States of Matter - Result Question 42
42. The ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is
(1981, 1M)
(a) $1.085: 1$
(b) $1: 1.086$
(c) $2: 1.086$
(d) $1.086: 2$
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Solution:
The two types of speeds are defined as;
Root mean square speed $\left(u _{rms}\right)=\sqrt{\dfrac{3 R T}{M}}$
$ \text { Average speed }\left(u _{av}\right)=\sqrt{\dfrac{8 R T}{\pi M}} $
For the same gas, at a given temperature, $M$ and $T$ are same, therefore
$ \begin{aligned} \dfrac{u _{rms}}{u _{av}} & =\sqrt{\dfrac{3 R T}{M}}: \sqrt{\dfrac{8 R T}{\pi M}} \\ & =\sqrt{3}: \sqrt{\dfrac{8}{\pi}}=\sqrt{3}: \sqrt{2.54}=1.085: 1 \end{aligned} $
BETA
