Kinetic Theory of Gases for JEE Main: Full Guide

The kinetic theory of gases connects the microscopic world of molecular speeds and collisions to the macroscopic gas laws we observe. JEE Main draws two to four questions from this chapter every session, covering speed distribution formulas, the equipartition theorem, and the ideal gas equation at the molecular level. The mathematics is light; the conceptual traps are the real challenge.

Molecular Speeds

Three characteristic speeds are tested: rms speed v_rms = √(3RT/M); average speed v_avg = √(8RT/πM); most probable speed v_mp = √(2RT/M). Their ratio is v_mp : v_avg : v_rms = √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22. All three are proportional to √T and inversely proportional to √M. At the same temperature, lighter molecules move faster. JEE uses these to compare speeds of different gases or to find the temperature at which a given speed equals a certain value. For the thermodynamics connections see our thermodynamics guide.

Ideal Gas Equation and Pressure Formula

PV = nRT = NkT (where n = moles, N = molecules, k = Boltzmann constant = 1.38 × 10⁻²³ J/K, R = kN_A = 8.314 J/(mol·K)). The microscopic pressure formula from kinetic theory: P = (1/3)ρv²_rms = (1/3)(Nm/V)v²_rms. This gives PV = (1/3)Nmv²_rms = NkT, confirming that ½mv²_rms = (3/2)kT — the average translational kinetic energy per molecule equals (3/2)kT, independent of the gas type.

Degrees of Freedom and Equipartition

The equipartition theorem assigns ½kT of average energy to each quadratic degree of freedom. A monatomic gas has 3 translational DOF: U = (3/2)nRT, C_v = (3/2)R, C_p = (5/2)R, γ = 5/3. A diatomic gas (at moderate T) has 3 translational + 2 rotational = 5 DOF: U = (5/2)nRT, C_v = (5/2)R, C_p = (7/2)R, γ = 7/5. A polyatomic gas has γ between 1.3–1.4. JEE regularly asks for γ or U of a given gas by counting DOF.

Mean Free Path

The mean free path λ_mfp = 1/(√2 π d² n) where d is molecular diameter and n = N/V is number density. At higher pressure or lower temperature, n increases and λ_mfp decreases. At low pressure (high vacuum), λ_mfp can exceed the container dimensions. JEE occasionally gives this formula in conceptual form: doubling the pressure halves the mean free path (at constant T). This inverse-pressure behaviour is the key testable fact.

Graham's Law of Diffusion

Graham's law: the rate of diffusion r ∝ 1/√M (at constant T and P). So r₁/r₂ = √(M₂/M₁). Lighter gases diffuse faster. This law allows JEE to set up problems where two gases are released simultaneously and you must find the position of the mixed front, or the time ratio for the same amount of gas to effuse. Pair this with the speed formulas (both scale as 1/√M at fixed T) to confirm consistency. After mastering all kinetic theory formulas, take a free mock test to consolidate.