Deviation from Grahams Law of Diffusion Due to Inert Gas Collisions

To determine the reason why the experimental distance $d$ traveled by gas $\mathbf{X}$ is smaller than the theoretical estimate calculated from Graham's law, we analyze the diffusion process in the presence of an inert gas:

1. Theoretical Estimate via Graham's Law: Graham's law of effusion/diffusion assumes that the gas molecules move through a vacuum (or an identical collision-free environment), where the rate of diffusion $r$ is inversely proportional to the square root of the molar mass $M$: $r \propto \frac{1}{\sqrt{M}}$ If $d$ is the distance traveled by gas $\mathbf{X}$ and $(L - d)$ is the distance traveled by gas $\mathbf{Y}$ through a tube of length $L$, the theoretical ratio of distances is: $\frac{d_{\text{theoretical}}}{L - d_{\text{theoretical}}} = \frac{r_{\mathbf{X}}}{r_{\mathbf{Y}}} = \sqrt{\frac{M_{\mathbf{Y}}}{M_{\mathbf{X}}}}$

2. Effect of Inert Gas Collisions (Real Diffusion): In reality, the tube contains an inert gas medium through which the molecules of $\mathbf{X}$ and $\mathbf{Y}$ must diffuse. The transport of molecules is governed by mutual diffusion, where collisions with the inert gas molecules impede the directed movement of the diffusing species.

   • The diffusion coefficient $D$ of a gas molecule through a medium is inversely related to its collision frequency $Z$ with the surrounding molecules (or directly proportional to its mean free path $\lambda$): $D \propto \lambda \propto \frac{1}{Z}$
   • If the collision frequency of gas $\mathbf{X}$ with the inert gas molecules is higher than that of gas $\mathbf{Y}$, molecules of $\mathbf{X}$ will undergo more frequent scattering events per unit time.
   • This increased collision frequency significantly retards the net forward diffusion rate of gas $\mathbf{X}$ relative to gas $\mathbf{Y}$.

3. Conclusion: Because gas $\mathbf{X}$ experiences a higher collision frequency with the inert gas than gas $\mathbf{Y}$, its actual rate of diffusion is suppressed more than predicted by Graham's law alone. Consequently, gas $\mathbf{X}$ travels a shorter distance before meeting gas $\mathbf{Y}$, making the experimentally observed value of $d$ smaller than the theoretical value.

Therefore, this deviation is attributed to an increased collision frequency of $\mathbf{X}$ with the inert gas as compared to that of $\mathbf{Y}$ with the inert gas.

Correct Answer: (D)