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Relating Translational KE and T

For a gas at ...

Question

For a gas at a temperature $T$ the root-mean-square speed $v_{r m s},$ the most probable speed $v_{m p}$ , and the average speed $v_{a v}$ obey the relationship

v_{m p} > v_{r m s} > v_{a v}

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v_{a v} > v_{r m s} > v_{m p}

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v_{r m s} > v_{a v} > v_{m p}

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v_{m p} > v_{a v} > v_{r m s}

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Solution

The correct option is C $v_{r m s} > v_{a v} > v_{m p}$
We know that $v_{r m s} = \sqrt{\frac{3 k T}{m}}, v_{r m s} = \sqrt{\frac{2 k T}{m}}, v_{a v} = \sqrt{\frac{8 k T}{π m}}$

Root mean square speed
$v_{r m s} = \sqrt{\frac{3 k T}{m}}$
Most probable speed
$v_{r m s} = \sqrt{\frac{2 k T}{m}}$
Average speed
$v_{a v} = \sqrt{\frac{8 k T}{π m}}$

$v_{m p} : v_{a v} : v_{r m s} = 1 : 1.128 : 1.224$
$v_{r m s} > v_{a v} > v_{m p}$

FINAL ANSWER: (b)

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