The Joule and Joule–Thomson Experiments

Consider an adiabatic system composed of two chambers, A and B, designed such that no heat flows into or out of the system. Initially, chamber A is filled with a gas at a fixed temperature T₁, pressure p₁, and volume V₁, while chamber B is evacuated. The gas is then gradually forced through a rigid, porous barrier to chamber B, ultimately reaching temperature T₂, pressure p₂, and volume V₂. A piston on the right side maintains a constant pressure (p₂), which is lower than p₁. The significant drop in pressure from p₁ to p₂ primarily occurs across the porous plug.

The piston on the left performs work (p₁V₁) on the gas, while the gas does work (-p₂V₂) on the right piston. The total work performed by the system after the first piston is fully pushed in is determined by the equation w_net = p₁V₁ - p₂V₂. As it's an adiabatic system, there's no heat transfer (q = 0), so the change in internal energy (∆U_net) equals the net work done. Equating these two expressions for w_net gives U₁ + p₁V₁ = U₂ + p₂V₂.

The sum of U + pV represents the enthalpy (H). In this experiment, the initial and final enthalpies of the gas are equal (H₁ = H₂), indicating no change in enthalpy (∆H = 0). Thus, the process is isenthalpic.

However, even though the change in enthalpy is zero, the temperature change is not. The temperature difference ΔT = T₂ - T₁ obtained from the Joule–Thomson experiment provides ΔT/Δp at constant H. The Joule-Thomson coefficient (µ), an intensive property, is defined as the change in temperature of a gas with respect to pressure at a constant enthalpy and is dependent on T, p, and the type of gas. For an ideal gas, µ equals zero, meaning the temperature remains unchanged during the Joule–Thomson expansion. However, for real gases, µ isn't zero, and the temperature changes during the isenthalpic process.

The sign of µ can be positive or negative. If it's negative, the temperature rises as pressure decreases (the gas heats upon expansion). If it's positive, the temperature falls as pressure decreases (the gas cools upon expansion). The temperature at which µ changes from negative to positive is known as the inversion temperature. To cool gases using the Joule-Thomson method, a gas must be below its inversion temperature. In Joule–Thomson liquefaction of gases, the porous plug is replaced by a narrow opening or a needle valve to produce the required pressure drop. An alternative liquefaction method uses a nearly reversible adiabatic expansion against a piston, which achieves cooling through work done by the gas.