Question

7.2 Derive an expression giving the diameter of the spherical particle that has an rms displacement in 1 second due to Brownian motion equal to its terminal settling velocity. Neglect slip correction.

Answer 1

We know from Einstein's theory that the mean square displacement is given by;

where is the diffusion coefficient which for a particle of radius in a medium of viscosity at temperature is equal to:

Here is the Boltzmann constant

Therefore,

For second, we would have, the rms displacement,

According to Stoke's law, for objects falling under gravity, the terminal vecloity is given from the relation,

Here is the force due to buoyancy. The final resulting force is expressed in terms of the density of the spherical particle and the density of the surrouding medium, therefore,

According to the problem, we should have

Thus,

Thus the required radius is