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THE REALITY OF MOLECULES 91

and m were also known. Thus Eqs. 5.7 and 5.8 represent one relation between
the two unknowns N and a.
The next thing that Einstein of course did (in the spirit of Loschmidt*) was
find a second connection between N and a. To this end, he used a reasoning which
is partly thermodynamic, partly dynamic. This argument is sketched in his thesis
and repeated in mbre detail in his first paper on Brownian motion [E2]. It is
extremely ingenious.
Consider first an ideal gas and a time-independent force K acting on its mole-
cules in the negative x direction. The force exerted per unit volume equals KpN/
m. In thermal equilibrium, the balance between this force and the gas pressure p
is given by

where R is the gas constant. Now, Einstein reasoned, according to van 't Hoff's

law, Eq. 5.9 should also hold for dilute solutions as long as the time-independent

force K acts only on the solute molecules.

Let K impart a velocity v (relative to the solvent) to the molecules of the solute.

If the mean free path of the solvent molecules is much less than the diameter of

the solute molecules, then (also in view of the boundary condition v = 0 on the

surface of the solute particles) we have the well-known Stokes relation

Then, from the thermal equilibrium condition (Eq. 5.9) and the dynamic equilib-

rium condition (Eq. 5.11)

*See Section 5a. The only nineteenth century method for finding N and a that Einstein discussed

in his 1915 review article on kinetic theory [E6] was the one by Loschmidt.

Observe that the force K has canceled out in Eq. 5.12. The trick was therefore to

use K only as an intermediary quantity to relate the diffusion coefficient to the

so that, under the influence of K, KpN/6irrjam solute molecules pass in the neg-

ative x direction per unit area per second. The resulting concentration gradient

leads to a diffusion in the x direction of DN/m. (dp/dx) particles/cm^2 /sec, where,

by definition, D is the diffusion coefficient. Dynamic equilibrium demands that

the magnitude of the diffusion current equal the magnitude of the current induced

by K: