States of Matte...

Collision cross-section is an area of an imaginary sphere of radius $$\sigma $$ around the molecule within which the center of another molecule cannot penetrate. The volume swept by a single molecule in unit time is,
$$V=(\pi \sigma ^{2}) \bar{u}$$, where $$\bar {u}$$ is the average speed.
If $$N^{*}$$ is the number of molecules per unit volume, then the number of molecules within the volume $$V$$ is 
$$N=VN^{*}=(\pi \sigma ^{2} \bar{u})N^{*}$$
Hence, the number of collision made by a single molecule in unit time will be,
$$Z=N=(\pi \sigma ^{2} \bar{u})N^{*}$$
In order to account for the movements of all molecules, we must consider the average velocity along the line of centres of two colliding molecules instead of the average velocity of a single molecule. If it is assumed that, on an average, molecules collide while approaching each other perpendicularly, then the average velocity along their centres is $$\sqrt{2} \bar {u}$$ as shown below,
Number of collision made by a single molecule with other molecules per unit time is given by,
$$Z_{1}=\pi \sigma ^{2} (\bar {u_{rel}}) N^{*}=\sqrt {2}\pi \sigma ^{2} \bar {u} N^{*}$$
The total number of bimolecular collisions $$Z_{11}$$ per unit volume per unit time is given by,
$$Z_{11}=\displaystyle \frac{1}{2}(Z_{1}N^{*})$$ or $$Z_{11}=\displaystyle \frac{1}{2}(\sqrt{2}\pi \sigma ^{2}\bar{u}N^{*})N^{*}=\displaystyle \frac{1}{\sqrt{2}}\pi \sigma ^{2}\bar{u}N^{*2}$$
If the collision involve two unlike molecules then the number of collisions $$Z_{12}$$ per unit volume per unit time is given as,
$$Z_{12}=\pi \sigma _{12}^{2}\left ( \sqrt{\displaystyle \frac{8kT}{\pi \mu }} \right )N_{1}N_{2}$$
where $$N_{1}$$ and $$N_{2}$$ are the number of molecules per unit volume of the two types of molecules, $$\sigma _{12}$$ is the average diameter of the two molecules and $$\mu$$ is the reduced mass. The mean free path is the average distance travelled by a molecule between two successive collisions. We can express it as follows;
$$\lambda =\displaystyle \frac{Average\: distance\: travelled\: per\: unit\: time}{No.\: of\: collisions\: made\: by\: a\: single\: molecule\: per\: unit\: time}=\displaystyle \frac{\bar{u}}{Z_{1}}$$
or $$\lambda =\displaystyle \frac{\bar {u}}{\sqrt{2}\pi \sigma ^{2}\bar {u}N^{*}}\Rightarrow \displaystyle \frac{1}{\sqrt{2}\pi \sigma ^{2}N^{*}}$$
Three ideal gas samples in separate equal volume containers are taken and following data are recorded :

	Pressure	Temperature	Mean free paths	Mol.wt
Gas A	1 atm	1600 K	0.16 nm	20
Gas B	2 atm	200 K	0.16 nm	40
Gas C	4 atm	400 K	0.04 nm	80