Pretend that you live in the $19$th century and don't know the value of Avogadro's number* (or of Boltzmann's constant or of the mass or size of any molecule). Show how you could make a rough estimate of Avogadro's number from a measurement of the thermal conductivity of gas, together with other measurements that are relatively easy.

The thermal conductivity and other macroscopic properties measured for an ideal gas can be used to generate a rough estimate of Avogadro's number. The formula for thermal conductivity is:

$k_t=\frac{C_V}{2V}\ell \overline{v}$ Let be equation ($1$)

Assume we've configured the system so that we have a volume box $V$and cross-sectional area $A$. We know that the average speed $\overline{v}$is approximately the RMS speed, which is:

$\overline{v}\approx v_{rms}=\sqrt{\frac{3kT}{m}}$

multiply with $\frac{\sqrt{N}}{\sqrt{N}}$, so:

Let the following be Equation ($2$)

localid="1650414641595" $$\begin{gathered} \overline{v}=\sqrt{\frac{3NkT}{Nm}} \\ =\sqrt{\frac{3NkT}{M}} \\ =\sqrt{\frac{3PV}{M}} \end{gathered}$$

where $\hspace{0.17em}M$is the total mass of the gas. Substitute from ($2$) into ($1$), so:

localid="1650414665465" $$\begin{gathered} k_t=\frac{C_V}{2V}\ell \overline{v} \\ =\frac{C_V}{2V}\ell \sqrt{\frac{3PV}{M}} \\ =\frac{C_V}{2}\ell \sqrt{\frac{3P}{MV}} \end{gathered}$$

$\ell =\frac{2}{C_V}{k}_t\sqrt{\frac{MV}{3P}}$

Schroeder's expression for $\ell$is based on the idea that the mean path length is equal to the length of a cylinder of radius equal to the diameter and volume of the molecule equal to the average volume per molecule $\frac{V}{N}$, so that:

$\ell =\frac{1}{4\pi r^2}\left(\frac{N}{V}\right)$

where $r$the radius of the molecule, To get $N$from this formula, we need to know $r$, but this is a microscopic quantity that we're assuming we don't know. I can't see any way of progressing from here unless we take a different value localid="1650296672007" $\ell$. Since we're after only a rough approximation of Avogadro's number, we can take $\ell$Instead of the average distance between collisions, it should be the average distance between molecules. That is to say:

Let be equation ($4$)

$\ell \approx {\left[\frac{V}{N}\right]}^{\frac{1}{3}}$

We can now combine equations ($4$) and ($3$), so:

${\left[\frac{V}{N}\right]}^{\frac{1}{3}}=\frac{2}{C_V}{k}_t\sqrt{\frac{MV}{3P}}$

apply the power $3$on both sides:

$\frac{V}{N}\approx {\left(\frac{2}{C_V}{k}_t\right)}^3{\left(\frac{MV}{3P}\right)}^{\frac{3}{2}}$

solve for $N$, so:

$\frac{N}{V}\approx {\left(\frac{C_V}{2k_t}\right)}^3{\left(\frac{3P}{MV}\right)}^{\frac{3}{2}}\to N\approx {\left(\frac{C_V}{2k_t}\right)}^3{\left(\frac{3P}{MV}\right)}^{\frac{3}{2}}V$

$N\approx {\left(\frac{C_V}{2k_t}\right)}^3{\left(\frac{3P}{M}\right)}^{\frac{3}{2}}\frac{1}{\sqrt{V}}$ let be equation ($5$)

Assuming we know the gas constant $R$, The ideal gas law can be used to calculate the number of moles:

$n=\frac{PV}{RT}$

And Avogadro number is given by:

$N_A=\frac{N}{n}=\frac{RT}{PV}N$

substitute from equation ($5$) so Avogadro's number is roughly:

$N_A\approx \frac{RT}{PV}\left[{\left(\frac{C_V}{2k_t}\right)}^3{\left(\frac{3P}{M}\right)}^{\frac{3}{2}}\frac{1}{\sqrt{V}}\right]$

$N_A\approx RT\sqrt{P}\left[{\left(\frac{C_V}{2k_t}\right)}^3{\left(\frac{3}{MV}\right)}^{\frac{3}{2}}\right]$