:Stokes radius

According to Stokes’ law, a perfect sphere traveling through a viscous liquid feels a drag force proportional to the frictional coefficient $f$:

$$F\_\backslash text\{drag\}\; =\; fs\; =\; (6\; \backslash pi\; \backslash eta\; a)s$$

where $\backslash eta$ is the liquid's viscosity, $s$ is the sphere's drift speed, and $a$ is its radius. Because ionic mobility $\backslash mu$ is directly proportional to drift speed, it is inversely proportional to the frictional coefficient:

$$\backslash mu\; =\; \backslash frac\{ze\}\{f\}$$

where $ze$ represents ionic charge in integer multiples of electron charges.

In 1905, Albert Einstein found the diffusion coefficient $D$ of an ion to be proportional to its mobility constant:

$$D\; =\; \backslash frac\{\backslash mu\; k\_\backslash text\{B\}\; T\}\{q\}\; =\; \backslash frac\{k\_\backslash text\{B\}\; T\}\{f\}$$

where $k\_\backslash text\{B\}$ is the Boltzmann constant and $q$ is electrical charge. This is known as the Einstein relation. Substituting in the frictional coefficient of a perfect sphere from Stokes’ law yields

$$D\; =\; \backslash frac\{k\_\backslash text\{B\}\; T\}\{6\; \backslash pi\; \backslash eta\; a\}$$

which can be rearranged to solve for $a$, the radius:

$$R\_H\; =\; a\; =\; \backslash frac\{k\_\backslash text\{B\}\; T\}\{6\; \backslash pi\; \backslash eta\; D\}$$

In non-spherical systems, the frictional coefficient is determined by the size and shape of the species under consideration.

Stokes radii are often determined experimentally by gel-permeation or gel-filtration chromatography.{{cite journal|last=Alamillo|first=J.|author2=Jacobo Cardenas |author3=Manuel Pineda |title=Purification and Molecular Properties of Urate Oxidase from Chlamydomonas Reinhardtii|journal=Biochimica et Biophysica Acta (BBA) - Protein Structure and Molecular Enzymology|year=1991|volume=1076|issue=2|pages=203–08|doi=10.1016/0167-4838(91)90267-4|pmid=1998721}}{{cite journal|last=Dutta|first=Samarajnee|author2=Debasish Bhattacharyya |title=Size of Unfolded and Dissociated Subunits versus That of Native Multimeric Proteins|journal=Journal of Biological Physics|year=2001|volume=27|issue=1|pages=59–71|doi=10.1023/A:1011826525684|pmid=23345733|pmc=3456399}}{{cite journal|last=Elliott|first=C.|author2=H. Joseph Goren |title=Adipocyte Insulin-binding Species: The 40 Å Stoke's Radius Protein|journal=Biochemistry and Cell Biology|year=1984|volume=62|issue=7|pages=566–70|doi=10.1139/o84-075|pmid=6383574 }}{{cite journal|last=Uversky|first=V.N.|title=Use of Fast Protein Size-exclusion Liquid Chromatography to Study the Unfolding of Proteins Which Denature through the Molten Globule|journal=Biochemistry|year=1993|volume=32|issue=48|pages=13288–98|doi=10.1021/bi00211a042|pmid=8241185}} They are useful in characterizing biological species due to the size-dependence of processes like enzyme-substrate interaction and membrane diffusion. The Stokes radii of sediment, soil, and aerosol particles are considered in ecological measurements and models.{{cite journal|last=Ellis|first=W.G.|author2=J.T. Merrill |title=Trajectories for Saharan Dust Transported to Barbados Using Stokes's Law to Describe Gravitational Settling|journal=Journal of Applied Meteorology and Climatology|year=1995|volume=34|issue=7|pages=1716–26|doi=10.1175/1520-0450-34.7.1716|bibcode=1995JApMe..34.1716E|doi-access=free}} They likewise play a role in the study of polymer and other macromolecular systems.

• Born equation
• Capillary electrophoresis
• Dynamic light scattering
• Equivalent spherical diameter
• Einstein relation (kinetic theory)
• Ionic radius
• Ion transport number
• Molar conductivity

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Category:Fluid dynamics

Category:Radii