Question

Let $\overline{v},v_{rms}$ and $v_p,$ respectively, denote the mean speed, root mean square speed and most probable speed of the molecules in an ideal monatomic 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. $v_p<\overline{v}<v_{rms}$
• D. the average kinetic energy of a molecule is $3/4\left(m{v}_p^2\right)$

Answer 1

Answer: (C), (D)

The correct options are
C $v_p<\overline{v}<v_{rms}$
D the average kinetic energy of a molecule is $3/4\left(m{v}_p^2\right)$

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