Stokes-Einstein relationship

Fig. 2 Plot of the dynamical correlation length h estimated from Z>c using the Einstein-Stokes relationship against PVA concentration C...

The fast mode was the diffusive mode for all solutions since Ft/q was independent of q. The cooperative diffusion coefficient >c could be estimated as an average of Ff/q and the dynamical correlation length iJh was estimated using the Einstein-Stokes relationship,... 

Similarly, the rotational diffusion coefficient at temperature r,Dr(T), is predicted to be coupled to rj(T) by the Debye-Stokes-Einstein (DSE) relationship ... 

Free Radical Self-Termination. The cage efficiencies and activation parameters for the phenylthiyl collisional cage pair provide the basis for illustrating some of the important features of equations (3)-(5) and for predicting the observed rates of self-termination of phenylthiyl free radicals. Application of the SW procedure to the completely diffusion controlled step of Scheme 1 (kj) ) for phenylthiyl free radicals in cis-decalin can be expressed by the transition state equation with a AH d of 3448 cal/mole and a AS d of -4.3 cal/mole-K. The corresponding activation enthalpy (AH d) from the Stokes-Einstein-Schmoluchowski relationship is 3685 cal/mole for cis-decalin) so that the a of equation (8) is 0.94. The micro-frictional multiplier (mf, equation 8 above), which is incorporated into the SW activation entropy (AS j)), is 2.4. The SW activation entropy for a truly diffusion controlled self-termination of phenylthiyl free radicals (2k obs -2kj), - 1 at... 

If E is measured as volts per meter, then dx/dt is called the mobility of the ion for the chosen experimental conditions. The coefficient / has a theoretical foundation in hydrodynamics, and a functional relationship between / and the coordinates of the particle can be derived for a few regular shapes (for a sphere it is known as the Stokes equation, but there are mathematical solutions for ellipsoids and cylinders). It is also known that / is inversely proportional to the randomizing effects of diffusion of a large number of ions (Einstein-Sutherland relationship). Essentially, Eq. (6) describes the movement of a single ion under the influence of an electrical field. Rarely, if ever, can one ion be studied experimentally, because at finite concentrations of ions there are > 10 ions per liter (a 10 ° M solution of 0.1 /xg liter for an ion of relative mass 10 contains 10 ions per liter). Diffusion of this population of ions spreads the boundary about an elec-trophoretically transported point called the centroid, and it is the velocity of this point that is described by Eq. (6) for experimental situations. The centroid or first moment (x) can be evaluated from Eq. (7) using a set of rectangular coordinates determined experimentally over an elec-trophoresing boundary. 

Since thermal agitation is the common origin of transport properties, it gives rise to several relationships among them, for example, the Nemst-Einstein relation between diffusion and conductivity, or the Stokes-Einstein relation between diffusion and viscosity. Although transport... 

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... 

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)... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

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]. 

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... 

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... 

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... 

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 ... 

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

As for Illustrative Example 18.2a (diffusivity of CFC-12 in air), these values agree fairly well with each other, except for the Stokes-Einstein relation, which was not meant to be a quantitative approximation but an expression to show qualitatively the relationship between diffusivity and other properties of both molecule and fluid. 

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]. 

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. 

Interpretation of temperature effects, however, remained uncertain. With polypeptides in particular the effect of a change in temperature was unpredictable. Certain ones followed rather closely the Stokes-Einstein relationship for free diffusion others deviated strongly. It was... 

Knowing these functions, the mean-square radius of gyration (S2)z and the translational diffusion coefficient Dz can easily be derived eventually by application of the Stokes-Einstein relationship an effective hydrodynamic radius may be evaluated. These five... 

Second, an equivalent hydrodynamic radius can be defined from the diffusion coefficient via a Stokes-Einstein relationship... 

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