Diffusion coefficients Stokes-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]. 

Now, substituting the Stokes-Einstein equation (Equation (14)) for the diffusion coefficient in the expression for kr leads to... 

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 ... 

Viscosity also affects the liquid-phase mass transfer coefficient through the Stokes-Einstein effect. Einstein proposed that the diffusion coefficient could be expressed as... 

When the probe polymer is larger than the network size, the Stokes-Einstein (S-B) type diffusion as shown in Eq. (7) becomes difficult and the polymer diffuses by reptation. In this case, the diffusion coefficient can be expressed by Eq. (8). Pajevic et al. [3] measured the diffusion coefficient D of polyst) ne (PS), which is included in the gel as a probe molecule, by using dynamic light scattering. The gel used was poly(methyl methacrylate) gel, which is composed of 12.5% (v/v) polymer and is swollen by toluene. The results are shown in Fig. 14 [3] where is the diffusion coefficient of PS at each molecular weight (Mf) in toluene and... 

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 ... 

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... 

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

To go further on with the dependence of bubbles radii with some few parameters, we can also replace in the latter equation the diffusion coefficient Do by its theoretical expression approached through the well-known Stokes-Einstein equation (Dq k d /Snpd). The following relationship expressed in the MKSA system was thus obtained ... 

To a fairly good approximation, the Stokes—Einstein relationship for the diffusion coefficient can be used [eqn. (28)], so that inverse recombination probability can be expressed 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. 

The use of the Stokes-Einstein equation (2) relating the diffusion coefficient (D) of a spherical solute molecule to its radius (r), the viscosity of the medium (tj) and the Boltzmann constant (k) permits the rate coefficient ( en) to be expressed in (3) in terms of the viscosity of the medium. In this derivation, the... 

The Stokes-Einstein relationship was used to express the diffusion coefficient through the viscosity of solvent D[cm2/s] = 0.894 x 10 5/r [cP]. All other parameters that change with viscosity, including Pekar s factor y, were borrowed or calculated from the same data sources [148]. They are listed in Table I. 

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. 

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 coefficient is a physical property that represents the speed of molecular diffusion. Recently it was shown that D changes during chemical reactions. Here, we describe the origin of the change in D. Intuitively, it may be easily understood that D decreases when molecular size increases because of the association reaction. In some cases, the relationship between D and the molecular size is well described by the Stokes-Einstein equation. The Stokes-Einstein equation is expressed by [9-13]... 

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. 

In experiments involving radiotracer measurements of diffusion in molten salt, the Stokes-Einstein equation has been found to be roughly applicable. For a series of ions, in molten salts it was found that the product D /T = 10 dyn mol . From this information, find whether the best form of the coefficient in this expression for this case is nearer to 6 or 4. 

Experimental Results. Effects of pH and Ionic Strength. Experiments showing the effects of pH and ionic strength on the configuration of NOM in solution are presented in Figure 3, taken from the work of Cornel et al. (14). Experiments were conducted with the 50-100 K apparent molecular weight fraction of a humic acid (HA). Results are expressed in terms of the equivalent Stokes-Einstein or hydrodynamic radius (rh) calculated from measurements of the diffusion coefficients of the HA fraction. 

The basis for the temperature dependence of the water diffusion coefficient arises from the Stokes-Einstein equation, given previously as Eq. (3). This can also be expressed as ... 

For particles 0.1 //ni in radius, the characteristic time (n,- + fl ) /( A + A) about 10 sec, and the use of the steady-state solution is justified in most cases of practical interest. When the Stokes-Einstein relation holds for the diffusion coefficient (Chapter 2) and dp )S> f, this expression becomes... 

The proportionality constant D is called the diffusion coefficient (S.I. unit m2-s ). Einstein also derived that D = kBT//, and taking Stokes s expression for the friction factor / for spheres, the relation becomes... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation Dx lkT)a = C holds, which forms the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the t5q)e of boundary conditions used in deriving Stokes law. It follows that the diffusion coefficient ratio is given by ... 

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