JEE Physics Notes: Kinetic Theory

Introduction

The Kinetic Theory of Gases explains the macroscopic properties of gases, such as pressure and temperature, in terms of the microscopic behavior of gas molecules. It provides a molecular-level interpretation of thermodynamics and helps derive important gas laws.

1. Assumptions of Kinetic Theory

• Gases consist of a large number of tiny particles (molecules) moving randomly in all directions.
• The volume of individual gas molecules is negligible compared to the total volume of the gas.
• There are no intermolecular forces between gas molecules except during collisions.
• Collisions between molecules and with the container walls are perfectly elastic.
• The average kinetic energy of gas molecules is proportional to the absolute temperature.

2. Pressure of an Ideal Gas

• Derivation: The pressure exerted by a gas on the walls of a container is due to the continuous collisions of gas molecules with the walls.
• Formula: P = (1/3) (N/V) m vrms², where:
  • P = Pressure
  • N = Number of molecules
  • V = Volume
  • m = Mass of one molecule
  • v_rms = Root mean square velocity

3. Kinetic Energy and Temperature

• Average Kinetic Energy: The kinetic energy of a gas molecule is directly proportional to the absolute temperature.
  • Formula: Ek = (3/2) kB T
  • Where k_B is the Boltzmann constant (1.38 × 10⁻²³ J/K).
• Relation Between Temperature and Molecular Speed:
  • Higher temperature means higher average kinetic energy and molecular speed.
  • Temperature is a measure of the average kinetic energy of molecules.

4. Degrees of Freedom and Equipartition of Energy

• Degrees of Freedom: The independent ways in which a molecule can store energy.
  • Monatomic gases: 3 translational degrees of freedom.
  • Diatomic gases: 5 degrees of freedom (3 translational + 2 rotational).
  • Polyatomic gases: 6 or more degrees of freedom.
• Equipartition of Energy: Energy is equally distributed among all degrees of freedom.
  • Energy per degree of freedom = (1/2) kB T.
  • Total energy for a system = (f/2) N kB T, where f is the total degrees of freedom.

5. Molecular Speeds

• Root Mean Square Speed (v_rms): vrms = √(3kB T / m) = √(3RT / M).
• Most Probable Speed (v_mp): vmp = √(2kB T / m) = √(2RT / M).
• Average Speed (v_avg): vavg = √(8kB T / πm) = √(8RT / πM).

6. Mean Free Path

• The average distance a molecule travels between successive collisions.
• Formula: λ = 1 / (√2 π d² n), where:
  • d = Diameter of the molecule
  • n = Number density of molecules

Conclusion

The kinetic theory provides a microscopic view of gas behavior, linking temperature, pressure, and energy. Understanding these concepts is essential for solving problems in thermodynamics and statistical mechanics in JEE Physics.