Diffusion Coefficient Calculator

The diffusion coefficient D measures how fast random thermal motion spreads one species through another, and it carries units of area per time because the mean square displacement of a particle grows linearly with elapsed time: in one dimension the average of x squared equals 2Dt. The Stokes-Einstein form D = kB*T/(6*pi*eta*r) is the Einstein relation D = kB*T/f evaluated with the Stokes drag coefficient f = 6*pi*eta*r, the friction on a rigid sphere of radius r creeping through a continuum of dynamic viscosity eta under no-slip boundary conditions. Since the 2019 SI revision the Boltzmann constant is fixed at exactly 1.380649e-23 J/K. In water near 25 degrees C, where eta is about 0.89 mPa.s, small solutes such as dissolved oxygen or glucose fall between roughly 6e-10 and 2e-9 m^2/s, globular proteins sit near 1e-11 to 1e-10 m^2/s, and gases at 1 atm are four to five orders of magnitude faster at about 1e-5 m^2/s. The radius r is a hydrodynamic radius that includes the hydration shell, so it always exceeds the crystallographic radius. For solutes comparable in size to the solvent molecules the continuum picture fails and a slip boundary condition, giving 4*pi*eta*r, fits better. The Wilke-Chang route is frankly empirical: the association parameter phi is 2.6 for water, 1.9 for methanol, 1.5 for ethanol and 1.0 for unassociated solvents, Vb is the Le Bas molar volume at the normal boiling point, and typical accuracy for dilute nonelectrolytes is about 10 percent.

Adolf Fick gave diffusion its macroscopic law in 1855, but the molecular meaning of D arrived fifty years later. Albert Einstein derived the relation in his 1905 Annalen der Physik paper on the motion of small particles suspended in a stationary liquid, and William Sutherland published essentially the same result that year in the Philosophical Magazine while working on albumin, which is why it is sometimes called the Sutherland-Einstein relation. Marian Smoluchowski reached it independently in 1906. The friction term came from George Gabriel Stokes, whose 1851 analysis of slow viscous flow past a sphere supplied the factor 6*pi*eta*r. Jean Perrin tested the predictions in Paris between 1908 and 1910 using gamboge granules and extracted a value for the Avogadro constant, work that earned him the 1926 Nobel Prize in Physics. Engineers needed estimates for solutes whose radius nobody had measured, so correlations followed. Charles Wilke and Pin Chang, at the University of California, Berkeley, fitted their expression to liquid-phase data and published it in the AIChE Journal in 1955; it still appears in Perry's Chemical Engineers Handbook. Direct measurement improved in parallel. Dynamic light scattering, practical once lasers arrived in the 1960s, yields D and lets workers back out a hydrodynamic radius, and the pulsed field gradient spin-echo method introduced by Stejskal and Tanner in 1965 made NMR a routine diffusion tool.