Stokes-Einstein diffusion equation

See Nernst-Plank equation, diffusion, Stokes-Einstein equation. 

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

Wilke-Chang This correlation for D°b is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it apphes to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. ( )b is an association factor of solvent B that accounts for hydrogen bonding. 

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. 

In connection with the earlier consideration of diffusion in liquids using tire Stokes-Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be T(exp(—Qvis/RT)) according to this equation, if the activation energy for viscous flow is included. 

Using the Stokes-Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals... 

By equating Fiek s seeond law and the Stokes-Einstein equation for diffusivity, Smoluehowski (1916,1917) showed that the eollision frequeney faetor takes the form... 

Very commonly Eq. (4-5) is combined with Eq. (4-6), the Stokes-Einstein equation relating the diffusion coefficient to the viscosity -q. 

Equations (4-5) and (4-7) are alternative expressions for the estimation of the diffusion-limited rate constant, but these equations are not equivalent, because Eq. (4-7) includes the assumption that the Stokes-Einstein equation is applicable. Olea and Thomas" measured the kinetics of quenching of pyrene fluorescence in several solvents and also measured diffusion coefficients. The diffusion coefficients did not vary as t) [as predicted by Eq. (4-6)], but roughly as Tf. Thus Eq. (4-7) is not valid, in this system, whereas Eq. (4-5), used with the experimentally measured diffusion coefficients, gave reasonable agreement with measured rate constants. 

T0 is a reference temperature which can be identified with T, and although the constant B is not related to any simple activation process, it has dimensions of energy. This form of the equation is derived by assuming an electrolyte to be fully dissociated in the solvent, so it can be related to the diffusion coefficient through the Stokes-Einstein equation. It suggests that thermal motion above T0 contributes to relaxation and transport processes and that... 

The same equation applies to other solvents. It is often easier to incorporate an expression for the diffusion coefficient than a numerical value, which may not be available. According to the Stokes-Einstein equation,6 diffusion coefficients can be estimated from the solvent viscosity by... 

Using the Stokes-Einstein equation of diffusion coefficient ... 

According to Stokes-Einstein equation, the diffusion coefficient is inversely proportional to the solution viscosity which increases with temperature. Hence, a lower diffusion coefficient corresponds to a lower size molecule. 

The Stokes-Einstein equation can be successfully used to explain diffusion under the following conditions [401], where (a) the diffusing molecule is large with respect to the molecules defining the medium, (b) the medium has a very low viscosity, and (c) no solute-solvent interactions occur. 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

Measurements of CuS04 molecular diffusivity by Cole and Gordon (C12a), referred to above, were carried out in diaphragm cells, mostly at 18°C. Their results were correlated by Fenech and Tobias (F3) using the Stokes-Einstein equation... 

The diffusivity, D, of a particle is inversely proportional to its radius, r, according to the Stokes-Einstein equation,... 

Substituting the diffusion coefficient D into its expression in the Stokes-Einstein equation, we... 

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... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

The Stokes-Einstein equation (Equation 9.7) is often used to describe the relationship between the diffusion coefficient of a solute and the viscosity of the solution... 

Here Tq is — C2 and is a prefactor proportional to which is determined by the transport coefficient (in this case at the given reference temperature. The constant B has the dimensions of energy but is not related to any simple activation process (Ratner, 1987). Eqn (6.6) holds for many transport properties and, by making the assumption of a fully dissociated electrolyte, it can be related to the diffusion coefficient through the Stokes-Einstein equation giving the form to which the conductivity, <7, in polymer electrolytes is often fitted,... 

Because PCS relies on the determination of the particle diffusion coefQcient, it is not a direct method for the determination of particle sizes. Information on the particle size can be obtained via the Stokes-Einstein equation... 

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