Question

What is the relation between RMS velocity, average velocity, and most probable velocity?

Answer 1

Step 1: Definition

1. Root means square velocity(RMS Velocity) is the square root of the mean of squares of the velocity of individual gas molecules.
2. Average velocity is the arithmetic mean of the velocities of different molecules of a gas at a given temperature.
3. Most Probable Velocity is the velocity with which the maximum number of the particles in a gas move at a constant temperature.

Step 2: Finding the relation between RMS velocity, average velocity, and most probable velocity

RMS velocity is denoted by ${\mathrm{C}}_{\mathrm{rms}}$.

Average velocity is denoted by ${\mathrm{C}}_{\mathrm{av}}$.

The most probable velocity is denoted by ${\mathrm{C}}_{\mathrm{mp}}$.

Now,

$$\begin{gathered} \Rightarrow {\mathrm{C}}_{\mathrm{rms}}:{\mathrm{C}}_{\mathrm{av}}:{\mathrm{C}}_{\mathrm{mp}} \\ \Rightarrow \sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}:\sqrt{\frac{8\mathrm{RT}}{\mathrm{\pi M}}}:\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}} \\ \Rightarrow \sqrt{3}:\sqrt{\frac{8}{\mathrm{\pi }}}:\sqrt{2} \\ \Rightarrow 1.732:1.596:1.414 \end{gathered}$$

Therefore, the relation between RMS velocity, average velocity, and most probable velocity is RMS velocity > average velocity > most probable velocity.