Question

The r.m.s. speed of hydrogen at S.T.P. is '\(u\)' \(\text{m/s}\). If the gas is heated at constant pressure till its volume becomes three times, the final temperature of the gas and the r.m.s. speed are respectively

• A. \(1092 \text{K}\) , \(3u \text{m/s}\)
• B. \(1092 \text{K}\) , \(\frac{u}{3} \text{m/s}\)
• C. \(819 \text{K}\) , \(\sqrt{3}\,u \text{m/s}\)
• D. \(819 \text{K}\) , \(\frac{u}{\sqrt{3}} \text{m/s}\)

Hint:

At constant pressure V is proportional to T, and r.m.s. speed is proportional to the square root of T.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
At constant pressure, Charles law gives \(\frac VT = \) constant. The r.m.s. speed is \(v_{rms} = \sqrt{\frac{3RT}{M}}\), so \(v_{rms} \propto \sqrt T\).

Step 2: Key Formula or Approach:
S.T.P. means \(T_1 = 273\) K. The volume becomes \(3V_1\).

Step 3: Detailed Explanation:
\(T_2 = 3T_1 = 3 \times 273 = 819\) K.
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt3 \]
So the new r.m.s. speed is \(\sqrt3\,u\).
Options A and B use \(1092\) K, which is \(4 \times 273\), and options B and D divide the speed instead of increasing it.

Final Answer:
The final temperature is \(819\) K and the speed is \(\sqrt{3}\,u\), option (C). \[ \boxed{819\ \text{K},\ \sqrt{3}\,u} \]