Question

The r.m.s. speed of gas molecules at 800 K will be

• A. same as at 200 K
• B. twice the value at 200 K
• C. four times the value at 200 K
• D. half the value at 200 K

Hint:

The r.m.s. speed of gas molecules is directly proportional to the square root of the temperature. If the temperature doubles, the r.m.s. speed increases by a factor of \( \sqrt{2} \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to compare the root-mean-square speeds of a gas at two different temperatures.
Step 2: Key Formula or Approach:
The r.m.s. speed is given by \( v_{rms} = \sqrt{\frac{3RT}{M}} \).
This implies \( v_{rms} \propto \sqrt{T} \).
Step 3: Detailed Explanation:
Let \( v_1 \) be the speed at \( T_1 = 200\text{ K} \) and \( v_2 \) be the speed at \( T_2 = 800\text{ K} \).
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{800}{200}} = \sqrt{4} = 2 \]
So, \( v_2 = 2 v_1 \).
Step 4: Final Answer:
The r.m.s. speed at 800 K is twice the value at 200 K.