Joule-Thomson Effect

Equivalent forms

\mu_{\text{JT}} = \frac{V}{C_p}(T\alpha - 1)

T_{\text{inv}} = \frac{2a}{Rb}

A pure thermodynamic-identity result: whether a gas cools or heats on throttling is decided entirely by the sign of Tα − 1.

Symbol	Name	SI unit	Dimension	Range
$μ_JT$	Joule-Thomson coefficientoutput Temperature change per unit pressure drop at constant enthalpy	K/Pa	Θ M⁻¹ L T²	-0.00001 – 0.00001
$T$	Temperature Inlet gas temperature	K	Θ	50 – 900
$C_p$	Heat capacity Isobaric heat capacity	J/K	M L² T⁻² Θ⁻¹	10 – 50
$α$	Expansion coefficient Isobaric thermal expansion (1/V)(∂V/∂T)_P	K⁻¹	Θ⁻¹	0 – 0.01

Unit systems

SI:: $\mu _JT$ in K/Pa
natural:: dimensionless in reduced units
CGS:: $\mu _JT$ in $K\cdot cm^{2}/dyn$

Where it holds

Constant-enthalpy throttling of a real gas; the inversion temperature depends on the

gas (\sim 620\,\mathrm{K}

for

N_{2}

,

\sim 40\,\mathrm{K}

for He, so He needs pre-cooling to liquefy).

Discovery

James Prescott Joule & William Thomson (Lord Kelvin) · 1852

Joule and Thomson's porous-plug experiments (1852–1862) measured the tiny temperature change of throttled gases, distinguishing real gases from ideal and founding cryogenics.

Try this

How do you liquefy air with no moving cold parts?

Push a real gas through a tiny throttle and it can cool itself — or heat up. Get below the inversion temperature and repeated throttling cascades all the way down to liquid nitrogen.

Research status: stable

Common misconceptions

It is not the same as adiabatic expansion doing work — throttling does no external work and exchanges no heat; the temperature change comes entirely from intermolecular forces (the a and b terms).

Derivation

Throttling is isenthalpic:

dH = 0

.

With

H = H(T

,P),

0 = C_p dT + (\partial H/\partial P)_T dP

.

Using the thermodynamic identity

(\partial H/\partial P)_T = V - T(\partial V/\partial T)_P

gives

\mu _JT = (\partial T/\partial P)_H = [T(\partial V/\partial T)_P - V]/C_p = (V/C_p)(T\alpha - 1)

.

For a van der Waals gas the inversion temperature is

T_inv = 2

a/Rb.

Limiting cases

Ideal gas⟶

T\alpha = 1

exactly,

so \mu _JT = 0

: no temperature change on throttling

T < T_inversion⟶

\mu _JT

> 0: gas cools on expansion (used to liquefy air,

N_{2}

,

O_{2})

T > T_inversion⟶

\mu _JT

< 0: gas heats on expansion (true for

H_{2} and

He at room temperature)

Van der Waals Equation→Heat Capacity at Constant Pressure→First Law of Thermodynamics→Clausius-Clapeyron Equation→