What is the Joule-Thomson coefficient?

μ_JT = (∂T/∂P)_H, the rate of temperature change with pressure at constant enthalpy. A positive value means the gas cools as pressure drops; negative means it warms. Ideal gases have μ_JT = 0.

What is the formula used?

μ_JT = V(Tα − 1) / Cp, where V is molar volume (m³/mol), T is temperature (K), α is the isobaric thermal expansion coefficient (1/K), and Cp is the molar heat capacity at constant pressure (J/(mol·K)).

What is the inversion temperature?

An inversion temperature is a state where μ_JT = 0. For the values entered here, the local zero condition is T·α = 1. Because α itself changes with temperature and pressure, 1/α at one state is not a complete real-gas inversion curve; use the calculated sign of μ_JT to determine cooling or warming at the entered state.

What are typical α values for real gases?

For an ideal gas α = 1/T, giving μ_JT = 0. For nitrogen near room temperature α ≈ 3.7×10⁻³ 1/K; for CO₂ at 300 K α ≈ 3.3×10⁻³ 1/K. Liquids have much smaller α values (water: ~2.6×10⁻⁴ 1/K at 25 °C).

Why does this tool use molar volume?

The thermodynamic formula is derived on a per-mole basis. Molar volume V_m = RT/P for an ideal gas; for real gases use the equation of state or experimental data. Units must be m³/mol for the result in K/Pa.

What units does the result have?

μ_JT is in K/Pa (kelvin per pascal). In practice it is often quoted in K/bar or K/MPa; 1 K/Pa = 10⁵ K/bar.

Joule-Thomson Coefficient Calculator

Enter Cp, temperature, molar volume, and thermal expansion coefficient to compute the Joule-Thomson coefficient and predict heating or cooling.

Inputs

Formula: μ_JT = V(Tα − 1) / C_p

C_p — Heat capacity at constant pressure

J/(mol·K)

T — Temperature

V_m — Molar volume

m³/mol

α — Thermal expansion coefficient

1/K

Load a preset gas

Result

Fill in the inputs and click Calculate to see results.

Formula reference

Joule-Thomson coefficient

μ_JT = V(Tα − 1) / C_p

Inversion temperature

T_inv = 1 / α (when μ_JT = 0)

Ideal gas limit

α = 1/T → μ_JT = 0

Summary

The Joule-Thomson coefficient (μ_JT) quantifies how a real gas's temperature changes when it expands at constant enthalpy — a process central to gas liquefaction and refrigeration. A positive μ_JT means the gas cools on expansion (below the inversion temperature); a negative value means it warms. This calculator uses the thermodynamic formula μ_JT = (T·α·V − V) / Cp = V(T·α − 1) / Cp, where α is the isobaric thermal expansion coefficient.

How it works

Enter the molar heat capacity at constant pressure (Cp) in J/(mol·K).
Enter the temperature (T) in Kelvin.
Enter the molar volume (V) in m³/mol.
Enter the isobaric thermal expansion coefficient (α) in 1/K.
Click Calculate — the tool applies μ_JT = V(Tα − 1) / Cp and shows the result with an interpretation.

Use cases

Determine whether a gas cools or heats during throttling in refrigeration cycles.
Estimate inversion temperature behavior for real gases like nitrogen or hydrogen.
Validate thermodynamic properties against van der Waals or Peng-Robinson equations of state.
Compare the Joule-Thomson coefficient across different gases at the same conditions.
Teach isenthalpic expansion concepts in physical chemistry courses.
Screen working fluids for heat pump or cryogenic liquefaction processes.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu