The Stokes radius or Stokes–Einstein radius of a solute is the radius of a hard sphere that diffuses at the same rate as that solute. Named after George Gabriel Stokes, 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 ion with weaker hydration. This is because the smaller ion drags a greater number of water molecules with it as it moves through the solution.[1]

Stokes radius is sometimes used synonymously with effective hydrated radius in solution.[2] Hydrodynamic radius, RH, can refer to the Stokes radius of a polymer or other macromolecule.

## Spherical case

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

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

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

${\displaystyle \mu ={\frac {ze}{f}}}$

where ${\displaystyle ze}$ represents ionic charge in integer multiples of electron charges.

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

${\displaystyle D={\frac {\mu k_{B}T}{q}}={\frac {k_{B}T}{f}}}$

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

${\displaystyle D={\frac {k_{B}T}{6\pi \eta a}}}$

which can be rearranged to solve for ${\displaystyle a}$, the radius:

${\displaystyle R_{H}=a={\frac {k_{B}T}{6\pi \eta D}}}$

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

## Research applications

Stokes radii are often determined experimentally by gel-permeation or gel-filtration chromatography.[3][4][5][6] They are useful in characterizing biological species due to the size-dependence of processes like enzyme-substrate interaction and membrane diffusion.[5] The Stokes radii of sediment, soil, and aerosol particles are considered in ecological measurements and models.[7] They likewise play a role in the study of polymer and other macromolecular systems.[5]

## References

1. Atkins, Peter; Julio De Paula (2006). Physical Chemistry (8 ed.). Oxford: Oxford UP. p. 766. ISBN 0-7167-8759-8.
2. Atkins, Peter; Julio De Paula (2010). Physical Chemistry (9 ed.). Oxford: Oxford UP.
3. 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.
4. Dutta, Samarajnee; Debasish Bhattacharyya (2001). "Size of Unfolded and Dissociated Subunits versus That of Native Multimeric Proteins". Journal of Biological Physics. 27: 59–71. PMC 3456399.
5. 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.
6. 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.
7. 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 Meteorology and Climatology. 34 (7): 1716–26. Bibcode:1995JApMe..34.1716E. doi:10.1175/1520-0450-34.7.1716.