Question

If $\overline{v},v_{rms}$ and $v_p$ represent the mean speed, root mean square and most probable speed of the molecules in an ideal monoatomic gas at temperature $T$ and if $m$ is mass of the molecule, then

• A. $v_p<\overline{v}<v_{rms}$
• B. No molecule can have a speed greater than $\sqrt{2v_{rms}}$
• C. No molecule can have a speed less than $v_p/\sqrt{2}$
• D. None of the above

Answer 1

The correct option is A $v_p<\overline{v}<v_{rms}$

Given that,

$\overline{v}$ is Mean speed of the gass,$v_{rms}$ is the root mean square speed of the gass and $v_p$ is the most probable speed of the gass,and $T$ is the temperature and $m$ is the mass of the molecule of the monoatomic gass.

We know that,

Mean speed, $\overline{v}=\sqrt{\frac{8kT}{\pi m}}=0.92v_{rms}$
rms speed, $v_{rms}=\sqrt{\frac{3kT}{m}}$
Most probable speed $v_p$
$=\sqrt{\frac{2kT}{m}}=0.816v_{rms}$

so
i.e., $v_p<\overline{v}<v_{rms}$