The Stokes-Einstein Relation

One can therefore define the absolute mobility u bs for diffusing particles by dividing the drift velocity by either the diffusional driving force or the equal and opposite Stokes viscous force 

The fundamental expression (4.172) relating the diffusion eoeffieient and the absolute mobility ean be written thus  

The Stokes-Einstein relation proved extremely useful in the classical work of Perrin. Using an ultramicroscope, he watched the random walk of a colloidal particle, and from the mean square distance traveled in a time t, he obtained the diffusion coefficient D from the relation (4.27) 

The weight of the colloidal particles and their density being known, their radius r was then obtained. Then the viscosity rf of the medium could be used to obtain the Boltzmann constant 

The use of Stokes law also permits the derivation of a very simple relation between the viscosity of a medium and the conventional electrochemical mobility conv Starting from the earlier derived equation (4.177) 

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In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns 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 type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

Dunning, JW Angus, JC, Particle-Size Measurement by Doppler-Shifted Laser Light, a Test of the Stokes-Einstein Relation, Journal of Applied Physics 39, 2479, 1968. 

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... 

Baxendale and Wardman (1973) note that the reaction of es with neutrals, such as acetone and CC14, in n-propanol is diffusion-controlled over the entire liquid phase. The values calculated from the Stokes-Einstein relation, k = 8jtRT/3jj, where 7] is the viscosity, agree well with measurement. Similarly, Fowles (1971) finds that the reaction of es with acid in alcohols is diffusion-controlled, given adequately by the Debye equation, which is not true in water. The activation energy of this reaction should be equal to that of the equivalent conductivity of es + ROH2+, which agrees well with the observation of Fowles (1971). 

Measurement of the translational diffusion coefficient, D0, provides another measure of the hydrodynamic radius. According to the Stokes-Einstein relation... 

With the help of the Stokes-Einstein relation, the translational diffusion coefficient may be calculated according to... 

These equations show that the ratio h/ Iu is proportional to the rate constant Hq for excimer formation. Assuming that the Stokes-Einstein relation (Eq. 4.12) is valid, ki is proportional to the ratio T /tj, tj being the viscosity of the medium. Application to the estimation of the fluidity of a medium will be discussed in Chapter 8. 

Once Dr is determined by fluorescence polarization measurements, the Stokes-Einstein relation can be used ... 

Various modifications of the Stokes-Einstein relation have been proposed to take into account the microscopic effects (shape, free volume, solvent-probe interactions, etc.). In particular, the diffusion of molecular probes being more rapid than predicted by the theory, the slip boundary condition can be introduced, and sometimes a mixture of stick and slip boundary conditions is assumed. Equation (8.3) can then be rewritten as... 

Changes in fluidity of a medium can thus be monitored via the variations of Jo/J — 1 for quenching, and Ie/Im for excimer formation, because these two quantities are proportional to the diffusional rate constant kj, i.e. proportional to the diffusion coefficient D. Once again, we should not calculate the viscosity value from D by means of the Stokes-Einstein relation (see Section 8.1). 

The changes in correlation time upon an external perturbation (e.g. temperature, pressure, additive, etc.) reflects well the changes in fluidity of a medium. It should again be emphasized that any microviscosity value that could be calculated from the Stokes-Einstein relation would be questionable and thus useless. 

The translational diffusion coefficient of micelles loaded with a fluorophore can be determined from the autocorrelation function by means of Eqs (11.8) or (11.9). The hydrodynamic radius can then be calculated using the Stokes-Einstein relation (see Chapter 8, Section 8.1) ... 

The self-translational diffusion coefficient D is related to f, by the Stokes-Einstein relation and is given by... 

On the other hand, the mobility of an ion is known to vary inversely with its solvation radius rt according to the Stokes—Einstein relation ... 

The z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

From the Stokes-Einstein relation, Equation (106), we then get... 

In the case of a polydisperse system, the overall decay of the function g,(s,/rf) is determined collectively by the decay rate (i.e., s2D) corresponding to each particle (notice that s2D varies with the particle size as evident from the Stokes-Einstein relation). In principle, the decay function in this case can be written formally in a simple manner as a weighted average of all possible decays ... 

The use of the Stokes-Einstein relation with the above value of the average diffusion coefficient leads to a hydrodynamic radius of roughly 30 nm, which is consistent with the specification of the manufacturer. ... 

We have made a note of the hydrodynamic interactions and other interactions to draw attention to an important fact. That is, the analysis of the DLS data is often quite complex, and a simple use of the single-exponential decay function and the Stokes-Einstein relation is not always sufficient, although many instruments available on the market use such an analysis and report an effective size for the particles in the dispersion. 

We can determine from Eq. (8.2) the values of D for solvated metal ions. The value of D changes with changes in solvent or solvent composition. The viscosity (rj) of the solution also changes with solvent or solvent composition. However, the relation between D and tj can be expressed by the Stokes-Einstein relation ... 

By comparing this result with Fick s first law (Eq. 18-6), we get the Stokes-Einstein relation between the diffusivity in aqueous solutions and the solution viscosity q ... 

In this expression, the primary environmental determinant is the viscosity in the denominator. Note that the exponential is slightly larger than in the Stokes-Einstein relation (Eq. 18-52). Since viscosity decreases by about a factor of 2 between 0°C and 30°C, D,w should increase by about the same factor over this temperature range. Furthermore, the influence of the molecule s size is also stronger in Eq. 18-53 than in 18-52 (note r, = constant V 173). In Box 18.4 experimental information on the temperature dependence of D,w is compared with the theoretical prediction from Eqs. 18-52 and 18-53. 

In the following table the different models are applied to CFC-11. Note the excellent correspondence between the temperature variation calculated by the Stokes-Einstein relation (Eq. 3) and the expression by Hayduk and Laudie (Eq. 4), although both models overestimate the temperature effect compared to the activation model derived from the experimental data (Eq. 2). 

In order to apply the Stokes-Einstein relation (Eq. 18-52) all variables have to be transformed into the correct units ... 

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. 

D = (1.8 0.6) x 10 12m2s 1. The encounter distance was estimated to be 1.32 nm, which, when used in the Stokes—Einstein relation for the mutual diffusion coefficient, eqn. (28), givesD as (1.1 0.03) x 10 I2m2 s-1, in reasonable agreement with the estimate from fitting experimental and theoretical decay curves (mentioned above). The germinate pair recombination probability at long times was measured and its increase correlates moderately well with T1/2/rj as noted for other systems (see Sect. 3.1) but was concave upwards (see ref. 22). 

In Fig. 3, the orientational diffusion time constants ror of the first solvation shell of the halogenie anions CD. Br, and D are presented as a function of temperature. From the observation that ror is shorter than rc, it follows that the orientational dynamics of the HDO molecules in the first solvation shell of the Cl ion must result from motions that do not contribute to the spectral diffusion, i.e. that do not affect the length of the O-H- -Cl hydrogen bond. Hence, the observed reorientation represents the orientational diffusion of the complete solvation structure. Also shown in Fig. 3 are fits to the data using the relation between ror and the temperature T that follows from the Stokes-Einstein relation for orientational diffusion ... 

Other than dynamical correlations, transport properties have also been derived using hydrodynamic theory. In hydrodynamics the diffusion of a tagged particle is defined by the Stoke-Einstein relation that is given by the following well-known expression ... 

Figure 8. The ratio of the self-diffusion coefficient of the solute (Di) to that of the solvent molecules (D ) plotted as a function of the solvent-solute size ratio (

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