Kinetic Theory of Gases — NEET Physics Notes

The kinetic theory of gases explains the behaviour of gases in terms of motion of molecules. This topic connects macroscopic gas laws (PV = nRT) with the microscopic molecular picture — a key conceptual chapter in NEET Physics.

Quick Revision

• Assumptions: Gas consists of large number of molecules, negligible volume, no intermolecular forces, elastic collisions, average KE ∝ temperature
• Pressure: P = (⅓) ρ v̄² = (⅓) (Nm/v) v̄²
• RMS Velocity: v_rms = √(3P/ρ) = √(3kT/m) = √(3RT/M)
• Average KE: (½) mv̄² = (3/2) kT per molecule
• Degrees of Freedom: f = 2 for monoatomic, 5 for diatomic, 6 for polyatomic
• Equipartition Theorem: Energy equally distributed among degrees of freedom — each degree gets (½) kT
• Gas Law: PV = nRT (universal gas constant R = 8.314 J/mol·K)

Standard Study

Kinetic Theory Assumptions

1. Large number of molecules moving in random directions
2. Volume of molecules is negligible compared to gas volume
3. No intermolecular forces (except during collisions)
4. Collisions between molecules are perfectly elastic
5. Time spent in collisions is negligible compared to time between collisions
6. Gas is in thermal equilibrium — uniform temperature throughout

Pressure Derivation

• Pressure P = (⅓) ρ v̄² (derived from momentum transfer during collisions)
• Alternatively: P = (Nm v̄²) / (3V)
• Combining with ideal gas equation: v_rms = √(3P/ρ)

Gas Velocities

RMS Velocity:

• v_rms = √(3kT/m) = √(3P/ρ)
• Depends only on temperature and molecular mass
• Lighter gases have higher RMS speed at same temperature

Most Probable Speed:

• v_mp = √(2kT/m) = √(2P/ρ)
• Speed at which maximum molecules are found

Average Speed:

• v_av = √(8kT/πm) = √(8RT/πM)
• Arithmetic mean of molecular speeds

Relation: v_mp : v_av : v_rms = √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22

Maxwell-Boltzmann Distribution

• Number of molecules with speed between v and v+dv
• Curve is skewed — most probable speed is at the peak
• As temperature increases, curve shifts right and flattens
• RMS speed always greater than most probable speed

Degrees of Freedom and Specific Heat

Monatomic Gas (He, Ne — 3 degrees of freedom):

• C_v = (3/2)R
• C_p = C_v + R = (5/2)R
• γ = C_p/C_v = 5/3 ≈ 1.67

Diatomic Gas (O₂, N₂ — 5 degrees of freedom at room temp):

• C_v = (5/2)R
• C_p = C_v + R = (7/2)R
• γ = C_p/C_v = 7/5 = 1.4

Polyatomic Gas (6+ degrees of freedom):

• C_v = f × (R/2)
• γ depends on f

Ideal Gas Equation

• PV = nRT = NkT
• P = pressure, V = volume, n = moles, N = molecules
• R = 8.314 J/mol·K, k = R/NA = 1.38 × 10⁻²³ J/K

Deep Study

Real Gases and Van der Waals Equation

• Real gases deviate from ideal behaviour at high pressure and low temperature
• Van der Waals equation: (P + a(n/V)²)(V − nb) = nRT
• a = accounts for intermolecular attraction
• b = excluded volume (molecular volume)

Mean Free Path

• Average distance travelled by a molecule between successive collisions
• λ = 1/(√2 × π × d² × n/V)
• d = molecular diameter, n/V = number density
• inversely proportional to pressure — lower pressure means longer mean free path

Diffusion and Effusion

• Graham’s Law: Rate of diffusion ∝ 1/√M
• Ratio of diffusion rates: r₁/r₂ = √(M₂/M₁)
• Effusion: Gas escaping through a small hole — same dependence on molecular weight

Behavior at Different Scales

• At very low pressures (high vacuum), mean free path is very large
• Gas molecules move in straight lines between collisions
• Collision frequency decreases with decreasing pressure

Exam Tips

1. rms speed formula: √(3kT/m) — depends on temperature and molecular mass
2. Average KE of gas molecule = (3/2) kT — depends only on temperature, not on gas type
3. Degrees of freedom determine specific heats — monatomic = 3 DOF, diatomic = 5 DOF
4. γ = C_p/C_v is greater for monoatomic than diatomic (5/3 > 7/5)
5. At same temperature, lighter molecules move faster (v_rms ∝ 1/√M)
6. Mean free path inversely proportional to pressure — important in vacuum applications
7. Ideal gas equation PV = nRT is universal — works for all gases

Common Pitfalls

• Confusing rms, average, and most probable speeds — remember their ratio
• Forgetting that at higher temperature, gas molecules move faster
• Not knowing which degrees of freedom contribute at room temperature vs high temperature
• Confusing specific heat at constant volume vs constant pressure (Cp = Cv + R)
• Applying kinetic theory assumptions to real gases without accounting for deviations
• Confusing mean free path with distance between molecules

Suggested Study Order

1. Kinetic theory assumptions and molecular picture
2. Derivation of pressure formula from kinetic theory
3. RMS, average, and most probable speeds
4. Maxwell-Boltzmann distribution (conceptual)
5. Degrees of freedom and equipartition theorem
6. Specific heats and γ relation
7. Real gases and van der Waals equation
8. Mean free path, diffusion, and effusion