Joule-Thomson Effect

The thought experiment

Imagine an insulated tube with a porous plug in the middle. Push gas in from the left at high pressure P₁; let it emerge on the right at lower pressure P₂. The walls are adiabatic — no heat enters or leaves. What happens to the temperature?

For an ideal gas: nothing. Temperature stays the same. Ideal gas molecules don't interact, so they don't care how spread out they are.

For a real gas: temperature usually drops (sometimes rises). This is the Joule-Thomson effect, named after James Joule and William Thomson (Lord Kelvin), who published their porous-plug experiment in 1852.

Why throttling is isenthalpic

Apply the first law to a parcel of gas crossing the plug. The pump on the left does work P₁V₁ on the parcel; the parcel does work P₂V₂ pushing into the right chamber. No heat is exchanged. Internal energy change:

ΔU = U₂ − U₁ = P₁V₁ − P₂V₂
⟹ U₁ + P₁V₁ = U₂ + P₂V₂
⟹ H₁ = H₂

Enthalpy H = U + PV is conserved. Throttling is isenthalpic. It's also irreversible (no work is extracted from the pressure drop), so entropy increases.

The Joule-Thomson coefficient

Define the slope of an isenthalpic process in the (P, T) plane:

μ_JT ≡ (∂T/∂P)_H

Since pressure drops on throttling (ΔP < 0):

• μ_JT > 0 ⟹ T falls (cooling).
• μ_JT < 0 ⟹ T rises (warming).
• μ_JT = 0 ⟹ ideal gas, or real gas at its inversion temperature.

A standard thermodynamic identity (cyclic rule + Maxwell relation) gives:

μ_JT = (1/C_p) · [T(∂V/∂T)_P − V]

Van der Waals predicts the inversion temperature

For a van der Waals gas (P + a/V²)(V − b) = RT, the low-pressure limit of μ_JT is:

μ_JT ≈ (1/C_p) · [ 2a/(RT) − b ]

Two competing terms:

• 2a/(RT) — attractive forces. Expansion does work against the attractions; kinetic energy and therefore T fall. This term dominates at low T.
• b — excluded volume (molecular size). Expansion lets molecules collide harder. This term dominates at high T.

Setting μ_JT = 0:

T_inv ≈ 2a/(Rb)

For N₂ (a = 1.37×10⁵ Pa·m⁶/mol², b = 3.87×10⁻⁵ m³/mol) this predicts T_inv ≈ 850 K. The actual maximum inversion temperature of N₂ is about 621 K — van der Waals is a rough but qualitatively correct guide.

Inversion temperatures for common gases

Gas	T_inv (max, K)	Boiling point (K)	μ_JT at 300 K, 1 atm (K/bar)	Cool or warm at 300 K?
N₂	≈ 621	77.4	+0.25	cools
O₂	≈ 765	90.2	+0.31	cools
Air	≈ 650	~78	+0.20	cools
CO₂	≈ 1500	194.7 (sublim.)	+1.10	cools strongly
CH₄	≈ 968	111.7	+0.40	cools
H₂	≈ 202	20.3	−0.03	warms (above T_inv)
He	≈ 51	4.2	−0.06	warms

The two outliers — hydrogen and helium — must be pre-cooled below their inversion temperatures (with liquid N₂ for H₂, with liquid H₂ for He) before throttling will cool them further. This is why early 20th-century low-temperature physics was a stepwise cascade.

Worked example — CO₂ snow from a fire extinguisher

A CO₂ extinguisher sits at room temperature (300 K) at internal pressure ~60 bar (the saturation pressure of CO₂ at 300 K). When you squeeze the trigger, gas flashes from 60 bar down to 1 atm across the valve.

ΔP = 1 − 60 = −59 bar
μ_JT(CO₂, 300 K) ≈ 1.1 K/bar
ΔT ≈ μ_JT · ΔP = 1.1 · (−59) ≈ −65 K

The exiting gas drops to roughly 235 K. CO₂ sublimates at 195 K (−78 °C) at 1 atm, so a fraction of the cold vapor crosses the sublimation line and freezes directly into solid CO₂ — the white "dry-ice snow" you see in fire-extinguisher discharges and stage-fog effects.

The Linde liquefaction cycle

Carl von Linde patented a beautifully simple cascade in 1895. The trick: feed the cold throttled gas back through a heat exchanger that pre-cools the incoming high-pressure gas. After enough cycles, the temperature drops below the boiling point and gas condenses to liquid.

1. Compressor raises gas to 200 bar; intercooler removes the heat of compression.
2. Counter-flow heat exchanger pre-cools the 200-bar stream with returning cold low-pressure gas.
3. JT valve drops pressure to ~1 bar; gas cools (or stays cool if already saturated).
4. Liquid fraction collects in a Dewar; vapor fraction returns through the heat exchanger.
5. Repeat. Steady state produces liquid air, N₂, or O₂ continuously.

Modern air-separation units produce upward of 3 000 tonnes of liquid oxygen per day per train. The ultimate origin of every steel mill's oxygen lance, every hospital's oxygen tank, every fertilizer plant's nitrogen feed.

JavaScript — JT calculations

const R = 8.314;  // J/(mol·K)

// Van der Waals JT coefficient (low-P limit)
function muJT_vdw(a, b, Cp, T) {
  // a in Pa·m⁶/mol², b in m³/mol, Cp in J/(mol·K)
  // returns K/Pa
  return (1 / Cp) * (2 * a / (R * T) - b);
}

// N₂ van der Waals constants
const N2 = { a: 1.37e5, b: 3.87e-5, Cp: 29.1 };

// At 300 K, return K/bar (× 1e5 Pa/bar)
const mu_N2_300K = muJT_vdw(N2.a, N2.b, N2.Cp, 300) * 1e5;
console.log(`N₂ μ_JT at 300 K: ${mu_N2_300K.toFixed(3)} K/bar`);  // ~+0.26

// Inversion temperature (vdw prediction)
function T_inversion_vdw(a, b) {
  return 2 * a / (R * b);
}
console.log(`N₂ T_inv (vdw): ${T_inversion_vdw(N2.a, N2.b).toFixed(0)} K`);  // ~850 K

// CO₂ snow from 60 bar to 1 bar
const CO2 = { a: 3.64e5, b: 4.27e-5, Cp: 37.1 };
const mu_CO2 = muJT_vdw(CO2.a, CO2.b, CO2.Cp, 300) * 1e5;
const dT = mu_CO2 * (-59);  // ΔP = -59 bar
console.log(`CO₂ valve drop: μ = ${mu_CO2.toFixed(2)} K/bar, ΔT ≈ ${dT.toFixed(0)} K`);

// Compare to ideal gas (should give zero)
function muJT_ideal() { return 0; }
console.log(`Ideal gas μ_JT: ${muJT_ideal()} (no effect)`);

// Linde cycle yield (Hampson-Linde approximation)
// Fraction of gas liquefied per pass:
//   y = (h₁ − h₂) / (h₁ − h_liq)
// where h₁, h₂ are inlet/outlet enthalpies of the warm stream
// and h_liq is enthalpy of saturated liquid.
function lindeYield(h1_warm, h2_warm, h_liquid) {
  return (h1_warm - h2_warm) / (h1_warm - h_liquid);
}
// Typical air liquefier: y ≈ 0.07 (7% per pass).
console.log(`Linde yield example: ${(lindeYield(300, 290, 150) * 100).toFixed(0)}%`);

Where the Joule-Thomson effect shows up

• Cryogenics. Liquid N₂, O₂, Ar, methane, hydrogen, helium — all produced at industrial scale via JT-based cascades or magnetic-then-JT hybrids.
• HVAC. The "expansion valve" in every air conditioner and refrigerator is a JT throttle. Refrigerant (R-134a, R-410a, propane) cools as it expands across the valve, becomes a two-phase mixture, and absorbs heat from your room.
• Natural-gas processing. JT plants drop pipeline gas pressure and condense out heavier hydrocarbons (ethane, propane) for recovery.
• Space cooling. JT mini-coolers cool infrared sensors on satellites and missiles to liquid-N₂ temperatures with no moving parts.
• Petroleum engineering. When natural gas flows up a wellbore the pressure drops; JT warming can melt hydrate plugs or, conversely, cause icing problems that require glycol injection.
• Helium-3 dilution refrigerators. Final stage of millikelvin cryostats uses JT throttling of a He-3/He-4 mixture.

Common mistakes

• Confusing isenthalpic with adiabatic. Throttling is both, but the defining constraint is constant H, not Q = 0. A reversible adiabatic expansion (isentropic) cools much more strongly than throttling because the gas does work on a piston.
• Assuming all gases cool on throttling. H₂ and He warm at room temperature. Linde's original air machine worked precisely because air's inversion temperature is well above 300 K; Dewar had to pre-cool H₂ to make it liquefy.
• Using ideal gas relations. μ_JT = 0 for ideal gas. The entire effect is a real-gas correction; if you start from PV = nRT you get zero cooling.
• Forgetting that μ_JT depends on T and P. The maximum inversion temperature is at low P; an inversion curve in the (P, T) plane bounds the region where cooling occurs. High enough pressure can put even N₂ outside its inversion region.
• Confusing throttling with free expansion (Joule expansion). Joule expansion is isothermal for an ideal gas; the two-chamber experiment with no plug is different from steady-state throttling through a constriction.