# The ratio of the root mean square speed, average speed and maximum possible speed for a gas will be -

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The ratio of the root mean square speed, average speed and maximum possible speed for a gas will be -
1. $\sqrt 3 \sim 2\sqrt 2 \sim 3\sqrt 3$
2. $\sqrt 2 \sim 2\sqrt 2 \sim 3\sqrt 3$
3. $\sqrt 3 \sim \sqrt 2 \sim 2\sqrt 2$
4. $\sqrt 3 \sim 2\sqrt 2 \sim \sqrt 2$

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Correct Answer - Option 4 : $\sqrt 3 \sim 2\sqrt 2 \sim \sqrt 2$

Concept:

• The gaseous particles are in continuous motion with each other.
• There are three measure parameters to measure speed of the gas.

Root - mean - square velocity of Gaseous particles:

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 by

$\Rightarrow {V_{rms}} = \sqrt{ {\frac{{3RT}}{M}} }$

Average velocity:

It is the arithmetic mean of the velocities of different molecules of a gas at a given temperature. It is given by

${V_{av}} = √ {\frac{{8RT}}{{\pi M}}}$

Most probable velocity:

It is the velocity possessed by maximum fraction of molecules at the same temperature. It is given by

${V_p} = √ {\frac{{2RT}}{M}}$

Explanation:

The required ratio between these parameters are

$√ {\frac{{3RT}}{M}} \;\;:√ {\frac{{8RT}}{{\pi M}}\;} \;:√ {\frac{{2RT}}{M}}$

By Canceling all constant terms from the above expression we have

√3 : √8 ; √2

Or

√3 : 2√2 ; √2

So, the correct option is √3 : 2√2 ; √2