Question

Let $\bar{v},v_{rms}$ and $v_{mp}$ 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
(1) $v_{mp}<\bar{v}<v_{rms}$
(2) no molecule can have a speed greater than $\sqrt{2}{v}_{rms}$
(3) the average kinetic energy of a molecule is $\frac{3}{4}m{v}_{mp}^2$
(4) no molecule can have a speed less than $v_{mp}/\sqrt{2}$

• A. 1, 2 and 3 are correct
• B. 1 and 2 are correct
• C. 2 and 4 are correct
• D. 1 and 3 are correct

Answer 1

Answer: (D)

$\bar{v}=\sqrt{\frac{8k_{B}T}{m\pi }}$
$v_{rms}=\sqrt{\frac{3k_{B}T}{m}},$
$v_{mp}=\sqrt{\frac{2k_{B}T}{m}}$
where $k_B$ is the Boltzmann's constant
$\therefore v_{rms}>\bar{v}>v_{mp}$
or$v_{mp}>\bar{v}<v_{rms}$
Option (1) is correct.
Average kinetic energy of a molecule is
$E=\frac{1}{2}m{v}_{rms}^2$
or $E=\frac{1}{2}m\frac{3k_{B}T}{m}$
$E=\frac{3}{4}m\frac{2k_{B}T}{m}$
or $E=\frac{3}{4}m{v}_p^2$
Option (3) is correct.
Again, according to kinetic theory of gases, a molecule of a gas can have speed such that it lies between 0 and $\infty$. ie, $0<v<\infty$.
Hence the options $(2)$ and (4) can never be correct: