[Physics] Home experiment to estimate Avogadro’s number

Question

How to get an approximation of Avogadro or Boltzmann constant through experimental means accessible by an hobbyist ?

Answer 1

Your best bet is to try to replicate the experiments of Perrin who first measured Avogadro's constant. This is a common lab in "Advanced Lab" courses in undergraduate or graduate courses, so you can probably find writeups and such via google.

The principle is to observe Brownian motion under a microscope and measure the diffusion constant.

Einstein's theory for Brownian motion relates the diffusion constant of a spherical particle to the temperature via the Einstein-Stokes law:

$D=\frac{k_BT}{6\pi\eta r}$

Here $D$ is the diffusion constant, $T$ is the temperature, $\eta$ is the viscosity, and $r$ is the radius of a small spherical particle. All of these properties should be measurable at least crudely with home equipment.

The way I did this in my lab course was as follows. We had access to a microscope with a CCD camera which allowed digital video recording, as well as samples of monodisperse polystyrene particles (which are commercially available, and labeled with their size).

Suspend the particles in water (the viscosity of water is well known) at room temperature and place on a slide (better, put a thermocouple in your sample).

Take video of the particles Brownian motion, and then using something like John Crocker, David Grier, and Eric Weeks's celebrated particle tracking code extract 2D (or maybe 3D?) particle trajectories (i.e. $x(t)$, $y(t)$.

Now plot the mean squared displacement of particles versus time. The slope of this curve is the diffusion constant, which then yields an estimate for $k_B$ via Stokes-Einstein.

To recover Avogadro's constant, you need the ideal gas constant $R$, which is measured through independent means; typically via macroscopic thermodynamic experiments which probe the slope in $pV=nRT$.