Stoke’s radius

Gel filtration is the last of the major chromatography techniques commonly applied in the resolving portion of a process. Of all the techniques discussed thus far in this chapter, gel filtration offers the lowest resolution. The separation is based solely on Stoke s radius of the protein molecule and is the most sensitive to flow rate and sample volume. To achieve significant resolution among sample components, the sample volume should be no greater that five percent of the column bed volume. 

Method used Stoke s radius (A) Frictional ratio (///o)... 

AA-gpjj. Conditionally, the ionic atmosphere is regarded as a sphere with radius r. The valnes of approach the size of colloidal particles, for which Stokes s law applies (i.e., the drag coefficient 9 = where r is the liquid s viscosity) when they... 

The radius R is related to the friction constant F in sedimentation by Stokes s equation ... 

Forced coalescence. Referring again to Stoke s equation. notice that the panicle radius occurs taken to (lie second power. II the particle can Ik- increased, the settling velocity increases by the square ol this change. Particle size then becomes the overriding parametei in the separation process. 

An expression for the friction factor, F, can be obtained from Stokes s law, which gives the force, F, exerted by a viscous medium on a sphere of radius, R, moving through it with a velocity, v, in the form [38]... 

STORE S LAW. (1) The rate at which a spherical particle will rise or fall when suspended in a liquid medium varies as the square of its radius the density of the particle and the density and viscosity of the liquid are essential factors, Stoke s lav/ is used in determining sedimentation of solids, creaming rale of fat particles in milk, etc. (2) In atomic processes, the wavelength of fluorescent radiation is always longer than that of the exciting radiation,... 

Use the s2o,w and the D2o,w and Co values obtained from DCDT+, as input parameters in the program SEDNTERP to calculate the Stoke s radii (RH) of the RNA molecules. A plot of RH versus cation concentration should show the opposite trend as compared to the s2o,w versus cation concentration plot, that is, the hydrodynamic radius of the RNA molecules decrease with increasing cation concentration. The axial ratio (a/b) is also calculated using SEDNTERP and when plotted as a function of cation concentration, should show the same trend as the RH plot (an example is provided in Fig. 10.3B). 

The relation between the mobility of the ion and the properties of the resin can be qualitatively examined with the aid of Stoke s law for the drift of a spherical object in a viscous medium (see, for example, Ref.27)). The mobility of a sphere of radius r, embedded in a medium of viscosity p and subjected to a force is... 

According to Stokes s law, / == Trrjru, where u is the steady velocity acquired by a particle of radius r under the influence of a force /. If the particle carries a charge Q and moves under a potential gradient of unity, F is equal to Q and u becomes the mobility u, hence, Q = QirrjrUe. The potential at a distance r from the charge Q is equal to Q/Dr, and if this is identified with f, equation (23) follows immediately. 

Since particles with different shapes fall with different velocities, the term equivalent or effective radius is used to overcome this difficulty in Stoke s law. Effective radius is defined as the radius of a sphere of the same materials which would fall with the same velocity as the particle in question. 

The frictional term x(l) is assumed to be governed by Stokes s Law, which states that the frictional force decelerating a spherical particle of radius a and mass m is... 

We will see in Chapter 11 that in 1850 the British physicist Sir George Gabriel Stokes (1819-1903) proposed a very simple relationship between the diffusion coefficient D and the radius r of the diffusing molecule, on the assumption that it is. spherical. Stokes s law is... 

The assumption that the molecule is spherical is, of course, an unsatisfactory one. If we have an alternative way of measuring the molecular weight we can make use of Stokes s law to obtain information about the actual shape of the molecule. From the molecular weight and the density we can calculate the radius on the assumption that the molecule is spherical Insertion of this radius into equation (3.16) then gives a hypothetical diffusion constant, which we can call D. If the directly determined diffusion constant D is the same as Do we have evidence that the molecule is in fact spherical. Often, however, D is found to be smaller than the Do value calculated from equation (3.16). A nonspherical molecule will diffuse more slowly than a spherical molecule of the same volume. 

Measurement of D in a solvent of known viscosity therefore permits a value of the radius r to be calculated. However, such a calculation would not be very satisfactory for macromoiecuies, for several reasons. In the first place, Stokes s law is based on the assumption of very large spherical particles and a continuous solvent, and therefore involves some error even for approximately spherical molecules. Secondly, the macromoiecuies may not be spherical, and this introduces an additional error. Furthermore, macromoiecuies are commonly solvated, and in moving through the solution they transport some of their solvation layer. In spite of these drawbacks, equation (11.66) has proved useful in providing approximate values of molecular sizes (see also pp. 99 i00). 

Similarly, if we compare the conductivity of the alkali metal ions (Table 31.5), in the light of Eq. (31.45) we would be forced to conclude that the radius of the lithium ion is larger than that of the cesium ion. Since the crystallographic radius of lithium ion is much smaller than that of cesium ion, this indicates a difficulty with the Stokes s law interpretation ofXi. 

Equation (31.60) is the Einstein relation between the mobility in unit electrical field and the diffusion coefficient. If we replace Wj in the Einstein equation by the Stokes s law value, Eq. (31.44), we obtain a relation between the diffusion coefficient, the ion radius, and the viscosity of the medium. 

The difficulty is that the frictional coefficient, /, may not be known. If the particles are spherical, then by Stokes s law, / = 6nria, in which t] is the coefficient of viscosity of the solvent and a is the radius of the macromolecule. The problem then reduces to that of determining the radius of the macromolecule. 

We can estimate a, the radius of the particle, from a knowledge of the diffusion coefficient. If the particle is not spherical, the frictional coefficient is larger than that given by the Stokes s law expression the nonspherical particles exert a frictional effect larger than that exerted by an equivalent spherical particle. 

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