Class 11 · Chapter 13

Kinetic Theory of Gases

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Kinetic Theory of Gases Short Notes

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Condensed revision points for Kinetic Theory of Gases — for quick recall before exams, not a substitute for the full notes.

Huge number of identical molecules, random motion, obey Newton's laws.
Molecular volume negligible vs container volume — molecules treated as point masses.
NO intermolecular forces except DURING collision. Collisions perfectly ELASTIC.
Time of collision << time between collisions. Straight-line motion between collisions.

Boyle's law: PV=const (T,n fixed). Charles' law: V/T=const (P,n fixed). Gay-Lussac's: P/T=const (V,n fixed).
Avogadro's law: equal V, same T & P ⇒ equal number of molecules, regardless of gas identity.
PV = nRT = NkT. k = R/N_A ≈ 1.38×10⁻²³ J/K (Boltzmann constant — R on a per-molecule basis).
Ideal gas behaviour is best approached at LOW pressure & HIGH temperature.

⟨KE⟩ per molecule = (3/2)kT — depends ONLY on T, NOT on mass/identity/P/V.
v_rms = √(3kT/m) = √(3RT/M). Lighter molecules (smaller M) ⇒ HIGHER v_rms at same T.
T=0 K ⇒ average KE → 0 (unreachable in practice).

v_rms = √(3RT/M); v_avg = √(8RT/πM); v_p (most probable) = √(2RT/M).
ALWAYS: v_rms > v_avg > v_p (fixed order for any gas, any T).
Maxwell distribution: rises from 0, peaks at v_p, long tail to high speed — NOT symmetric.
Higher T ⇒ curve shifts right & flattens; area under curve (total molecules) stays the same.

Equipartition law: average energy per degree of freedom = (1/2)kT. Total ⟨E⟩ = (f/2)kT.
Monoatomic: f=3 (translation only). Diatomic (no vibration): f=5 (3 trans+2 rot).
Diatomic WITH vibration (high T): f=7. Non-linear polyatomic: f=6 (3 trans+3 rot).
Each VIBRATIONAL mode adds 2 dof (KE + PE of the 'spring'), not 1 — common slip point.

λ = 1/(√2 π d²n). d=molecular diameter, n=number density.
λ ∝ 1/n — compressing gas (↑n) ⇒ shorter λ. Low-pressure gas ⇒ long λ.
√2 factor arises because BOTH colliding molecules are moving (relative speed), not just the tracked one.

(P + a/V²)(V−b) = RT (per mole). 'a' → intermolecular ATTRACTION correction. 'b' → finite molecular SIZE (excluded volume).
Real gas ≈ ideal gas at LOW pressure + HIGH temperature (molecules far apart, fast-moving).
Above the critical temperature, a gas CANNOT be liquefied no matter how much pressure is applied.

Brownian motion: random zig-zag of suspended particles from molecular bombardment — direct visual evidence for molecular motion.
More vigorous at: HIGHER temperature, SMALLER suspended particle size.
N_A ≈ 6.022×10²³ /mol — bridges molecular (k) and molar (R) descriptions: k = R/N_A.

Average translational KE per molecule (3/2)kT depends ONLY on T — never on the gas's molar mass. Don't assume heavier gas molecules carry more KE at the same T (they don't; they just move SLOWER).
v_rms, v_avg, v_p are NOT equal — always v_rms > v_avg > v_p. A common slip is treating all three as the same 'average speed'.
Vibrational degrees of freedom contribute 2 each (KE+PE), not 1 — under-counting this is a frequent f/γ error.
γ = 1+2/f is INVERSE in f — more degrees of freedom give a SMALLER γ, not larger.
λ ∝ 1/n (inversely proportional to number density), not directly proportional — compressing a gas SHORTENS the mean free path.
Van der Waals 'a' (attraction) and 'b' (size) correct DIFFERENT assumptions — don't mix up which constant fixes which idealisation.
PV=(2/3)E uses only TRANSLATIONAL kinetic energy, even for polyatomic gases with rotational/vibrational energy too — don't substitute total internal energy here.

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