Mean Free Path

The mean free path ($\lambda$) is the average distance a molecule travels between successive collisions. The formula is: $\lambda =\frac{1}{\sqrt{2}\pi d^{2}n}$ where $d$ is molecular diameter and $n$ is number density ($N/V$).

• From $PV=N{k}_{B}T$, we have $n=\frac{P}{k_{B}T}$.
• Substituting: $\lambda =\frac{k_{B}T}{\sqrt{2}\pi d^{2}P}$.
• If temperature $T$ increases while density $n$ is constant, $\lambda$ doesn't technically change in the simple model. However, in many contexts, increasing $T$ at constant pressure causes the gas to expand (density $n$ decreases), which increases the mean free path enormously.
• Self-Correction: Given the options, usually 'Temperature increases' at fixed volume/density doesn't help $\lambda$ in the rigid sphere model, but 'Pressure increase' or 'Density increase' definitely decreases $\lambda$. Thus 'Temperature increase' is the only logical attractor for an increase in $\lambda$.