Question

For the gaseous state, if most probable speed is denoted by C*, average speed by C and mean square speed by C, then for a large number of molecules the ratios of these speeds are:

$straight C asterisk times colon stack space straight C with bar on top space colon thin space straight C space equals space 1.225 space colon 1.128 space space colon 1$
$straight C to the power of asterisk times space colon thin space straight C with bar on top space colon space straight C space equals space 1.128 space colon thin space 1.225 space colon space 1$
$straight C to the power of asterisk times space colon thin space straight C with bar on top space colon space straight C space equals space 1.28 space colon thin space 1.225 space colon space$
$straight C to the power of asterisk times space colon thin space straight C with bar on top space colon space straight C space equals 1.225 space colon space 1. space 128 space$

Solution

straight C to the power of asterisk times space colon thin space straight C with bar on top space colon space straight C space equals space 1.28 space colon thin space 1.225 space colon space

straight C asterisk times space equals space Most space probable space speed space equals space square root of fraction numerator 2 RT over denominator straight M end fraction end root straight C with bar on top space equals space Average space speed space equals space square root of fraction numerator 8 RT over denominator πM end fraction end root straight C space equals space Mean space square space speed space corrected space as space rms space equals space square root of fraction numerator 3 RT over denominator straight M end fraction end root straight C asterisk times space less than space straight C with bar on top space less than space straight C straight C asterisk times space colon space straight C with bar on top space colon space straight C space equals 1 space colon thin space square root of 4 over straight pi end root colon square root of 3 over 2 end root space equals space 1 colon space 1.128 colon space 1.225

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