Posts Tagged ‘William Rankine’


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:


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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

– – – –


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


St. Peter’s Parish Church, Bradford

The year is 1845. The location is the vicarage of St. Peter’s Parish Church, Bradford in Northern England. Inside the vicarage two men, one a 55 year-old retired sea captain and the other a 26 year-old brewery manager, have been conducting a series of experiments using a powerful magneto-electric apparatus. The results that they would publish the following year in Philosophical Magazine would prove beyond doubt that the electric motor was not the perpetual motion miracle that many believed, and put an end to the ‘electrical euphoria’ that had swept through Europe and the United States for a decade.

The retired sea captain, now the vicar of the parish and the owner of the massive magneto, was William Scoresby Jnr, a renowned Arctic explorer and an authority on whale fishery and compass navigation.

The young brewery manager, who had already published a number of scientific articles in the Annals of Electricity and who represented the ‘brains’ behind the experiments at the vicarage, was James Joule.

– – –

It was Michael Faraday’s discovery of electromagnetic induction in 1831 that started it all. The magneto was invented that same year, and with the additional invention of the commutator by William Sturgeon and others, the electric motor became a reality in 1832. Inventors in Europe and America , sensing its possibilities, flocked to develop it.

William Sturgeon 1783-1850, inventor of the electromagnet and commutator

William Sturgeon 1783-1850, inventor of the electromagnet and commutator

In 1835, the euphoria began in earnest when Professor Moritz von Jacobi, a German professor of engineering at the Imperial University of Dorpat in Russia, published a paper on the physics of the electric motor in which he pointed out that if frictional forces and the counter-electromotive force (back emf) could be reduced to small proportions by ingenious design, a near-infinite velocity of rotation could be achieved and with it, virtually limitless power.

Moritz von Jacobi 1801-1874, whose 1835 paper triggered electrical euphoria

Moritz von Jacobi 1801-1874, whose 1835 paper triggered electrical euphoria

Jacobi’s paper was translated into English and appeared in 1836 in the Annals of Electricity, a British journal newly founded and edited by William Sturgeon, who besides independently inventing the commutator had also invented the electromagnet (in 1824). Among those who read Jacobi’s paper in the Annals was James Joule, an 18 year old lad from Manchester who had a fascination with magnetism and electricity.

Born into a wealthy family whose business was beer brewing, young James had plenty of free time to follow his favourite pursuit – electrical experiments. In Jacobi’s paper he saw the opportunity to turn his practical skills and ingenuity to good effect by designing and building an electric motor that would realise the promise of unbounded power. He worked for three solid years on this task, and got absolutely nowhere. But it was not a total waste of time, because along the way the teenager from Manchester made a significant discovery.

– – – –

The 1830s were important years for the emerging science of electricity. The magneto, the electric motor and the tangent galvanometer were all invented in this decade. And in 1834, the man who influenced the direction of scientific enquiry in Britain more than any other – Michael Faraday – discovered the laws of electrolysis. Unusually for Faraday, whose genius otherwise expressed itself in conceptual, qualitative relationships between things, his laws of electrolysis were quantitatively exact.

The effect of this on Joule cannot be understated. Although he had no training and little interest in mathematics, the young experimenter from Manchester was a natural hunter for numerical relationships between measurable quantities. With Faraday’s quantitative discoveries to spur him on, and with a home laboratory full of accurate (and expensive) devices for measuring temperature and electric current, Joule was well-equipped for discovery. And it was not long in coming.

In December 1840, at the age of 21, Joule sent to the Royal Society a paper entitled “On the production of heat by Voltaic electricity”. In it, he stated that the heat generated by the passage of an electric current through a wire is proportional to the square of the current and the resistance of the wire. He further claimed that the law held irrespective of the shape, size or form of the circuit or the type of wire used. In other words, he was enunciating a general law. The Royal Society rejected the paper for the prestigious Transactions, but allowed a short abstract of it to appear in the more lowly Proceedings for that month. What we now know as Joule’s first law (ohmic heating) made a modest entry into this world.

– – – –

William Scoresby (1789-1857), owner of the great magneto-electric machine

William Scoresby (1789-1857), owner of the great magneto-electric machine

James Joule first met Captain the Reverend William Scoresby at a scientific meeting in Manchester in 1842. Scoresby was a colourful character by all accounts; having attended Edinburgh University he decided to follow his father’s calling and go to sea, becoming the master of a whaling ship in true pre-mechanised, Moby Dick fashion. Following the death of his wife, Scoresby gave up the sea to take Holy Orders and was now installed as the Vicar of Bradford.

Unlike most whalers of his era, he had the benefit of a scientific education, and his time at sea had given him a strong interest in terrestrial magnetism and its application in compass navigation. It was a common interest in magnetism that brought Scoresby and Joule together, and it was at this meeting that Joule learned of Scoresby’s ‘great magneto-electric machine’ now housed in the vicarage at Bradford. It was a mighty machine with enormous magnets, capable of producing a hefty current. Joule now had an acquaintance who owned the mother of all magnetos.

– – – –

Joule’s discovery of ohmic heating during his futile attempts to build an ingeniously designed electric motor introduced the first seeds of doubt in his mind about the validity of Professor Jacobi’s bold assertion. The unavoidable heating of the circuit during power production clearly implied a waste of zinc in the Daniell cell which Joule had used a source of voltaic electricity. And since zinc cost 60-70 times the same weight of coal, things didn’t look quite so promising, at least from an economic perspective.

From a purely scientific perspective the heating effect also set Joule wondering, like Count Rumford before him: Where does the heat come from? If heat were a substance, as the scientific establishment insisted it was, the heating effect could only be explained by transference, i.e. the magneto acting as a heat pump to move heat from the armature to the outside circuit and being necessarily cooled in the process. This was something that could be ascertained by careful experiment, and Joule appointed himself to the task.

The summer of 1842 was a hectic time for Joule. Besides visiting Scoresby in Bradford to experiment with the massive magneto in the vicarage, he was also slaving away at home in Manchester on those careful experiments which would eventually bring him to prominence as a great man of science. But years of struggle lay ahead of him before he achieved that goal.

Suffice to say that in his 1843 paper “On the calorific effects of magneto-electricity and on the mechanical value of heat”, read before the British Association for the advancement of science, Joule demonstrated that heat was not a substance; it was generated by the chemical action of the battery when the magneto-electric machine was operated as a motor, and by mechanical work when operated as a magneto. And he went on to propound the thesis that not only were work and heat interconvertible, but that a precise numerical relationship existed between them – what he called the mechanical value of heat.

In 1843, no-one listened. The scientific establishment dismissed Joule’s youthful paper – he was only 24 – as the work of a misguided and uneducated amateur.

– – – –

While the British Association was busy ignoring Joule, Joule was busy thinking about zinc, the anode metal that was consumed in the Daniell cell which he had used in his experiments. The idea was forming in Joule’s mind that the work obtained from an electric motor must arise from the conversion of heat due to the chemical reaction by which zinc was consumed in the cell. Since a precise numerical relation existed between work and heat, there must be a limit to the work obtainable from the consumption of a given amount of zinc. In Joule’s mind, Professor Jacobi’s golden vision of the electric motor as a limitless source of power was fading fast.

Joule followed up on his new thinking with a truly inspired idea. He reasoned that since the heat released from the complete combustion of zinc in a calorimeter must represent the maximum heat attainable, he could calculate – by means of the mechanical equivalent – the maximum work attainable from the consumption of a given amount of zinc by any means whatsoever. Using this yardstick, he could rate the performance of any design of electric motor.

Joule wasted no time in doing the calorimetry and determining the maximum work using his best estimate of the mechanical equivalent of heat. He was now ready to test his idea on an actual magneto-electric machine, the biggest one he knew of, which sat in the home of Captain the Reverend William Scoresby.

– – – –

I mentioned at the start of this post that Joule represented the ‘brains’ behind the experiments conducted at the vicarage of St. Peter’s Church Bradford in 1845. Without any disrespect to Scoresby’s undoubted competences as a scientific thinker and writer, the work that was accomplished here and published under their joint names the following year was very much a synthesis of all that Joule had learned from his own efforts. Indeed, Joule notes in the introduction to their paper that he wrote it almost entirely himself.

Joule used three Daniell cells to power Scoresby ‘s machine as an electric motor, and his purpose in conducting the experiment was to determine the quantity of heat (arising from the consumption of zinc) that was converted into useful work. His method was simple, but brilliantly conceived.

Joule first connected the battery to the motor at rest (i.e. with the rotor prevented from turning), and with a carefully calibrated tangent galvanometer measured the standing current (a). He then allowed the rotor to turn, and when the motor was running steadily, he measured the driving current (b). Joule knew from his own discovery that ohmic heating would be generated by the current in the circuit, and from Faraday’s law that the measured current was in direct proportion to the quantity of zinc consumed in the battery. The difference of the currents, a – b, “will therefore represent the quantity of heat converted by the engine into useful mechanical effect” as Joule put it.

It was now a simple matter to compute the mechanical effect of the motor. Joule had already determined by calorimetry that if all the heat from the combustion of a grain of zinc could be converted into work, it would be equal to raising 158 pounds through a distance of 1 foot.
Since (a – b) represented the heat actually available for conversion, and (a) the amount theoretically available, the work obtainable from the motor per grain of zinc was:

W = 158 x (a – b)/a.

– – – –

It was this formula, and the generalisation it implied, that provided the final refutation of Professor Jacobi’s heady suggestion of infinite power and put an end to electrical euphoria. For it showed that the work obtainable from a source of electrical power is finite, and limited by the heat (=energy) in the fuel.

Furthermore, it showed that the maximum amount of work can only be achieved when (b) becomes vanishingly small. But since (b) is the driving current, the power will then also tend to zero, and it would take an infinitely long time to accomplish the work (students of classical thermodynamics will no doubt detect a familiar principle here).

Besides electric motors, Joule and Scoresby also discussed the fuel economy of steam engines and horses in their paper, and came to a firm conclusion on which of them was the most efficient. But for now we must leave them, sipping their brandy by the fireside and contemplating what they had discovered.

For Joule, it would be a further two years before anyone paid any attention at all to the fact that he had discovered an important electrical heating law, laid the foundations for the first law of thermodynamics, revealed the limits imposed on power production by the second law, and saved gullible investors from wasting their life savings on electrical dream machines.

But he was only 26 years old. He had time on his side.

– – – –

8 years later…

William John Macquorn Rankine (1820-1872)

William John Macquorn Rankine (1820-1872)

One day in 1853, a letter arrived for the now famous James Joule. It was from William Rankine, a Scottish engineer and physicist, who was at the time playing a central role in laying the theoretical foundations of thermodynamics.

In the letter, Rankine drew attention to the fact that the efficiency of an electric motor, given by the fraction (a – b)/a in the Joule-Scoresby formula


was exactly analagous to the thermodynamic expression for the efficiency of a heat engine

(q1 – q2)/q1

where q1 is the heat received, q2 is the heat rejected, and the difference between them is the heat converted into useful work. This expression had been deduced from complex theoretical considerations by Rankine, Rudolf Clausius and William Thomson, all skilled mathematical physicists.

It is a testament to Joule’s abilities as an experimental physicist that, without the benefit of his contemporaries’  mathematical training, he nonetheless found his way to the same mathematical truth.

An engraving from the earliest known portrait of James Joule, painted 18 years after the experiments with William Scoresby in Bradford

An engraving from the earliest known portrait of James Joule, painted 18 years after the experiments with William Scoresby in Bradford

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Links to the original scientific papers cited in this post

Professor Jacobi’s paper, published in the Annals of Electricty in 1836:

James Joule’s first published work, a letter to the Annals of Electricity written in 1838 at the age of 19:

An abstract of Joule’s rejected 1840 paper “On the production of heat by Voltaic electricity”, in which he states the law of ohmic heating:

Joule’s 1843 paper “On the calorific effects of magneto-electricity and on the mechanical value of heat”, ignored by science until William Thomson (later Lord Kelvin) suddenly seized on its significance in 1847:

Joule & Scoresby’s joint paper, published in Philosophical Magazine in 1846:

– – – –

photo credits

Moritz von Jacobi / wikipedia
William Sturgeon /
William Scoresby /
William Rankine / Wikipedia

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P Mander October 2013