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

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

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]