Mean Free Path Dependence

The mean free path of a gas obeys $\lambda =\frac{k_{B}T}{\sqrt{2}\pi d^{2}P}$, so for a fixed molecular diameter it is directly proportional to absolute temperature and inversely proportional to pressure, $\lambda \propto T/P$. The cleanest way to see the combined effect is to recognise that the path ultimately depends on number density $n=P/(k_{B}T)$ through $\lambda \propto 1/n$. When the temperature is doubled, $\lambda$ would double; when the pressure is halved, $\lambda$ would double again. Combining the two independent effects multiplies the path by $2\times 2=4$. Hence the mean free path increases by a factor of four. The value 2 accounts for only one of the two changes. The value 1 incorrectly assumes the effects cancel, as if $\lambda \propto T\times P$. The value 0.25 inverts the dependence, treating $\lambda$ as proportional to $P/T$. This applies the NCERT kinetic-theory expression for mean free path, where heating spreads molecules apart at fixed pressure and lowering pressure reduces number density. As a plausibility check, the molecular number density $n=P/(k_{B}T)$ falls to one-quarter of its original value under these changes, and since $\lambda \propto 1/n$, the mean free path must rise four-fold, in exact agreement with the result.