States of Matter 1 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) $1.086: 2$
Objective Questions II
(One or more than one correct option)
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Solution:
- The two types of speeds are defined as;
Root mean square speed $\left(u_{\mathrm{rms}}\right)=\sqrt{\frac{3 R T}{M}}$
$$ \text { Average speed }\left(u_{\mathrm{av}}\right)=\sqrt{\frac{8 R T}{\pi M}} $$
For the same gas, at a given temperature, $M$ and $T$ are same, therefore
$$ \begin{aligned} \frac{u_{\mathrm{rms}}}{u_{\mathrm{av}}} & =\sqrt{\frac{3 R T}{M}}: \sqrt{\frac{8 R T}{\pi M}} \ & =\sqrt{3}: \sqrt{\frac{8}{\pi}}=\sqrt{3}: \sqrt{2.54}=1.085: 1 \end{aligned} $$
