Diffusion Einstein expression

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 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]. 

Axial diffusion variance. The variance due to longitudinal diffusion is expressed by the Einstein equation [24,27-30]... 

The Peclet Number can be interpreted as the ratio between the transport time over the distance L by diffusion and by advection, respectively. Transport time by diffusion is expressed by the relation of Einstein and Smoluchowski (Eq. 18-8) ... 

The failure of the model to reproduce satisfactorily the dynamics of PeMe (Figure 3B) can be attributed to its slower dynamics. Diffusional processes then become more relevant and our rough estimation of kr, by means of the Stokes-Einstein expression, is probably not good enough. Much better agreement can be obtained when the translational diffusion coefficient is calculated with the semiempirical expression of Spemol and Wirtz [5]. 

Self diffusion coefficients can be obtained from the rate of diffusion of isotopically labeled solvent molecules as well as from nuclear magnetic resonance band widths. The self-diffusion coefficient of water at 25°C is D= 2.27 x 10-5 cm2 s 1, and that of heavy water, D20, is 1.87 x 10-5 cm2 s 1. Values for many solvents at 25 °C, in 10-5 cm2 s 1, are shown in Table 3.9. The diffusion coefficient for all solvents depends strongly on the temperature, similarly to the viscosity, following an Arrhenius-type expression D=Ad exp( AEq/RT). In fact, for solvents that can be described as being globular (see above), the Stokes-Einstein expression holds ... 

This is the Slokes-Einstein expression for the coefficient of diffusion, ft relates D to the properties of the fluid and the particle through the friction coefficient discussed in the next section,... 

This factor amounts to about 10-20 for mono-valent ions with radia r+ + r > 5 A when they associate in ethereal solvents of dielectric constant D = 5-8 i.e. their rates of encounters are by order of magnitude larger than of similar neutral particles. Using the Stokes-Einstein expression for the diffusion constants 2) = kT/6jn/r, one finds... 

Equation (44) can be transformed further by making a few more assumptions. Writing Rjj = r- + r-, and introducing the Stokes-Einstein expression for the diffusion constant of spheres in a continuous medium of viscosity rj... 

Often, the results of diffusion experiments in dilute solution are reported in terms of the hydrodynamic radius i h, defined through the Stokes-Einstein expression by solving for R after Do is measured. The reporting of i h is common even when solute shape is far from spherical. 

The coefficients ci(0), i = 1. .. n) represent the intensity weights of the different particles wiA diffusion coefficients Di = Fj/q, i = 1. .. n at scattering angle 0. In the third step the set diffusion coefficients Di) are related to the set particle sizes (di). To this end the Stokes-Einstein expression for the diffusion coefficient is used... 

In the case of the intermolecular contribution to relaxation times of proton-containing molecules and for the effect of pressure on self-diffusion one can use the Stokes-Einstein expression, which relates the shear viscosity, rj, to the self-diffusion coefficient D ... 

The Nernst-Einstein expression can also be applied to ionic conduction in liquids. However, the derivation of an expression for the mobility from first principles is much more difficult, because there is no regular lattice. Furthermore, in a liquid any conceivable, though irregular, site network is itself continuously rearranging via self-diffusion. 

Einstein expressed the diffusion coefficient D in terms of the particle radius a, fluid dynamic viscosity t, absolute temperature T, gas constant R, and Avogadro s Number Na (Einstein 1905) ... 

For particles of any shape at an absolute temperature T, Einstein showed that f is related to the experimental diffusion coefficient D by the expression... 

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... 

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. 

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... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

The mobility ratio equal to the diffusion ratio in this equation would naturally follow from application of the Nemst-Einstein equation, Eq. (88), to transport gels. Since the Nemst-Einstein equation is valid for low-concentration solutes in unbounded solution, one would expect that this equation may hold for dilute gels however, it is necessary to establish the validity of this equation using a more fundamental approach [215,219]. (See a later discussion.) Morris used a linear expression to fit the experimental data for mobility [251]... 

The three summands found in the right-hand side of expression (5.10) correspond to the three major channels (ways) of EEP losses the first summand characterizes the gaseous-phase de-excitation due to collisions, the second one stands for the gaseous-phase de-excitation on account of spontaneous radiation, and the third summand characterizes the heterogeneous decay of EEPs. A possible contribution of the radiative term to the value of ) D can be done a priori. With the radiative time of EEP lifetime r,ad known from the spectroscopy, one can easily estimate (by the formula of Einstein) the diffusion length over which the radiative decay of EEP will be perceptible ... 

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

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... 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

This condition expresses that the mass flux has to preserve the potential energy. The simplest way to account for this is to add to Tick s law another gradient term representing the tendency for the stars to drift to a deeper potential, a bit like in Einstein-Smolukowski law of diffusion in an external potential. This involves adding to Tick s law a contribution proportional to the gradient of the energy density. This changes Eq. (9) into... 

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