Stokes radius

Stokes radius is sometimes used synonymously with effective hydrated radius in solution. Hydrodynamic radius, R_H, 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 ${\displaystyle f}$:

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

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:

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

where ${\displaystyle k_{\text{B}}}$ 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

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

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

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

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