What is the ratio of the number of molecules having speeds in the range of $2 u_{m p}$ and $2 u_{m p} + d u$ to the number of molecules having speeds in the range of $u_{m p}$ and $u_{m p} + d u$ ?
(Answer should be closest integer to the ratio)

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Solution

Let $d N_{1}$ represents the number of molecules in the speed range $u_{m p}$ to $u_{m p} + d u$ .
$d N_{2}$ represents the number of molecules in the speed range $2 u_{m p}$ to $2 u_{m p}$
According to maxwell distribution,
$\frac{d N_{1}}{N} = 4 π (\frac{M}{2 π R T})^{1.5} u_{m p}^{2} e x p (- M u_{m p}^{2} / 2 R T) d u$ and

$\frac{d N_{2}}{N} = 4 π (\frac{M}{2 π R T})^{1.5} 2 u_{m p}^{2} e x p (- 4 M u_{m p}^{2} / 2 R T) d u$

Hence, $\frac{d N_{2}}{d N_{1}} = 4 \frac{e x p (- 4 M u_{m p}^{2} / 2 R T)}{e x p (- M u_{m p}^{2} / 2 R T)} = 4 e x p (- 3 M u_{m p}^{2} / 2 R T)$

But, $u_{m p}^{2} = \frac{2 R T}{M}$

Hence, $\frac{d N_{2}}{d N_{1}} = 4 e^{- 3} = 0.199$ .
Hence answer is 0.

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