Question

The temperature of an ideal gas is increased from 100 K to 400 K. If \( v \) is the R.M.S. velocity of its molecules at 100 K, it becomes

• A. \( \frac{v}{\sqrt{2}} \)
• B. \( 2v \)
• C. \( 3v \)
• D. \( 4v \)

Hint:

The R.M.S. velocity of gas molecules is proportional to the square root of the temperature. If the temperature increases by a factor, the R.M.S. velocity increases by the square root of that factor.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
RMS velocity of gas molecules is directly proportional to the square root of its absolute temperature.
Step 2: Key Formula or Approach:
\( v_{rms} \propto \sqrt{T} \).
Step 3: Detailed Explanation:
Given \( T_1 = 100\text{ K} \) and \( T_2 = 400\text{ K} \).
Velocity at 100 K is \( x \).
Velocity at 400 K is \( v_2 \).
\[ \frac{v_2}{x} = \sqrt{\frac{400}{100}} = \sqrt{4} = 2 \]
\[ v_2 = 2x \]
Step 4: Final Answer:
The RMS velocity becomes 2x.