Learning Objectives
- Understand the molecular picture of matter and gas behaviour
- Derive the kinetic theory equation for pressure
- Relate temperature to kinetic energy of gas molecules
- Understand degrees of freedom and the law of equipartition of energy
- Study mean free path and transport phenomena
Key Concepts
Ideal Gas and Gas Laws
Ideal gas equation: PV = nRT = NkT, where k = R/Nₐ = 1.38 × 10⁻²³ J/K (Boltzmann constant).
Boyle's Law: PV = constant (at constant T).
Charles's Law: V/T = constant (at constant P).
Gay-Lussac's Law: P/T = constant (at constant V).
Dalton's Law: P_total = P₁ + P₂ + ... (sum of partial pressures).
Kinetic Theory of Gases - Assumptions
- Gas consists of a large number of tiny molecules in random motion.
- Molecules are point particles (size negligible compared to intermolecular distances).
- Collisions are perfectly elastic.
- No intermolecular forces except during collisions.
- Time of collision is negligible compared to time between collisions.
Pressure from Kinetic Theory
Pressure: P = (1/3)ρv²_rms = (1/3)(mN/V)v²_rms
where v_rms is the root mean square speed.
This gives: PV = (1/3)Nmv²_rms
Kinetic Energy and Temperature
Average KE per molecule: (1/2)mv²_rms = (3/2)kT
Average KE per mole: (3/2)RT
RMS speed: v_rms = √(3kT/m) = √(3RT/M)
Average speed: v_avg = √(8kT/πm) = √(8RT/πM)
Most probable speed: v_mp = √(2kT/m) = √(2RT/M)
Relation: v_mp < v_avg < v_rms (ratio ≈ 1 : 1.128 : 1.224)
Degrees of Freedom
Number of independent ways a molecule can store energy:
- Monatomic gas (He, Ne, Ar): 3 translational DOF → f = 3
- Diatomic gas (H₂, O₂, N₂): 3 translational + 2 rotational DOF → f = 5 (at room temp). At high temp, +2 vibrational → f = 7.
- Triatomic linear (CO₂): f = 7. Triatomic non-linear (H₂O): f = 6.
Law of Equipartition of Energy
Each degree of freedom contributes (1/2)kT energy per molecule or (1/2)RT per mole.
Total energy: U = (f/2)nRT
Specific heats: Cᵥ = (f/2)R, Cₚ = (f/2 + 1)R, γ = Cₚ/Cᵥ = 1 + 2/f
Monatomic: Cᵥ = (3/2)R, γ = 5/3. Diatomic: Cᵥ = (5/2)R, γ = 7/5.
Mean Free Path
Average distance a molecule travels between successive collisions.
λ = 1/(√2 × πd²n), where d is molecular diameter and n is number density (N/V).
Using PV = NkT: λ = kT/(√2 × πd²P). Mean free path increases with temperature and decreases with pressure.
Summary
The kinetic theory provides a molecular explanation for gas properties. Pressure arises from molecular collisions with container walls. Temperature is a measure of average translational kinetic energy. The law of equipartition distributes energy equally among all degrees of freedom. Specific heat capacities can be predicted from degrees of freedom. Mean free path describes the average collision-free distance.
Important Terms
- RMS Speed: Square root of mean of squared speeds, v_rms = √(3RT/M)
- Degrees of Freedom: Independent modes of energy storage for a molecule
- Equipartition of Energy: Each DOF gets (1/2)kT energy per molecule
- Mean Free Path: Average distance between successive molecular collisions
- Boltzmann Constant: k = R/Nₐ = 1.38 × 10⁻²³ J/K
Quick Revision
- PV = nRT = NkT; P = (1/3)ρv²_rms
- KE per molecule = (3/2)kT; per mole = (3/2)RT
- v_rms = √(3RT/M); v_avg = √(8RT/πM); v_mp = √(2RT/M)
- DOF: monatomic 3, diatomic 5, γ = 1 + 2/f
- Cᵥ = (f/2)R, Cₚ = Cᵥ + R
- Mean free path: λ = kT/(√2 πd²P)