Stokes-Einstein equation modifications

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. 

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... 

For small bubbles, a modification of the Stokes-Einstein equation has been proposed, leading to a 50% increase in the contribution from the Ps bubble part in the equation of the reaction rate constant [99] ... 

All correlations proposed for estimating the diffusivity at infinite dilution are modifications of the original Stokes-Einstein equation... 

The dendrimer size in solution can he obtained employing diffusion NMR spectroscopy [pulsed gradient spin-echo (PGSE), diffusion ordered spectroscopy (DOSY)] that allows the determination of the self-diffusion coefficient of a molecule, which is related to the hydrodynamic radius through the Stokes-Einstein equation. Using this technique, the flexible structure of dendrimers in solution was demonstrated in PAMAM derivatives, which swell or shrink with pH modification. This size variation in solution should be taken into account if interactions with nucleic acids are being studied. 

The Stokes-Einstein equation, one of tire earliest theoretical expressions for liquid diffusivities, viewed the diffusion process as a hydrodynamic phenomenon in which the thermal motion of the molecules is resisted by a Stokesian drag force. This theory, along with subsequent modification by Sutherland and Eyring, established the following proportionality for the diffusion coefficient ... 

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... 

Importantly, Yoshizaki and Yamakawa [25] found that, in contrast to /, [77] of a wormlike cylinder undergoes significant end surface effects until the axial ratio p reaches about 50, on the basis of numerical solutions to the Navier-Stokes equation with the no-slip boundary condition for spheroid cylinders, spheres, and prolate and oblate ellipsoids of rotation. They constructed an empirical interpolation formula for [ y] of a spheroid cylinder which reduces to eq 2.37 for p > 1 and to the Einstein value at p = 1. Then, with its aid, Yamakawa and Yoshizaki [4] formulated a modified theory of [77] for wormlike cylinders which agrees with the Yamakawa-Fujii theory [3] for Lj lq > 2.278 and with the Einstein value at Ljd = 1, regardless of dj2q smaller than 0.1. However, no formulation has as yet been made for L/2q < 2.278 and d/2q > 0.1, i.e., for short flexible cylinders. In what follows, the Yamakawa-Yoshizaki modification is referred to as the Yamakawa-Fujii-Yoshizaki theoiy. 

Two examples of a theoretical approach to the problem of the prediction of diffusion coefficients in fluid media are the equations postulated in 1905 by Einstein and in 1936 by Eyring. The former is based on kinetic theory and a modification of Stokes law for the movement of a particle in a fluid, and is most conveniently expressed in the form... 

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