For a given gas at 1 atm pressure,
Question:

For a given gas at $1 \mathrm{~atm}$ pressure, rms speed of the molecules is $200 \mathrm{~m} / \mathrm{s}$ at $127^{\circ} \mathrm{C}$. At 2 atm pressure and at $227^{\circ} \mathrm{C}$, the rms speed of the molecules will be:

1. (1) $100 \mathrm{~m} / \mathrm{s}$

2. (2) $80 \sqrt{5} \mathrm{~m} / \mathrm{s}$

3. (3) $100 \sqrt{5} \mathrm{~m} / \mathrm{s}$

4. (4) $80 \mathrm{~m} / \mathrm{s}$

Correct Option: , 3

Solution:

(3) $V_{r m s}=\sqrt{\frac{3 R T}{M}}$

$\frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}=\sqrt{\frac{(273+127)}{(273+227)}}$

$=\sqrt{\frac{400}{500}}=\sqrt{\frac{4}{5}}=\frac{2}{\sqrt{5}}$

$\therefore v_{2}=\frac{\sqrt{5}}{2} v_{1}=\frac{\sqrt{5}}{2} \times 200=100 \sqrt{5} \mathrm{~m} / \mathrm{s}$