Kinetic Theory of Gases - Detailed Explanation

Kinetic Theory of Gases (KTG)

1. Introduction

The Kinetic Theory of Gases explains the behavior of gases by assuming that they are made up of a large number of small particles (molecules) in constant random motion. It connects microscopic particle behavior to macroscopic properties like pressure, temperature, and volume.

2. Assumptions of Kinetic Theory

  • Gases consist of tiny particles called molecules.
  • Molecules are in constant random motion.
  • Collisions between molecules and with the container are perfectly elastic.
  • Volume of gas molecules is negligible compared to the container.
  • No intermolecular forces act between gas molecules.
  • Pressure is due to collisions of molecules with container walls.

3. Important Terms and Definitions

Pressure: Force exerted per unit area due to molecular collisions.

Temperature: Measure of average kinetic energy of gas molecules.

Root Mean Square Speed (vrms): A measure of average speed of gas particles.

vrms = √(3RT / M)

R: Universal gas constant = 8.314 J/mol·K

4. Kinetic Energy and Temperature

The kinetic energy of a gas is directly proportional to its temperature. For a single molecule:

KE = (3/2) kT

k: Boltzmann constant = 1.38 × 10-23 J/K

T: Absolute temperature in Kelvin

For n moles of gas, the total kinetic energy:

KE = (3/2) nRT

5. Derivation of Pressure Formula

For a cubic container of volume V and gas of N molecules:

P = (1/3) × (N/V) × m × v2

Where:

  • N = number of molecules
  • m = mass of one molecule
  • v2 = mean of square of molecular speeds

6. Degrees of Freedom

The number of independent ways in which a molecule can possess energy. It depends on the structure:

  • Monoatomic gas: 3 degrees
  • Diatomic gas: 5 degrees (3 translational + 2 rotational)
  • Polyatomic gas: 6 degrees (3 translational + 3 rotational)
KE per molecule = (f / 2) × kT
KE per mole = (f / 2) × RT

7. Maxwell-Boltzmann Distribution

Describes how the speeds of gas molecules are distributed. Most molecules have speeds around the most probable value, but some are much faster or slower.

  • vmp = Most probable speed
  • vavg = Average speed
  • vrms = Root mean square speed
vmp = √(2RT/M), vavg = √(8RT/πM), vrms = √(3RT/M)

8. Real Gas vs Ideal Gas

Ideal Gas: Follows all postulates of kinetic theory perfectly. No intermolecular forces.

Real Gas: Deviates at high pressure and low temperature due to molecular size and attraction.

Example: CO₂ behaves as an ideal gas at room temperature but shows deviation at high pressure.

9. Daily Life Examples of Kinetic Theory

1. Air Pressure in Tires: The pressure inside a tire is due to gas molecules hitting the walls. Higher temperature increases molecular speed, increasing pressure.
2. Perfume in a Room: Molecules of perfume spread out in air due to random motion explained by KTG (diffusion).
3. Cooking in Pressure Cooker: Increased pressure raises boiling point of water. Explained by molecular motion and collisions.
4. Balloons Burst in Heat: On heating, gas molecules move faster and expand the balloon until it bursts.
5. Cooling with Refrigerant Gases: Cooling occurs when high-speed molecules escape, taking away energy.

10. Summary Table

Term Formula Description
Pressure P = (1/3)(N/V)mv² Due to collisions of molecules
KE (per molecule) KE = (3/2)kT Depends on temperature only
RMS Speed vrms = √(3RT/M) Average speed of molecules
Degrees of Freedom f Ways in which energy is stored
Distribution Maxwell-Boltzmann Spread of molecular speeds