Translational Kinetic Energy and Pressure of Ideal Gas

To determine the correctness of the statements and the relationship between them:

Evaluation of Statement-1:
According to the kinetic theory of gases, the pressure $P$ exerted by an ideal gas of volume $V$ containing $N$ molecules, each of mass $m$, is given by:
$P = \frac{1}{3} \frac{N m}{V} v_{\text{rms}}^2$

The total translational kinetic energy ($K_{\text{trans}}$) of all the molecules of the gas is:
$K_{\text{trans}} = \frac{1}{2} N m v_{\text{rms}}^2$

Rearranging the expression for pressure in terms of $K_{\text{trans}}$:
$P = \frac{2}{3} \frac{K_{\text{trans}}}{V} \implies K_{\text{trans}} = \frac{3}{2} PV = 1.5 \, PV$

This relation holds true for any ideal gas, irrespective of its atomicity (monatomic, diatomic, or polyatomic), because it considers only the translational degrees of freedom (which is always $3$). Thus, Statement-1 is True.

Evaluation of Statement-2:
In a gas, molecules are in continuous, random motion. They undergo frequent collisions with one another (intermolecular collisions) as well as with the walls of the container. These collisions are assumed to be elastic, resulting in a continuous redistribution of momentum and energy, which causes the velocities (both magnitude and direction) of the individual molecules to change constantly. Thus, Statement-2 is True.

Relationship between Statement-1 and Statement-2:
The relation $K_{\text{trans}} = \frac{3}{2}PV$ is fundamentally a consequence of the momentum imparted to the container walls by the molecules during collisions with the walls along the three spatial dimensions (degrees of freedom). In fact, the ideal gas equation of state and the relation for translational kinetic energy can be derived even by neglecting collisions between molecules altogether (treating them as point masses that only collide elastically with the walls). Therefore, intermolecular collisions are not the reason why $K_{\text{trans}} = 1.5 \, PV$.

Thus, Statement-2 is NOT the correct explanation for Statement-1.

Correct Option: (B)