Kinetic Theory of Gases NEET Notes: Complete Guide with Formulas & Solved Examples

Kinetic Theory of Gases
for NEET Physics

Q: What is the difference between ideal gas and real gas?

Ideal gas obeys PV = nRT at all conditions and has no intermolecular forces. Real gases deviate at high pressure and low temperature due to finite molecular size and attraction.

Q: Why is CP greater than CV?

At constant pressure, gas expands and does work, so additional heat is needed to supply that work. At constant volume, no work is done, so all heat goes to increase internal energy.

Q: What is the physical significance of mean free path?

It is the average distance a molecule travels between collisions. It determines transport properties like diffusion and thermal conductivity.

Q: Does the rms speed depend on the mass of the molecule?

Yes, at a given temperature, vrms ∝ 1/√M. Lighter molecules move faster.

Master the microscopic behavior of gases — ideal gas laws, pressure derivation, RMS speed, degrees of freedom, equipartition of energy, mean free path, and specific heat capacities. Essential for NEET 2026.

Ideal Gas Gas Laws RMS Speed Degrees of Freedom Equipartition Mean Free Path

📌 What you'll learn: NEET-level problem solving • All essential formulas • Step-by-step numericals • Quick revision hacks

🔍 Detailed concept explanation

🧪 1. Ideal Gas

An ideal gas is a gas that follows the ideal gas equation PV = nRT under all conditions of temperature and pressure. It obeys the kinetic gas postulates perfectly. Real gases behave ideally at low pressure and high temperature.

📜 2. Gas Laws

Boyle's Law: At constant T, PV = constant.
Charles's Law: At constant P, V ∝ T.
Gay-Lussac's Law: At constant V, P ∝ T.
Avogadro's Law: At constant P, T, V ∝ n.
Ideal Gas Equation: PV = nRT = (N/Nₐ)RT = Nk_BT, where k_B = R/Nₐ = 1.38×10⁻²³ J/K.

⚙️ 3. Work Done in Compressing a Gas

For reversible isothermal process: W = nRT ln(V₂/V₁) = nRT ln(P₁/P₂). For isobaric: W = PΔV. For adiabatic: W = (P₁V₁ – P₂V₂)/(γ–1).

📉 4. Equation of a Real Gas (Van der Waals)

(P + a/V²)(V – b) = RT (for one mole). Here 'a' corrects for intermolecular attraction, 'b' corrects for finite molecular volume.

📦 5. Postulates of Kinetic Theory

Gas consists of a large number of tiny, identical molecules in random motion.
Volume of molecules is negligible compared to container volume.
No intermolecular forces except during elastic collisions.
Collisions with walls are perfectly elastic.
Duration of collision is negligible.
Pressure is due to collisions with walls.

📐 6. Pressure of a Gas using Kinetic Theory

P = (1/3) (N/V) m v²_rms = (1/3) ρ v²_rms. Also, P = (2/3) E, where E is translational KE per unit volume.

🌡️ 7. Kinetic Interpretation of Temperature

From PV = (1/3) N m v²_rms and PV = Nk_BT, we get (1/2) m v²_rms = (3/2) k_BT. Thus, average translational KE per molecule = (3/2)k_BT. Temperature is a measure of average KE.

Root mean square speed: v_rms = √(3RT/M) = √(3k_BT/m).

Most probable speed: v_mp = √(2RT/M). Average speed: v_avg = √(8RT/πM).

🔄 8. Degrees of Freedom

Number of independent coordinates required to describe the motion of a molecule. Monatomic (He, Ar): f = 3 (translational). Diatomic (N₂, O₂) at ordinary temp: f = 5 (3 trans + 2 rotational). At high temp, vibrational modes add (f=7). Polyatomic nonlinear: f = 6.

Gas Type	Degrees of Freedom (f)	Examples
Monatomic	3	He, Ne, Ar
Diatomic (rigid)	5	N₂, O₂, H₂
Diatomic + vibrational	7	at high T
Polyatomic nonlinear	6	CO₂, H₂O

⚖️ 9. Law of Equipartition of Energy

For a system in thermal equilibrium, the total energy is equally distributed among all degrees of freedom, each contributing (1/2)k_BT per molecule (or ½RT per mole).

Thus, U = (f/2) nRT for n moles. For monatomic gas: U = (3/2)nRT. For diatomic (rigid): U = (5/2)nRT.

🔥 10. Specific Heat Capacity

Molar specific heat at constant volume: C_V = (1/n) dU/dT = (f/2)R.

Molar specific heat at constant pressure: C_P = C_V + R = (f/2 + 1)R.

Ratio γ = C_P/C_V = 1 + 2/f.

For monatomic: γ = 5/3 ≈ 1.67; diatomic: γ = 7/5 = 1.4; polyatomic: γ ≈ 4/3 = 1.33.

🏃 11. Mean Free Path (λ)

Average distance traveled by a molecule between successive collisions. λ = 1/(√2 π d² n), where d is molecular diameter, n is number density. Also, λ ∝ T/(P d²).

📋 Complete formula sheet

Ideal gas eq. PV = nRT = Nk_BT

Pressure (kinetic) P = (1/3)ρv²_rms

v_rms = √(3RT/M) = √(3k_BT/m)

v_avg = √(8RT/πM)

v_mp = √(2RT/M)

Avg KE/molecule = (f/2)k_BT

U (n moles) = (f/2)nRT

C_V = (f/2)R

C_P = (f/2 + 1)R

γ = C_P/C_V = 1 + 2/f

Mean free path λ = 1/(√2 π d² n)

✏️ Solved NEET-level examples

Example 1 (RMS speed)

Calculate the rms speed of oxygen molecules at 27°C. (Molecular mass of O₂ = 32 g/mol, R = 8.314 J/mol·K)

1 T = 27 + 273 = 300 K, M = 32×10⁻³ kg/mol.

2 v_rms = √(3RT/M) = √(3×8.314×300 / 0.032) = √(7482.6 / 0.032) = √(233831.25) ≈ 483.6 m/s.

Example 2 (Pressure from kinetic theory)

Find the pressure exerted by 10²³ gas molecules each of mass 10⁻²⁶ kg in a container of volume 0.1 m³. The rms speed is 200 m/s.

1 P = (1/3)(N/V) m v²_rms = (1/3) × (10²³ / 0.1) × 10⁻²⁶ × (200)².

2 N/V = 10²⁴. So P = (1/3) × 10²⁴ × 10⁻²⁶ × 40000 = (1/3) × 10⁻² × 4×10⁴ = (1/3) × 400 = 133.3 Pa.

Example 3 (Degrees of freedom & C_V)

For a diatomic gas (rigid), find C_V, C_P, and γ. Also find total kinetic energy of 2 moles at 300 K.

1 For diatomic (rigid), f = 5. C_V = (f/2)R = (5/2)R = 20.775 J/mol·K (R=8.31).

2 C_P = C_V + R = (7/2)R = 29.085 J/mol·K. γ = C_P/C_V = 7/5 = 1.4.

3 U = (f/2)nRT = (5/2)×2×8.31×300 = 5×8.31×300 = 12465 J.

Example 4 (Mean free path)

For nitrogen at STP, number density n = 2.7×10²⁵ molecules/m³ and molecular diameter d = 3.0×10⁻¹⁰ m. Calculate mean free path.

1 λ = 1/(√2 π d² n) = 1/(1.414 × 3.14 × (9×10⁻²⁰) × 2.7×10²⁵).

2 First, πd² = 3.14×9×10⁻²⁰ = 2.826×10⁻¹⁹ m².

3 √2 π d² n = 1.414×2.826×10⁻¹⁹×2.7×10²⁵ = 1.414×2.826×2.7×10⁶ ≈ 1.414×7.63×10⁶ ≈ 10.78×10⁶ = 1.078×10⁷ m⁻¹.

4 λ = 1/(1.078×10⁷) ≈ 9.28×10⁻⁸ m = 92.8 nm.

📈 Important graphs & key points

Maxwell-Boltzmann distribution : plot of fraction of molecules vs speed; peak shifts right with T, area constant
v_mp < v_avg < v_rms always
PV vs P for real gases: shows deviations from ideal
λ vs T : λ ∝ T at constant P

⭐ Translational KE depends only on temperature, not on type of gas.

⚡ Quick revision box

Ideal gas PV = nRT, R = 8.314 J/mol·K, k_B = 1.38×10⁻²³ J/K

Speeds v_rms:√(3RT/M), v_avg:√(8RT/πM), v_mp:√(2RT/M)

Energy & heat U = (f/2)nRT, C_V=(f/2)R, γ=1+2/f

Mean free path λ = 1/(√2 π d² n)

⚠️ Common mistakes to avoid

Using M in g/mol instead of kg/mol in speed formulas.
Forgetting that C_V and C_P are molar specific heats (per mole).
Confusing degrees of freedom for diatomic (vibrational only at high T).
Assuming all gases have same C_V — depends on f.
Not converting temperature to Kelvin in gas law equations.
Thinking that v_rms is the actual speed of molecules — it's root mean square.

🧠 Exam strategy tips

Memorize the three speeds: v_rms, v_avg, v_mp and their ratios.
Remember that for all ideal gases, average translational KE per molecule = (3/2)k_BT.
For mixture problems, use weighted averages for degrees of freedom.
Practice Maxwell-Boltzmann curve interpretation.
Relate λ with P, T, d — λ ∝ T/(P d²).

❓ Frequently asked questions

📌 What is the difference between ideal gas and real gas?