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 we can introduce the energy density p (v m) to transform this result into the Einstein coefficient of absorption, viz. the probability that the molecule (or atom) will absorb a quantum in unit time under unit radiation density. The probability of absorption in the Einstein expression is given by B m p (v m). Under the influence of the radiation polarized In x-directions, the relationship between the field strength E in x-direction and the radiation density is deduced as follows ... 

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,... 

The former is directly proportional to the square of the droplet size according to the Stokes-Einstein expression. In spite of the broad droplet size distribution and the changing droplet size with time, measuring the creaming rate can still provide a useful method for determining the relative droplet sizes of different emulsion products. The ultimate emulsion stability is not solely controlled by... 

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... 

The Debye theory assumes that there is a continuous distribution of frequencies from V = 0 to a certain maximum value v = Vj). The final expression obtained for the heat capacity is complicated, but succeeds in interpreting the heat capacity of many solids over the entire temperature range rather more accurately than the Einstein expression. At low temperatures, the Debye theory yields the simple result... 

Using the Stokes-Einstein expression for the difllusivity Dq and Eq. (110) one obtains ... 

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. 

One reason for the interest in computing absorption intensity correlations is that, once the parameters of the Judd-Ofelt theory are evaluated, they can be used to compute the radiative lifetime of any excited state of interest via the Einstein expression... 

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) ... 

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