Posts Tagged ‘William Thomson’

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:

whence

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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

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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.

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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.

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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)

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.

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P Mander May 2014

Photo credit: Scientific American

On Monday 3 December 1877, the French Academy of Sciences received a letter from Louis Cailletet, a 45 year-old physicist from Châtillon-sur-Seine. The letter stated that Cailletet had succeeded in liquefying both carbon monoxide and oxygen.

Liquefaction as such was nothing new to 19th century science, it should be said. The real news value of Cailletet’s announcement was that he had liquefied two gases previously considered ‘non condensable’.

While a number of gases such as chlorine, carbon dioxide, sulfur dioxide, hydrogen sulfide, ethylene and ammonia had been liquefied by the simultaneous application of pressure and cooling, the principal gases comprising air – nitrogen and oxygen – together with carbon monoxide, nitric oxide, hydrogen and helium, had stubbornly refused to liquefy, despite the use of pressures up to 3000 atmospheres. By the mid-1800s, the general opinion was that these gases could not be converted into liquids under any circumstances.

But in 1869, a paper appeared in a British journal which caused the scientific community to rethink its view.

The paper, entitled “On the Continuity of the Gaseous and Liquid States of Matter” and published in the Philosophical Transactions of the Royal Society, was written by 55-year-old Thomas Andrews, the vice-president of Queen’s College Belfast in Northern Ireland.

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

In addition to his administrative role, Thomas Andrews was also professor of chemistry at Queen’s College Belfast. From the start of his long professorial career he took an interest in gases, beginning with a study of ozone conducted jointly with the Scottish mathematical physicist Peter Guthrie Tait. Then in the summer of 1860, Professor Andrews turned his attention to the liquefaction of gases, a subject that the influential Michael Faraday had brought into the scientific spotlight during the 1820s; Faraday had been the first to liquefy chlorine gas in 1823.

Not surprisingly perhaps, Thomas Andrews went for the big prize in his initial experiments, in which he attempted to liquefy the ‘non condensable’ gases. And not surprisingly, he got absolutely nowhere – none of these gases showed any willingness to liquefy. Andrews then refocused his research on the liquefaction of carbon dioxide [called carbonic acid in his day], and in 1863 made the observation that would set him on the path to fame.

He wrote: “On partially liquefying carbonic acid by pressure alone, and gradually raising at the same time the temperature to 88° Fahr. [31.1°C], the surface of demarcation between the liquid and gas became fainter, lost its curvature, and at last disappeared. The space was then occupied by a homogeneous fluid, which exhibited, when the pressure was suddenly diminished or the temperature slightly lowered, a peculiar appearance of moving or flickering striæ [stripes] throughout its entire mass. At temperatures above 88° no apparent liquefaction of carbonic acid, or separation into two distinct forms of matter, could be effected, even when a pressure of 300 or 400 atmospheres was applied.”

Andrews had discovered the existence of a fundamental property of gases, which he called the “critical temperature” – the temperature above which no gas could be liquefied by pressure alone. If all gases had a critical temperature, then all gases could be liquefied if cooled below that temperature. The gases deemed ‘non condensable’ were simply gases whose critical temperatures were lower than the lowest achievable temperature at that time, which was around –110°C. What was needed was a new cooling principle to enable lower temperatures to be reached.

The good news was that the new cooling principle was already known to science. It had been discovered by James Joule and William Thomson (later Lord Kelvin) in a Manchester cellar a decade earlier.

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

In May 1852, James Joule and William Thomson conducted a famous experiment in the basement of Joule’s home in Salford, Manchester, England, in which they pumped pressurised air at a steady rate through a coil of lead pipe which was narrowly constricted at a certain point along its length and open to the atmosphere at its far end.

The apparatus was equipped with thermometers to measure the temperature of the airflow on either side of the constriction, which was insulated to prevent heat exchange with the surroundings.

Joule and Thomson observed a lowering of temperature. The air was cooled as it flowed through the narrowed section of the pipe, from a region of higher pressure to a region of lower pressure.

The discovery of this cooling effect, called the Joule-Thomson effect in their honour, was a landmark moment in the history of physical science and opened the way to cryotechnological applications of great scientific and commercial importance.

The hand pump which formed part of the original apparatus used by Joule and Thomson in 1852 is now in the collection of the Museum of Science and Industry in Manchester.

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Good fortune has smiled on many along the path of scientific discovery, and Thomas Andrews was among the fortunate. In 1869 – the year he published his paper – the Royal Society chose his work as the Bakerian Lecture of that year and thereby brought Andrews and his singular study to prominence. Across the scientific world, important people took notice.

In Scotland, it made a deep impression on James Clerk Maxwell, who was busy writing his textbook Theory of Heat (1871) and devoted several pages to analysing Andrews’ findings.

In the United States, it made a deep impression on Josiah Willard Gibbs, who cited Andrews’ experiments on carbonic acid as supporting evidence for a free energy function in the 1873 paper “A method of geometrical representation of the thermodynamic properties of substances by means of surfaces”.

In the Netherlands it set a physicist thinking. His name was Johannes van der Waals.

And in the rest of Europe, it set off a race among enterprising engineers to liquefy the gases hitherto termed non condensable. As we have seen, that race was won by Louis Cailletet, although in fairness it should be stated that a Swiss physicist called Raoul Pictet also succeeded in liquefying oxygen within days of Cailletet.

The two men used different cooling principles: Pictet opted for enthalpic cooling using liquid SO2 and CO2 while Cailletet employed Joule-Thomson cooling. The advantage of the latter method, as Joule and Thomson had shown during their pioneering experimental work in the 1850s, was that it allowed recirculation of gas cooled by previous passage through the throttle.

Joule and Thomson’s recirculation design from 1853. The red arrow shows the location of the throttle.

This self-intensifying cooling technique was the key to the first large-scale gas liquefaction method developed by William Hampson (1895) and by Carl von Linde (1895), in which the gas was recirculated through a heat exchanger in order to lower the temperature of incoming gas:

In the early days of liquid oxygen production from air, the biggest use by far for the gas was the oxyacetylene torch, invented in France in 1904, which revolutionized metal cutting and welding in the construction of ships, skyscrapers, and other iron and steel structures.

Cylinders of oxygen being loaded on a tractor-trailer truck (1914) owned by the Linde Air Products Company. Courtesy Praxair, Inc.

Another method of commercial liquefaction of air, which employed adiabatic cooling as well as the Joule-Thomson effect, was developed by Georges Claude (1901) in France:

A by-product of the air liquefaction process was neon, which spawned a lucrative new industry in the shape of neon lighting. The first public demonstration of neon lights was at the Paris Motor Show of 1910.

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Related blog posts

Joule, Thomson, and the birth of big science
The story of how Joule and Thomson’s extraordinary collaboration in the 1850s propelled experimental research into the modern era. The second part of this post also explains the thermodynamics of the Joule-Thomson effect.

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.

Thomas Andrews, “On the Continuity of Gaseous and Liquid States of Matter”, Phil. Trans. R. Soc. Lond. 1869;159:545-590
The paper that challenged the prevailing notion of non condensable gases, and opened the way to a new era of cryogenic science. It also led to deeper understanding of the thermodynamics of real gases [to be explored in Part II of this blogpost]

“Liquefaction of gases – Cailletet’s Experiments” Scientific American, Vol. XXXVIII No.8, February 23, 1878
A detailed contemporaneous account of Cailletet’s experimental apparatus, method and results.

Andrea Sella, “Pictet’s liquefier”
Raoul Pictet deservedly had joint priority with Louis Cailletet for the first liquefaction of a ‘non condensable’ gas, namely oxygen. Here is his story, as told by Professor Andrea Sella of University College London. This is one of Professor Sella’s Classic Kit series on the Royal Society of Chemistry website.

T. O’Conor Sloane, “Liquid Air and the Liquefaction of Gases” (1899)
A wonderful period piece made available online at archive.org by the Omania University in Hyderabad, India. The picture reproduction is awful, but for pure historical interest it’s well worth delving into. Sloane’s writing style is a fascination in itself.

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P Mander February 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.

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.

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, johncassidy.org.uk

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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”.

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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.

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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Acknowledgements
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.

(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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Joule, Thomson, and trouble with the neighbours

Posted: September 1, 2012 in thermodynamics
Tags: , , , , , ,

The house in Manchester where neighbour trouble broke out. Photo credit: Charlie Hulme

We all have our problems. And those engaged in the business of scientific discovery are no strangers to it. Budget cuts, lack of resources, experiments that fail, theories that fall apart, and so on. Rarely though does trouble with the neighbours derail one’s research programme.

Yet curiously, this is exactly what happened to the research conducted in the 1850s by James Joule and William Thomson [later Lord Kelvin] into the expansion of gases – pioneering work that led to the discovery of the Joule-Thomson effect before their endeavours were brought to an abrupt end.

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In the late 1840s, Henri Regnault in France published the most accurate data so far attained of the pressure volume relationship of air maintained at constant temperature. Although the data substantially confirmed Boyle’s law, or Mariotte’s law as Regnault would have called it, the results nevertheless revealed a small discrepancy, in that the density of air increased slightly more than expected in relation to pressure.

Hardly a big deal you might think, but to those involved in the newly-forming, cutting-edge science of Thermo-dynamics as it was spelled then, the discrepancy suggested that there should be a similarly small cooling effect when air expands through a throttle into a region of lower pressure. Here was an opportunity to study the behaviour of real gases in motion and explore their differences, however slight, from ideal behaviour. At Glasgow University in Scotland, the maverick Professor William Thomson designed an innovative steady state experimental method, for which he had a collaborator of proven experimental skill in mind – Mr James Joule of Manchester.

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James Joule, the son of a brewery owner, had a passion for experimentation and from an early age had been conducting scientific experiments at home which had convinced him that heat could be created by mechanical means, and that there was a precise quantitative relationship between the two. This was in stark opposition to the tenets of the caloric theory, which at the time was the prevailing belief of the scientific establishment. But Joule was not a member of the scientific establishment, and this freed him from peer pressure and allowed him to pursue his own ideas and increasingly refined experimentation.

There was a downside to Joule’s non-membership of Britain’s scientific club, however. None of its members took any notice of him, or what he had to say. To them he was an uneducated commoner, an amateur, an outsider.

Joule’s collected papers, which can be read online at archive.org, chart his tireless yet fruitless efforts to get either the Royal Society or the British Association for the Advancement of Science to take notice of his discovery. He kept at it, without luck, for four years. And then one day, in 1847, his luck changed. As Joule himself wrote in a subsequent note:

“It was in the year 1843 that I read a paper “On the Calorific Effects of Magneto-Electricity and the Mechanical Value of Heat” to the Chemical Section of the British Association assembled at Cork … the subject did not excite much general attention, so that when I brought it forward again at the meeting [in Oxford] in 1847, the chairman suggested that I should not read my paper but confine myself to a short verbal description of my experiments. This I endeavoured to do, and discussion not being invited, the communication would have passed without comment if a young man had not risen in the section, and by his intelligent observations created lively interest in the new theory. The young man was William Thomson, who had two years previously passed the University of Cambridge with the highest honour, and is now [1855] probably the foremost scientific authority of the age.”

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Foremost scientific authority he may later have been, but the fact is that William Thomson was at the time of the Oxford meeting a staunch supporter of the yet-to-be-discredited caloric theory, and history shows that his conversion to the dynamical theory that James Joule espoused was somewhat slow – some would even say reticent. But by 1852, Thomson was sufficiently convinced that heat and work were interconvertible, in the quantitative ratio that Joule had discovered.

Now, with Regnault’s discoveries waiting to be tested,  the two men were ready to embark on their first collaborative work (Joule having been admitted to the Royal Society in the meantime, thus making him eligible for research grants). Some accounts suggest that their collaboration was conducted by correspondence, with Thomson in Glasgow devising experiments and analysing results while Joule did the experimental work in Manchester. But an account of Joule’s life by Osborne Reynolds (he of the Reynolds number) shows that Thomson made several trips to Manchester and seems to have enjoyed his visits. In one recollection, Thomson writes of Mr and Mrs Joule that “both she and he showed me the greatest kindness during my visits to them in Manchester for our experiments on the thermal effects of fluid in motion”.

The experiments involved forcing a gas under constant pressure through a tube, narrowed at one point along its length to act as a throttle, and open to the atmosphere at the distal end. Thermometers were positioned in the flow to measure the temperature of the gas on entering and exiting the throttle. The first experiments, using air,  were conducted in one of the cellars of the Joule’s family home at 1 Acton Square, Manchester, opposite what is now the campus of the University of Salford.

Although some cooling was detected, the effect was very small and it became quickly clear that the apparatus needed to be scaled up, and the pressures increased. So with a forcing pump furnished by a grant from the Royal Society, the experiments were moved to the family’s brewery in New Bailey Street Manchester. This was later followed by yet another round of upscaling, this time with the installation of a full size steam engine to drive the pressure pump, again financed by the Royal Society. The design of the throttle was refined with the use of a porous plug rather than a single small orifice, and the gases used in the experiments now included hydrogen and what was referred to as carbonic acid, which we now call carbon dioxide.

It must have been a sight to see Joule and Kelvin, who would ultimately have their names commemorated in the units of energy and absolute temperature respectively, up to their elbows in engine oil and slaving away in the dank cellars of a Manchester brewery to keep their hugely pressurised apparatus in a steady state and obtain meaningful measurements. If someone had taken a photograph of them, it would be worth gold now.

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The next few years brought personal tragedy to Joule, with the death of his newborn second son, closely followed by the death of his wife. During this painful time he sold the brewery and moved back to his father’s house with his young son and daughter, and his steam engine. His elder brother Benjamin writes of how Joule threw himself into continued experimental work at this time:

“My brother was very busy with experiments, many of which were decidedly dangerous owing to the pressure he made use of. During this period, for some months he could not find time to take his meals properly – just ran in and out again. The experiments were so delicate that many were carried out in the night, because a [horse-drawn] cab or cart passing along the road disturbed them, though the laboratory was at the back of the stables.”

Then tragedy struck again with the death of Joule’s father. The house was sold, and Joule moved on yet again, this time buying a house in Old Trafford (the district where Manchester United’s stadium is today). He took with him all the Joule-Thomson experimental apparatus, including the steam engine.

And it was at this point that the neighbour trouble appeared.

There was an obscure item in the house purchase deed that prohibited any steam engine being used at the property. This fact was known to the occupant of the neighbouring property, who insisted upon its strict observance despite it being an obsolete clause, and despite protests from other more lenient and understanding local residents.

Joule’s experiments juddered to a sudden halt, and although he reacted by putting his newly acquired home up for sale, it was an empty gesture and the incident seemed to deflate him. His brother Benjamin wrote of Joule suffering  “a great and lasting disappointment”, and noting that “what really affected him was the refusal to be allowed the use of his one-horse power steam engine … My brother was anticipating a series of important experiments in conjunction with W. Thomson, for which a grant had been obtained.”

In his collected papers, Joule himself couches his disappointment in more detached language:

“[William Thomson and I] pursued the discussion of the thermal effects of fluids in motion until the experiments were interrupted by the action of the owners of the adjacent property, who on the strength of an obsolete clause in the deeds of conveyance, threatened legal proceedings, the cost of which I did not feel disposed to incur”.

Joule and Thomson, the latter having lately suffered a debilitating fall,  experimented no more after this setback. The threat of legal action by a perverse individual ended an historically important piece of research.

But the work performed by the dynamic duo, and the series of joint papers they published, established the Joule-Thomson effect and the knowledge that throttled (real) gases can cool or heat on expansion depending on whether they are below or above their inversion point. This knowledge, and the equations that attach to it, are of great practical importance to today’s chemical engineers in the liquefaction of gases, refrigeration and many other fields of application.

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Related blog posts

Joule, Thomson, and the birth of big science
The story of how Joule and Thomson’s extraordinary collaboration in the 1850s propelled experimental research into the modern era. The second part of this post also explains the thermodynamics of the Joule-Thomson effect.

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.