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Kinetic Theory & Statistical Physics

Gases as swarms of molecules, distributions and statistical mechanics. 42 topics, from beginner to advanced. Open the interactive version with labs and quizzes

Kinetic theory explains the behaviour of gases and liquids by treating them as huge numbers of moving molecules. It derives the ideal gas law from molecular collisions, links temperature to average kinetic energy, and introduces probability distributions such as the Maxwell-Boltzmann distribution. Statistical mechanics extends these ideas to explain why macroscopic laws are so reliable.

How the topics connect

Kinetic theory explains gases, liquids and solids as collections of moving molecules. Temperature becomes a measure of average molecular energy, pressure comes from collisions, and statistics explain why the large-scale behaviour is so predictable even though individual molecules are not. This is the bridge between thermodynamics and quantum statistics.

Real-world uses

Gas laws in engineering

Predict how pressure in a tyre or a gas cylinder changes with temperature, and design containers that stay within safe limits.

Atmospheric science

Model how air density and pressure fall with altitude, which affects weather, aviation and mountain climbing.

Chemistry and materials

Explain diffusion, evaporation and reaction rates by relating them to molecular speeds.

Computational physics

Use statistical methods and the Boltzmann distribution to simulate large systems of particles, such as plasmas or liquids.

Industrial gases and chemistry

Chemists use ideal and real gas equations to predict how much gas a reaction produces and how it behaves under pressure.

Electronics and materials

Engineers use Fermi energy and quantum statistics to understand conduction in metals and semiconductors.

Beginner

Molecular picture of matter

All matter is made of moving particles.

All matter is made of atoms and molecules in constant motion with forces between them. Solids have particles vibrating in fixed positions, liquids move past each other, gases fly freely. This picture explains pressure, temperature and diffusion from mechanics.

Worked example

A drop of ink spreads in water without stirring.

Common mistake

Atoms aren't static.

Beginner

Brownian motion

Random jiggling of small particles evidences molecular motion.

Pollen grains in water jitter randomly because molecules hit them unevenly. Einstein's 1905 analysis of this motion gave decisive evidence for atoms and let Perrin measure Avogadro's number.

Worked example

Smoke particles dancing in a light beam.

Common mistake

The grain, not the molecules, is what we see.

Beginner

Ideal gas assumptions

Point particles with elastic collisions and no forces.

Key relationgas

The model treats molecules as point particles with negligible volume, no intermolecular forces except instantaneous elastic collisions, moving randomly. It works well at low density and high temperature.

Worked example

Air at normal conditions follows it within 1%.

Common mistake

Fails near condensation.

Intermediate

Pressure from molecular collisions

Gas pressure arises from momentum transfer to walls.

Key relationP = ⅓ρ⟨v²⟩

Each molecule hitting a wall transfers momentum 2mv_x. Averaging over huge numbers gives P = ⅓ρ⟨v²⟩ = ⅓(N/V)m⟨v²⟩.

Worked example

Air at 1 atm: about 10²⁷ collisions per m² each second.

Common mistake

Pressure is from the average, not a single hit.

Intermediate

Kinetic interpretation of temperature

Temperature is proportional to mean translational kinetic energy.

Key relation⟨K⟩ = (3/2)k_BT

Comparing P = ⅓nm⟨v²⟩ with PV = NkT gives ⟨½mv²⟩ = (3/2)kT: temperature is proportional to the mean translational kinetic energy.

Worked example

At 300 K the average kinetic energy is 6.2×10⁻²¹ J (0.039 eV).

Common mistake

It applies to translational energy, independent of molecule mass.

Intermediate

Boltzmann constant

Connects energy and temperature.

Key relationk_B = 1.38×10⁻²³ J/K

k_B = 1.381×10⁻²³ J/K links temperature to energy per particle: R = N_A k_B. kT is the characteristic thermal energy.

Worked example

kT at 300 K = 4.14×10⁻²¹ J = 26 meV.

Common mistake

Per molecule, not per mole.

Intermediate

RMS speed

Root-mean-square molecular speed.

Key relationv_rms = √(3k_BT/m)

v_rms = √(3kT/m) = √(3RT/M). Lighter molecules move faster at the same temperature.

Worked example

Nitrogen at 300 K: 517 m/s; hydrogen: 1930 m/s.

Common mistake

Not the same as the average speed.

Intermediate

Mean free path

Average distance between collisions.

Key relationλ = 1/(√2 π d² n)

The average distance between collisions is λ = 1/(√2 π d²n). At STP air λ ≈ 70 nm; in a vacuum chamber it can be metres.

Worked example

Nitrogen at 1 atm: 68 nm.

Common mistake

Decreases with pressure and molecular size.

Advanced

Collision frequency

Number of collisions per second.

Each molecule makes about v/λ collisions per second: roughly 5×10⁹ per second in air at STP.

Worked example

Ensures rapid thermal equilibrium.

Common mistake

Increases with density and temperature.

Intermediate

Degrees of freedom

Independent ways a molecule stores energy.

The independent ways a molecule can store energy: 3 translational, up to 3 rotational, and vibrational modes. A monatomic gas has 3, diatomic 5 (room temperature).

Worked example

Helium f = 3; oxygen f = 5.

Common mistake

Vibrations are 'frozen out' at low temperature.

Intermediate

Equipartition theorem

Each quadratic degree of freedom has ½kT of energy.

Key relationE = ½k_BT per dof

In thermal equilibrium each quadratic term in the energy gets ½kT on average. Thus U = (f/2)NkT and C_v = (f/2)R.

Worked example

Diatomic gas: C_v = 5R/2.

Common mistake

Fails when quantum effects freeze modes.

Intermediate

Specific heats of gases

Cv depends on degrees of freedom.

Key relationC_v = (f/2)R

C_v = (f/2)R, C_p = C_v + R, γ = 1 + 2/f. Measured values test the number of degrees of freedom.

Worked example

Monatomic γ = 5/3, diatomic 7/5.

Common mistake

Hydrogen's C_v drops at low T due to quantised rotation.

Advanced

Maxwell–Boltzmann distribution

Distribution of molecular speeds.

Key relationf(v) ∝ v²e^{−mv²/2kT}

The fraction of molecules with speed v is f(v) ∝ v²e^{−mv²/2kT}: few slow, few very fast, with a peak and a long tail. Higher T broadens and shifts the peak.

Worked example

The tail explains evaporation and chemical reaction rates.

Common mistake

It describes speeds of a large sample in equilibrium.

Advanced

Most probable speed

Peak of the speed distribution.

Key relationv_p = √(2kT/m)

The peak of f(v) is v_p = √(2kT/m).

Worked example

N₂ at 300 K: 422 m/s.

Common mistake

Smaller than v_rms and the mean.

Advanced

Average, rms and most probable speeds

Ordered v_p < v_avg < v_rms.

Ordering: v_p : v_avg : v_rms = 1 : 1.128 : 1.225. They differ because the distribution is skewed.

Worked example

For O₂ at 300 K: 395, 446, 484 m/s.

Common mistake

Use the one that fits the question: v_rms for energy.

Advanced

Law of atmospheres

Density falls exponentially with height.

Key relationn = n₀e^{−mgh/kT}

In a uniform-temperature atmosphere density falls as n = n₀e^{−mgh/kT}, scale height kT/mg ≈ 8.5 km for air.

Worked example

At 8.5 km air density is about 37% of sea level.

Common mistake

Real atmosphere has temperature variation.

Intermediate

Diffusion

Spreading of particles from high to low concentration.

Particles spread from high concentration to low by random motion. The mean squared distance grows linearly with time: ⟨x²⟩ = 2Dt.

Worked example

Perfume spreads slowly across a room by diffusion, much faster by convection.

Common mistake

Diffusion is slow over large distances.

Advanced

Fick's law

Diffusion flux is proportional to the concentration gradient.

Key relationJ = −D dn/dx

Diffusion flux is proportional to the concentration gradient: J = −D dn/dx. D for gases is ~10⁻⁵ m²/s, for liquids ~10⁻⁹.

Worked example

Oxygen diffusing into the blood in the lungs.

Common mistake

Flux runs down the gradient.

Intermediate

Graham's law of effusion

Rate of effusion is inversely proportional to the square root of molar mass.

The rate of effusion through a tiny hole is inversely proportional to √M. Light gases escape faster. It was used to separate uranium isotopes.

Worked example

H₂ effuses 4 times faster than O₂.

Common mistake

Valid when the hole is smaller than the mean free path.

Intermediate

Real gases

Molecular volume and attraction make real gases deviate from ideal.

Molecules have finite volume and attract each other, so real gases deviate from PV = nRT at high pressure and low temperature, and can liquefy.

Worked example

CO₂ liquefies at 5 atm and 20 °C.

Common mistake

Attraction lowers the pressure; volume raises it.

Advanced

Van der Waals equation

Corrected gas equation.

Key relation(P + a/V²)(V − b) = RT

(P + a/V²)(V − b) = RT per mole corrects for attraction (a) and finite molecular volume (b). It predicts a critical point.

Worked example

CO₂: a = 0.364 Pa·m⁶/mol², b = 4.27×10⁻⁵ m³/mol.

Common mistake

It's an approximation.

Advanced

Compressibility factor

Z = PV/nRT measures non-ideality.

Z = PV/nRT equals 1 for an ideal gas; Z < 1 where attraction matters, Z > 1 at high pressure where size matters.

Worked example

Methane at 100 atm: Z ≈ 0.8.

Common mistake

Plot Z vs P to see non-ideality.

Advanced

Liquefaction of gases

Gases must be cooled below the critical temperature.

A gas can be liquefied only below its critical temperature; techniques include compression, cooling and Joule–Thomson expansion. Nitrogen boils at 77 K, helium at 4.2 K.

Worked example

Linde process for air.

Common mistake

Above T_c no pressure suffices.

Advanced

Microstates and macrostates

Many microscopic arrangements give the same macroscopic state.

A macrostate (P, V, T) is realised by an enormous number of microstates (positions and momenta). The most probable macrostate is the one with the most microstates.

Worked example

Coin tosses: 50% heads is more likely than 100% heads.

Common mistake

Equilibrium is overwhelmingly probable.

Advanced

Boltzmann distribution

Probability falls exponentially with energy.

Key relationP ∝ e^{−E/kT}

At temperature T, the probability of a state with energy E is proportional to e^{−E/kT}. Lower-energy states are more populated; the ratio of populations of two levels is e^{−ΔE/kT}.

Worked example

Excited state with ΔE = 10 kT is 4.5×10⁻⁵ as populated.

Common mistake

Includes degeneracy factors.

Advanced

Partition function

Sum over states from which all thermodynamic quantities follow.

Key relationZ = Σe^{−E/kT}

Z = Σe^{−E_i/kT} sums over all states and provides all thermodynamic quantities: F = −kT ln Z, U = −∂ln Z/∂β.

Worked example

Two-level system Z = 1 + e^{−ΔE/kT}.

Common mistake

Z normalises probabilities.

Advanced

Canonical ensemble

System in contact with a heat bath.

A system in contact with a heat bath at fixed T, exchanging energy but not particles, follows the Boltzmann distribution. Energy fluctuates around the mean.

Worked example

A beaker of water in a room.

Common mistake

Fluctuations are tiny for macroscopic systems.

Advanced

Fermi–Dirac statistics

Distribution for fermions obeying Pauli exclusion.

Key relationf(E) = 1/(e^{(E−μ)/kT}+1)

Identical fermions (electrons, protons) can't share a quantum state, so the occupation is f(E) = 1/(e^{(E−μ)/kT} + 1). At T = 0 states fill up to the Fermi energy.

Worked example

Electrons in metals.

Common mistake

Never greater than one per state.

Advanced

Bose–Einstein statistics

Distribution for bosons; leads to condensation.

Key relationf(E) = 1/(e^{(E−μ)/kT}−1)

Bosons (photons, helium-4) can pile into the same state: f(E) = 1/(e^{(E−μ)/kT} − 1). At low T many occupy the ground state.

Worked example

Lasers rely on photon bunching.

Common mistake

Needs μ below the lowest energy.

Advanced

Bose–Einstein condensate

Atoms collapse into the lowest quantum state near absolute zero.

Below a critical temperature, a macroscopic fraction of bosons occupies the lowest quantum state, acting as one matter wave. First made in 1995 with rubidium at 170 nK.

Worked example

Superfluid helium is related.

Common mistake

Requires extremely cold dilute gases.

Advanced

Fermi energy

Highest filled energy at absolute zero.

The energy of the highest filled state at T = 0: E_F = (ħ²/2m)(3π²n)^{2/3}. For copper ≈ 7 eV, so electrons move at 10⁶ m/s even at 0 K.

Worked example

Only electrons near E_F take part in conduction and heat capacity.

Common mistake

Much larger than kT at room temperature.

Advanced

Specific heat of solids

Dulong–Petit law and Debye model.

Key relationC = 3R

Classically each atom has 6 quadratic terms, so C = 3R ≈ 25 J/mol·K (Dulong–Petit), valid at high T. At low temperatures C falls, which classical theory couldn't explain.

Worked example

Copper 24.5 J/mol·K at room temperature.

Common mistake

Diamond is an exception at room T.

Advanced

Debye model

Phonons explain low-temperature heat capacity.

Key relationC ∝ T³

Treats lattice vibrations as quantised sound waves (phonons) with a cutoff frequency. It predicts C ∝ T³ at low temperature.

Worked example

Matches measurements of insulators near 10 K.

Common mistake

Electrons add a linear term in metals.

Advanced

Photon gas

Radiation behaves as a gas of photons.

Thermal radiation in a cavity behaves as a gas of photons with energy density u ∝ T⁴ and pressure P = u/3. Its entropy and heat capacity follow from Bose statistics.

Worked example

Radiation pressure inside stars.

Common mistake

Photon number is not conserved.

Advanced

Planck distribution

Spectrum of thermal radiation.

Key relationu(ν) ∝ ν³/(e^{hν/kT}−1)

The spectral energy density of blackbody radiation is u(ν) ∝ ν³/(e^{hν/kT} − 1), which cures the ultraviolet catastrophe.

Worked example

At low frequency it matches Rayleigh–Jeans; at high it falls exponentially.

Common mistake

Led to the birth of quantum physics.

Advanced

Random walks

Steps in random directions give diffusion.

Key relation⟨x²⟩ = 2Dt

A particle taking random steps of length l moves a typical distance l√N after N steps. This links microscopic randomness to diffusion.

Worked example

After 10⁶ steps, only 1000 step lengths from the origin.

Common mistake

Net distance grows like √t, not t.

Advanced

Equation of state of ideal gas in statistical terms

PV = NkT from counting states.

Counting the microstates of N non-interacting particles in volume V and using S = k ln Ω gives P/T = ∂S/∂V, resulting in PV = NkT.

Worked example

Shows that kT sets thermal energy.

Common mistake

Correct counting needs 1/N! for identical particles.

Advanced

Heat conduction at microscopic level

Phonons and electrons carry heat.

In metals free electrons carry most heat (hence Wiedemann–Franz law); in insulators, phonons carry it. Impurities and boundaries scatter carriers and reduce conductivity.

Worked example

Diamond conducts heat very well via phonons.

Common mistake

Thermal conductivity depends on mean free path.

Advanced

Transport phenomena

Viscosity, conduction and diffusion in gases.

Diffusion (mass), viscosity (momentum) and heat conduction (energy) all arise from molecules carrying properties across layers. In gases η ≈ ⅓ρv̄λ, independent of pressure.

Worked example

Gas viscosity increases with temperature, unlike liquids.

Common mistake

All three share the same underlying mechanism.

Advanced

Phase transitions

Abrupt changes in matter properties.

Abrupt changes in properties at critical conditions: first-order (latent heat: melting) or continuous (no latent heat: ferromagnet at the Curie point). They arise from a competition between energy and entropy.

Worked example

Water freezing, iron losing magnetism at 770 °C.

Common mistake

Near the transition fluctuations dominate.

Advanced

Ising model

A lattice of spins explaining magnetism and phase transitions.

A lattice of spins ±1 with neighbour interactions is the simplest model of a magnet, showing spontaneous magnetisation below a critical temperature. Solved exactly in 1D and 2D.

Worked example

Also used for neural networks and alloys.

Common mistake

No phase transition in 1D at T > 0.

Advanced

Critical phenomena

Fluctuations and universality near critical points.

Near a critical point, correlation length diverges, and quantities follow power laws with universal exponents shared by very different systems (universality).

Worked example

Water-steam and a magnet behave similarly at their critical points.

Common mistake

Described by the renormalisation group.

Frequently asked questions

Why does a gas exert pressure on a container wall?

Molecules hit the wall and change their momentum. The sum of these many small impacts per second, spread over the wall area, gives the macroscopic pressure.

Is temperature the same as the speed of molecules?

Temperature is proportional to the average kinetic energy of molecules, not their average speed. Lighter molecules move faster than heavier ones at the same temperature.

Why does Brownian motion matter?

It is visible evidence that molecules are in constant random motion. Its study by Einstein helped confirm the existence of atoms and molecules.

Why does a gas feel hotter when molecules move faster?

Temperature measures the average kinetic energy of molecules, so faster molecules hit your skin harder and more often, which you feel as heat.

When does the ideal gas law fail?

At high pressure or low temperature, when molecules are close together and the attractions between them and their volumes matter.

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