Rankers Physics
Topic: Thermal Physics
Subtopic: Kinetic Theory of Gases

Which of the following gases possesses maximum rms velocity, all being at the same temperature :
Oxygen
Air
Carbon dioxide
Hydrogen

Solution:

The root mean square (r.m.s.) velocity \( v_{\text{rms}} \) of gas molecules is given by:

\[
v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}
\]

where \( m \) is the molar mass of the gas. Since all gases are at the same temperature, \( v_{\text{rms}} \) is inversely proportional to the square root of their molar mass:

\[
v_{\text{rms}} \propto \frac{1}{\sqrt{m}}
\]

Among air, oxygen (\( \text{O}_2 \)), carbon dioxide (\( \text{CO}_2 \)), and hydrogen (\( \text{H}_2 \)), hydrogen has the lowest molar mass. Therefore, it has the highest r.m.s. velocity.

Answer: Hydrogen has the maximum r.m.s. velocity.

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