A) \[4\,{{\exp }^{-4}}\]
B) \[4\,{{\exp }^{-3}}\]
C) \[3\,{{\exp }^{-3}}\]
D) \[3\,{{\exp }^{-4}}\]
Correct Answer: B
Solution :
[b] Let \[dN\Rightarrow \] Number of molecules in the speed range \[{{u}_{mp}}\]to \[({{u}_{mp}}+du)\] \[d{{N}_{2}}\Rightarrow \] Number of molecules in the speed range 2u to \[(2{{u}_{mp}}+du)\]. According to Maxwell distribution, \[\frac{d{{N}_{1}}}{N}=4\pi {{\left( \frac{M}{2\pi RT} \right)}^{3/2}},\] \[{{u}^{2}}_{mp}\exp (-M{{u}^{2}}_{mp}/2RT)du\] and \[\frac{d{{N}_{2}}}{N}=4\pi {{\left( \frac{M}{2\pi RT} \right)}^{3/2}}\] \[{{(2{{u}_{mp}})}^{2}}\exp (-4M{{u}^{2}}_{mp}/2RT)du\] \[\frac{d{{N}_{2}}}{d{{N}_{1}}}=4\left( \frac{\exp (-Mu_{mp}^{2}/2RT}{\exp (-4Mu_{mp}^{2}/2RT} \right)\] \[=4\exp \,(-3M{{u}^{2}}_{mp}/2RT\] Now since, \[{{u}^{2}}_{mp}=\frac{2RT}{M}\] \[\frac{d{{N}_{2}}}{d{{N}_{1}}}=4{{e}^{-3}}=4{{\exp }^{-3}}\]
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