Question

The root mean square velocity, ${v_{rms}}$, the average velocity ${v_{av}}$and the most probable velocity, ${v_{mp}}$of the molecules of the gas are in the order.
A. ${v_{mp}} > {v_{av}} > {v_{rms}}$
B. ${v_{rms}} > {v_{av}} > {v_{mp}}$
C. ${v_{av}} > {v_{mp}} > {v_{rms}}$
D. ${v_{mp}} > {v_{rms}} > {v_{av}}$

Answer 1

Hint Calculate the values of ${v_{mp}}$, ${v_{rms}}$ and ${v_{av}}$of the molecules of the gas from:
${v_{av}} = \sqrt {\dfrac{{8RT}}{{m\pi }}}$
${v_{rms}} = \sqrt {\dfrac{{3RT}}{m}}$
${v_{mp}} = \sqrt {\dfrac{{2RT}}{m}}$
Solving these formulas, we get the values of velocities and then compare them.

Complete step-by-step solution :
Here,
${v_{rms}} =$root mean square velocity
${v_{av}} =$average velocity
${v_{mp}} =$probable velocity
$m =$molar mass of the gas
$R =$gas constant
$T =$Temperature in kelvin
$\pi =$
Root mean square velocity is used to measure the velocity of a particle in gas. It is given by:
${v_{rms}} = \sqrt {\dfrac{{3RT}}{m}} = 1.73\sqrt {\dfrac{{RT}}{m}} \Rightarrow \left( i \right)$
Most probable speed, is the speed most likely to be possessed by any molecule in the system. It is given by:
${v_{mp}} = \sqrt {\dfrac{{2RT}}{m}} = 1.41\sqrt {\dfrac{{RT}}{m}} \Rightarrow \left( {ii} \right)$
Average velocity is the arithmetic mean of the velocities of different molecules of gas at a temperature. It is given by:
${v_{av}} = \sqrt {\dfrac{{8RT}}{{m\pi }}} = 1.59\sqrt {\dfrac{{RT}}{m}} \Rightarrow \left( {iii} \right)$
From equations $\left( i \right),\left( {ii} \right)$and $\left( {iii} \right)$, we can conclude that
${v_{rms}} > {v_{av}} > {v_{mp}}$
Therefore, ${v_{rms}}$is greater than ${v_{av}}$and ${v_{av}}$is greater than ${v_{mp}}$.
So, option (B) is correct.

Note:- Solve the expressions of root mean square velocity, probable velocity and average velocity and then compare each equation with each other. There is no need to put the values of $R$, $T$ and $m$.