Ratio of rms speed and average speed of molecules is 2 ( 2 N + 1 ) 3 ( N + 1 )

Ratio of rms speed and average speed is ( 2 N + 1 ) ( N + 1 ) 6 N

Ratio of rms speed and average speed is ( 2 N + 1 ) ( N + 1 ) 6

Ratio of rms speed and average speed of molecules is 2 ( 2 N + 1 ) 3 ( N + 1 )

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$N (< 100)$ molecules of a gas have velocities 1, 2, 3… N $k m / s$ respectively. Then

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rms speed and average speed of molecules are same

Ratio of rms speed and average speed is $\sqrt{\frac{(2 N + 1) (N + 1)}{6 N}}$

Ratio of rms speed and average speed is $\sqrt{\frac{(2 N + 1) (N + 1)}{6}}$

Ratio of rms speed and average speed of molecules is $\sqrt{\frac{2 (2 N + 1)}{3 (N + 1)}}$

answer is D.

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Detailed Solution

$v_{rms} = \sqrt{\frac{1^{2} + 2^{2} + 3^{2} + \dots \dots + N^{2}}{N}} \Rightarrow v_{rms} = \sqrt{\frac{N (N + 1) (2 N + 1)}{6 N}} = \sqrt{\frac{(N + 1) (2 N + 1)}{6}}$

$v_{avg} = \frac{1 + 2 + \dots + N}{N} = \frac{N (N + 1)}{2 N} = \frac{N + 1}{2}$

$∴ \frac{v_{rms}}{v_{avg}} = \sqrt{\frac{(N + 1) (2 N + 1)}{6}} \times \frac{2}{(N + 1)} = \sqrt{\frac{2}{3} (\frac{2 N + 1}{N + 1})}$

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