Question

Let $\overrightarrow{v}$, v${}_{rms}$ and v${}_p$ respectively denote the mean speed, root mean square speed and most probable speed of the molecules in an ideal monoatomic gas at absolute temperature T. The mass of a molecule is m. Then

• A. no molecule can have a speed greater than $\sqrt{2}{v}_{rms}$
• B. no molecule can have speed less than $v_p\sqrt{2}$
• C. $v_p<\overrightarrow{v}<v_{rms}$
• D. the average kinetic energy of a molecule is $\frac{3}{4}m{v}_p^2$

Answer 1

Answer: (C), (D)

$v_{rms}=\sqrt{\frac{3RT}{M}},\overrightarrow{v}=\sqrt{\frac{8}{\pi }.\frac{RT}{M}}=\sqrt{\frac{2.5RT}{M}}$
and $v_p=\sqrt{\frac{2RT}{M}}$
From these expressions we can see that
$v_p<\overrightarrow{V}<v_{rms}$
Secondly $v_{rms}=\sqrt{\frac{3}{2}}{v}_p$
and average kinetic energy of a gas molecule
$=\frac{1}{2}m{v}_{rms}^2=\frac{1}{2}m(\sqrt{\frac{3}{2}{v}_p}{)}^2=\frac{3}{4}m{v}_p^2$