AIIMS 2006 Physics Kinetic Theory of Gases Assertion Reason Question

To tackle the question, let's break down both the assertion and the reason provided.

Assertion: The root mean square and most probable speeds of the molecules in a gas are the same.

Reason: The Maxwell distribution for the speed of molecules in a gas is symmetrical.

Understanding the Concepts

1. Most Probable Speed ($v_{mp}$): This is the speed at which the maximum number of gas molecules are moving. It can be derived from the Maxwell-Boltzmann distribution of speeds, where the formula is given by:

   $v_{mp} = \sqrt{\frac{2kT}{m}}$

   where:

   • $k$ is the Boltzmann constant,
   • $T$ is the temperature in Kelvin,
   • $m$ is the mass of a gas molecule.

2. Root Mean Square Speed ($v_{rms}$): This is a measure of the average speed of gas molecules given by:

   $v_{rms} = \sqrt{\frac{3kT}{m}}$

Comparison of Speeds

• From the formulas, we can see that $v_{mp}$ and $v_{rms}$ are not equal. Specifically, we find that:

  $v_{mp} = \sqrt{\frac{2kT}{m}} \quad \text{and} \quad v_{rms} = \sqrt{\frac{3kT}{m}}$

• Therefore, $v_{mp}$ is less than $v_{rms}$ since $\sqrt{2} < \sqrt{3}$.

Thus, the assertion is false.

1. Symmetry of Maxwell Distribution: The reason states that the Maxwell distribution is symmetrical. However, the Maxwell-Boltzmann distribution for speeds is actually skewed; it has a peak (the most probable speed) and tails on both ends. This distribution is not symmetrical around the most probable speed, but rather, it has a specific shape where the most probable speed is less than the average speed and root mean square speed.

Conclusion

Given the above insights:

1. The assertion is false because the root mean square speed and the most probable speed are not the same.
2. The reason is also false because the Maxwell distribution is not symmetrical; it is skewed.

Final Answer

Thus, the correct answer is D) Both Assertion and Reason are false.

Why Other Options are Incorrect

• Option A: Incorrect since both the assertion and reason are false.
• Option B: Incorrect as the assertion is false and the reason is also false; hence the reason cannot be a correct explanation.
• Option C: Incorrect since both are false, not just the reason.

This comprehensive breakdown clarifies the misconceptions surrounding the speeds of gas molecules and their distribution as per the kinetic theory of gases.