Joule–Thomson effect
Encyclopedia
In thermodynamics
, the Joule–Thomson effect or Joule–Kelvin effect or Kelvin–Joule effect describes the temperature
change of a gas or liquid when it is forced through a valve
or porous plug while kept insulated so that no heat is exchanged with the environment. This procedure is called a throttling process or Joule–Thomson process. At room temperature, all gases except hydrogen
, helium
and neon
cool upon expansion by the Joule–Thomson process.
The effect is named for James Prescott Joule
and William Thomson, 1st Baron Kelvin
who discovered it in 1852 following earlier work by Joule on Joule expansion
, in which a gas undergoes free expansion in a vacuum
.
(no heat exchanged) expansion of a gas may be carried out in a number of ways. The change in temperature experienced by the gas during expansion depends not only on the initial and final pressure, but also on the manner in which the expansion is carried out.
A throttling process proceeds along a constant-enthalpy line in the direction of decreasing pressure, which means that the process occurs from left to right on a T-P diagram. As we proceed along a constant-enthalpy line from high enough pressures the temperature increases, until the inversion temperature. After this, as the fluid continues its expansion, the temperature drops. If we do this for several constant enthalpies and join the inversion points a line called the inversion line is obtained. This line intersects the T-axis at some temperature, named the maximum inversion temperature.
For hydrogen this temperature is -68°. In Vapour-compression refrigeration
we need to throttle the gas and cool it at the same time. This poses a problem for substances whose maximum inversion temperature is well below room temperature. Thus hydrogen must be cooled below its inversion temperature if any cooling is achieved by throttling.
s grows. Because of intermolecular attractive force
s (see Van der Waals force
), expansion causes an increase in the potential energy
of the gas. If no external work is extracted in the process and no heat is transferred, the total energy of the gas remains the same because of the conservation of energy
. The increase in potential energy thus implies a decrease in kinetic energy
and therefore in temperature.
A second mechanism has the opposite effect. During gas molecule collisions, kinetic energy is temporarily converted into potential energy. As the average intermolecular distance increases, there is a drop in the number of collisions per time unit, which causes a decrease in average potential energy. Again, total energy is conserved, so this leads to an increase in kinetic energy (temperature). Below the Joule–Thomson inversion temperature, the former effect (work done internally against intermolecular attractive forces) dominates, and free expansion causes a decrease in temperature. Above the inversion temperature, gas molecules move faster and so collide more often, and the latter effect (reduced collisions causing a decrease in the average potential energy) dominates: Joule–Thomson expansion causes a temperature increase.
with respect to pressure 
in a Joule–Thomson process (that is, at constant enthalpy 
) is the Joule–Thomson (Kelvin) coefficient 
. This coefficient can be expressed in terms of the gas's volume 
, its heat capacity at constant pressure 
, and its coefficient of thermal expansion 
as:
See the Derivation of the Joule–Thomson (Kelvin) coefficient below for the proof of this relation. The value of
is typically expressed in °C/bar
(SI units: K
/Pa
) and depends on the type of gas and on the temperature and pressure of the gas before expansion. Its pressure dependence is usually only a few percent for pressures up to 100 bar.
All real gases have an inversion point at which the value of
changes sign. The temperature of this point, the Joule–Thomson inversion temperature, depends on the pressure of the gas before expansion.
In a gas expansion the pressure decreases, so the sign of
is negative by definition. With that in mind, the following table explains when the Joule–Thomson effect cools or warms a real gas:
Helium
and hydrogen
are two gases whose Joule–Thomson inversion temperatures at a pressure of one atmosphere
are very low (e.g., about 51 K (−222 °C) for helium). Thus, helium and hydrogen warm up when expanded at constant enthalpy at typical room temperatures. On the other hand nitrogen
and oxygen
, the two most abundant gases in air, have inversion temperatures of 621 K (348 °C) and 764 K (491 °C) respectively: these gases can be cooled from room temperature by the Joule–Thomson effect.
For an ideal gas,
is always equal to zero: ideal gases neither warm nor cool upon being expanded at constant enthalpy.
(usually a valve
) which must be very well insulated to prevent any heat transfer to or from the gas. No external work is extracted from the gas during the expansion (the gas must not be expanded through a turbine
, for example).
The effect is applied in the Linde technique
as a standard process in the petrochemical industry, where the cooling effect is used to liquefy gases, and also in many cryogenic applications (e.g. for the production of liquid oxygen, nitrogen, and argon
). Only when the Joule–Thomson coefficient for the given gas at the given temperature is greater than zero can the gas be liquefied at that temperature by the Linde cycle. In other words, a gas must be below its inversion temperature to be liquefied by the Linde cycle. For this reason, simple Linde cycle liquefiers cannot normally be used to liquefy helium, hydrogen, or neon
.
remains constant. To prove this, the first step is to compute the net work done by the gas that moves through the plug. Suppose that the gas has a volume of V1 in the region at pressure P1 (region 1) and a volume of V2 when it appears in the region at pressure P2 (region 2). Then the work done on the gas by the rest of the gas in region 1 is P1V1. In region 2 the amount of work done by the gas is P2V2. So, the total work done by the gas is
The change in internal energy plus the work done by the gas is, by the first law of thermodynamics, the total amount of heat absorbed by the gas (here it is assumed that there is no change in kinetic energy). In the Joule–Thomson process the gas is kept insulated, so no heat is absorbed. This means that
where
and 
denote the internal energy of the gas in regions 1 and 2, respectively.
Using the definition of enthalpy
, the above equation then implies that:
where
and 
denote the enthalpy of the gas in regions 1 and 2, respectively.
for the Joule–Thomson (Kelvin) coefficient.
The partial derivative of T with respect to P at constant H can be computed by expressing the differential of the enthalpy dH in terms of dT and dP, and equating the resulting expression to zero and solving for the ratio of dT and dP.
It follows from the fundamental thermodynamic relation that the differential of the enthalpy is given by:
(here, 
is the entropy
of the gas).
Expressing dS in terms of dT and dP gives:
Using
(see Specific heat capacity), we can write:
The remaining partial derivative of S can be expressed in terms of the coefficient of thermal expansion via a Maxwell relation as follows. From the fundamental thermodynamic relation, it follows that the differential of the Gibbs energy is given by:
The symmetry of partial derivatives of G with respect to T and P implies that:
where
is the coefficient of thermal expansion. Using this relation, the differential of H can be expressed as
Equating dH to zero and solving for dT/dP then gives:
It is easy to verify that for an ideal gas the thermal expansion coefficient
is 
, and so an ideal gas does not experience a Joule–Thomson effect. The cooling of a gas by pure isentropic expansion is not Joule-Thomson cooling, although it is sometimes erroneously called J-T cooling by some laboratory practitioners.
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, the Joule–Thomson effect or Joule–Kelvin effect or Kelvin–Joule effect describes the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
change of a gas or liquid when it is forced through a valve
Valve
A valve is a device that regulates, directs or controls the flow of a fluid by opening, closing, or partially obstructing various passageways. Valves are technically pipe fittings, but are usually discussed as a separate category...
or porous plug while kept insulated so that no heat is exchanged with the environment. This procedure is called a throttling process or Joule–Thomson process. At room temperature, all gases except hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
, helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
and neon
Neon
Neon is the chemical element that has the symbol Ne and an atomic number of 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in either low-voltage neon glow lamps or...
cool upon expansion by the Joule–Thomson process.
The effect is named for James Prescott Joule
James Prescott Joule
James Prescott Joule FRS was an English physicist and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work . This led to the theory of conservation of energy, which led to the development of the first law of thermodynamics. The...
and William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
who discovered it in 1852 following earlier work by Joule on Joule expansion
Joule expansion
The Joule expansion is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container , with the other side of the container being evacuated; the partition between the two parts of the container is then opened, and the gas fills the whole...
, in which a gas undergoes free expansion in a vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
.
Description
The adiabaticAdiabatic process
In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...
(no heat exchanged) expansion of a gas may be carried out in a number of ways. The change in temperature experienced by the gas during expansion depends not only on the initial and final pressure, but also on the manner in which the expansion is carried out.
- If the expansion process is reversibleReversible process (thermodynamics)In thermodynamics, a reversible process, or reversible cycle if the process is cyclic, is a process that can be "reversed" by means of infinitesimal changes in some property of the system without loss or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic...
, meaning that the gas is in thermodynamic equilibriumThermodynamic equilibriumIn thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
at all times, it is called an isentropic expansion. In this scenario, the gas does positive workMechanical workIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
during the expansion, and its temperature decreases. - In a free expansionFree expansionFree expansion is an irreversible process in which a gas expands into an insulated evacuated chamber.Real gases experience a temperature change during free expansion...
, on the other hand, the gas does no work and absorbs no heat, so the internal energy is conserved. Expanded in this manner, the temperature of an ideal gasIdeal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
would remain constant, but the temperature of a real gas may either increase or decrease, depending on the initial temperature and pressure. - The method of expansion discussed in this article, in which a gas or liquid at pressure P1 flows into a region of lower pressure P2 via a valve or porous plug under steady state conditions and without change in kinetic energy, is called the Joule–Thomson process. During this process, enthalpyEnthalpyEnthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
remains unchanged (see a proof below).
A throttling process proceeds along a constant-enthalpy line in the direction of decreasing pressure, which means that the process occurs from left to right on a T-P diagram. As we proceed along a constant-enthalpy line from high enough pressures the temperature increases, until the inversion temperature. After this, as the fluid continues its expansion, the temperature drops. If we do this for several constant enthalpies and join the inversion points a line called the inversion line is obtained. This line intersects the T-axis at some temperature, named the maximum inversion temperature.
For hydrogen this temperature is -68°. In Vapour-compression refrigeration
Vapor-compression refrigeration
Vapor-compression refrigeration is one of the many refrigeration cycles available for use. It has been and is the most widely used method for air-conditioning of large public buildings, offices, private residences, hotels, hospitals, theaters, restaurants and automobiles...
we need to throttle the gas and cool it at the same time. This poses a problem for substances whose maximum inversion temperature is well below room temperature. Thus hydrogen must be cooled below its inversion temperature if any cooling is achieved by throttling.
Physical mechanism
As a gas expands, the average distance between moleculeMolecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s grows. Because of intermolecular attractive force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s (see Van der Waals force
Van der Waals force
In physical chemistry, the van der Waals force , named after Dutch scientist Johannes Diderik van der Waals, is the sum of the attractive or repulsive forces between molecules other than those due to covalent bonds or to the electrostatic interaction of ions with one another or with neutral...
), expansion causes an increase in the potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
of the gas. If no external work is extracted in the process and no heat is transferred, the total energy of the gas remains the same because of the conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
. The increase in potential energy thus implies a decrease in kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
and therefore in temperature.
A second mechanism has the opposite effect. During gas molecule collisions, kinetic energy is temporarily converted into potential energy. As the average intermolecular distance increases, there is a drop in the number of collisions per time unit, which causes a decrease in average potential energy. Again, total energy is conserved, so this leads to an increase in kinetic energy (temperature). Below the Joule–Thomson inversion temperature, the former effect (work done internally against intermolecular attractive forces) dominates, and free expansion causes a decrease in temperature. Above the inversion temperature, gas molecules move faster and so collide more often, and the latter effect (reduced collisions causing a decrease in the average potential energy) dominates: Joule–Thomson expansion causes a temperature increase.
The Joule–Thomson (Kelvin) coefficient
The rate of change of temperature with respect to pressure in a Joule–Thomson process (that is, at constant enthalpy ) is the Joule–Thomson (Kelvin) coefficient . This coefficient can be expressed in terms of the gas's volume , its heat capacity at constant pressure , and its coefficient of thermal expansion as:See the Derivation of the Joule–Thomson (Kelvin) coefficient below for the proof of this relation. The value of
is typically expressed in °C/barBar (unit)
The bar is a unit of pressure equal to 100 kilopascals, and roughly equal to the atmospheric pressure on Earth at sea level. Other units derived from the bar are the megabar , kilobar , decibar , centibar , and millibar...
(SI units: K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
/Pa
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
) and depends on the type of gas and on the temperature and pressure of the gas before expansion. Its pressure dependence is usually only a few percent for pressures up to 100 bar.
All real gases have an inversion point at which the value of
changes sign. The temperature of this point, the Joule–Thomson inversion temperature, depends on the pressure of the gas before expansion.In a gas expansion the pressure decreases, so the sign of
is negative by definition. With that in mind, the following table explains when the Joule–Thomson effect cools or warms a real gas:| If the gas temperature is | then  is | since  is | thus  must be | so the gas |
|---|---|---|---|---|
| below the inversion temperature | positive | always negative | negative | cools |
| above the inversion temperature | negative | always negative | positive | warms |
Helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
and hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
are two gases whose Joule–Thomson inversion temperatures at a pressure of one atmosphere
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...
are very low (e.g., about 51 K (−222 °C) for helium). Thus, helium and hydrogen warm up when expanded at constant enthalpy at typical room temperatures. On the other hand nitrogen
Nitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...
and oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...
, the two most abundant gases in air, have inversion temperatures of 621 K (348 °C) and 764 K (491 °C) respectively: these gases can be cooled from room temperature by the Joule–Thomson effect.
For an ideal gas,
is always equal to zero: ideal gases neither warm nor cool upon being expanded at constant enthalpy.Applications
In practice, the Joule–Thomson effect is achieved by allowing the gas to expand through a throttling deviceThrottle
A throttle is the mechanism by which the flow of a fluid is managed by constriction or obstruction. An engine's power can be increased or decreased by the restriction of inlet gases , but usually decreased. The term throttle has come to refer, informally and incorrectly, to any mechanism by which...
(usually a valve
Valve
A valve is a device that regulates, directs or controls the flow of a fluid by opening, closing, or partially obstructing various passageways. Valves are technically pipe fittings, but are usually discussed as a separate category...
) which must be very well insulated to prevent any heat transfer to or from the gas. No external work is extracted from the gas during the expansion (the gas must not be expanded through a turbine
Turbine
A turbine is a rotary engine that extracts energy from a fluid flow and converts it into useful work.The simplest turbines have one moving part, a rotor assembly, which is a shaft or drum with blades attached. Moving fluid acts on the blades, or the blades react to the flow, so that they move and...
, for example).
The effect is applied in the Linde technique
Carl von Linde
Professor Doctor Carl Paul Gottfried von Linde was a German engineer who developed refrigeration and gas separation technologies...
as a standard process in the petrochemical industry, where the cooling effect is used to liquefy gases, and also in many cryogenic applications (e.g. for the production of liquid oxygen, nitrogen, and argon
Argon
Argon is a chemical element represented by the symbol Ar. Argon has atomic number 18 and is the third element in group 18 of the periodic table . Argon is the third most common gas in the Earth's atmosphere, at 0.93%, making it more common than carbon dioxide...
). Only when the Joule–Thomson coefficient for the given gas at the given temperature is greater than zero can the gas be liquefied at that temperature by the Linde cycle. In other words, a gas must be below its inversion temperature to be liquefied by the Linde cycle. For this reason, simple Linde cycle liquefiers cannot normally be used to liquefy helium, hydrogen, or neon
Neon
Neon is the chemical element that has the symbol Ne and an atomic number of 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in either low-voltage neon glow lamps or...
.
Proof that enthalpy remains constant in a Joule–Thomson process
In a Joule–Thomson process the enthalpyEnthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
remains constant. To prove this, the first step is to compute the net work done by the gas that moves through the plug. Suppose that the gas has a volume of V1 in the region at pressure P1 (region 1) and a volume of V2 when it appears in the region at pressure P2 (region 2). Then the work done on the gas by the rest of the gas in region 1 is P1V1. In region 2 the amount of work done by the gas is P2V2. So, the total work done by the gas is
The change in internal energy plus the work done by the gas is, by the first law of thermodynamics, the total amount of heat absorbed by the gas (here it is assumed that there is no change in kinetic energy). In the Joule–Thomson process the gas is kept insulated, so no heat is absorbed. This means that
where
and denote the internal energy of the gas in regions 1 and 2, respectively.Using the definition of enthalpy
, the above equation then implies that:where
and denote the enthalpy of the gas in regions 1 and 2, respectively.Derivation of the Joule–Thomson (Kelvin) coefficient
A derivation of the formulafor the Joule–Thomson (Kelvin) coefficient.
The partial derivative of T with respect to P at constant H can be computed by expressing the differential of the enthalpy dH in terms of dT and dP, and equating the resulting expression to zero and solving for the ratio of dT and dP.
It follows from the fundamental thermodynamic relation that the differential of the enthalpy is given by:
(here, is the entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of the gas).
Expressing dS in terms of dT and dP gives:
Using
(see Specific heat capacity), we can write:The remaining partial derivative of S can be expressed in terms of the coefficient of thermal expansion via a Maxwell relation as follows. From the fundamental thermodynamic relation, it follows that the differential of the Gibbs energy is given by:
The symmetry of partial derivatives of G with respect to T and P implies that:
where
is the coefficient of thermal expansion. Using this relation, the differential of H can be expressed asEquating dH to zero and solving for dT/dP then gives:
It is easy to verify that for an ideal gas the thermal expansion coefficient
is , and so an ideal gas does not experience a Joule–Thomson effect. The cooling of a gas by pure isentropic expansion is not Joule-Thomson cooling, although it is sometimes erroneously called J-T cooling by some laboratory practitioners.See also
- Critical temperature
- Ideal gasIdeal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
- EnthalpyEnthalpyEnthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
and Isenthalpic - RefrigerationRefrigerationRefrigeration is a process in which work is done to move heat from one location to another. This work is traditionally done by mechanical work, but can also be done by magnetism, laser or other means...
- Reversible process (thermodynamics)Reversible process (thermodynamics)In thermodynamics, a reversible process, or reversible cycle if the process is cyclic, is a process that can be "reversed" by means of infinitesimal changes in some property of the system without loss or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic...