Perrin, Einstein, and Avogadro's Number

Chapter 4 of Intermediate Physics for Medicine and Biology includes a homework exercise (Problem 12) about Jean Perrin’s experiment to determine Avogadro’s number. Perrin measured the equilibrium distribution of small particles suspended in water as a function of height, fit his data to a Boltzmann factor to determine the Boltzmann constant, and then calculated Avogadro’s number via the gas constant. I like that homework problem because it combines a mini history lesson with a physics exercise, and the numbers aren’t made up; they came from Perrin’s book Brownian Movement and Molecular Reality.

Perrin didn’t use just one method to determine Avogadro’s number; he used several. Below I present a new homework problem describing another technique of Perrin’s. Again I draw data from his book.

Section 4.6

Problem 12 ½. Jean Perrin used a relationship between diffusion and viscosity to determine Avogadro’s number. He recorded the variance of the displacement, σ², as a function of time, t, for small particles suspended in water. The particles had a radius, a, of 0.212 μm, and the viscosity of water, η, was 0.0012 N s/m² at a temperature, T, of 17 °C.

(a) Use the data below and Eq. 4.77 to estimate the diffusion constant, D, of the particles.

t  (s)	σ2  (μm2)
30	45
60	86.5
90	140
120	195

Either use the least squares method of Sec. 11.1 to fit the data, or estimate an average value of D by trial and error.

(b) Use the Einstein relationship, Eq. 4.23, to determine Boltzmann’s constant, k_B, from the diffusion constant found in part (a).

(c) Use your result from part (b), along with the gas constant, R, and Eq. 3.31, to calculate Avogadro’s number, N_A. Your result may not be the same as the currently accepted value of N_A, but it should be close.

For those of you who don’t have your copy of IPMB at your side, Eq. 4.77 is

Eq. 4.23 is

and Eq. 3.31 is

Although Perrin was the first to perform this experiment, Albert Einstein initially proposed the idea during his miraculous year, 1905. The story behind this method can be found in Abraham Pais’s magnificent biography ‘Subtle is the Lord…’. Pais writes

One never ceases to experience surprise at this result, which seems, as it were, to come out of nowhere: prepare a set of small spheres which are nevertheless huge compared with simple molecules, use a stopwatch and a microscope, and find Avogadro’s number.

During the first decade of the twentieth century, the research by Perrin and Einstein confirmed the existence of atoms.

I’ll give Perrin the last word by quoting from the conclusion of Brownian Movement and Molecular Reality.

I think it is impossible that a mind, free from all preconception, can reflect upon the extreme diversity of phenomena which thus converge to the same result, without experiencing a very strong impression, and I think that it will henceforth be difficult to defend by rational arguments a hostile attitude to molecular hypotheses.