Inventing Temperature

by Hasok Chang

  In a typical Watt engine, steam was generated in a boiler and led into a cylinder, pushing a piston and thereby generating mechanical work (see fig. 4.5). That is the easy part to understand. Watt's innovations concerned subtler points.26 The steam, having filled the cylinder, needed to be condensed (back into the liquid state), so that the piston could return to its original position and the whole operation could be repeated. All steam engineers knew the necessity of this condensing phase,27 but Watt was the first one to study the physics of that phase carefully. At that time, the condensation was normally achieved by spraying cold water into the cylinder. Watt found that the exiting engines used an excessive amount of cold water. That not only condensed the steam but also cooled down the cylinder, which resulted in a waste of heat because in the next cycle fresh steam was

  25. See Schofield 1963 on Watt's interaction with the Lunar Society.

  26. There are many expositions of Watt's innovations, but here I rely mostly on Cardwell 1971, 42-52.

  27. In fact, in earlier steam engines due to Savery and Newcomen, it was the contractive phase that performed the useful work, in the form of mechanical pull. Watt's work on the steam engine began when he was asked to fix a Newcomen engine used for lecture demonstrations at the University of Glasgow.

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  Figure 4.5. A diagram illustrating the workings of a Watt engine, adapted from Cardwell 1971, 49, and Sandfort 1964, 31. Hot steam enters the upper part of the main cylinder, pushing down the piston and performing work; then the piston is pulled back up; at the same time the bypass valve is opened to allow the steam to pass to the part of the cylinder below the piston; from there it gets sucked into the condenser, where a jet of cold water cools it and turns it into liquid water.

  needed to heat up the cylinder all over again. Watt reduced the amount of cold water, putting in only as much cold water as required for condensing most of the steam. (He had an experimental knowledge of latent heat, which Black helped him understand theoretically.)

  This innovation certainly cut down on the consumption of fuel; however, Watt found that it also reduced the power of the engine. The problem was that maintaining the cylinder at a relatively high temperature also meant that there still remained steam at a relatively high pressure, which made it difficult to push the piston back in when the contractive operation took place. In order to tackle this problem, Watt invented the "separate condenser," a vessel into which the used

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  steam was evacuated in order to be condensed (see fig. 4.5). As cold water was sprayed into the separate condenser, the steam that had moved into it became condensed and created a relative vacuum, causing more steam to flow into the condenser from the main cylinder. All the while the main cylinder itself remained at a high temperature, but most of the hot steam got sucked away into the condenser. Watt's separate condenser allowed a great increase in efficiency, and it comes down as one of the landmarks in the history of technology.

  However, Watt was not completely satisfied yet. He noticed that the steam rushed into the condenser from the steam vessel and reckoned that there must be useful work wasted in that rushing. This dissatisfaction led him to the "expansive principle" in 1769. Watt realized that steam introduced into the cylinder was capable of doing further work by expanding by its own force, until its pressure became equal to the external atmospheric pressure. Therefore he decided to cut off the supply of steam from the boiler well before the piston was pushed out all the way, in order to extract all the work that the steam was capable of doing. The significance of this innovation cannot be stressed too strongly. As D. S. L. Cardwell puts it (1971, 52): "By his invention of the expansive principle (1769) this meticulous Scotsman foreshadowed the progressive improvement of heat-engines and the postulation by Sadi Carnot of a general theory of motive power of heat. With astonishing insight Watt had laid one of the cornerstones of thermodynamics."

  This, finally, brings us to Sadi Carnot. Carnot's theoretical investigation into the workings of heat engines, too, began with a concern about efficiency. In order to have a theoretically tidy measure, Carnot decided that efficiency should be reckoned in a cycle of operations. In the cycle originally conceived by Carnot, the engine receives a certain amount of caloric from a hot place, performs a certain amount of mechanical work by the agency of the caloric, and then releases the caloric into a cooler place (such as Watt's condenser); at the end of the process the engine returns to its original state. Carnot's metaphor for understanding the heat engine was the waterwheel, which produces mechanical work by harnessing the water falling from a higher place to a lower place; likewise, caloric performed work by falling from a place of higher temperature to a place of lower temperature. Efficiency in such a cycle is measured as the ratio of work performed to the amount of caloric passed through the engine. The task that Carnot set himself was to understand the factors that affect this efficiency.

  The peculiar greatness of Carnot's work was to extract the essence of the functioning of all heat engines in a highly abstract form. The abstractness is also what makes Carnot's theory somewhat bewildering and difficult to grasp for most of those who come across it for the first time. As shown in figure 4.6, Carnot imagined a heat engine with an unspecified "working substance" enclosed in a cylinder throughout the cycle (rather than injected into the cylinder and then evacuated, as in a Watt engine). The working substance receives heat and performs work by expanding; then it has to be compressed and cooled in order to return to its original state if the cycle is to be complete; the latter process of condensation requires some work to be done to the substance, and some heat to be taken away from it at the same time. A steam engine operating in that way would be a cylinder closed in by a piston, containing in it a water-steam mixture in equilibrium, under a certain

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  Figure 4.6. A schematic representation of a possible form of the Carnot engine, adapted from Thomson [1849] 1882, 121.

  amount of pressure; heat is supplied to the system, generating more steam and pushing the piston out; afterwards heat is taken away and the piston pushed back into its original position, forcing some of the steam to condense back to a liquid state; hopefully the compression requires less work than the work generated by the expansion, so that we can have a positive net performance of work.

  To finalize the shape of Carnot's cycle, we need to add one more element, which I can only imagine was inspired by Watt's "expansive principle" as Cardwell suggests, namely the insight that steam can do further work after the supply of heat

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  is cut off. That is a process that modern physicists call "adiabatic expansion," namely expansion without any communication of heat from (or to) an external source. It was well known by Carnot's time that an adiabatic expansion of a gas resulted in the lowering of its temperature, though there were serious disputes about the underlying cause of this phenomenon.28 Now we can see how Carnot's famous cycle is put together in its modern four-stroke form (refer to fig. 4.6):

  First stroke. The working substance, at initial temperature S, receives a quantity of heat (H) from the "heat reservoir" A, also kept at temperature S. The working substance maintains its temperature at S, and expands, pushing the piston out from position EF to position E 1 F 1 , doing a certain amount of work, which we denote by W 1 .

  Second stroke. The heat reservoir is removed, and the working substance expands further on its own (adiabatically), cooling down from temperature S to temperature T. The piston is pushed out further, from E 1 F 1to E 2 F 2 , and further work, W 2 , is done.

  Third stroke. The working substance is now compressed, staying at the lower temperature T and releasing some heat to the heat reservoir B, also kept at temperature T. In this process the piston is pushed in from E 2 F 2to E 3 F 3 , and some work, W 3 , is done to the working substance. (In Carnot's original conception, this stroke is continued until the amount of heat released is same as H, the amount of heat absorbed in the first stroke.)

 
; Fourth stroke. The working substance is compressed further, adiabatically (without contact with the heat reservoir). More work, W 4 , is done to it in the compression, in which the piston moves from E 3 F 3back to its original position EF. The temperature of the working substance goes up from T to S. (It is assumed that the cycle will "close" at the end of the fourth stroke, by returning the working substance exactly to its original state.)

  The efficiency of this cycle of operations is defined as the ratio W/H, where W is the net work performed by the working substance (W 1 + W 2 − W 3 − W 4 ), and H is the amount of heat absorbed in the first stroke (which is also the amount of heat released in the third stroke, in Carnot's original theory). Through theoretical reasoning about such a cycle, Carnot derived the very important result that the engine efficiency only depended on the temperatures of the cold and hot reservoirs (S and T in the description of the four-stroke cycle).

  Finally, it will be helpful to explain the abstract graphic representation of the Carnot cycle that is almost universally encountered in textbooks. That representation originates from the 1834 exposition of Carnot's theory by Émile Clapeyron (1799-1864), an engineer trained at the École Polytechnique. Thomson initially learned Carnot's theory from Clapeyron's article and adopted the latter's graphic presentation. Most likely, Clapeyron's representation was inspired by another practice of Watt's, namely his use of the "indicator diagram." Watt monitored the performance of his steam engines by a mechanical device that automatically plotted the changing values of pressure and volume inside the steam vessel. According to

  28. See, for example, Dalton 1802a, which contains some of the pioneering experimental results and a nice Irvinist explanation.

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  Figure 4.7. The indicator-diagram representation of the Carnot cycle, for the case of an air engine, adapted from Clapeyron [1834] 1837, 350.

  Cardwell (1971, 80-81), the indicator was invented in 1796 by Watt's assistant John Southern, and it was a closely guarded trade secret. It is not clear how Clapeyron would have got access to this secret, but he plotted the Carnot cycle as a closed curve in this pressure-volume representation, which helped a great deal in creating a visual image of what was going on in the abstract engine. One very nice feature of the pressure-volume diagram is that the amount of mechanical work involved in a process is straightforwardly represented as the area under the curve representing the process (the work being the integral ∫pdv). For a closed cycle of operations, the net work produced is represented as the area enclosed in the closed curve. Figure 4.7 is an example of the Carnot cycle plotted this way, representing an engine filled with a simple gas rather than the steam-water mix. The isothermal line representing the first stroke (AA 1 ) is a curve following Boyle's law (pv = constant); the adiabatic line representing the second stroke (A 1 A 2 ) is a curve following the adiabatic gas law (pvγ = constant, where γ is the ratio between the specific heat of the gas at constant pressure and the specific heat at constant volume).

  With that background, we can now make full sense of Thomson's conception of absolute temperature. (For a complete understanding of the rest of the narrative part of this chapter, some familiarity with elementary physics and calculus is required; however, readers without such technical background will still be able to understand the gist of the arguments by following the verbal parts of the exposition. The analysis part refrains from technical discussions.) Initially in 1848, Thomson's basic idea was

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  to define the interval of one degree of temperature as the amount that would result in the production of unit amount of mechanical work in a Carnot engine operating in that temperature interval. More precisely, in Thomson's own words: The characteristic property of the scale which I now propose is, that all degrees have the same value; that is, that a unit of heat descending from a body A at the temperature T° of this scale, to a body B at the temperature (T − 1)°, would give out the same mechanical effect, whatever be the number T. This may justly be termed an absolute scale, since its characteristic is quite independent of the physical properties of any specific substance. (Thomson [1848] 1882, p. *104; emphasis added)

  This definition is what I will refer to as Thomson's "first absolute temperature." It is very important to note that Thomson's sense of "absolute" here has nothing to do with counting temperature from the absolute zero. In fact, as Thomson clarified later, his 1848 temperature did not have a zero point at all. The common notion of "absolute zero" that survives into modern conceptions is in fact much older than Thomson's work. It can be traced back to Guillaume Amontons's idea that an objective scale of temperature could be obtained if the zero point were found by extrapolating the observed pressure-temperature relation of air until the pressure became zero, since zero pressure would indicate a complete absence of heat. I will refer to this notion of temperature as the Amontons air temperature, or simply Amontons temperature. It is quite close to what people commonly mean by "absolute temperature" nowadays if they have not studied carefully what is meant by the notion in thermodynamic theory. As we shall see shortly, Thomson later modified his absolute temperature concept to bring it more into line with Amontons temperature, and from that point on the two different senses of "absolute" (not being related to particular materials, and having an absolute zero) became forever conflated.

  Thomson's Second Absolute Temperature

  Interestingly, almost as soon as Thomson advanced his initial concept of absolute temperature, he began to abandon the entire theoretical framework in which that concept was couched. This was in large part a consequence of his encounter with James Prescott Joule (1818-1889), the gentleman scientist from a Manchester family of brewers, who is credited with a crucial role in establishing the principle of conservation of energy.29 (The course of Joule and Thomson's collaboration is well known to historians of science, so I will not repeat many details here.30 ) When Thomson heard Joule present his idea about the interconvertibility of heat and mechanical work at the 1847 meeting of the British Association for the Advancement of Science in Oxford, he was interested but skeptical. After reading Thomson's 1848 article on absolute temperature, Joule wrote urging him to reformulate his idea on the basis of the interconvertibility of heat and work, rather than holding on to Carnot's assumption that heat passed through the heat engine intact: "I dare say

  29. For further details about Joule's life and work, see Cardwell 1989 and Smith 1998.

  30. See, for example, Cardwell 1989, chs. 5 and 8.

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  they [your ideas] will lose none of their interest or value even if Carnot's theory be ultimately found incorrect." Thomson sent a congenial reply, but he was not quite ready to renounce Carnot's theory, which he proceeded to elaborate in an article published in 1849.31

  By early 1851, however, Thomson had committed himself to a serious modification of Carnot's theory in light of Joule's ideas about the interconversion of heat and work. The few years after Thomson's conversion to interconversion were both highly unsettled and highly productive. The entire basis on which he had defined absolute temperature in 1848 had to be changed, because the understanding of the Carnot engine had to be revised fundamentally if heat was no longer considered to be a conserved quantity, and the generation of mechanical effect was seen as the conversion of a part of the heat input into work, rather than a by-product of the movement of heat.32

  It would be fascinating to follow all the twists and turns that Thomson took in reshaping his concept of absolute temperature, but in the interest of clarity and accessibility I will only present a streamlined account here. There were three major steps, through which Thomson arrived at a very simple and pleasing definition of temperature (expressed as T 1 /T 2= Q 1 /Q 2 , as I will explain shortly), which was very different from the original definition but related to it.

  The first step was to reformulate Carnot's theory itself so that it was compatible with energy conservation; this was achieved in a series of articles on the "dynamical theory of h
eat," starting in March 1851 (Thomson [1851a, b, c] 1882). The most important part of this reformulation, for the purpose of thermometry, was the concept of "Carnot's function." Recall that the original definition of temperature was based on the amount of mechanical effect produced in a Carnot cycle, for a given amount of heat passing through the engine. A crucial factor in such consideration of engine efficiency was what Thomson called "Carnot's coefficient" or "Carnot's multiplier," the parameter μ in the following relation: