The average distance travelled by the molecule between two successive collisions is called mean free path
Mean free path \(\lambda=\frac{k T}{\sqrt{2} \pi d^{2} P}\)
Where K - Boltzmann's constant
T - Temperature
d - diameter of the molecule
P - Pressure

Molar mass of nitrogen m = 0.028 kg/mol
Temperature T = 20 + 273 = 293 K
Universal gas constant \( R=8.314 \mathrm{~J} / \mathrm{mol} / \mathrm{k}\)
RMS speed of nitrogen (N₂),
\(v_{r m s} =\sqrt{\frac{3 R T}{m}}  \)
\(\left(\mathrm{~N}_{2}\right), v_{m s} =\sqrt{\frac{3 \times 8.31 \times 273}{0.028}}  \)
\(\left(\mathrm{~N}_{2}\right), v_{r m s} =\sqrt{2610 \times 10^{2}}=511 \mathrm{~m} / \mathrm{s}\)
Molar mass of Oxygen = 0.032 kg / mol
RMS speed of oxygen (O₂),
\(v_{m s}=\sqrt{\frac{3 R T}{m}}\)
RMS speed of (O₂) \(v_{m s} =\sqrt{\frac{3 \times 8.31 \times 293}{0.032}} \)
\(=\sqrt{2280 \times 10^{2}}\)
RMS speed of \(\left(\mathrm{O}_{2}\right), \quad v_{m s}=478 \mathrm{~m} / \mathrm{s}\)

RMS velocity \(\mathrm{C} =\sqrt{\frac{3 R T}{M}} \)
\(\mathrm{T}=\mathrm{T}_{1} =273 \mathrm{~K}  \)
\(\therefore C =\sqrt{\frac{3 R \times 273}{M}}\)
When the RMS velocity is tripled
\(3 \mathrm{C}=\sqrt{\frac{3 R T}{M}}\)
Dividing equation is (2) by (1) we get
\(\frac{3 C}{C} =\sqrt{\frac{3 R T / M}{3 R \times 273 / M}}  \)
\(3 =\sqrt{\frac{T}{273}}  \)
\(\therefore 9 =\frac{T}{273}  \)
\(\therefore T =273 \times 9=2457 \mathrm{~K}\)
\(\therefore\) Temperature \(\mathrm{T}_{2}=2457 \mathrm{~K}, \mathrm{~T}_{1}=273 \mathrm{~K}\)

We get \(P V=\frac{2}{3} U\) But the internal energy of an ideal gas is equal to N times the average kinetic energy \((\epsilon)\) of each molecule.
\(\mathrm{U}=\mathrm{N} \in\)
For a fixed temperature, the average translational kinetic energy \(\in\) will remain constant It implies that
\(\mathrm{PV} =\frac{2}{3} \mathrm{~N} \in  \)
\(\text {Thus } \mathrm{PV} =\text { constant }\)
Therefore, pressure of a given gas is inversely proportional to its volume provided the temperature remains constant. This is Boyle's law.

(i) Temperature of the gas
(ii) Pressure of the gas
(iii) Diameter of the gas molecules