Physics Fundamentals

Thermal
Escape

Heat a gas until particles escape their container.
Real Maxwell-Boltzmann distributions. 8 levels.

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🔥 Thermodynamics Game

Thermal Escape — Free Online Kinetic Theory & Maxwell-Boltzmann Physics Game

Thermal Escape simulates a real gas — each particle's speed is drawn from the Maxwell-Boltzmann distribution. Drag the temperature slider and watch the distribution shift as fast particles find the escape gaps. This is exactly how evaporation, atmospheric escape, and stellar wind work.

The physics behind the game

Maxwell-Boltzmann distribution

f(v) = 4π(m/2πkT)^(3/2) · v² · exp(−mv²/2kT)

At any temperature, particles have a range of speeds — not all the same. The distribution has a peak (most probable speed) but a long tail of fast particles. Raising temperature shifts the whole distribution to higher speeds.

Thermal kinetic energy

KE_avg = ½mv² = (3/2)kT

Temperature is literally a measure of average kinetic energy per particle. Double the absolute temperature, double the average KE, increase average speed by √2.

Escape rate — Arrhenius equation

k = A·exp(−E_a/kT)

The fraction of particles exceeding the escape energy grows exponentially with temperature. This is why small temperature increases can dramatically speed up evaporation.

Atmospheric escape (Jeans escape)

Escape if v > v_esc = √(2gR)

Earth retains nitrogen and oxygen but slowly loses hydrogen — because H₂ molecules are light enough that even at 300K, enough of them reach escape velocity. Mars lost its atmosphere this way.

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Kinetic Theory and Maxwell-Boltzmann Distributions

Temperature as Kinetic Energy

Temperature T (kelvin) measures average molecular kinetic energy: ⟨KE⟩ = (3/2)k_BT per molecule (k_B = 1.381×10⁻²³ J/K). Total internal energy of n moles of ideal monatomic gas: U = (3/2)nRT. Diatomic gases (N₂, O₂) at room temperature: U = (5/2)nRT (two rotational modes active). This molecular picture explains specific heat capacities from first principles: C_V = (3/2)R ≈ 12.5 J mol⁻¹ K⁻¹ for monatomic, (5/2)R ≈ 20.8 for diatomic ideal gas.

Maxwell-Boltzmann Distribution

Gas molecules have a distribution of speeds, not all the same. Three characteristic speeds: most probable v_p = √(2k_BT/m); mean ⟨v⟩ = √(8k_BT/πm); RMS v_rms = √(3k_BT/m). Higher temperature shifts the distribution to higher speeds and broadens it. Lighter molecules move faster at the same T (v_rms ∝ 1/√m). At 300 K: N₂ v_rms ≈ 517 m/s, H₂ v_rms ≈ 1,934 m/s. The high-speed tail of the distribution determines which molecules can escape a planet's gravity.

Jeans Escape and Planetary Atmospheres

Thermal (Jeans) escape: molecules in the high-speed tail exceeding escape velocity leave the atmosphere permanently. Earth (v_esc = 11.2 km/s) retains N₂ and O₂ but slowly loses H₂ and He — their v_rms is much closer to v_esc. Mars (v_esc = 5.0 km/s, no magnetic field): lost most atmosphere over 4 Gyr to Jeans escape + solar wind stripping. Jupiter (v_esc = 59.5 km/s): retains all gases. Rule of thumb: retain gas long-term if v_esc/v_rms > 6.

ThermalEscape