Question

The ratio among most probable velocity, mean velocity, and root mean square velocity is given by:

• A. $1:2:3$
• B. $1:\sqrt{2}:\sqrt{3}$
• C. $\sqrt{2}:\sqrt{3}:\sqrt{8\pi }$
• D. $\sqrt{2}:\sqrt{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}:\sqrt{3}$

Answer 1

The correct option is D

$\sqrt{2}:\sqrt{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}:\sqrt{3}$

Step 1: Concept :

1. Average velocity is defined as total displacement in total time.
2. The mathematical expression for it is ${\mathrm{v}}_{\mathrm{Avg}}=\sqrt{\frac{8\mathrm{RT}}{\mathrm{\pi M}}}$, where $v_{avg}$ = average velocity, $\mathrm{R}$ = gas constant, $\mathrm{T}$ = temperature, $\mathrm{M}$= molar mass.
3. Most probable velocity is defined as velocity possessed by the maximum fraction of gaseous molecules at a particular time.
4. Mathematical expression for most probable velocity ${\mathrm{v}}_{\mathrm{mp}}=\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}}$, where$v_{mp}$= most probable velocity, $\mathrm{R}$ = gas constant, $\mathrm{M}$= molar mass.
5. The root-mean-square velocity is the value of the square root of the sum of the squares of the stacking velocity values divided by the number of values.
6. The mathematical expression for root-mean square velocity ${\mathrm{v}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$, where $v_{rms}$= root-mean square velocity, $\mathrm{R}$ = gas constant, $\mathrm{M}$= molar mass and $\mathrm{T}$= temperature.

Step 2: Calculation :

The ratio among most probable velocity, mean velocity, and root mean square velocity will be :

$$\begin{gathered} {\mathrm{v}}_{\mathrm{mp}}:{\mathrm{v}}_{\mathrm{Avg}}:{\mathrm{v}}_{\mathrm{rms}}=\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}}:\sqrt{\frac{8\mathrm{RT}}{\mathrm{\pi M}}}:\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}} \\ {\mathrm{v}}_{\mathrm{mp}}:{\mathrm{v}}_{\mathrm{Avg}}:{\mathrm{v}}_{\mathrm{rms}}=\sqrt{2}:\sqrt{\left(\frac{8}{\mathrm{\pi }}\right)}:\sqrt{3} \end{gathered}$$

Hence option $\mathrm{D}$ is the correct option.