Debye-Stokes equation

Tr for spherical molecules can be estimated from the Debye-Stokes equation (Eq.24) ... 

An extended form of the Debye-Huckel equation is the hydration one of Robinson and Stokes (11). It contains two adjustable parameters, ap and h, where h is rilated to the hydration number. It can be fitted to y for several electrolytes for concentrations in excess of 1 m. Their equation has the valuable feature of describing not only the salting-in but also the salting-out part of the y+ versus m curve. It should be noted, however, that the... 

The Debye-Stokes-Einstein (DSE) equation (60) predicts that the orientational correlation time of a spherical object in a continuum liquid is... 

The Co longitudinal relaxation rates of tris(acetylacetonato)cobalt(lIl) in dichrolo-methane and benzene were measured over the concentration range between 20 and 110 mol m 3 at several temperatures. The dependence of the relaxation rate on the temperature and the complex concentration is primarily attributable to the change in the viscosity of solutioa The values of eQqlh were calculated from the concentration dependence of the Co relaxation rate by using the Debye-Stokes-Einstein equation and the Einstein relationship between molar volume and viscosity B coefflcienL... 

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 > ... 

Observed molar conductivities were analyzed by assuming the ion association (ion-pair formation) between the complex ions and the counter ions in the same manner as described previously. The closest distances of approach of ions (a) in the Robinson-Stokes conductivity equation and in the Debye-HUckel equation were taken as 6.8 and 7.3 A for chlorides and perchlorates of the tris(phen) complexes 6.6 and 7.1 A for those of the tris(bpy) complex, respectively, using the effective ionic radii of the complex ions, shown in T le 1, and those of Cl (1.81 A) and C104 (2.30A). The values of ref were estimated from the ionic partial molar volumes (f i°°) by use of Glueckauf equation. > ... 

The rotational diffusion coefficient, Dr, of a probe molecule in a glass-former follows the temperature dependence of the Debye-Stokes-Einstein (DSE) equation [167-171],... 

Thus, the combined SE and the DSE equations predict that the product Dtxc = (A Tc)sedse should equal 2r /9. Measurements of probe translational diffusion and rotational diffusion made in glass-formers have found that the product Dtr can be much larger than this value, revealing a breakdown of the Stokes-Einstein (SE) relation and the Debye-Stokes-Einstein (DSE) relation. There is an enhancement of probe translational diffusion in comparison with rotational diffusion. The time dependence of the probe rotational time correlation functions tit) is well-described by the KWW function,... 

This is not the Stokes-Robinson equation, though some common elements may be discerned. Our result is different because we substituted I for I in the Debye-Hiickel equation for the excess Gibbs energy, whereas Stokes and Robinson (1) made the substitution in the Debye-Hiickel equation for the activity coefficient of a formally hydrated solute (In y ,a)- Differentiation of our equation for the excess Gibbs energy would yield a somewhat different equation for y + A-... 

Assume that the resistance to the cylinder motion is due to the shear stress associated with the electroosmotic flow that is generated, so that the Navier-Stokes equation reduces to a balance between viscous and electrical forces. Show that the solution for the electrophoretic velocity of the cylinder is the same as that for a sphere of the same zero potential with the Debye length small. 

As explained in section 3.6.1, many modifications have been proposed for the Debye-Hiickel relationship for estimating the mean ionic activity coefficient 7 of an electrolyte in solution and the Davies equation (equation 3.35) was identified as one of the most reliable for concentrations up to about 0.2 molar. More complex modifications of the Debye-Huckel equation (Robinson and Stokes, 1970) can greatly extend the range of 7 estimation, and the Bromley (1973) equation appears to be effective up to about 6 molar. The difficulty with all these extended equations, however, is the need for a large number of interacting parameters to be taken into account for which reliable data are not always available. 

Robinson and Stokes [Ro 59] calculated the solvation numbers of electrolytes from the difference between the experimentally determined activity coefficients and those calculated on the basis of the Debye-Hiickel equation. Their method was... 

Outside of the Debye layer, the fluid is electrically neutral. Inside the Debye layer, the fluid obeys the Navier-Stokes equation with an electrical body force... 

Now we consider the electrokinetic behavior of soft particles, i.e., colloidal particles covered with a polymer layer (Figure 2.2). A number of theoretical studies have been made [34-46] on the basis of the model of Debye and Bueche [47], which assumes that the polymer segments are regarded as resistance centers distributed in the polymer layer, exerting frictional forces y on the liquid flowing in the polymer layer, where u the liquid flow velocity and y a frictional coefficient. The Navier-Stokes equation for the liquid flow inside the polymer layer is thus given by... 

Whereas the Debye-Stokes-Einstein equation may be applicable for macromolecules in a low molecular weight solvent it is apparently not a realistic model for molecular motion in a neat liquid. A modificatior of the Debye-Stokes-Einstein relation was proposed by Gierer and Wirtz [67] who tried to take into account the discontinuous nature of the liquid. For a spherical molecule they obtained... 

The dominant forces that determine deviations from ideal behaviour of transport processes in electrolytes are the relaxation and electrophoretic forces [16]. The first of these forces was discussed by Debye [6, 17]. When the equilibrium ionic distribution is perturbed by some external force in an ionic solution, electrostatic forces appear, which will tend to restore the equilibrium distribution of the ions. There is also a hydrodynamic effect. It was first discussed by Onsager [2, 3]. Different ions in a solution will respond differently to external forces, and will thus tend to have different drift velocities The hydrodynamic (friction) forces, mediated by the solvent, will tend to equalize these velocities. The electrophoretic ( hydrodynamic) correction can be evaluated by means ofNavier-Stokes equation [18, 19]. Calculating the relaxation effect requires the evaluation of the electrostatic drag of the ions by their surroundings. The time lag of this effect is known as the Debye relaxation time. 

The assumption of a diffusive motion entails also the validity of hydrodynamics for the reorientation. If this is the case, then we are justified to some extent to use the Debye-Stokes-Einstein equation ... 

The correlation time data can be further rationalized by the use of the Debye-Stokes-Einstein equation 35, as displayed in figure 13. 

The time for rotational diffusion trot can be related to the viscosity t) using the modified Stokes-Debye-Einstein equation (115) ... 

Rg. 2.7 Experimental activity coefficients ( ) and their ionic strength dependence shown schematically. DHL = Oebye-Huckel limiting law EOHL = Extended Debye-HQckel law RS s Robinson and Stokes equation. 

The above model has been used by Bull et al. (102) to analyse the chloride NMR relaxation data for several proteins. The overall correlation time(s) was estimated from the Debye-Stokes-Einstein equation or, when available, was taken from dielectric relaxation measurements. In order to perform the calculations, also the "true" value of the chloride quadrupole coupling constant in the site is needed. This is, however, not known and therefore Bull et al. estimated a value for Cl bound to a NH3 group of 3.6 MHz based on the electrostatic model of Cohen and Reif (103). In this way it was possible to calculate the values shown in Table 6. Obviously the model used is an oversimplification however, it is noteworthy that all internal correlation times come out with a reasonable value of about 1 ns. 

Intcrmolecular dipole-dipole relaxation depends on the correlation time for translational motion rather than rotational motion. Intermolecular dipole-dipole interactions arise from the fluctuations which are caused by the random translational motions of neighboring nuclei. The equations describing the relaxation processes are similar to those used to describe the intramolecular motions, except is replaced by t, the translation correlation time. The correlation times are expressed in terms of diffusional coefficients (D), and t, the rotational correlation time and the translational correlation time for Brownian motion, are given by the Debye-Stokes-Einstein theory ... 

A number of attempts have been made to account for the discrepancies between correlation times calculated from the Debye-Stokes-Einstein equation and the correlation times observed by magnetic resonance spectroscopy. A micro viscosity correction factor (/ ) has been introduced in the Debye-Stokes-Einstein equation in which rj is reduced by... 

Ti measurements performed on cyclic dipeptides in dimethyl-sulfoxide-dg revealed a correlation between the molecular weight of the dipeptide and the value of the a-carbon of optically active amino acid residues. These are shown in Figure 26 (Deslauriers et al, 1975b). According to the Debye-Stokes-Einstein equation, the correlation plotted in dotted lines is to be expected. The discrepancy between observed and calculated values is worse for the lower-molecular-weight dipeptides. For dipeptides... 

The Debye-Stokes-Einstein relation assumes a particle to be spherical. For a nonspherical particle an alternate equation must be chosen that takes into consideration the effect of molecular shape on the diffusion properties of the particle. 

It was established that CDs labelled with TEMPO derivatives, Le. spin labelled CDs 32-34 undergo interaction with PEG 600 (PEG = polyethylene glycol) and with PPG 425 (PPG = polypropylene glycol) in concentrated aqueous solutions. The EPR spectra of 32-34 are changed when they are complexed with PEG or PPG. It was observed that the relationship between rotational correlation times (t) and solvent viscosity caimot be described by the Debye-Stokes-Einstein equation, this fact being due to self-aggregation of alkylene glycols in concentrated solutions. However the use of the fractional Debye-Stokes-Einstein equation, i.e. the relationship between relative x values and relative viscosity is in accordance with the experimental data [77]. 

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