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Q.
If average velocity becomes $4 $ times then what will be the effect on rms velocity at that temperature?
Bihar CECEBihar CECE 2008Kinetic Theory
Solution:
Ratio of $v_{a v} / v_{r m s}$ remains constant.
Average speed is the arithmetic mean of the speeds of molecules in a gas at a given temperature, ie,
$v_{a v}=\left(v_{1}+v_{2}+v_{3}+\ldots\right) / N$
and according to kinetic theory of gases,
$v_{a v}=\sqrt{\frac{8 R T}{M \pi}}\,\,\,...(i)$
Also, rms speed (root mean square speed) is defined as the square root of mean of squares of the speed of different molecules, ie,
$v_{r m s}=\sqrt{\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+\ldots\right) / N} $$
=\sqrt{(\bar{v})^{2}}$
and according to kinetic theory of gases,
$v_{r m s}=\sqrt{\frac{3 R T}{M}}\,\,\,...(ii) $
From Eqs. (i) and (ii), we get
$v_{a v}=\sqrt{\left(\frac{8}{3 \pi}\right)} v_{r m s}=0.92 v_{r m s} \,\,\,...(iii)$
Therefore, $\frac{v_{a v}}{v_{r m s}}=$ constant
Hence, root mean square-velocity is also become $4$ times.