Question

A $25\times 10^{-3}{m}^3$ volume cylinder is filled with $1$ mol of $O_2$ gas at room temperature $(300K)$.
The molecular diameter of $O_2$, and its root mean square speed, are found to be $0.3nm$, and $200m/s$, respectively. What is the average collision rate (per second) for an $O_2$ molecule ?

• A. $\sim 10^{11}$
• B. $\sim 10^{13}$
• C. $\sim 10^{10}$
• D. $\sim 10^{12}$

Answer 1

Answer: (D)

$v=\frac{V_{av}}{\lambda }$
$\lambda =\frac{RT}{\sqrt{2}\pi {\sigma }^2{N}_{A}P}$
$\sigma =2\times .3\times 10^{-9}$
$P=\frac{RT}{V}$
$\Rightarrow \lambda =\frac{V}{\sqrt{2}\pi {\sigma }^2{N}_A}{V}_{av}=\sqrt{\frac{8}{3\pi }}\times V_{rms}$
$\therefore v=\frac{200\times \sqrt{2}\pi \times {\sigma }^2{N}_A}{25\times 10^{-3}}\times \sqrt{\frac{8}{3\pi }}$
$=17.68\times 10^8/sec$.
$=1768\times 10^{10}/sec.\sim 10^{10}$