Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.65.
We can get a rough estimate of Avogadro’s number from the thermal conductivity and other macroscopic quantities measurable for an ideal gas. Schroeder’s derivation of the thermal conductivity formula gives the result
Suppose we set up the system so that we have a box of volume and cross-sectional area . We know that the average speed is approximately the rms speed, which is
where is the total mass of the gas. If we measure the thermal conductivity and heat capacity, we can then get the mean free path:
Schroeder’s expression for is based on the idea that the mean path length is equal to the length of a cylinder of radius equal to the molecule’s diameter and volume equal to the average volume per molecule , so that
where is the radius of the molecule. To get from this formula, we need to know , but this is a microscopic quantity which we’re assuming we don’t know. I can’t see any way of progressing from here unless we take a different value for . Since we’re after only a rough approximation of Avogadro’s number, we can take to be the average distance between molecules, rather than the average distance between collisions. That is
We can now combine this with 3 to get an estimate for the number of molecules :
Assuming we know the gas constant , we can get the number of moles from the ideal gas law
so Avogadro’s number is roughly
As a check on this formula, we can verify that it’s dimensionless. has the dimensions of , so dimensionally, the quantity is equivalent to
Thus the dimensions check out.
I don’t know if this is the solution Schroeder had in mind, or whether it’s possible to get using the formula 4 for the mean free path. If we’re not allowed to know a priori any microscopic quantities, that means we don’t know (Boltzmann’s constant), (the mass of a single molecule) or (the radius of a molecule). Together with , that makes 4 unknowns, so we need 4 independent equations to find them. Comments welcome.