Question

At $101.325kPa$ and $300K$ , The fraction of $N_2$ molecules will have speeds in the range of $(u_{mp}-0.005u_{mp})$ to $(u_{mp}+0.005u_{mp})$ is $p\times 10^{-3}$ . What is the value of $p$ ?
(Give answer after rounding off to the nearest integer value.)

Answer 1

Answer: 8

$\frac{d{N}_u}{N}=4\pi {\left(\frac{M}{2\pi RT}\right)}^{3/2}{e}^{-\frac{M{u}^2}{2RT}}{u}^{2}du$
The most probable speed $u_{mp}$ can be calculated as follows,
$\begin{array}{l}{u}_{mp}=\sqrt{\frac{2RT}{M}}\\ du=\left(u_{mp}+0.005u_{mp}\right)-\left(u_{mp}-0.005u_{mp}\right)=0.01u_{mp}\\ =\frac{4}{{\pi }^{1/2}}\times {\left(\frac{M}{2RT}\right)}^{3/2}{e}^{-\frac{M2RT}{2MT}}\left(\frac{2RT}{M}\right)0.01\sqrt{\frac{2RT}{M}}\\ =\frac{4}{1.772}\times \frac{1}{e}\times 10^{-2}\\ =\frac{4}{1.772}\times \frac{1}{2.7182}\times 10^{-2}\cong 8\times 10^{-3}\end{array}$