Question

Since molecules undergo frequent collisions with each other in a volume of a gas, they move in straight lines at a constant speed only between one collision and the next. The average distance traversed between successive collisions is called the mean free path, $\lambda$. Among the given options, contemplate and choose the option(s) on which $\lambda$ should not explicitly depend

• A. $\lambda \propto V$ (V : volume)
• B. $\lambda \propto N$ (N: number of particles in a fixed V)
• C. $\lambda \propto d$ (d: diameter of the molecule)
• D. $\lambda \propto m$ (m: mass of the molecules).

Answer 1

Answer: (B), (C), (D)

$\lambda \propto m$ (m: mass of the molecules).

At a fixed temperature, the average speed of the molecules of a monoatomic gas in a given volume will be a constant. If the molecules are tightly packed, the frequency of inter-molecular collisions for a particular molecule will increase, which will reduce the mean free pathλ. The number density can be denoted by $N=\left(\frac{N}{V}\right)$. Also, a larger molecule, i.e., with a large cross-sectional area has higher chances of encountering another stray molecule, compared to a smallerone. Combining all these arguments, we can say -

$\lambda \propto \frac{1}{\rho N}$,

$\Rightarrow \lambda \propto \frac{V}{N}$.

For a fixed N,

$\lambda \propto V$ ---------- (1)

For a fixed V,

$\lambda \propto \frac{V}{N}$ ---------- (2)

Also, $\lambda \propto \frac{1}{d^2}$ (where d is the diameter is proportional to the cross- sectional area) ----------- (3)

The mass of the molecule seems to play no role in the explicitly.