AIIMS 2004 Physics Maxwellian Distribution MCQ Question

Certainly! Let's delve into the velocities of gas molecules as described by the Maxwellian velocity distribution and clarify the relationships among the root mean square speed ($v_{\text{rms}}$), average speed ($v_{\text{av}}$), and most probable speed ($v_{\text{mp}}$).

Concept Overview

In kinetic theory, the speeds of gas molecules follow a Maxwellian distribution, which is characterized by three key types of speeds:

1. Most Probable Speed ($v_{\text{mp}}$): This is the speed at which the maximum number of molecules are moving. It can be calculated using the formula: $v_{\text{mp}} = \sqrt{\frac{2kT}{m}}$ where $k$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $m$ is the mass of a gas molecule.

2. Average Speed ($v_{\text{av}}$): This is the arithmetic mean of the speeds of all gas molecules. The formula for average speed is given by: $v_{\text{av}} = \sqrt{\frac{8kT}{\pi m}}$

3. Root Mean Square Speed ($v_{\text{rms}}$): This represents the square root of the average of the squares of the speeds. It is calculated as: $v_{\text{rms}} = \sqrt{\frac{3kT}{m}}$

Relationships Among $v_{\text{rms}}$, $v_{\text{av}}$, and $v_{\text{mp}}$

From these formulas, we can derive the relationships among these three speeds. For an ideal gas, it is known that:

• The most probable speed $v_{\text{mp}}$ is less than the average speed $v_{\text{av}}$.
• The average speed $v_{\text{av}}$ is less than the root mean square speed $v_{\text{rms}}$.

Thus, the order of the speeds is: $v_{\text{mp}} < v_{\text{av}} < v_{\text{rms}}$

Verification of the Correct Answer

Given the order of the velocities, we can analyze the options:

• Option A: $v_{\text{rms}} < v_{\text{av}} < v_{\text{mp}}$ (Incorrect)
• Option B: $v_{\text{rms}} > v_{\text{av}} > v_{\text{mp}}$ (Correct)
• Option C: $v_{\text{mp}} < v_{\text{rms}} < v_{\text{av}}$ (Incorrect)
• Option D: $v_{\text{mp}} > v_{\text{rms}} > v_{\text{av}}$ (Incorrect)

Conclusion

Thus, the correct answer is B: $v_{\text{rms}} > v_{\text{av}} > v_{\text{mp}}$.

This relationship is fundamental in kinetic theory and reflects how molecular speeds are distributed in a gas. Understanding these concepts is crucial for problems related to gas behavior in thermodynamics and statistical mechanics.