Question

The root mean square velocity of a gas molecule is $100\text{ ms}^{-1}$. The mass of the molecule is increased four times keeping the temperature constant. Then, the root mean square velocity is

• A. 25 ms$^{-1}$
• B. 50 ms$^{-1}$
• C. 75 ms$^{-1}$
• D. 2500 ms$^{-1}$
• E. 125 ms$^{-1}$

Hint:

Heavier gas molecules move slower than lighter ones at the same temperature. Increasing mass by 4 times reduces the rms velocity by a factor of $\sqrt{4} = 2$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The root mean square (rms) velocity of a gas molecule is directly proportional to the square root of the absolute temperature and inversely proportional to the square root of the mass of the molecule.
Key Formula or Approach:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
Since $T$ is constant, $v_{rms} \propto \frac{1}{\sqrt{m}}$.

Step 2: Detailed Explanation:
Given initial velocity $v_1 = 100\text{ ms}^{-1}$ for mass $m_1 = m$.
Final mass $m_2 = 4m$.
The ratio of velocities is:
\[ \frac{v_2}{v_1} = \sqrt{\frac{m_1}{m_2}} \]
\[ \frac{v_2}{100} = \sqrt{\frac{m}{4m}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]
\[ v_2 = 100 \times \frac{1}{2} = 50\text{ ms}^{-1} \]

Step 3: Final Answer:
The new root mean square velocity is 50 ms$^{-1}$.