States of Matter 1 Question 12
12. For gaseous state, if most probable speed is denoted by $C^{*}$, average speed by $\bar{C}$ and root square speed by $C$, then for a large number of molecules, the ratios of these speeds are
(a) $C^{*}: \bar{C}: C=1.225: 1.128: 1$
(2013 Main)
(b) $C^{*}: \bar{C}: C=1.128: 1.225: 1$
(c) $C^{*}: \bar{C}: C=1: 1.128: 1.225$
(d) $C^{*}: \bar{C}: C=1: 1.225: 1.128$
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Solution:
- $C^{*}=$ Most probable speed $=\sqrt{\frac{2 R T}{M}}$
$$ \bar{C}=\text { Average speed }=\sqrt{\frac{8 R T}{\pi M}} $$
$C=$ Root square speed corrected as root means square speed, i.e.
rms $=\sqrt{\frac{3 R T}{M}}$ and as we know $\dot{\hat{C}}<\bar{C}<C$
$$ \stackrel{*}{C}: \bar{C}: C=1: \sqrt{\frac{4}{\mathrm{p}}}: \sqrt{\frac{3}{2}}=1: 1.128: 1.225 $$
NOTE
As no option correspond to root square speed, it is understood as misprint. It should be root mean square speed.
