Diffusion coefficients Stokes-Einstein equation

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

In connection with the earlier consideration of diffusion in liquids using tire Stokes-Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be T(exp(—Qvis/RT)) according to this equation, if the activation energy for viscous flow is included. 

Very commonly Eq. (4-5) is combined with Eq. (4-6), the Stokes-Einstein equation relating the diffusion coefficient to the viscosity -q. 

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. 

T0 is a reference temperature which can be identified with T, and although the constant B is not related to any simple activation process, it has dimensions of energy. This form of the equation is derived by assuming an electrolyte to be fully dissociated in the solvent, so it can be related to the diffusion coefficient through the Stokes-Einstein equation. It suggests that thermal motion above T0 contributes to relaxation and transport processes and that... 

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... 

Using the Stokes-Einstein equation of diffusion coefficient ... 

According to Stokes-Einstein equation, the diffusion coefficient is inversely proportional to the solution viscosity which increases with temperature. Hence, a lower diffusion coefficient corresponds to a lower size molecule. 

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

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

Solutes diffuse in proportion to their diffusion coefficient, which is inversely proportional to their hydrodynamic radii, ah, (Stokes-Einstein equation) ... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

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... 

Here Tq is — C2 and is a prefactor proportional to which is determined by the transport coefficient (in this case at the given reference temperature. The constant B has the dimensions of energy but is not related to any simple activation process (Ratner, 1987). Eqn (6.6) holds for many transport properties and, by making the assumption of a fully dissociated electrolyte, it can be related to the diffusion coefficient through the Stokes-Einstein equation giving the form to which the conductivity, <7, in polymer electrolytes is often fitted,... 

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 ... 

The form of Eq. (4.84) should look familiar. For large, spherical particles in a low-molecular-weight solvent, f = 671 fir, where fi is the viscosity of the pure solvent and r is the large particle radius, and Eq. (4.84) becomes Equation (4.70) a form of the Stokes-Einstein equation, which gives the binary diffusion coefficient, Dab ... 

Wilke (W8) has developed a correlation for diffusion coefficients on the basis of the Stokes-Einstein equation. His results may be summarized by the approximate analyti-... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

The final section (Section 5.8) introduces dynamic light scattering with a particular focus on determination of diffusion coefficients (self-diffusion as well as mutual diffusion), particle size (using the Stokes-Einstein equation for the diffusion coefficient), and size distribution. 

Here, D, which is the quantity we seek from gi(s,td), is the diffusion coefficient of the particle (and 5 is the magnitude of the scattering vector defined in Equation (57)). We can now use the Stokes-Einstein equation (see Equation (2.32) and the accompanying comment) to obtain the particle radius R from D ... 

What is the difference between the self-diffusion coefficient and the mutual diffusion coefficient Which of these two is described by the Stokes-Einstein equation ... 

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

This involves the diffusion coefficients DA and DB for the two colliding partners and the encounter radii RAB = aA + Cb sum of the interaction radii (Figure 6.1), p is a probability factor per encounter. The diffusion coefficients are given by the Stokes-Einstein equation... 

On the other hand, if the rate constant for the quenching step exceeds that expected for a diffusion-controlled process, a modification of the parameters in the Debye equation is indicated. Either the diffusion coefficient D as given by the Stokes-Einstein equation is not applicable because the bulk viscosity is different from the microviscosity experienced, by the quencher (e.g. quenching of aromatic hydrocarbons by O, in paraffin solvents) or the encounter radius RAb is much greater than the gas-kinetic collision radius. In the latter case a long-range quenching... 

The first molecular system studied with this type of spectroscopy was suspensions of PSL spheres (22) whose spherical shape allowed theoretical calculation of the diffusion coefficient according to the Stokes-Einstein equation (58)... 

Diffusion coefficients can be related to molecular weight in three ways first by application of the Stokes-Einstein equation, second by combination with sedimentation data, and third by consideration of homologous polymer solutions. In the first method, an equivalent spherical size of the molecules is calculated from Dt, and an approximate molecular weight is found by combining these data with the appropriate density. In the second method, diffusion measurements are coupled with those of sedimentation velocity to give molecular weights, and in the third method, molecular weights may be determined directly from measurements of diffusion coefficients alone once a calibration has been... 

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