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 \(R_{\text { mfp }}\) , like the ideal-gas constant) and define units for \(R_{\operatorname{mfp}}\) .

According to the first relationship, the mean free path of a gas molecule is directly proportional to temperature at constant pressure. Mathematically, this can be represented as: \[ \lambda \propto T \]

According to the second relationship, the mean free path of a gas molecule is inversely proportional to pressure at constant temperature. Mathematically, this can be represented as: \[ \lambda \propto \frac{1}{P} \]

According to the third relationship, the mean free path of a gas molecule is inversely proportional to the square of the diameter of the gas molecules at the same temperature and pressure. Let's represent the diameter of the gas molecules as \(d\). Therefore, we have: \[ \lambda \propto \frac{1}{d^2} \]

To combine these three relationships into a single formula, we will multiply the proportionalities together: \[ \lambda \propto \frac{T}{Pd^2} \]

To convert the proportionality into an equation, we need to introduce a proportionality constant \(R_{\text { mfp }}\): \[ \lambda = R_{\text { mfp }} \frac{T}{Pd^2} \]

Now let's analyze the units of \(R_{\text { mfp }}\) to get the final answer. We will use SI units: - Mean free path \(\lambda\) has the unit of length: meters (m). - Temperature \(T\) is measured in Kelvin (K). - Pressure \(P\) is measured in pascals (Pa). - Diameter \(d\) is measured in meters (m). Thus, the units for \(R_{\text { mfp }}\) can be written as: \[ \text{meters} \cdot \frac{\text{pascals} \cdot \text{meters}^2}{\text{Kelvin}} = \frac{\text{Pa} \cdot \text{m}^2}{\text{K}} \] So, the formula for the mean free path of a gas molecule is: \[ \lambda = R_{\text { mfp }} \frac{T}{Pd^2} \] where \(R_{\text { mfp }}\) is the proportionality constant with the unit \(\frac{\text{Pa}\cdot\text{m}^2}{\text{K}}\).