The diffusion constant…is closely related to the viscosity, as was first pointed out by Albert Einstein. This is not surprising, since diffusion is caused by the random motion of particles under the bombardment of neighboring atoms, and viscous drag is also caused by the bombardment of neighboring atoms.All this random bombardment is also related to Brownian motion: the random movement of small particles when they collide with many water molecules.
What is typically referred to as the Einstein relationship is given by our Eq. 4.22,
D = kT/β, (4.22)
where D is the diffusion constant, k is Boltzman’s constant, T is the absolute temperature, and β is a factor relating the viscous force to the drift velocity, sometimes called the frictional drag coefficient. Essentially, β is like the reciprocal of the mobility. This equation doesn’t contain the viscosity, but if you use Stokes’ law for β you get
D = kT/6πηa, (4.23)
where a is the radius of the particle being considered and η is the coefficient of viscosity.
To answer that question, I consulted my favorite biography of Einstein, Abraham Pais’s Subtle is the Lord. Pais writes
If Marian Ritter von Smolan-Smoluchowski had been only an outstanding theoretical physicist and not a fine experimentalist as well, he would probably have been the first to publish a quantitative theory of Brownian motion.
Smoluchowski, born to a Polish family, spent his early years in Vienna, where he also received a university education. After finishing his studies in 1894, he worked in several laboratories abroad, and then returned to Vienna, where he became Privatdozent. In 1900 he became professor of theoretical physics in Lemberg (now Lvov), where he stayed until 1913. In that period he did his major work. In 1913 he took over the directorship of the Institute for Experimental Physics at the Jagiellonian University in Cracow. There he died in 1917, victim of a dysentery epidemic.
It is quite remarkable how often Smoluchowski and Einstein simultaneously and independently pursued similar if not identical problems. In 1904 Einstein worked on energy fluctuations, Smoluchowski on particle number fluctuations of an ideal gas. Einstein completed his first paper on Brownian motion in May 1905; Smoluchowski his in July 1906.
Smoluchowski began his 1906 paper by referring to Einstein’s two articles of 1905: “The findings [of those papers] agree completely with some results which I had… obtained several years ago and which I consider since then as an important argument for the kinetic nature of this phenomenon.” Then why had he not published earlier? “Although it has not been possible for me till now to undertake an experimental test of the consequences of this point of view, something I originally intended to do, I have decided to publish these considerations…”Apparently he wanted to get experimental support for his ideas, and by waiting he got scooped.
Both Einstein and Smoluchowski went on to independently study critical opalescence: how the scattering of light passing through a gas increases in the neighborhood of a critical point. Pais concludes
Smoluchowski’s last contribution to this problem [of critical opalescence] was experimental: he wanted to reproduce the blue of the sky in a terrestrial experiment. Preliminary results looked promising, and he announced that more detailed experiments were in progress. He did not live to complete them.
After Smoluchowski’s death, Sommerfeld and Einstein wrote obituaries in praise of a good man and a great scientist. Einstein called him an ingenious man of research and a noble and subtle human being.
