Kinetic Theory
'The theory of gases is the bridge between the macroscopic and the microscopic worlds.' — Physics of Gases
1. Chapter Overview
KINETIC THEORY provides a MICROSCOPIC explanation for the MACROSCOPIC behaviour of gases. It treats gases as LARGE NUMBERS of molecules in RANDOM motion and derives their observable properties (pressure, temperature, specific heats) from molecular motion alone. This chapter connects the ATOMIC world with the world of PV diagrams and thermometers.
2. Assumptions of Kinetic Theory of Gases
- Gas consists of TINY, identical molecules (point masses, volume negligible compared to container)
- Molecules are in CONSTANT, random motion
- Collisions with walls and other molecules are PERFECTLY ELASTIC (no KE loss)
- No INTERMOLECULAR forces except during collisions (ideal gas assumption)
- The duration of a collision is negligible compared to time between collisions
- Newton's laws of motion APPLY to molecular motion
- Pressure is due to molecular COLLISIONS with the container walls
3. Pressure of an Ideal Gas (Kinetic Derivation)
- Pressure P = (1/3)ρv²_rms (where ρ = density, v_rms = root-mean-square speed)
- OR: P = (1/3)(nmv²_rms) = (1/3)(Mv²_rms/V)
- OR: PV = (1/3)Mv²_rms (M = total mass of gas)
- Since nRT = (1/3)Mv²_rms: v_rms = √(3RT/M) = √(3kT/m)
Root Mean Square Speed
- v_rms = √(3RT/M)
- Depends on temperature and MOLAR MASS, not on pressure
- For H₂ at 300 K: v_rms ≈ 1930 m/s (very fast!)
- For O₂ at 300 K: v_rms ≈ 483 m/s
4. Gas Laws from Kinetic Theory
| Law | Equation | Condition | Statement |
|---|---|---|---|
| Boyle's | P₁V₁ = P₂V₂ | T, n constant | P ∝ 1/V |
| Charles' | V₁/T₁ = V₂/T₂ | P, n constant | V ∝ T |
| Gay-Lussac's | P₁/T₁ = P₂/T₂ | V, n constant | P ∝ T |
| Avogadro's | V ∝ n | T, P constant | Equal volumes → equal molecules |
Ideal Gas Equation
- PV = nRT
- R = 8.314 J/mol·K (UNIVERSAL gas constant)
- k_B = R/N_A = 1.38 × 10⁻²³ J/K (Boltzmann constant)
- In molecular form: PV = Nk_BT (N = number of molecules)
5. Kinetic Energy and Temperature
- Average KE per molecule: (1/2)mv²_rms = (3/2)kT
- Average KE per mole: (1/2)Mv²_rms = (3/2)RT
- Total KE of n moles: KE = (3/2)nRT
Key Insight
Temperature is a DIRECT measure of the average kinetic energy of molecules. NO motion (absolute zero) → T = 0 K = -273.15°C.
Worked Problem
Q: Find v_rms of oxygen (M = 32 g/mol) at 27°C (R = 8.314 J/mol·K). A: v_rms = √(3RT/M) = √(3×8.314×300/0.032) = √(7482.6/0.032) = √233831 = 483.6 m/s.
6. Degrees of Freedom
- Definition: The number of INDEPENDENT ways a molecule can possess energy
Degrees of Freedom for Different Gases
| Gas Type | Example | Translational | Rotational | Vibrational | Total f |
|---|---|---|---|---|---|
| Monatomic | He, Ar, Ne | 3 | 0 | 0 | 3 |
| Diatomic | H₂, O₂, N₂ | 3 | 2 | 2* | 7* |
| Polyatomic (linear) | CO₂ | 3 | 2 | varies | varies |
| Polyatomic (non-linear) | H₂O, CH₄ | 3 | 3 | varies | varies |
*Vibrational modes are often NOT excited at ordinary temperatures (contribute at high T only). At room temperature, diatomic has effectively 5 degrees of freedom (3 trans + 2 rot).
Law of Equipartition of Energy
- Each degree of freedom contributes (1/2)kT per molecule (or ½RT per mole)
- Total energy per molecule: f×(½kT) = fkT/2
- For monatomic: U = (3/2)nRT
- For diatomic (room temp): U = (5/2)nRT
7. Specific Heats of Gases
Using Equipartition
| Gas Type | C_v | C_p | γ = C_p/C_v |
|---|---|---|---|
| Monatomic | (3/2)R | (5/2)R | 5/3 = 1.67 |
| Diatomic (room T) | (5/2)R | (7/2)R | 7/5 = 1.40 |
| Diatomic (high T) | (7/2)R | (9/2)R | 9/7 = 1.29 |
- C_p — C_v = R always (Mayer's formula)
- These values MATCH experimental data → strong validation of kinetic theory
Worked Problem
Q: Find C_v and C_p for argon (monatomic). (R = 8.314 J/mol·K) A: C_v = 3R/2 = 12.47 J/mol·K. C_p = 5R/2 = 20.785 J/mol·K. γ = 5/3 = 1.67.
8. Mean Free Path (λ)
- Definition: The AVERAGE distance a molecule travels BETWEEN collisions
- λ = 1/(√2 × n × πd²) = kT/(√2 × πd² × P)
- n = number density (molecules/m³)
- d = molecular diameter
- λ ∝ T/P: increasing temperature OR decreasing pressure INCREASES mean free path
- At STP: λ ≈ 10⁻⁷ m for air molecules
Collision Frequency
- f = v_avg/λ (average collisions per second)
- At STP: f ≈ 5 × 10⁹ collisions per second!
9. Common Mistakes
- v_rms ≠ average speed: v_rms = √(3kT/m), v_avg = √(8kT/πm). v_rms > v_avg
- All molecules do NOT have the same speed: The MAXWELL-BOLTZMANN distribution gives a range of speeds
- Equipartition applies only at THERMAL EQUILIBRIUM: each mode gets equal average energy
- Ideal gas assumptions break down at HIGH pressure and LOW temperature: Real gases deviate
- C_v for diatomic is NOT always 5R/2: At very high temperatures, vibrational modes activate
10. CBSE Exam Focus
- Derivation of pressure of an ideal gas from kinetic theory (5-mark)
- RMS speed, average speed, most probable speed — definitions and formulas
- Degrees of freedom and specific heats (3-mark)
- Mean free path derivation (3-mark)
- Equipartition of energy theorem — applications
- Kinetic energy-temperature relationship
11. Key Formulas
- P = (1/3)ρv²_rms
- v_rms = √(3RT/M) = √(3kT/m)
- v_avg = √(8RT/πM), v_mp = √(2RT/M)
- KE per molecule = (3/2)kT, per mole = (3/2)RT
- U = (f/2)nRT
- C_v = (f/2)R, C_p = (f/2 + 1)R, γ = 1 + 2/f
- λ = 1/(√2πnd²) = kT/(√2πd²P)
12. Self-Test (5+ Q&A)
Q1: Find v_rms of hydrogen (M = 2 g/mol) at 27°C. A: v_rms = √(3RT/M) = √(3×8.314×300/0.002) = √(7482.6/0.002) = √3,741,300 = 1934 m/s.
Q2: What are the degrees of freedom of a diatomic gas at room temperature? A: 5 (3 translational + 2 rotational). Vibrational modes are frozen at room temperature.
Q3: Find C_v for helium (monatomic). R = 8.314 J/mol·K. A: C_v = 3R/2 = 12.47 J/mol·K.
Q4: Calculate mean free path of air at STP (d = 3.5 × 10⁻¹⁰ m, T = 273 K, P = 1.013×10⁵ Pa). A: λ = kT/(√2πd²P) = (1.38×10⁻²³×273)/(1.414×π×(3.5×10⁻¹⁰)²×1.013×10⁵) = 3.77×10⁻²¹/(5.5×10⁻¹⁴) ≈ 6.8×10⁻⁸ m.
Q5: Why does mean free path increase with temperature? A: At higher T, molecules move faster (higher v_rms), which increases the distance travelled between collisions. But more importantly, if P is constant, higher T means lower number density (n = P/kT), so there are fewer molecules per unit volume to collide with.
13. Conclusion
Kinetic theory builds a MICROSCOPIC model that explains MACROSCOPIC observations. Pressure is due to molecular collisions, temperature measures average KE, and specific heats depend on molecular structure (degrees of freedom). The mean free path quantifies how far molecules travel between collisions — crucial for understanding transport phenomena (diffusion, thermal conductivity, viscosity). This theory beautifully BRIDGES the atomic world with everyday observables.
