In Joule's free-expansion experiment, a gas expanded into a vacuum and no work was performed because there was no opposing pressure. A more practical and technologically important process occurs when a gas is forced through a constriction such as a porous plug, valve, or throttle.
This process was studied by James Joule and William Thomson (later Lord Kelvin) and is known as the Joule-Thomson effect.
In a typical experiment, gas flows continuously from a high-pressure region to a low-pressure region through a constriction:
High pressure → Valve / Porous Plug → Low pressure
The temperatures on both sides of the constriction are measured to determine whether the gas warms or cools during the expansion.
The apparatus is designed so that heat transfer between the gas and its surroundings is negligible. Under these conditions, the expansion occurs at approximately constant enthalpy:
\[ dH = 0 \]
Such a process is called an isenthalpic expansion.
Unlike Joule's free expansion, the gas performs work as it flows through the constriction. However, the balance of energy transfers is such that the enthalpy remains essentially unchanged.
The temperature change accompanying an isenthalpic pressure change is described by the Joule-Thomson coefficient, \(\mu_{JT}\):
\[ \mu_{JT} = \left( \frac{\partial T} {\partial p} \right)_H \]
This derivative measures how much the temperature changes when the pressure changes during an isenthalpic process.
| Value of \(\mu_{JT}\) | Behavior upon Expansion |
|---|---|
| \(\mu_{JT} > 0\) | Gas cools as pressure decreases |
| \(\mu_{JT} < 0\) | Gas warms as pressure decreases |
| \(\mu_{JT} = 0\) | No temperature change |
Most gases at room temperature and pressure cool when they undergo a Joule-Thomson expansion.
As the gas expands, the average distance between molecules increases. Intermolecular attractions must therefore be overcome, requiring energy. That energy comes from the thermal motion of the molecules, causing the temperature to decrease.
Some gases, notably hydrogen and helium near room temperature, exhibit the opposite behavior and warm during expansion. These gases have \(\mu_{JT}<0\) under those conditions.
The temperature at which \(\mu_{JT}\) changes sign is called an inversion temperature. Above the inversion temperature a gas warms during expansion, while below it the gas cools.
For an ideal gas, the enthalpy depends only on temperature:
\[ H = H(T) \]
Consequently, an isenthalpic process cannot change the temperature:
\[ \mu_{JT} = \left( \frac{\partial T} {\partial p} \right)_H = 0 \]
The Joule-Thomson effect is therefore a consequence of deviations from ideal behavior and provides direct evidence of intermolecular interactions in real gases.
Big picture: The Joule-Thomson effect describes the temperature change that accompanies an isenthalpic pressure change. Most real gases cool when expanded through a valve or porous plug, making the Joule-Thomson effect one of the fundamental principles underlying modern refrigeration and gas-liquefaction technologies.
Carbon dioxide at room temperature has a Joule-Thomson coefficient of
\[ \mu_{JT} = 1.1\ \mathrm{K\,atm^{-1}} \]
Estimate the temperature change when the gas undergoes an isenthalpic expansion from \(10.0\ \mathrm{atm}\) to \(1.00\ \mathrm{atm}\).
The Joule-Thomson coefficient is defined as
\[ \mu_{JT} = \left( \frac{\partial T} {\partial p} \right)_H \]
Assuming that \(\mu_{JT}\) remains approximately constant over this pressure range,
\[ dT = \mu_{JT}\,dp \]
Integrating between the initial and final pressures gives
\[ \Delta T = \int_{p_1}^{p_2} \mu_{JT}\,dp \]
and therefore
\[ \Delta T = \mu_{JT} (p_2-p_1) \]
Substituting the values:
\[ \Delta T = (1.1\ \mathrm{K\,atm^{-1}}) (1.00-10.0\ \mathrm{atm}) \]
\[ \Delta T = -9.9\ \mathrm{K} \]
Therefore,
\[ \boxed{ \Delta T = -9.9\ \mathrm{K} } \]
If the gas initially had a temperature of \(298\ \mathrm{K}\), its final temperature would be
\[ T_f = 298 - 9.9 = 288\ \mathrm{K} \]
Physical interpretation: Because carbon dioxide has a positive Joule-Thomson coefficient under these conditions, it cools as it expands. The pressure drop of \(9.0\ \mathrm{atm}\) produces a temperature decrease of approximately \(10\ \mathrm{K}\), illustrating why the Joule-Thomson effect is useful in refrigeration and gas-liquefaction processes.
Big picture: The Joule-Thomson effect connects pressure changes to temperature changes during isenthalpic expansions. Because real gases can cool significantly when expanded through a valve or porous plug, the Joule-Thomson effect forms the basis of many refrigeration and gas liquefaction technologies.