Einstein relationship

Einstein relationships hold for other transport properties, e.g. the shear viscosity, the bu viscosity and the thermal conductivity. For example, the shear viscosity t] is given by ... 

Fig. 13. Relative viscosity vs volume fraction for a typical dispersion ( where (-) represents the Einstein relationship [77] = 2.5, and (—...

The absolute mobiUty is given in equation 10, which is again the Einstein relationship. 

Equation 11 gives the conductivity for a particular ion having a transference number, in a crystal, which is the Nemst-Einstein relationship. 

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient is introduced by the Einstein relationship... 

One of the most popular applications of molecular rotors is the quantitative determination of solvent viscosity (for some examples, see references [18, 23-27] and Sect. 5). Viscosity refers to a bulk property, but molecular rotors change their behavior under the influence of the solvent on the molecular scale. Most commonly, the diffusivity of a fluorophore is related to bulk viscosity through the Debye-Stokes-Einstein relationship where the diffusion constant D is inversely proportional to bulk viscosity rj. Established techniques such as fluorescent recovery after photobleaching (FRAP) and fluorescence anisotropy build on the diffusivity of a fluorophore. However, the relationship between diffusivity on a molecular scale and bulk viscosity is always an approximation, because it does not consider molecular-scale effects such as size differences between fluorophore and solvent, electrostatic interactions, hydrogen bond formation, or a possible anisotropy of the environment. Nonetheless, approaches exist to resolve this conflict between bulk viscosity and apparent microviscosity at the molecular scale. Forster and Hoffmann examined some triphenylamine dyes with TICT characteristics. These dyes are characterized by radiationless relaxation from the TICT state. Forster and Hoffmann found a power-law relationship between quantum yield and solvent viscosity both analytically and experimentally [28]. For a quantitative derivation of the power-law relationship, Forster and Hoffmann define the solvent s microfriction k by applying the Debye-Stokes-Einstein diffusion model (2)... 

Towards the middle of 1929, Mark was clearly close to establishing a viscosity equation. He and H. Fikentscher published a somewhat complex relationship of viscosity, and molecular volume (33). It was based on the Einstein relationship of viscosity and solute concentration. 

At about the same time, Staudinger derived his well known "law of viscosity". His work was formulated in 1929 and published in 1930 (34, 35). Also based on the Einstein relationship, Staudinger s equation was a direct relationship between the specific viscosity and the polymer molecular weight. 

The last point to be made is the famous Stokes-Einstein relationship that was found by Einstein by comparing the Brownian motion with common diffusion processes [66,67]. Accordingly the translational diffusion was found to depend... 

Proton conductivity, c7h+/ can be related through the Nernst-Einstein relationship to the activity of protons ( h+) in the membrane as well as to the mobility (Uh+) of those protons ... 

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... 

It may be shown (Ratner, 1987) that assuming the Nernst-Einstein relationship (Eqn (6.4)), a free volume expression for the conductivity in the form of Eqn (6.7) may be derived, provided the electrolyte is fully dissociated and... 

Selected entries from Methods in Enzymology [vol, page(s)] Electron paramagnetic resonance [effect on line width, 246, 596-598 motional narrowing spin label spectra, 246, 595-598 slow motion spin label spectra, 246, 598-601] helix-forming peptides, 246, 602-605 proteins, 246, 595 Stokes-Einstein relationship, 246, 594-595 temperature dependence, 246, 602, 604. 

Wilke and Chang (1955) developed an empirical relationship that was based on the temperature and viscosity characterization of the Stokes-Einstein relationship. It deviates from the equivalent diameter characterization by using another parameter, and incorporates the size of the solvent molecule and a parameter for polarized solvents. It is the most generally used of the available equations (Lyman et al., 1990) and is given as... 

Person 2 Estimate the nnsolvated Stake s sphere friction coefficient, fo, nsing the Stokes-Einstein relationship for spheres, fo = 6jrp,r. 

Solution Equation (4.41) gives the Einstein relationship between [r/] and , the volume fraction occupied by the dispersed spheres. The volume fraction that should be used in this relationship is the value that describes the particles as they actually exist in the dispersion. In this case this includes the volume of the adsorbed layer. For spherical particles of radius R covered by a layer of thickness 8R, the total volume of the particles is (4/3) + 4ttR2 8R. Factoring out the volume of the dry particle gives Vdfy(1 + 38RJRS), which shows by the second term how the volume is increased above the core volume by the adsorbed layer. Since it is the dry volume fraction that is used to describe the concentration of the dispersion and hence to evaluate [77], the Einstein coefficient is increased above 2.5 by the factor (1 + 36/Vfts) by the adsorbed layer. The thickness of adsorbed layers can be extracted from experimental [77] values by this formula. ... 

MD calculations may be used not only to gain important insight into the microscopic behavior of the system but also to provide quantitative information at the macroscopic level. Different statistical ensembles may be generated by fixing different combinations of state variables, and, from these, a variety of structural, energetic, and dynamic properties may be calculated. For simulations of diffusion in zeolites by MD methods, it is usual to obtain estimates of the diffusion coefficients, D, from the mean square displacement (MSD) of the sorbate, (rfy)), using the Einstein relationship (/) ... 

Subsequent calculations (53) for methane in ZK4 took account of the applicability of the Einstein relationship to determine the diffusion coefficient, by examining the probability density of finding a particle at a given... 

Since the Stokes—Einstein relationship between the diffusion coefficient, D, and coefficient of viscosity, 77, of eqn. (28) for a spherical species of radius a is... 

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... 

Up to now, only hydrodynamic repulsion effects (Chap. 8, Sect. 2.5) have caused the diffusion coefficient to be position-dependent. Of course, the diffusion coefficient is dependent on viscosity and temperature [Stokes—Einstein relationship, eqn. (38)] but viscosity and temperature are constant during the duration of most experiments. There have been several studies which have shown that the drift mobility of solvated electrons in alkanes is not constant. On the contrary, as the electric field increases, the solvated electron drift velocity either increases super-linearly (for cases where the mobility is small, < 10 4 m2 V-1 s-1) or sub-linearly (for cases where the mobility is larger than 10 3 m2 V 1 s 1) as shown in Fig. 28. Consequently, the mobility of the solvated electron either increases or decreases, respectively, as the electric field is increased [341— 348]. 

Baird et al. [350]). In the following analysis, the functional forms, p(E), which have been proposed (see below) to represent the field-dependence of the drift mobility are used for electric fields up to 1010Vm 1. The diffusion coefficient of ions is related to the drift mobility. Mozumder [349] suggested that the escape probability of an ion-pair should be influenced by the electric field-dependence of both the drift mobility and diffusion coefficient. Baird et al. [350] pointed out that the Nernst— Einstein relationship is not strictly appropriate when the mobility is field-dependent instead, the diffusion coefficient is a tensor D [351]. Choosing one orthogonal coordinate to lie in the direction of the electric field forces the tensor to be diagonal, with two components perpendicular and one parallel to the electric field. 

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