Question

Assertion: Mean free path of gas molecules varies inversely as the density of the gas.
Reason: Mean free path varies inversely as the pressure of the gas.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect and Reason is correct.

Answer 1

Hint: Using the expressions that relate the parameters such as, the mean free path, the density, temperature, pressure, the diameter of the molecules of the gas and Boltzmann’s constant, we will explain the relationship between the assertion and the reason statements.

Formula used:
\[\begin{align}
  & \lambda =\dfrac{kT}{\sqrt{2}\pi {{\sigma }^{2}}\rho } \\
 & PV=nRT \\
\end{align}\]

Complete step-by-step answer:
From the given information, we have the data as follows.
Assertion: Mean free path of gas molecules varies inversely as the density of the gas.
Reason: Mean free path varies inversely as the pressure of the gas.
The mean free path of a gas molecule equals the average distance between the successive collisions. The mathematical representation of the same is,
\[\lambda =\dfrac{kT}{\sqrt{2}\pi {{\sigma }^{2}}\rho }\]
Where \[k\] is Boltzmann’s constant, \[T\] is the temperature, \[\sigma \] is the diameter of the gas molecules and \[\rho \] is the density of the molecules.
Considering the other parameters constant, we get the relation between the mean free path and the density of the gas.
\[\lambda \propto \dfrac{1}{\rho }\]
Therefore, the “Assertion: Mean free path of gas molecules varies inversely as the density of the gas.” is correct.
Again considering the other parameters constant, we get the relation between the mean free path and the density of the temperature.
\[\lambda \propto T\]
Now, we will replace the temperature with the pressure using the gas equation, that is, \[PV=nRT\].
So, we get the relation between the mean free path and the pressure.
\[\lambda \propto \dfrac{1}{P}\]
Therefore, the “Reason: Mean free path varies inversely as the pressure of the gas” is correct.
But, the assertion and the reason statements are not related to each other in terms of the explanation.
\[\therefore \] B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

So, the correct answer is “Option B”.

Note: The reason statement explains the relation of the mean free path with the parameter pressure, whereas, the assertion statement defines the relation between the mean free path with the parameter density. So, these statements are not related.