The Stokes-Einstein equation is the equation first derived by Einstein in his Ph.D thesis for the diffusion coefficient of a "Stokes" particle undergoing Brownian Motion in a quiescent fluid at uniform temperature. The result was formerly published in Einstein's (1905) classic paper on the theory of Brownian motion (it was also simultaneously derived by Sutherland (1905) using an identical argument). Einstein's result for the diffusion coefficient D of a spherical particle of radius a in a fluid of dynamic viscosity h at absolute temperature T is:
where is the gas constant and NA is Avogadro's Number. The formula is historically important since it was used to make the first absolute measurement of NA so confirming molecular theory. Although the formula can be derived alternatively using the Langevin equation of motion for a Brownian particle [Chandrasekhar (1943)] the derivation of Einstein is a powerful and ingenious one, correct even when the Langevin equation is only approximate. Einstein assumed that van't Hoff's law for the osmotic pressure exerted by solute molecules in a solvent fluid at equilibrium was equally applicable to the pressure p associated with a suspension of Brownian particles at equilibrium in the same fluid, i.e.,
where nM is the number of gram moles of fluid per unit volume and f the 'molar fraction' defined here as the ratio of the number of particles to the number of fluid molecules. Einstein then argued that a suspension of Brownian particles at equilibrium under their own weight could be viewed in two ways both of which were equivalent: a balance between the net weight of the particles and the gradient of the particle pressure in the direction of gravity; or a balance between the diffusion flux and the settling flux due to gravity. Using the Stokes drag formula for the settling velocity (see Stokes Law) and the formula for p above gives the formula for D given above. A similar argument allows one to deduce a form for the particle pressure in a turbulent gas knowing the form for the particle turbulent diffusion coefficient (see Particle Transport in Turbulent Fluids).
Chandrasekhar, S. (1943) Rev. Mod. Phys., 15, 1.
Einstein, A. (1905) Ann. der Physik, 17, 549.
Sutherland, W. (1905) Phil. Mag., 9. 781.