On Thursday 19 January 1854, a 33-year-old Scottish engineer and physicist named William Rankine read a paper before the Royal Society in London, of which he had just recently been elected a fellow.

Rankine’s paper, entitled *“On the geometrical representation of the expansive action of heat, and the theory of thermodynamic engines”*, came at crucial time in the development of the new science of heat and work. James Joule and William Thomson had begun publishing the results of their important experiments in Manchester (see my previous post for more on this), and Rudolf Clausius in Germany had recently given a mathematical statement of the first law, equating the change in the internal energy of a thermodynamic system to the heat received and the mechanical work done:

ΔU = Q – W.

Rankine was in the right place at the right time to build on the theoretical foundations that Thomson and Clausius were already fashioning, but he did no such thing. He was not one for following anyone else’s lead. Like Kipling’s Cat, He Walked by Himself. He had his own way of looking at things; he did things his way.

The engineer from Edinburgh certainly had the necessary attributes for doing so. He had plenty of practical experience as a railway and waterway engineer, he had a thorough grounding in higher mathematics, dynamics and physics, and he was the possessor of a remarkable scientific imagination – a characteristic that was to prove not altogether advantageous, as we shall see.

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Indications of Rankine’s different way of looking at things were evident right from the start of his paper. Apart from using the pressure volume diagram as his geometrical framework, there is little if anything in common with the ‘Carnot Cycle’ analytical approach previously taken by Clapeyron and Clausius.

For a start, Rankine dispenses completely with isothermal curves in Fig.3, where OX is the line of no pressure and OY is the line of no volume. His ‘cycle’ consists of an upper and lower curve of no transmission of heat [i.e. adiabatic curves] which are infinitely extended, a line of constant volume, and a curvilinear line ACB representing an arbitrary succession of volumes and pressures through which the working substance is supposed to pass in changing from state A to state B.

He draws ordinates AV_{A} and BV_{B} from points A and B down to the line of no pressure, and calls the area V_{A}ACBV_{B} the ‘expansive power’ developed during the operation ACB. He shows by an astute piece of reasoning that the indefinitely-prolonged area MACBN is exactly equal to the ‘heat received’ (H_{A,B}) by the working substance during the operation ACB. He then advances the theorem that the difference between the ‘heat received’ and the ‘expansive power':

depends simply on the initial and final points A and B, and not on the form of the curve ACB. The above expression is regarded by Rankine as the energy *stored up* (his italics) in the working substance during the operation ACB. He identifies this stored energy as the sum of two quantities – the increase of what he calls the ‘actual energy of heat’ (Q) of the substance in passing from state A to state B:

ΔQ = Q_{B} – Q_{A}

and the change in what he calls the ‘potential energy of molecular action’ (S) [not to be confused with the modern symbol for entropy] in passing from state A to state B:

ΔS = S_{B} – S_{A}

Thus:

This is a form of what he calls the general equation of the expansive action of heat, which was the object of his geometrical reasoning. Rankine’s approach is thus wholly different to that of Clausius, who gained his insights into path-independent thermodynamic functions by considering the operations of a complete cycle. Rankine instead arrives at a path-independent result by considering an arbitrary curve on a diagram of energy.

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Rankine’s ‘potential energy of molecular action’ needed to be expressed in terms of measurable quantities in order to give a definite form to equation 2, and in the next section of the paper he does this, at the same time introducing the symbol Ψ to denote the sum of the actual energy of heat and the potential energy of molecular action present in the working substance in any state:

Rankine assigns no name to Ψ, and although it is clear from the theorem already advanced that a change in this quantity between initial and final states is independent of path, he draws no attention to the fact. Perhaps Rankine simply didn’t recognise at the time that he had defined the thermodynamic state function known as internal energy, albeit on his own terms.

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As he continued with the reading of his paper before the Royal Society on that January day in 1854, neither Rankine nor his audience were aware that a defining moment in the history of thermodynamics was shortly to take place.

It began at the third corollary to the general equation of the expansive action of heat, where Rankine observes that for two adiabatic curves infinitely close together, “the ratio of the heat consumed in passing from one of those curves to the other, to the actual heat present, will be the same”. He expresses this ratio as:

and proceeds to show that F, which he simply labels “a thermodynamic function”, has a constant value for any adiabatic curve. This was the second thermodynamic function that Rankine had introduced in as many pages, although again he seems not to have attached particular significance to this.

In the fourth section of his paper, having introduced the relation

between the ‘actual energy of heat’ of a substance, its specific heat, and the absolute temperature, Rankine converts F into a more convenient thermodynamic function Φ by defining:

The fortunate effect of this conversion is to remove the functional relationship with the dubiously-defined Q and give Φ the dimensions of energy (as heat received) per degree of absolute temperature. To students of classical thermodynamics, those units will no doubt sound familiar.

In a theorem under Proposition XII, Rankine then introduces both Ψ and Φ into his general equation of the expansive action of heat, with the following result:

I have no way of knowing what reaction this equation elicited from those assembled at the Royal Society when Rankine wrote it on the blackboard, or recited it, or whatever. What I do know is that when I first saw it in a volume of Rankine’s miscellaneous papers held at the University of Wisconsin and publicly available on archive.org, I nearly fell off my chair.

Exchanging Rankine’s symbols for their modern equivalents, the equation reads:

This is the fundamental relation of thermodynamics.

Eleven years before Clausius, Rankine gave mathematical expression to the thermodynamic state function we now call entropy, from a consideration of infinitely close adiabatic curves (incidentally it was Rankine who introduced the term ‘adiabatic’). And by putting both internal energy (Ψ) and entropy (Φ) together in equation 40A, Rankine made the first mathematical statement that combined both the first and second laws of thermodynamics.

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Although Φ appeared in print more than a decade before Clausius arrived at another symbolic expression for the same thing and called it entropy, Rankine’s thermodynamic function gained little from its antecedence. The reason derives principally from the fact that Rankine was a model-based thinker with a fanciful imagination. He formed intricate mental pictures of molecular motion to explain the phenomenon of heat, and when the math he applied appeared to produce sensible results, he assumed that his model – which he called the hypothesis of molecular vortices – was correct.

Up to this point in the article I have deliberately refrained from mentioning this aspect, so as not to obscure the (surprisingly) hypothesis-independent sequence of deduction by which Rankine reaches equation 40A. But there is no escaping the fact that much of the paper read by Rankine before the Royal Society in January 1854 is laden with hypothetical apparatus, accompanied by a lexicon of abstruse terminology such as ‘actual energy of heat’, to which it is difficult to attach any distinct meaning.

It proved altogether too much for Rankine’s contemporaries to swallow. When Clausius presented his conception of entropy in 1865, they found it much more palatable, and Rankine’s Φ was forgotten.

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Rankine may have failed to convince colleagues of his molecular vortices, but in no way did this dampen his academic enthusiasm. In 1855 he wrote the very first formal treatise on thermodynamics for Nichol’s Cyclopedia, and in 1857 he penned an important treatise on shipbuilding as well as his famous Manual of Applied Mechanics. He followed this up with his Manual of Civil Engineering in 1861, and by 1865 had become a consulting engineer. In 1869, another great engineering treatise, Machinery and Millwork, appeared. During his time as a fellow of the Royal Society, Rankine published no fewer than 150 papers on mathematical, thermodynamic and engineering subjects, yet still found time to study botany, learn to play the cello and piano, and develop other creative aspects of his intellect, including the writing of humorous songs. One of his early efforts was “The mathematician in love”, in the following stanzas of which Rankine wittily propounds his theory of love and marriage:

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William Rankine never found the time to test this theory in practice. He died a bachelor on Christmas Eve 1872, at the age of 52, of overwork.

Re figure 3: Could we reconcile Rankine’s explanation with the closed cycle argument if we consider his figure a closed cycle comprising three parts – but the two adiabatic lines would close at infinity?

Concerning figure 3, your approach and Rankine’s actually coincide, in the way I understand it. Rankine reached his conclusion by supposing the vertical line DD’ (proportional to heat emitted) removed in the direction of X until it becomes vanishingly small, which is equivalent to your proposition of adiabatic lines closing at infinity.