Class 11 · Notes

Kinetic Theory of Gases— Notes, Formulas & Revision

Complete revision notes and formulas for Kinetic Theory of Gases (Class 11). Curated for JEE, NEET, AP Physics, SAT, and CUET. Tap any topic to open the live simulation and full PYQ set.

Random Molecular Motion

Particles bounce inside a box, faster as T rises.

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Kinetic theory: gas molecules are in constant random motion, colliding elastically with each other and walls.

Random direction at any instant; speeds follow a Maxwell-Boltzmann distribution.

Brownian motion (1827): tiny particles suspended in fluid show random jiggle — direct evidence of molecular collisions.

No preferred direction in a gas at equilibrium — isotropic.

Average velocity of a gas at rest = 0 (vectors cancel). Average SPEED ≠ 0.

Diffusion: net transport of molecules from high to low concentration due to random motion.

Effusion (Graham's law): rate of effusion ∝ 1/√M — lighter gases escape faster.

Random walks: in time t, average displacement scales as √t (not t).

Average velocity (vector)

Equal probability in all directions ⇒ vector sum zero.

Average speed (scalar)

Non-zero, depends on T and mass.

Random walk distance

Diffusion grows linearly in time.

Graham's law

Effusion rate inversely proportional to √M.

At equilibrium: vector velocity sums to zero — gas at rest macroscopically.

Brownian motion is direct visual evidence of random molecular collisions.

Random motion is fundamental to gas behavior — explains pressure, temperature, diffusion.

Diffusion is SLOW: small molecules in air diffuse ~cm in seconds; perfumes in still air over ~minutes.

In a sealed container: molecules continuously bounce — pressure is uniform on average.

Lighter gases (H₂, He) move faster at same T ⇒ diffuse and effuse faster.

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Wall Collisions → Pressure

Δp = 2mv per elastic hit — pressure builds.

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Gas molecules collide with container walls — these collisions create pressure.

Assume collisions are ELASTIC: kinetic energy and momentum conserved.

When molecule hits wall, momentum component perpendicular to wall REVERSES (Δp = 2m·v_perp).

Number of collisions per unit time per unit area = (1/4)·n·⟨v⟩, where n = number density.

Total force on wall = rate of momentum transfer = pressure × area.

Pressure P = (1/3)ρ·v_rms² — derived from kinetic theory.

Higher T ⇒ higher v_rms ⇒ higher pressure (at constant V, n).

Reflection assumed specular (angle of incidence = angle of reflection); real walls may have more complex interactions.

Momentum change per collision

Perpendicular velocity component reverses.

Collision rate per unit area

Standard kinetic-theory result.

Pressure (kinetic theory)

Connects microscopic motion to macroscopic pressure.

Force on wall

Wall feels macroscopic force from countless molecular impacts.

Pressure arises from MOMENTUM TRANSFER during collisions — not from molecular weight.

Each collision transfers 2m·v_perp of momentum to wall.

More molecules, faster molecules, or larger m ⇒ more pressure.

Pressure is uniform throughout a contained gas at equilibrium (Pascal's principle).

Walls feel a fluctuating force, but on macroscopic scales it averages to a steady pressure.

Elastic collision with wall ⇒ no energy lost ⇒ gas pressure stable indefinitely.

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Pressure from Molecular Theory

P = ⅓ ρ⟨v²⟩ — kinetic origin of pressure.

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Kinetic-theory derivation: P = (1/3)·n·m·⟨v²⟩, where n = number density, m = molecular mass, ⟨v²⟩ = mean-square speed.

Equivalently: P = (1/3)·ρ·v_rms², where ρ = mass density.

Derived from: molecules in random motion, elastic collisions with walls, momentum transfer 2m·v.

Connects MICROSCOPIC (molecular motion) to MACROSCOPIC (pressure).

Implies pressure depends ONLY on density and speed — independent of detailed molecular interactions (in ideal gas).

At fixed T: ⟨v²⟩ = 3k_BT/m ⇒ P = nk_BT (ideal gas law form).

Total KE per molecule: ⟨KE⟩ = (3/2)k_BT (in 3D, monatomic).

Temperature is a measure of MEAN MOLECULAR KE.

Kinetic pressure formula

From molecular dynamics: momentum transfer rate.

Ideal gas connection

Combining with ⟨v²⟩ = 3k_BT/m gives PV = NkT.

Average KE per molecule

Direct from kinetic theory.

RMS speed

Defines the molecular speed scale.

P = (1/3)ρv_rms² connects molecular motion to macroscopic pressure.

Doubling number density doubles pressure (same v_rms).

Doubling v_rms ⇒ quadrupling pressure.

Combining with ⟨v²⟩ = 3k_BT/m gives PV = Nk_BT — ideal gas law DERIVED from kinetic theory.

Temperature is microscopically defined: T ∝ ⟨KE⟩.

Real gases deviate from ideal at high P/low T (intermolecular forces, finite molecular size).

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RMS / Avg / Most-Probable Speeds

Compare H₂, He, N₂, O₂, CO₂.

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RMS (Root Mean Square) speed: v_rms = √⟨v²⟩.

v_rms = √(3k_BT/m) = √(3RT/M). Lighter molecule or higher T ⇒ faster v_rms.

Three commonly cited speeds: v_rms = √(3RT/M); v_mean = √(8RT/πM); v_mp (most probable) = √(2RT/M).

Ratio: v_mp : v_mean : v_rms = √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22.

Sound speed in gas ≈ v_rms/√3 — same order of magnitude as molecular speeds.

v_rms in air at 300 K: oxygen ~480 m/s, nitrogen ~510 m/s, hydrogen ~1930 m/s, helium ~1370 m/s.

Maxwell-Boltzmann distribution: f(v) ∝ v²·exp(−mv²/2kT) — gives all three speeds from one curve.

RMS speed

M = molar mass.

Mean speed

Average of |v|.

Most probable speed

Peak of Maxwell-Boltzmann distribution.

Ratio

Three speeds always in this ratio for ideal gas.

v_rms is what enters kinetic-theory pressure formula.

Higher T ⇒ faster v_rms. Doubling T ⇒ × √2.

Lighter mass ⇒ faster speed. H₂ moves ~4× faster than O₂ at same T.

Sound speed ≈ v_rms/√3 — comparable to molecular speeds.

Highest mean KE = highest v_rms (heavier vs lighter at same T).

Maxwell distribution gives full range of speeds — most molecules near v_mp, some much faster.

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Temperature ↔ Kinetic Energy

⟨KE⟩ = (3/2) k_B T — linear in T.

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Temperature is a measure of AVERAGE KINETIC ENERGY of molecules.

For an ideal gas (monatomic): ⟨KE⟩ = (3/2)k_BT per molecule.

Per mole: ⟨KE⟩ × N_A = (3/2)RT (using k_B·N_A = R).

Linear relation: T ∝ ⟨KE⟩. Absolute T = 0 ⇒ molecules at rest (classical limit).

For diatomic gas (5 DoF): U = (5/2)nRT — translation + rotation.

Polyatomic: more DoF, more energy at same T.

Equipartition theorem: each quadratic degree of freedom gets ½k_BT per molecule.

Temperature is a STATISTICAL concept — needs many molecules to be meaningful.

Mean KE per molecule (3D)

Monatomic ideal gas — only translation.

Mean KE per mole

Per mole basis.

Equipartition theorem

Holds for each independent DoF.

Diatomic at room T

3 translational + 2 rotational DoF.

T is THE measure of average molecular KE — fundamental definition.

Higher T ⇒ faster molecules ⇒ more KE ⇒ more pressure (at constant V).

Absolute zero (T = 0 K) ⇒ zero molecular motion (classical limit; quantum gives zero-point energy).

Different gases at same T have SAME mean KE per molecule (regardless of m). Heavier ones move slower.

Translation: 3 DoF. Rotation: 2 DoF (for linear molecules) or 3 (for non-linear). Vibration: 2 DoF per mode.

At low T, vibrational modes FREEZE OUT (quantum effect) — only translation remains.

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Maxwell-Boltzmann Distribution

Speed distribution f(v) — peaks shift with T.

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Maxwell-Boltzmann distribution: probability density of molecular speeds in a gas at thermal equilibrium.

f(v) = 4π·n·(m/(2πkT))^(3/2)·v²·exp(−mv²/2kT). Peaks at v_mp.

Tail: very few molecules at extreme speeds, but always non-zero probability.

Distribution depends only on T and m — not on V, P, or n (those just rescale).

Higher T: distribution SHIFTS RIGHT (faster speeds), BROADENS (wider spread).

Heavier gases: distribution NARROWER, lower peak speed.

Maxwell-Boltzmann is correct for CLASSICAL gases at moderate T. Quantum gases (e.g., electrons in metals) need Fermi-Dirac instead.

Underlies effusion (Graham's law), reaction kinetics (Arrhenius), evaporation, atmospheric escape.

Maxwell speed distribution

Probability density per unit speed.

Most probable speed (peak)

Where f(v) is maximum.

Mean speed

Average ⟨v⟩.

RMS speed

√⟨v²⟩.

Fraction with speed > v₀

High-speed tail; exponentially suppressed.

Distribution is HEAVY-TAILED but exponentially DECAYING — few molecules at very high speeds.

Increase T: peak shifts right and broadens.

Increase m: peak shifts left and narrows.

Reaction rates often depend on the fast tail (high-energy molecules) — explains Arrhenius behavior.

Atmospheric escape: light gases (H, He) escape Earth's gravity because their high-speed tail exceeds escape velocity.

Three peaks (v_mp, v_mean, v_rms) all visible on the same plot in fixed ratios.

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Boyle's Law

PV = const at fixed T.

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Boyle's law: at constant temperature, pressure × volume = constant. PV = const.

Equivalently: P₁V₁ = P₂V₂ for an ideal gas at constant T.

Inverse proportionality between P and V.

On PV diagram: hyperbola (also an isotherm).

Experimentally verified by Robert Boyle (1662) using mercury and air in a J-tube.

Microscopically: smaller V ⇒ same molecules hit walls more often ⇒ higher pressure.

Real gases deviate at high P (intermolecular forces, molecular size).

Special case of ideal gas law (PV = nRT) at constant n and T.

Boyle's law

P ∝ 1/V at fixed T and n.

Two-state form

Same gas, same T ⇒ PV invariant.

Connection to ideal gas law

Boyle's law follows by fixing T and n.

PV = const at FIXED T. If T changes, both can change.

Doubling P halves V (or vice versa).

Air in a sealed syringe: compress it, pressure rises predictably.

Microscopic reason: smaller V ⇒ same number of molecules ⇒ more frequent collisions ⇒ higher P.

Real gases at high P: deviation due to molecular size and attractions (van der Waals corrections).

Used in diving (Boyle's law explains why divers must equalize ear pressure during descent).

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Charles's Law

V/T = const at fixed P.

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Charles's law: at constant pressure, V/T = constant for an ideal gas (T in Kelvin).

Equivalently: V₁/T₁ = V₂/T₂ at constant P.

DIRECT proportionality between V and absolute T.

On V-T graph: straight line through origin.

Extrapolating V to zero gives T = 0 K = −273.15°C = ABSOLUTE ZERO.

Microscopically: higher T ⇒ faster molecules ⇒ more momentum transfer ⇒ gas expands at constant P.

Discovered by Jacques Charles (1787) and later refined by Joseph Gay-Lussac.

Special case of ideal gas law at constant P and n.

Charles's law

V ∝ T at constant P.

Two-state form

Same gas at constant P.

From ideal gas law

V linear in T at constant P.

Volume expansion coefficient (gas)

Linear T dependence.

Charles's law uses ABSOLUTE temperature (Kelvin) — NOT Celsius.

Doubling T (in K) at constant P doubles V.

V vs T (Kelvin): straight line through origin. V vs T (Celsius): straight line, not through origin.

Extrapolation gives absolute zero: T = 0 K where V → 0 (classical limit).

Hot air balloons: heated air expands, becomes less dense, balloon rises.

Real gases deviate at low T (close to liquefaction) where molecular forces matter.

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Pressure vs Temperature

P/T = const at fixed V (Gay-Lussac).

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Gay-Lussac's law: at constant volume, pressure × T = constant. P/T = const.

Equivalently: P₁/T₁ = P₂/T₂ for an ideal gas at constant V.

Direct proportionality between P and absolute T.

On P-T graph: straight line through origin (in Kelvin).

Extrapolating P to zero gives T = 0 K = absolute zero.

Microscopic: higher T ⇒ faster molecules ⇒ stronger wall collisions ⇒ higher P.

Pressure cookers use this: heating sealed container raises P, which raises boiling point ⇒ faster cooking.

Special case of ideal gas law at constant V and n.

Gay-Lussac's law

P ∝ T at constant V.

Two-state form

Same gas, same V at two different T.

From ideal gas law

Linear in T at constant V, n.

P-T law uses ABSOLUTE T (Kelvin) — NOT Celsius.

Doubling T (K) at constant V doubles P.

Aerosol cans warn against heating because P can rise enough to burst the can.

Tire pressure: rises as temperature rises (hot days, after driving).

Pressure cookers: heating raises P, which raises water's boiling point ⇒ food cooks faster.

Real gas deviations at low T (near phase change) and very high P.

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Mean Free Path

λ = 1/(√2 πd²n) — distance between collisions.

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Mean free path λ: average distance a molecule travels between successive collisions.

λ = 1/(√2·π·d²·n), where d = molecular diameter, n = number density.

At 1 atm and 25°C (n ~ 2.5 × 10²⁵ /m³, d ~ 3.7 Å): λ ~ 70 nm in air. Very short.

λ inversely proportional to n and to d². Lower density (higher altitude) ⇒ longer λ.

λ in space (interstellar medium, n ~ 10⁵/m³): kilometers to light-years.

Collision frequency f = ⟨v⟩/λ ~ 10¹⁰ collisions/second in atmosphere.

Vacuum quality determined by λ: 'molecular flow' regime when λ exceeds container size.

Important for diffusion, viscosity, thermal conductivity calculations.

Mean free path

d = molecular diameter; n = number density.

Collision frequency

Collisions per molecule per second.

Air at STP (typical values)

Reference values.

Vacuum regimes

Kn < 0.01: continuum flow. Kn > 1: molecular flow.

λ inversely proportional to n and d².

λ in atmosphere at sea level ~ 70 nm — very short.

At 100 km altitude: λ ~ meters. In interstellar space: λ ~ light-years.

Diffusion coefficient D ~ λ·⟨v⟩/3 — small λ ⇒ slow diffusion.

Vacuum quality: 'high vacuum' (~10⁻⁸ atm) gives λ ~ km, allowing molecular flow.

λ governs viscosity and thermal conductivity in gases.

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Degrees of Freedom

Mono / diatomic / polyatomic — Cv, Cp, γ.

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Degrees of freedom (DoF) f: independent ways a molecule can store energy.

Monatomic gas (He, Ar, Ne): 3 translational ⇒ f = 3.

Diatomic at room T (N₂, O₂): 3 translation + 2 rotation = 5. (Vibrations frozen out at room T.)

Diatomic at high T: vibrations contribute 2 more ⇒ f = 7.

Polyatomic non-linear (H₂O, NH₃): 3 translation + 3 rotation = 6 (room T). Vibrations add more at high T.

Equipartition theorem: each quadratic DoF gets ½k_BT per molecule.

Specific heat per mole: C_v = (f/2)R, C_p = (f/2 + 1)R. γ = C_p/C_v = 1 + 2/f.

Monatomic γ = 5/3. Diatomic (room T) γ = 7/5 = 1.4. Polyatomic γ = 4/3 (approximately).

Equipartition (per molecule)

Each quadratic degree of freedom gets ½kT.

Internal energy

f-fold larger than monatomic for same T.

Specific heats

Per mole values.

Adiabatic index γ

Monatomic 5/3; diatomic 7/5; polyatomic 4/3.

f counts INDEPENDENT QUADRATIC energy storage modes.

Translation always contributes 3 DoF.

Rotation: 2 for linear molecules (no rotation along symmetry axis), 3 for non-linear.

Vibration: each mode = 2 DoF (KE + PE), but FROZEN OUT at low T (quantum effect).

Adiabatic index γ decreases as f increases — diatomic γ = 1.4 vs monatomic 5/3.

At very high T, more vibrational modes activate ⇒ C_v approaches Dulong-Petit limit for solids and high-T behavior for gases.

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