At constant pressure, the mean free path \((\lambda)\) of a gas molecule is directly proportional to temperature. At constant temperature, \(\lambda\) is inversely proportional to pressure. If you compare two different gas molecules at the same temperature and pressure, \(\lambda\) is inversely proportional to the square of the diameter of the gas molecules. Put these facts together to create a formula for the mean free path of a gas molecule with a proportionality constant (call it \(R_{\mathrm{mfp}}\), like the ideal-gas constant) and define units for \(R_{\mathrm{mfp}}\).

The mean free path \((\lambda)\) of a gas molecule is directly proportional to temperature. Mathematically, this means that when temperature increases, the mean free path increases. This can be represented as follows: \[ \lambda \propto T\]

At constant temperature, the mean free path \((\lambda)\) is inversely proportional to pressure. This means that when pressure increases, the mean free path decreases. This can be represented as follows: \[ \lambda \propto \frac{1}{P}\]

For two different gas molecules at the same temperature and pressure, the mean free path \((\lambda)\) is inversely proportional to the square of the diameter of the gas molecules. If we use \(d\) to represent the diameter, this relationship can be expressed as: \[ \lambda \propto \frac{1}{d^2}\]

Now, we will combine the three relationships to create a formula for the mean free path \((\lambda)\). This can be done by multiplying the relationships, since the proportionality is for all the same variable, in our case \(\lambda\): \[ \lambda \propto T \times \frac{1}{P} \times \frac{1}{d^2} \]

To make the above equation an equality, we need to introduce a proportionality constant \((R_{\mathrm{mfp}})\): \[ \lambda = R_{\mathrm{mfp}} \times \frac{T}{Pd^2} \] The units of \(R_{\mathrm{mfp}}\) must be such that the left side of the equation (\(\lambda\)) has the same units as the right side. Since mean free path has units of length, \([\lambda] = L\), and we know the units of pressure \(P\) are \([P] = M L^{-1} T^{-2}\) and the diameter \([d] = L\), we can find the units of \(R_{\mathrm{mfp}}\) as follows: \[ [R_{\mathrm{mfp}}] = [\lambda] \times \frac{[P][d^2]}{[T]} = L \times \frac{M L^{-1} T^{-2} \times L^2}{T} = \frac{M L^2}{T^3} \] Therefore, the proportionality constant \(R_{\mathrm{mfp}}\) has units of \(\frac{M L^2}{T^3}\). The final formula for the mean free path of a gas molecule and the units for the proportionality constant \(R_{\mathrm{mfp}}\) is: \[ \lambda = R_{\mathrm{mfp}} \times \frac{T}{Pd^2} \quad , \quad [R_{\mathrm{mfp}}] = \frac{M L^2}{T^3} \]