Question

At $101.325kPa$ and $300K$ , The fraction of $N_{2}$ molecules will have speeds in the range of $\left(\right.u_{mp}-0.005u_{mp}\left.\right)$ to $\left(\right.u_{mp}+0.005u_{mp}\left.\right)$ is $p\times 10^{- 3}$ . What is the value of $p$ ?
(Give answer after rounding off to the nearest integer value.)

Answer 1

$ \frac{ dN _{ u }}{ N }=4 \pi\left(\frac{ M }{2 \pi RT }\right)^{3 / 2} e ^{-\frac{ Mu ^2}{2 RT }} u ^2 du $
The most probable speed $u_{ mp }$ can be calculated as follows,
$ \begin{array}{l} u _{ mp }=\sqrt{\frac{2 RT }{ M }} \\ du =\left( u _{ mp }+0.005 u _{ mp }\right)-\left( u _{ mp }-0.005 u _{ mp }\right)=0.01 u _{ mp } \\ =\frac{4}{\pi^{1 / 2}} \times\left(\frac{ M }{2 RT }\right)^{3 / 2} e ^{-\frac{ M 2 RT }{2 MT }}\left(\frac{2 RT }{ M }\right) 0.01 \sqrt{\frac{2 RT }{ M }} \\ =\frac{4}{1.772} \times \frac{1}{ e } \times 10^{-2} \\ =\frac{4}{1.772} \times \frac{1}{2.7182} \times 10^{-2} \cong 8 \times 10^{-3} \end{array} $