than the prediction of the Stokes-Einstein law with the crystallographic radius for the slip boundary condition. 

Extending the Stokes-Einstein law to the high-frequency regime, we expect... 

Generally, mean size and size distribution of nanoparticles are evaluated by quasi-elastic light scattering also named photocorrelation spectroscopy. This method is based on the evaluation of the translation diffusion coefficient, D, characterizing the Brownian motion of the nanoparticles. The nanoparticle hydro-dynamic diameter, is then deduced from this parameter from the Stokes Einstein law. 

The molecular mass of 6 was estimated to be (1.0 + 0.5) x 10 from the data on the rates of sedimentation of its aqueous solution on ultracentrifugation, by use of the Stokes Einstein law. - " This method is analogous to the technique used for determination of the molecular mass of the cluster Au55(PPh3)i2Cl6 in the pioneering work by G. Schmid. 

Miller CC. The Stokes Einstein law for diffusion in solution. Proc R Soc Land SerA 1924 106(740) 724-49. 

In fact, two expressions of this type are very well known, the Stokes-Einstein law for the self-diffusion coefficient, and the Debye law for the rotational diffusion coefficient. Both of these laws for the appropriate diffusion coefficient D are derived by hydrodynamics and have Darj, where 17 is the coefficient of shear viscosity, a transport coefficient. The Stokes-Einstein and Debye laws were reconciled with formal theory with the use of mode-mode... 

Of course, one must always be careful not to read too much into explanations like that just given nevertheless, we do feel that a considerable amount of truth is contained in the droplet picture. We hasten to point out that the 6 in Eq. (136a), which occurs in the usual Stokes-Einstein law with stick boundary conditions, only occur because we have used O-Z theory this has been stressed by Swinney et In addition, note that droplet diffusion is a... 

We now wish to illustrate another of the three main consequences of mode-mode coupling mentioned in the introduction, that of the existence of expressions for transport coefiidents in terms of other transport coefficients such expressions could not exist if classical theory held. Of course, the results derived in the previous section are examples of such expressions. However, we wish to discuss cases that have nothing to do with critical phenomena. The best known example of the expressions of interest is the Stokes-Einstein law for the self-diffusion coefficient of a large spherical particle. 

The diffusion coefficients are related to the viscosity of the solvent, T], and hydrodynamic radius, r, of the diffusing species by the Stokes-Einstein Law ... 

The time scale t2 Y is set by the inverse of the decay rate y of the number of big droplets in the system due to coalescence. When the Stokes-Einstein law is applied, t2 depends on the distance between large droplets and on... 

The general model was applied to the ion cluster structure of fluorinated ion exchange membrane shown in Eigure 5.9. The friction constant of free ions is given by the Stokes-Einstein law ... 

Stokes s law is known from the following type of experiment. Put a ball of radius a and mass m in a liquid. The gravitational force is mg. Measure its velocity. The ratio of gravitational force to velocity is 5- Figure 18.13 gives friction factors for particles that aren t spherical. Combining Equations (18.49) and (18.50) gives the Stokes-Einstein law of diffusion for spherical particles,... 

Because a sphere has a w-eight that is proportional to its volume m a the Stokes-Einstein law implies D m O/ Figure 18.14 confirms that diffusion... 

The Stokes-Einstein law gives the diffusion constant, kT 1.38 X 10- 3jk-i X 300K... 

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