Chapter 9 – Kinetic Theory of Gases
1. Introduction to Kinetic Theory of Gases
The kinetic theory of gases explains the macroscopic properties of gases
such as pressure, temperature, and volume by considering the microscopic
motion of gas molecules.
This theory bridges **thermodynamics** and **molecular motion**.
2. Assumptions of Kinetic Theory
- Gas consists of a large number of identical molecules
- Molecules are in continuous random motion
- Volume of molecules is negligible compared to gas volume
- No intermolecular force except during collisions
- Collisions are perfectly elastic
- Newton’s laws of motion are obeyed
3. Concept of Ideal Gas
An ideal gas is a hypothetical gas that strictly obeys gas laws
at all temperatures and pressures.
Ideal Gas Equation:
$$PV = nRT$$
4. Pressure of a Gas (Kinetic Theory Expression)
Pressure of a gas arises due to collisions of gas molecules with the walls of the container.
$$P = \frac{1}{3}\rho \overline{c^2}$$
Where:
- $\rho$ = density of gas
- $\overline{c^2}$ = mean square speed
5. Mean, RMS and Most Probable Speed
Mean speed:
$$\bar{c} = \sqrt{\frac{8RT}{\pi M}}$$
Root Mean Square speed:
$$c_{rms} = \sqrt{\frac{3RT}{M}}$$
Most probable speed:
$$c_{mp} = \sqrt{\frac{2RT}{M}}$$
Relation:
$$c_{rms} > \bar{c} > c_{mp}$$
6. Kinetic Energy of Gas Molecules
Average kinetic energy per molecule:
$$\overline{E} = \frac{3}{2}kT$$
This shows that **temperature is a measure of average kinetic energy**.
7. Degrees of Freedom
Degrees of freedom are the number of independent ways
in which a molecule can possess energy.
| Molecule | Degrees of Freedom |
|---|---|
| Monoatomic | 3 |
| Diatomic (Rigid) | 5 |
| Polyatomic | 6 or more |
8. Law of Equipartition of Energy
Each degree of freedom contributes $\frac{1}{2}kT$ energy per molecule.
Total energy per molecule:
$$E = \frac{f}{2}kT$$
Where $f$ = degrees of freedom.
9. Specific Heat Capacities of Gases
At constant volume:
$$C_v = \frac{f}{2}R$$
At constant pressure:
$$C_p = C_v + R$$
Ratio:
$$\gamma = \frac{C_p}{C_v}$$
10. Mean Free Path
Mean free path is the average distance travelled by a gas molecule
between two successive collisions.
$$\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$$
11. Effect of Temperature on Molecular Speed
As temperature increases, molecular speed increases,
leading to higher pressure or volume.
Speed $\propto \sqrt{T}$
12. Comparison of Real Gas and Ideal Gas
| Ideal Gas | Real Gas |
|---|---|
| No intermolecular force | Forces exist |
| Obeys gas laws always | Deviates at high pressure |
13. Limitations of Kinetic Theory
- Cannot explain liquefaction of gases
- Fails at very high pressure and low temperature
- Assumes point-sized molecules
14. Important JEE Formula Summary
$$P = \frac{1}{3}\rho c_{rms}^2$$
$$c_{rms} = \sqrt{\frac{3RT}{M}}$$
$$E = \frac{f}{2}kT$$
15. Final Exam Tips
- Remember all three molecular speeds
- Use degrees of freedom carefully
- Units consistency is crucial
- Understand derivations conceptually