A guerrilla version of Einstein's elegant derivation

If the concentration of solute on one side of the membrane is one mole / L (one mole = N, about 6 10²³ molecules) , and it's zero on the other, R the gas constant, and T the temperature:

[4]    P = R T

If the membrane were suddenly removed, diffusion would cause a flow J, of solute of (the following equation is the definition of D, the diffusion coefficient):

[5]    J = D c

Now think about the individual solute molecules, which are moving with velocity V across the surface that used to be blocked by the membrane. The flow will just be proportional to the velocity of the molecules:

[6]    J = c V

This V is due to a force F generated by thermal motion. A number of scientists, particularly Stokes in the late 1800's had investigated the motion of objects in a liquid, finding that the velocity increased in direct proportion to the force on the object. They thus defined a Frictional Coefficient, f , for the object and fluid:

[7]    V = F / f

Now substitute [7] into [6] to get:

[8]    J = c F / f

But Einstein had already guessed what F was in [4], the pressure per unit area of the membrane per mole (N) was R T, or per molecule, R T / N :

[9]    J = c R T / N f

Now look back at [5], the definition of D, and we see that D must be:

[10]    D = R T / N f

Since Stokes ( Sir George Gabriel Stokes, 1819 -1903) had already shown that for a sphere of radius r, moving through a fluid with viscosity v , the frictional coefficient was:

[11]    f = 6 pi v r

we can substituting this value for f into [10] to get:

[12]    D = R T / 6 pi N v r    Q. E. D.