Stokes radius

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The Stokes radius or Stokes-Einstein radius (named after George Gabriel Stokes) of a solute is the radius of a hard sphere that diffuses at the same rate as that solute. It is closely related to solute mobility, factoring in not only size but also solvent effects. A smaller ion with stronger hydration, for example, may have a greater Stokes radius than a larger but weaker ion.

Stokes radius is sometimes used synonymously with effective hydrated radius in solution.[1] Hydrodynamic radius, RH, can refer to the Stokes radius of a polymer or other macromolecule.

Spherical Case[edit]

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

F_{drag}=fs=(6 \pi \eta a)s

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

 \mu = \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:

 D = \frac{\mu k_B T}{q} = \frac{k_b T}{f}

where  k_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 = \frac{k_b T}{6 \pi \eta a}

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

 R_H = a = \frac{k_b T}{6 \pi \eta D}

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

Research Applications[edit]

Stokes radii are often determined experimentally by gel-permeation or gel-filtration chromatography.[2][3][4][5] They are useful in characterizing biological species due to the size-dependence of processes like enzyme-substrate interaction and membrane diffusion.[4] The Stokes radii of sediment, soil, and aerosol particles are considered in ecological measurements and models.[6] They likewise play a role in the study of polymer and other macromolecular systems.[4]

See also[edit]


  1. ^ Atkins, Peter; Julio De Paula (2010). Physical Chemistry (9 ed.). Oxford: Oxford UP. 
  2. ^ Alamillo, J.; Jacobo Cardenas; Manuel Pineda (1991). "Purification and Molecular Properties of Urate Oxidase from Chlamydomonas Reinhardtii". Biochimica Et Biophysica Acta (BBA) - Protein Structure and Molecular Enzymology 1076 (2): 203–08. doi:10.1016/0167-4838(91)90267-4. 
  3. ^ Dutta, Samarajnee; Debasish Bhattacharyya (2001). "Size of Unfolded and Dissociated Subunits versus That of Native Multimeric Proteins". Journal of Biological Physics 27: 59–71. 
  4. ^ a b c Elliott, C.; H. Joseph Goren (1984). "Adipocyte Insulin-binding Species: The 40 Å Stoke's Radius Protein". Biochemistry and Cell Biology 62 (7): 566–70. doi:10.1139/o84-075. 
  5. ^ Uversky, V.N. (1993). "Use of Fast Protein Size-exclusion Liquid Chromatography to Study the Unfolding of Proteins Which Denature through the Molten Globule". Biochemistry 32 (48): 13288–98. doi:10.1021/bi00211a042. 
  6. ^ Ellis, W.G.; J.T. Merrill (1995). "Trajectories for Saharan Dust Transported to Barbados Using Stokes's Law to Describe Gravitational Settling". Journal of Applied Meterology 34 (7): 1716–26. doi:10.1175/1520-0450-34.7.1716.