Nitrogen and oxygen are held in a $0\mathrm{.}5d{m}^3$ container maintained at $27C$ at partial pressure of $P_{N2}=0\mathrm{.}8$atm and $P_{O2}=0\mathrm{.}2$atm. Assuming that the collision diameters are ${}_{N_2}=0\mathrm{.}374nm$ and $d_{O_2}=0\mathrm{.}357nm$ and the values for $d$ and the molecule numbers do not change. At $30C,(\frac{?}{\text{b}\text{a}\text{r}}(u{)}_{N^2}^2\frac{+}{b}ar(u{)}_{02}^2{)}^{\frac{1}{2}}=615m{s}^{-1}\frac{,}{b}ar(u{)}_{N2}=450m{s}^{-1}\frac{,}{b}ar(u{)}_{02}=420m{s}^{-1}.$ Considering that all kinds of collision $($collision between like$-$molecules and collision between unlike$-$molecules$)$ in the container are taken into account, calculate

a$.$ the average number of collision experienced in unit time by one molecule of $N_2$ and by one molecule of $O_2$ :

b$.$ the total number of collision per unit volume and per unit time :

c$.$ the mean free paths for like$-$molecules and for the unlike$-$molecules.

$(6$ points$)$

$(8$ points$)$

$(6$ points$)$