Question

If c̅, C and α represent the mean velocity, root mean square velocity and the most probable velocity of a gas at a temperature T then: 
1. α > c̅ > C
2. α < C < c̅
3. α > C > c̅
4. α < c̅ < C

Answer 1

Correct Answer - Option 4 : α < c̅ < C

CONCEPT:

Mean velocity of the gas

• Mean velocity is the arithmetic mean of the velocities of different molecules of a gas at a given temperature.
• It is given as,

\(⇒ v_{av}=\sqrt{\frac{8RT}{\pi M}}\)

Root mean square velocity of the gas

• Root mean square velocity (RMS value)is the square root of the mean of squares of the velocity of individual gas molecules.
• It is given as,

​\(⇒ v_{rms}=\sqrt{\frac{3RT}{M}}\)

The most probable velocity of a gas

• The most probable velocity at which the most molecules in gas travel.

• It is given as,

\(⇒ v_{mp}=\sqrt{\frac{2RT}{M}}\)

​EXPLANATION:

Mean velocity	Root mean square velocity	The most probable velocity
\(v_{av}=\sqrt{\frac{8RT}{\pi M}}\)	\(v_{rms}=\sqrt{\frac{3RT}{M}}\)	\(v_{mp}=\sqrt{\frac{2RT}{M}}\)

• From the formula of the above three velocities of the gas at a particular temperature, it is clear that,

⇒ α < c̅ < C

• Hence, option 4 is correct.