Stokes-Einstein equation correlation

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

Wilke-Chang This correlation for D°b is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it apphes to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. ( )b is an association factor of solvent B that accounts for hydrogen bonding. 

Measurements of CuS04 molecular diffusivity by Cole and Gordon (C12a), referred to above, were carried out in diaphragm cells, mostly at 18°C. Their results were correlated by Fenech and Tobias (F3) using the 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... 

The reorientational correlation time can be predicted for spherical rigid particles, according to the Stokes Einstein equation (75-77) ... 

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-... 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

The water proton NMRD profile of Cu(II) aqua ion at 298 K [108] (Fig. 5.36) is in excellent accordance with what expected from the dipole-dipole relaxation theory, as described by the Solomon equation (Eq. (3.16)). The best fitting procedure applied to a configuration of 12 water protons bound to the metal ion provides a distance between water protons and the paramagnetic center equal to 2.7 A, and a correlation time equal to 2.6 x 10 11 s, which defines the position of the cos dispersion. The correlation time is determined by rotation as expected from the Stokes-Einstein equation (Eq. (3.8)). The electron relaxation time is in fact expected to be one order of magnitude longer (see Table 5.6). This also ensures... 

Diffusions NMR spectroscopy (e.g. PGSE = Pulsed Gradient Spin Echo STE = Stimulated Echo DOSY = Diffusion Ordered Spectroscopy) is a straightforward and accurate method for determination of the self-diffusion coefficient of a molecule. Its principal use in dendrimer chemistry is for size determination of dissolved dendrimers since the self-diffusion coefficient is directly correlated with the hydrodynamic radius of the molecule via the Stokes-Einstein equation [24]. Although one-dimensional and multidimensional diffusion NMR experiments can thus make an important contribution to structural characterisation of dendrimers, they have been used comparatively rarely until recently [25, 26]. 

In general, diffusion coefficients in gases can be often be predicted accurately. Predictions of diffusion coefficients in liquids are also possible using the Stokes-Einstein equation or its empirical parallels. On the contrary in solids and polymers, models allow coefficients to be correlated but predictions are rarely possible. 

As a correlation function is recorded, the correlator offers the decay time (x), particles diffusion coefficient (D), and particles mean radius (R). The two latters are related with by Stokes-Einstein equation [6],... 

The temperature dependencies of the viscosity (Figure 5.6) and the summation of the self-diffusion coefficient (Dcation + Oanion) (Figure 5.4) interestingly show the contrasted profiles with the indication of inverse relationship between viscosity and self-diffiision coefficient. This can be explained in terms of the Stokes-Einstein equation, which correlates the self-diffusion coefficient (Dcation Danion) with viscosity (q) by the following relationship ... 

Since the rotation of the complex molecule will be isotropic as in the cases of several tiis(did iate)cobalt(III) complexes, and the mdecule can be regarded as a sphere, we assume that the rotational correlation time is expressed by the Debye-Stokes-Einstein equation > ... 

Dynamically raised processes in the dispersion, such as Brownian molecular motion, cause variations in the intensities of the scattered light with time, which is measured by PCS. Smaller the particle, higher the fluctuations by Brownian motion. Thus, a correlation between the different intensities measured is only possible for short time intervals. In a monodisperse system following first-order kinetics, the autocorrelation function decreases rather fast. In a half logarithmic plot of the auto correlation function, the slope of the graph enables the calculation of the hydrodynamic radius by the Stokes-Einstein equation. With the commercial PCS devices the z-average is determined, which corresponds to the hydrodynamic radius. 

To measure the droplet size distribution of the primary emulsion (W/O in W/O/W or O/W in O/W/O) that has a micron range (with an average radius of 0.5-1.0 pm), a dynamic light-scattering technique (also referred to as photon correlation spectroscopy PCS) can be apphed. Details of this method are described in Chapter 19. Basically, the intensity fluctuation of scattered light by the droplets as they undergo Brownian diffusion is measured from this, the diffusion coefficient of the droplets can be determined, and in turn the radius can be obtained by using the Stokes-Einstein equation. 

The effect of wQ on the enzyme was examined by comparing the EPR spectra of lysine-labeled tryptophanase. Although the spin label is nonspecific, it can still provide molecular-level information about the enzyme under different conditions. EPR spectra of the spin-labeled enzyme in bulk water and in reverse micelles are shown in Figure 6. Much broader spectra were obtained in reverse micelles, and the calculated rotational correlation time of the attached label (101 increased with w0. Thus, the enzyme-bound spin label became more constrained as the water content of the reverse micelle decreased. Since the rotational correlation time of the entire protein in bulk water calculated from the Stokes-Einstein equation is about 200 nsec, the motion of the spin label was still rapid relative to the tumbling rate of the enzyme. Therefore, broadening of the spectrum was apparently caused by a change in the local dynamics of tryptophanase rather than by a decrease in the enzyme s overall rotation rate. The tumbling rate of the enzyme could have decreased as well, however. 

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