Question

The rms speed of oxygen molecule at some temperature is $150 \text{ ms}^{-1}$. Then the rms speed of hydrogen molecule at the same temperature is

• A. $400 \text{ ms}^{-1}$
• B. $600 \text{ ms}^{-1}$
• C. $200 \text{ ms}^{-1}$
• D. $800 \text{ ms}^{-1}$

Hint:

Lighter gas molecules move much faster! Since hydrogen is 16 times lighter than oxygen, its molecules move $\sqrt{16} = 4$ times faster at any given temperature.

Answer 1

The Correct Option is B

Step 1: rms speed formula.
The root-mean-square speed is $v_{rms} = \sqrt{\dfrac{3RT}{M}}$, where $M$ is the molar mass.

Step 2: Same temperature link.
At the same $T$, $v_{rms} \propto \dfrac{1}{\sqrt{M}}$. Lighter gas moves faster.

Step 3: Write the ratio.
\[ \frac{v_{H}}{v_{O}} = \sqrt{\frac{M_{O}}{M_{H}}} \]
Step 4: Put in the molar masses.
Oxygen $M_{O}=32$ and hydrogen $M_{H}=2$. \[ \frac{v_{H}}{v_{O}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]
Step 5: Use the given oxygen speed.
$v_{O} = 150$ m s$^{-1}$, so $v_{H} = 4\times 150$.

Step 6: Final value.
\[ v_{H} = 600 \text{ m s}^{-1} \]This is option 2.
\[ \boxed{v_{H} = 600 \text{ m s}^{-1}} \]