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  4. A 25 × 10-3 m3 volume cylinder is filled with 1 mol of O2 gas at room temperature (300K). The molecular diameter of O2, and its root mean square speed, are found to be 0.3 nm, and 200 m/s, respectively. What is the average collision rate (per second) for an O2 molecule ?

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Q. 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 $0.3\, nm$, and $200\, m/s$, respectively. What is the average collision rate (per second) for an $O_2$ molecule ?

JEE MainJEE Main 2019Kinetic Theory

Solution:

$v = \frac{V_{av}}{\lambda}$
$ \lambda = \frac{RT}{\sqrt{2} \pi\sigma^{2} N_{A}P}$
$ \sigma= 2\times.3 \times10^{-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 \times10^{-3}} \times\sqrt{\frac{8}{3\pi}} $
$= 17.68 \times 10^8 /sec$.
$= 1768 \times 10^{10} /sec. \sim 10^{10}$