Posts Tagged ‘Joule-Thomson coefficient’


Historical background

If you received formal tuition in physical chemistry at school, then it’s likely that among the first things you learned were the 17th/18th century gas laws of Mariotte and Gay-Lussac (Boyle and Charles in the English-speaking world) and the equation that expresses them: PV = kT.

It may be that the historical aspects of what is now known as the ideal (perfect) gas equation were not covered as part of your science education, in which case you may be surprised to learn that it took 174 years to advance from the pressure-volume law PV = k to the combined gas law PV = kT.


The lengthy timescale indicates that putting together closely associated observations wasn’t regarded as a must-do in this particular era of scientific enquiry. The French physicist and mining engineer Émile Clapeyron eventually created the combined gas equation, not for its own sake, but because he needed an analytical expression for the pressure-volume work done in the cycle of reversible heat engine operations we know today as the Carnot cycle.


The first appearance in print of the combined gas law, in Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the Motive Power of Heat, 1834) by Émile Clapeyron

Students sometimes get in a muddle about combining the gas laws, so for the sake of completeness I will set out the procedure. Beginning with a quantity of gas at an arbitrary initial pressure P1 and volume V1, we suppose the pressure is changed to P2 while the temperature is maintained at T1. Applying the Mariotte relation (PV)T = k, we write

The pressure being kept constant at P2 we now suppose the temperature changed to T2; the volume will then change from Vx to the final volume V2. Applying the Gay-Lussac relation (V/T)P = k, we write

Substituting Vx in the original equation:


– – – –

Differences of opinion

In the mid-19th century, the ideal gas equation – or rather the ideal gas itself – was the cause of no end of trouble among those involved in developing the new science of thermodynamics. The argument went along the lines that since no real gas was ever perfect, was it legitimate to base thermodynamic theory on the use of a perfect gas as the working substance in the Carnot cycle? Joule, Clausius, Rankine, Maxwell and van der Waals said yes it was, while Mach and Thomson said no it wasn’t.

With thermometry on his mind, Thomson actually got quite upset. Here’s a sample outpouring from the Encyclopaedia Britannica:

“… a mere quicksand has been given as a foundation of thermometry, by building from the beginning on an ideal substance called a perfect gas, with none of its properties realized rigorously by any real substance, and with some of them unknown, and utterly unassignable, even by guess.”


Joule (inset) and Thomson may have had their differences, but it didn’t stop them from becoming the most productive partnership in the history of thermodynamics

It seems strange that the notion of an ideal gas, as a theoretical convenience at least, caused this violent division into believers and disbelievers, when everyone agreed that the behavior of all real gases approaches a limit as the pressure approaches zero. This is indeed how the universal gas constant R was computed – by extrapolation from pressure-volume measurements made on real gases. There is no discontinuity between the measured and limiting state, as the following diagram demonstrates:


Experiments on real gases show that

where v is the molar volume and i signifies ice-point. The universal gas constant is defined by the equation

so for real gases

The behavior of n moles of any gas as the pressure approaches zero may thus be represented by

The notion of an ideal gas is founded on this limiting state, and is defined as a gas that obeys this equation at all pressures. The equation of state of an ideal gas is therefore

– – – –


William Thomson, later Lord Kelvin, in the 1850s

Testing Mayer’s assumption

The notion of an ideal gas was not the only thing troubling William Thomson at the start of the 1850s. He also had a problem with real gases. This was because he was simultaneously engaged in a quest for a scale of thermodynamic temperature that was independent of the properties of any particular substance.

What he needed was to find a property of a real gas that would enable him to
a) prove by thermodynamic argument that real gases do not obey the ideal gas law
b) calculate the absolute temperature from a temperature measured on a (real) gas scale

And he found such a property, or at least he thought he had found it, in the thermodynamic function (∂U/∂V)T.

In the final part of his landmark paper, On the Dynamical Theory of Heat, which was read before the Royal Society of Edinburgh on Monday 15 December 1851, Thomson presented an equation which served his purpose. In modern notation it reads:

This is a powerful equation indeed, since it enables any equation of state of a PVT system to be tested by relating the mechanical properties of a gas to a thermodynamic function of state which can be experimentally determined.

If the equation of state is that of an ideal gas (PV = nRT), then

This defining property of an ideal gas, that its internal energy is independent of volume in an isothermal process, was an assumption made in the early 1840s by Julius Robert Mayer of Heilbronn, Germany in developing what we now call Mayer’s relation (Cp – CV = PΔV). Thomson was keen to disprove this assumption, and with it the notion of the ideal gas, by demonstrating non-zero values for (∂U/∂V)T.

In 1845 James Joule had tried to verify Mayer’s assumption in the famous experiment involving the expansion of air into an evacuated cylinder, but the results Joule obtained – although appearing to support Mayer’s claim – were deemed unreliable due to experimental design weaknesses.


The equipment with which Joule tried to verify Mayer’s assumption, (∂U/∂V)T = 0. The calorimeter at the rear looks like a solid plate construction but is in fact hollow. This can be ascertained by tapping it – which the author of this blogpost has had the rare opportunity to do.

Thomson had meanwhile been working on an alternative approach to testing Mayer’s assumption. By 1852 he had a design for an apparatus and had arranged with Joule to start work in Manchester in May of that year. This was to be the Joule-Thomson experiment, which for the first time demonstrated decisive differences from ideal behavior in the behavior of real gases.

Mayer’s assumption was eventually shown to be incorrect – to the extent of about 3 parts in a thousand. But this was an insignificant finding in the context of Joule and Thomson’s wider endeavors, which would propel experimental research into the modern era and herald the birth of big science.

Curiously, it was not the fact that (∂U/∂V)T = 0 for an ideal gas that enabled the differences in real gas behavior to be shown in the Joule-Thomson experiment. It was the other defining property of an ideal gas, that its enthalpy H is independent of pressure P in an isothermal process. By parallel reasoning

If the equation of state is that of an ideal gas (PV = nRT), then

Since the Joule-Thomson coefficient (μJT) is defined

and the second term on the right is zero for an ideal gas, μJT must also be zero. Unlike a real gas therefore, an ideal gas cannot exhibit Joule-Thomson cooling or heating.

– – – –

Finding a way to define absolute temperature

But to return to Thomson and his quest for a scale of absolute temperature. The equation he arrived at in his 1851 paper,


besides enabling any equation of state of a PVT system to be tested, also makes it possible to give an exact definition of absolute temperature independently of the behavior of any particular substance.

The argument runs as follows. Given the temperature readings, t, of any arbitrary thermometer (mercury thermometer, bolometer, whatever..) the task is to express the absolute temperature T as a function of t. By direct measurement, it may be found how the behavior of some appropriate substance, e.g. a gas, depends on t and either V or P. Introducing t and V as the independent variables in the above equation instead of T and V, we have

where (∂U/∂V)t, (∂P/∂t)V and P represent functions of t and V, which can be experimentally determined. Separating the variables so that both terms in T are on the left, the equation can then be integrated:

Integrating between the ice point and the steam point

This completely determines T as a function of t.

But as we have already seen, there was a catch to this argumentation – namely that (∂U/∂V) could not be experimentally determined under isothermal conditions with sufficient accuracy.

– – – –

The Joule-Thomson coefficient provides the key

Thomson’s means of circumventing this problem was the steady state Joule-Thomson experiment, which measured upstream and downstream temperature and pressure, and enabled the Joule-Thomson coefficient, μJT = (∂T/∂P)H, to be computed.

It should be borne in mind however that when Joule and Thomson began their work in 1852, they were not aware that their cleverly-designed experiment was subject to isenthalpic conditions. It was the Scottish engineer and mathematician William Rankine who first proved in 1854 that the equation of the curve of free expansion in the Joule-Thomson experiment was d(U+PV) = 0.

William John Macquorn Rankine (1820-1872)

William John Macquorn Rankine (1820-1872)

As for the Joule-Thomson coefficient itself, it was the crowning achievement of a decade of collaboration, appearing in an appendix to Joule and Thomson’s final joint paper published in the Philosophical Transactions of the Royal Society in 1862. They wrote it in the form


where the upper symbol in the derivative denotes “thermal effect”, and K denotes thermal capacity at constant pressure of a unit mass of fluid.

The equation is now usually written

By the method applied previously, this equation can be expressed in terms of an empirical t-scale and the absolute T-scale:

where C’P is the heat capacity of the gas as measured on the empirical t-scale, i.e. C’P = CP(dT/dt). Cancelling (dT/dt) and separating the variables so that both terms in T are on the left, the equation becomes:

Integrating between the ice point and the steam point

This completely determines T as a function of t, with all the terms under the integral capable of experimental determination to a sufficient level of accuracy.

– – – –

P Mander May 2014


Future Nobel Prize winners both. Kamerlingh Onnes and Johannes van der Waals in 1908.

On Friday 10 July 1908, at Leiden in the Netherlands, Kamerlingh Onnes succeeded in liquefying the one remaining gas previously thought to be non-condensable – helium – using a sequential Joule-Thomson cooling technique to drive the temperature down to just 4 degrees above absolute zero. The event brought to a conclusion the race to liquefy the so-called permanent gases, following the revelation that all gases have a critical temperature below which they must be cooled before liquefaction is possible.

This crucial fact was established by Dr. Thomas Andrews, professor of chemistry at Queen’s College Belfast, in his groundbreaking study of the liquefaction of carbon dioxide, “On the Continuity of the Gaseous and Liquid States of Matter”, published in the Philosophical Transactions of the Royal Society of London in 1869.

As described in Part I of this blog post, Andrews’ discovery of the critical temperature (aided and abetted by Joule and Thomson’s earlier discovery of isenthalpic cooling) opened the way to cryotechnological advances of great commercial importance, and gave birth to the industrial gases industry which played such a significant role in shaping the 20th century.

This fact alone was enough to ensure Dr. Andrews’ study a place in the history of physical science. But there was another aspect to his paper – a theoretical one – which had equally far-reaching effects and is the subject of the remainder of this post.

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Dr. Thomas Andrews FRS (1813-1885). Photograph taken in Paris 1875 when Andrews was 62.

Thomas Andrews was a scientist whose experimental skills were evidently comparable to those of the illustrious James Joule. Ten years of care and devotion went into Andrews’ study of the liquefaction of carbon dioxide (called carbonic acid in his day), the essential results of which are contained in this diagram taken from his 1869 paper.


It is a pressure-volume diagram (with the line of no volume to the right) upon which are drawn isothermal carbon dioxide curves for temperatures ranging from 13.1°C to 48.1°C, pressures ranging from 50 to 100 atmospheres. Isothermal air curves are included in the upper left quadrant to illustrate the degree of deviation of the carbon dioxide curves from the rectangular hyperbola associated with ideal gas behavior.

The lowest isothermal curve (13.1°C) shows that at a pressure of around 47 atmospheres, condensation occurs. The compressed gas separates into two distinct coexistent portions – vapor and liquid – along a line of constant pressure, with further compression driving the conversion to the liquid form until finally the whole is converted to liquid, at which point compressibility becomes markedly reduced.

In the next isothermal curve (21.5°C), where condensation takes place at a pressure of about 60 atmospheres, gas and liquid are closer still in density, the liquid occupying nearly a third of the volume of the gas. As James Clerk Maxwell put it in Theory of Heat, written in 1871, “the exceedingly dense gas is approaching in its properties to the exceedingly light liquid”.

These properties eventually coincide at the isopycnic point* (the point of inflexion on the critical isotherm, marked X in the figure below) corresponding to a critical pressure of 72.8 atmospheres and a critical temperature of 31°C.

*isopycnic means ‘of equal density’. The isopycnic point (sometimes called the critical point) is where the densities of vapor and liquid coincide; this occurs under the conditions of critical temperature Tc and critical pressure pc.

Above the critical temperature, isothermals do not show any discontinuity; it is not possible to detect the point at which a liquid becomes a gas or vice versa. If liquid carbon dioxide, represented by point Z in the figure below, is heated at constant pressure until its temperature reaches 48°C, its condition at different temperatures will be represented by the line ZY. At Z the substance concerned is a liquid; at Y it is a gas. The change has taken place smoothly and continuously, representing continuity of state.


It was this revelation provided by Andrews’ data that started theoreticians thinking about how to reconcile the idea of continuity of state with the discontinuous change observed experimentally within the confines of the dotted parabola shown in Fig.12.


The history of science is full of coincidences – Belfast in Northern Ireland was the birthplace of not only Thomas Andrews, but also William Thomson (later Lord Kelvin) and his elder brother James. William frequently discussed thermodynamics with James, who just happened to be professor of civil engineering at Queen’s College Belfast exactly at the time when Thomas Andrews was conducting his famous experiments there.


James Thomson (1822-1892), physicist and engineer, whose achievements were largely overshadowed by his equally hairy brother William Thomson, also known as Lord Kelvin. Photo credit: Wikipedia

Not surprisingly, James Thomson with his practised skills in thermodynamics took an active interest in Andrews’ remarkable results, and in 1871 proposed a highly original solution to the problem of reconciling the discontinuous isotherms below the isopycnic point with the continuous isotherms above it.

Thomson’s thesis was that the gaseous and liquid parts of a discontinuous isotherm (AB and CD in the above diagram) were only apparently discontinuous, and were actually parts of one smooth curve shown in dotted lines in the diagram below. Every isotherm, according to Thomson, was a continuous curve.


Diagram from “Considerations on the Abrupt Change at Boiling or Condensing in reference to the Continuity of the Fluid State of Matter” by Professor James Thomson, LL.D., Queen’s College, Belfast. Communicated to The Royal Society of London by Dr. Andrews. Received July 4, 1871.

The task that now confronted theoreticians was to find a satisfactory mathematical equation for this curve. Thomson’s curve gave some useful clues, as Lewis and Randall subsequently observed in their classic textbook Thermodynamics and the Free Energy of Chemical Substances:

“It is evident that the equation for such a complete curve must be of odd degree in V, for V increases with diminishing P at both ends of the curve. Furthermore the equation must be of at least the third degree in V, since a certain pressure may correspond to more than one volume. At lower temperatures three roots of the equation are real, at the critical point the three coincide, and at higher temperatures two of them become imaginary.”

The first to provide a solution was a physics student in the Netherlands. His name was Johannes van der Waals.

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The title page of Johannes van der Waals’ doctoral thesis

At Leiden University on Saturday 14 June 1873, from 12 noon to 3 pm, Johannes van der Waals defended his doctoral thesis, which sought to explain Thomas Andrews experimental results on the basis of kinetic theory, and whose title “Over de Continuiteit van den Gas en Vloeistoftoestand (On the Continuity of the Gaseous and Liquid State)” was almost exactly the same as Andrews’ 1869 paper.

In his thesis, van der Waals introduced the concepts of molecular attraction and molecular volume, and derived the equation of state which bears his name:

where a and b are gas-specific constants related to molecular attraction and molecular volume respectively; the term a/V2 identifies with the derivative (∂U/∂V)T while b turns out to be equal to a third of the critical volume.

Multiplying out the van der Waals equation gives

Since this expression equals zero, it follows that

Multiplying out and rearranging terms

A cubic equation in V is thus obtained. For any given values of p and T, there will be three values of V, since a, b and R are constants for one mole of a given gas.

The cubic form of van der Waals’ equation produces curves like those shown below. They are very similar to the isothermal curves hypothesized by James Thomson (cf. above), and give three values of V along the line of first-order phase transition where all the roots A,B,C are real; on the critical isotherm the roots are coincident at the isopycnic point. At higher temperatures two of the roots become imaginary as the curves become increasingly hyperbolic.


The van der Waals equation of state modifies the ideal gas equation, and is an improvement on it in accounting for the shape of pressure-volume curves above the critical isotherm. At lower temperatures it is also qualitatively reasonable for the liquid state and the low-pressure gaseous state.

During first-order phase transition (A↔C) however, the equation is clearly at variance with the empirically determined fact that the pressure remains constant. The reason why the van der Waals equation fails to describe the behavior of real substances within the dotted region in the above figure is precisely because it assumes continuity of state. It cannot therefore account for the fact that the substance, by separating into two coexistent phases – liquid and saturated vapor – is rendered more stable than in the homogeneous state.

It should be noted that under certain conditions, states corresponding to the portions AA’ and B’C are respectively realizable as superheated liquid and supersaturated vapor (both portions representing states stable with respect to infinitesimal variations but unstable relative to the coexistent liquid-vapor system). The portion of the curve A’B’, on the other hand, represents states that are absolutely unstable since

and according to the energy test of stability in rational mechanics (not to mention common sense), such states where the volume and pressure increase and diminish together are never realizable.

– – – –


Thomas Andrews’ discovery of the critical temperature provided Johannes van der Waals with the starting point for yet further theoretical insight.

It is evident from the van der Waals equation that at the critical temperature, all three values of V are identical. At the isopycnic point (p=pc, V=Vc, T=Tc) the volume V can therefore be set equal to the critical volume Vc, so that


The cubic form of the van der Waals equation gives, on dividing terms by pc:

Identifying like terms

Divide the third term by the second term to get Vc:

Substitute Vc in the third term to get pc:

Substitute Vc and pc in the first term to get Tc:

The term a/V2 in the van der Waals equation identifies with the derivative (∂U/∂V)T, since it follows from the fundamental relation of thermodynamics (dU =TdS – pdV) that

Using the Maxwell relation



Taking the van der Waals equation

and differentiating with respect to temperature at constant volume

Substituting in (1)

The derivative (∂U/∂V)T can be computed from the experimentally determined Joule-Thomson coefficient:

Cp may be obtained calorimetrically or spectroscopically, while (∂V/∂P)T and (∂(PV)/∂P)T can be obtained from data on the compressibility of the gas at constant temperature.

Hence (∂U/∂V)T = a/V2 can be computed for any gas, enabling the constants a and b in the van der Waals equation to be determined. This in turn allows the three critical constants, Vc, pc, Tc, to be calculated.

These critical data – for which van der Waals provided further means of estimation in 1880 with his “principle of corresponding states” – were invaluable in helping Dewar’s determination of the method of liquefying hydrogen (Tc = 33K) in 1898, and Onnes’ determination of the method of liquefying helium (Tc = 5.2K) in 1908.

The van der Waals equation constants a and b also proved useful in the early days of the industrial gases industry. Taking the equation deduced in 1862 by Joule and Thomson for the temperature change ΔT when a gas is subjected to a pressure drop Δp under isenthalpic conditions:

If we apply the van der Waals equation as the equation of state, the Joule-Thomson equation becomes

If a and b become sufficiently small

For most gases the expression in brackets is positive at not-too-high temperatures. A cooling effect is therefore obtained, since Δp is always negative. Carl von Linde, who in 1895 established the first large-scale air liquefaction plant, based the construction of his Joule-Thomson cooling machine on this fact.

It is only when the attractive forces between gas molecules are very small at ordinary temperatures, and thus the constant a becomes minuscule – as is the case for hydrogen and the inert gases helium and neon – that the expression in brackets becomes negative and ΔT becomes positive, i.e. heating occurs. For cooling to occur, the temperature must be lowered below the inversion point, which according to the above equation is

– – – –


JD van der Waals 1837 – 1923

The Nobel Prize in Physics 1910 was awarded to Johannes Diderik van der Waals “for his work on the equation of state for gases and liquids”.


HK Onnes 1853 – 1926

The Nobel Prize in Physics 1913 was awarded to Heike Kamerlingh Onnes “for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”.

photo credits:

– – – –

P Mander March 2014


William Thomson, later Lord Kelvin (left), James Joule and the famous hand-operated pump. The Joule-Thomson effect is named after them, as are the SI units of thermodynamic temperature (kelvin) and energy (joule).

Historical background

In early May 1852, in the cellar of a house in Acton Square, Salford, Manchester (England), two men began working a mechanical apparatus which consisted of the above hand-operated forcing pump attached to a coiled length of lead piping equipped with a stopcock at its far end to act as a throttle.

The two men were the owner of the house, 33-year-old James Joule, a Manchester brewer who was rapidly making a name for himself as a first-rate experimental scientist, and 27-year-old William Thomson (later Lord Kelvin), a maverick theoretician who was already a professor of natural sciences at Glasgow University. Over a period of 10 days, they were to conduct a series of experiments with this highly original apparatus which would serve to crank experimental research into the modern era and herald the birth of what we would now call big science.

What Joule and Thomson were looking for was a slight cooling of the expanded gas as it emerged from the throttle – and they found it. But the results were unsatisfactory due to the modest size of the apparatus; fluctuations in the ambient conditions exerted too much influence on the measurements for the data to be considered reliable.

The remedy was clear to them. As they wrote in their first joint paper, read to the British Association for the Advancement of Science on Friday 3 September 1852: “… the authors are convinced that, without apparatus on a much larger scale, and a much more ample source of mechanical work than has hitherto been available to them, they could not get as complete and accurate results as are to be desired…”

It was easy enough to read between the lines. Joule and Thomson were fishing for money from sponsors to fund the scaling up of their apparatus. And they got it. In fact, a repeating pattern of funding and upscaling over the following years would become a key characteristic of Joule and Thomson’s collaboration.

From modest beginnings in the cellar at Acton Square, the experiments conducted in Manchester by Joule and Thomson were expanded in size and scope on an unprecedented scale by these two increasingly influential scientists. By the time they ended their work a decade later, they were using massive, highly pressurised equipment driven by steam engines and financed by substantial government grants.

The first round of scaling up. Joule and Thomson's experimental equipment was longer and taller than a London Bus.

The first round of scaling up. Joule and Thomson’s experimental equipment was longer and taller than a London Bus.

Scaling-up continues. By 1854, the Joule-Thomson equipment stretched the length of 3 London buses.

Scaling-up continues. By 1854, the Joule-Thomson equipment stretched the length of 3 London buses.

This was light years beyond what published experimental science in Britain had been up to that point – a desultory activity for individual gentleman-devotees of an enquiring disposition, undertaken at their own expense and largely for their own amusement.

Another paradigm shift was that Joule and Thomson published all their work jointly, in a total of eight papers. It should be borne in mind that in the mid-19th century, a joint paper was a rarity. A whole series of joint papers was unheard of; even in the scientific centres of Europe, there had never been anything like it.

While Europe’s laboratories were mainly occupied with the accurate determination of physical constants, the work going on in Manchester represented something radically new. It was the systematic and continued exploration of the unknown by talented individuals of complementary ability. Not surprisingly it led to discoveries, which turned out to have commercial as well as purely scientific value, since the Joule-Thomson effect made possible the liquefaction of gases and opened the way to important applications of throttling such as refrigeration.

Refrigerators, liquefied gases for industrial and domestic use, and air conditioning systems. All these technological advances resulted directly from the discovery of the Joule-Thomson effect.

Refrigerators, liquefied gases for industrial and domestic use, and air conditioning systems. All these technological advances resulted directly from the discovery of the Joule-Thomson effect.

Joule and Thomson’s sustained and extraordinary work provided a new model of collaborative scientific enquiry that would help the scientific establishment (and governments) to recognize the need for research institutions with both the physical and financial resources to undertake advanced and ambitious scientific programmes in order to gain valuable knowledge, scientific pre-eminence and commercial advantage.

All this took time, and of course there were other factors involved, but the original driving force that led to the establishment in 1874 of Britain’s first big science institution – the Cavendish Laboratory in Cambridge – can be traced back directly to the handle of the forcing pump that William Thomson cranked in a Manchester cellar in the spring of 1852.

The birthplace of big science. It was in the cellar of this house in Acton Square, Manchester, that Joule and Thomson began the work that would propel experimental research into the modern era. The cellar layout is practically unaltered to this day, and the house is now owned and occupied by the University of Salford. Photo credit: Charlie Hulme,

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The Joule-Thomson Experiment explained

Ok, so now let’s turn our attention to the thermodynamic characteristics of the Joule-Thomson experiment. The elegantly simple steady-state experimental set-up is easy enough to describe, but not so easy to understand.

Briefly stated, Joule and Thomson’s apparatus allowed a pressurised gas to flow along a tube which narrowed at a certain point, after which the gas expanded into a region of lower (atmospheric) pressure. The pressurised gas upstream of the throttle was kept at a constant temperature, and precautions were taken to ensure that the gas did not gain heat from, or lose heat to, the throttling section of the apparatus before it emerged on the other side and the temperature of the expanded gas was measured.

When the first experiments were conducted using air, a slight cooling on expansion was detected. This result was interpreted by a number of contemporary scientific observers as the familiar cooling observed when a gas does external work under adiabatic conditions. But this is not what happens in the apparatus; the cooling has a different explanation.

The Joule-Thomson effect can be understood by looking at a diagrammatic representation of the experiment, in which the flow of a given mass of gas from the high pressure region to the low pressure region is equated to work done by, and on, conceptual pistons:


Upstream of the throttle in the high pressure region, the flow of the gas can be equated to work done on the gas by the conceptual piston at left against the resistance of the throttle. The force-distance work done by the conceptual piston is equivalent to the pressure-volume product P1V1 of the gas.

Downstream of the throttle in the low pressure region, the flow of the gas can be equated to work done by the gas against the resistance of the conceptual piston at right, which is under lower pressure (atmospheric pressure in the original Joule-Thomson experiment). The force-distance work on the conceptual piston is equivalent to the pressure-volume product P2V2 of the gas.

The net work W done on the given mass of gas is therefore

In the Joule-Thomson experiment, no heat is gained or lost by the gas (Q = 0) as it flows from the higher to the lower pressure region. Applying the first law of thermodynamics to this transition:

[the + sign shows that W represents work done on the gas]

U + PV happens to be a thermodynamic state function, called enthalpy. The Joule-Thomson experiment therefore involves a constant-enthalpy, or isenthalpic, expansion. Cooling observed under these conditions has a fundamentally different thermodynamic aetiology to the cooling observed when a gas does external work under (reversible) adiabatic conditions, which is a constant-entropy, or isentropic, expansion.

When Joule and Thomson began their work “on the thermal effects of fluid in motion” in 1852, they were not aware that their cleverly-designed experiment was subject to isenthalpic conditions. The recognition of enthalpy as a thermodynamic state function did not come until 1875 when J. Willard Gibbs in America introduced it as “the heat function for constant pressure”, although the Scottish engineer and theoretician William Rankine – who was well aware of Joule and Thomson’s work – showed in his 1854 paper “On the geometrical representation of the of the expansive action of heat” that the equation of the curve of free expansion in their experiment was d(U+PV) = 0. Similarly, James Clerk Maxwell in his 1871 book Theory of Heat recognised that in the Joule-Thomson experiment “the sum of the intrinsic energy and the product of the volume and the pressure remains the same after passing through the plug [throttle], provided no heat is lost or gained from external sources”.

– – – –

The Joule-Thomson Coefficient

In Joule and Thomson’s first experiments with their apparatus, they varied the upstream pressure and temperature, while the downstream pressure was always atmospheric since the end of the pipe was open to the air. These were pathfinding experiments – they were exploring the unknown.

It was later realised that more useful results are obtained if the upstream pressure and temperature are held constant, and the downstream  pressure is held at several decreasing values at each of which the downstream temperature is measured. The data is then recorded on a temperature-pressure plot. Each point on the plot represents a state for which the enthalpy is equal to the initial (upstream) enthalpy. By joining up the points, a constant-enthalpy line, or isenthalpic curve, is obtained.


The slope at any point on an isenthalpic curve is known as the Joule-Thomson coefficient μJT. The maximum point of the curve, at which the coefficient is zero, is called the inversion point for the isenthalpic curve in question. By joining up the inversion points on each isenthalpic curve, an inversion curve is obtained. In the region within the inversion curve where μJT is positive, cooling will occur. In the region outside the inversion curve where μJT is negative, heating will occur.

The upper intersection of the inversion curve with the line of no pressure denotes the maximum inversion temperature for the given gas. Above this temperature the Joule-Thomson effect cannot produce cooling at any pressure.


The Joule-Thomson coefficient is defined:

Since the initial pressure is always greater than the final pressure in the Joule-Thomson experiment, ∂P is always negative. So a positive slope (μJT >0) on the isenthalpic curve means that ∂T is also negative i.e. cooling will occur. Conversely a negative slope (μJT <0) on the isenthalpic curve means that ∂T is positive i.e. heating will occur. 

Joule-Thomson cooling vs. Adiabatic cooling

The Joule-Thomson coefficient also provides a convenient mathematical means of showing the fundamental difference between Joule-Thomson cooling and adiabatic cooling.

In the Joule-Thomson experiment, the thermodynamic relationship between the independent state variables P and T, and the state function enthalpy (H), is such that we may write the implicit function:

Making use of the triple product rule, one of the useful relationships among first partial derivatives that follows directly is:

The second term on the right has particular significance for a perfect gas, since just as the internal energy is independent of volume at constant temperature, so by parallel reasoning the enthalpy is independent of pressure at constant temperature:

For a perfect gas therefore, μJT is zero. Which means that unlike a real gas, a perfect gas cannot exhibit Joule-Thomson cooling. This is in direct contrast to the familiar adiabatic expansion of a gas doing external work, where a perfect gas would indeed exhibit cooling, just like a real gas.

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Related blog posts (click on title, opens in separate window)

Joule, Thomson, and trouble with the neighbours
The story of how Joule and Thomson came to form their historic partnership, their increasingly ambitious research in Manchester, and the unfortunate circumstance that derailed it.

The Liquefaction of Gases – Part I
The story of how the scientific community came to realise that the term ‘non condensable’ gases was a misnomer, and the role that the Joule-Thomson effect had in enabling the commercial liquefaction of air.

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I am indebted to Charlie Hulme, Kenneth Letherman President of the Manchester Literary and Philosophical Society, Professor Keith Ross of the University of Salford, Professor Nigel Mellors of the University of Salford, and John Beckerson, Senior Curator of the Museum of Science and Industry, Manchester, for their engagement and input during the preparation of this article.

Header photo credits
(left) William Thomson / Smithsonian Libraries (centre) The forcing-pump used by Joule and Thomson in their first experiments of May 1852, now preserved by the Museum of Science and Industry, Manchester / MOSI (right) James Joule / Manchester Libraries

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