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.

### Like this:

Like Loading...

*Related*

snailmecream.ruToday, Avogadro’s number is formally defined to be the number of carbon-12 atoms in 12 grams of unbound carbon-12 in its rest-energy electronic state. The current state of the art estimates the value of N

JesseWhen I worked through this problem, I had the same issue, not knowing of a way to eliminate the microscopic parameters. I’m not sure if this is what Schroeder had in mind, but I ended up deciding that the ‘b’ parameter in the Van der Waal equation of state was suitably macroscopic and measurable. This equation was discussed in problem 1.17 so I figured it’s fair game.

You can do a quick “proof” that b = V/A, where V is the Van der Waal volume associated with the Van der Was radius and A is Avogadro’s number. Just show that this is the limit of the volume per molecule at high pressure. This lets you eliminate r in the mean free path and then you can solve the ideal gas conductivity equation for Avogadro’s number.

Plugging in values that I looked up for helium at 20°C and atmospheric pressure, I calculated A=5.23×10^23 using only macroscopic quantities, which seems pretty good for all the approximations involved!

JesseEDIT: There is a typo in my previous reply, that should read b = V*A, not b = V/A.