Inventing Temperature

by Hasok Chang

  where W is the mechanical work produced in the cycle, H the amount of heat passing through the engine, and T 1and T 2are the absolute temperatures of the hot and cold reservoirs.33 When Thomson revised Carnot's theory, he preserved a very

  31. Joule to Thomson, 6 October 1848, and Thomson to Joule, 27 October 1848, Kelvin Papers, Add. 7342, J61 and J62, University Library, Cambridge.

  32. Although Thomson clearly preserved as much as he could from the old analyses in the formal sense, the following claim made many years later seems to me like unhelpful bravado: "This paper [of 1848] was wholly founded on Carnot's uncorrected theory. … [T]he consequently required corrections … however do not in any way affect the absolute scale for thermometry which forms the subject of the present article." What Thomson did demonstrate was that there was a simple numerical conversion formula linking his two definitions of absolute temperature: T1 = 100(logT2 − log273)/(log373 − log273). Note that T2 = 0 puts T1 at negative infinity, and T1 is set at 0 when T2 is 273. See the retrospective note attached to Thomson [1848] 1882, 106.

  33. I have deduced this relation from formula (7) given in Thomson [1849] 1882, 134, §31, by assuming that μ is a constant, which is a consequence of Thomson's first definition of absolute temperature.

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  similar factor, still denoted by μ and called "Carnot's function."34 This was similar to the old Carnot coefficient, but there were two important differences. First, because heat was no longer a conserved quantity, H in equation (1) became meaningless; therefore Thomson substituted for it the heat input, namely the amount of heat absorbed in the first stroke (isothermal expansion). Second, this time μ was a function of temperature defined for an infinitesimal Carnot cycle, with an infinitesimal difference between the temperatures of the two heat reservoirs. With those adjustments, Thomson defined μ through the following work-heat relation parallel to equation (1):

  where dT is the infinitesimal temperature difference and Q is the heat input.

  Thomson's second step, simple yet crucial, was to liberalize the theoretical concept of temperature. Carnot's function was related to engine efficiency, to which Thomson still wanted to tie the temperature concept. But he now realized that nothing theoretical actually dictated the exact relation that Carnot's function should bear to temperature; his initial notion of 1848, defining temperature so that μ became a constant, was too restrictive for no compelling reason. Much more freely, Thomson wrote in an article of 1854 co-authored with Joule: "Carnot's function (derivable from the properties of any substance whatever, but the same for all bodies at the same temperature), or any arbitrary function of Carnot's function, may be defined as temperature."35

  The third step, made possible by the second one, was to find a function of μ that matched existing practical temperature scales reasonably well. In a long footnote attached to an article of 1854 on thermo-electric currents, Thomson admitted a shortcoming of his first definition of absolute temperature, namely that the comparison with air-thermometer temperature showed "very wide discrepancies, even inconveniently wide between the fixed points of agreement" (as shown in table 4.1 in the next section). The most important clue in improving that shortcoming came from Joule, quite unsolicited: "A more convenient assumption has since been pointed to by Mr Joule's conjecture, that Carnot's function is equal to the mechanical equivalent of the thermal unit divided by the temperature by the air thermometer from its zero of expansion" (Thomson [1854] 1882, 233, footnote). What Thomson called "the temperature by the air thermometer from its zero of expansion" is Amontons temperature, introduced at the end of the last section. What he called "Mr. Joule's conjecture" can be expressed as follows:

  where t c

  34. The following derivation of the form of Carnot's function is taken from Thomson [1851a] 1882, 187-188, §21. Here and elsewhere, I have changed Thomson's notation slightly for improved clarity in relation with other formulae.

  35. Joule and Thomson [1854] 1882, 393; emphasis added. It is stated there that this liberal conception was already expressed in Thomson's 1848 article, but that is not quite correct.

  end p.184

  is temperature on the centigrade scale, t ais Amontons temperature, and J is the constant giving the mechanical equivalent of heat.36 Thomson called this proposition Joule's "conjecture" because he had serious doubts about its rigorous truth. In fact, Thomson considered it a chief objective of the "Joule-Thomson experiment" to determine the value of μ empirically, which in itself would imply that he considered any empirical proposition regarding the value of μ as an as-yet unverified hypothesis. However, he thought Joule's conjecture was probably approximately true, and therefore capable of serving as a point of departure in finding a concept of absolute temperature closely aligned with practical temperature scales. Thus, Thomson used Joule's unverified conjecture as a heuristic device "pointing to" a new theoretical definition of temperature. In a joint article with Joule, Thomson wrote: Carnot's function varies very nearly in the inverse ratio of what has been called "temperature from the zero of the air-thermometer" [Amontons temperature] … and we may define temperature simply as the reciprocal of Carnot's function. (Joule and Thomson [1854] 1882, 393-394; emphasis added)

  This new idea can be expressed mathematically as follows:

  where T denotes the theoretical absolute temperature (compare with equation (3), which has the same form but involves Amontons temperature (t a ), which is defined operationally by the air thermometer).

  After giving this definition, the Joule-Thomson essay added another formulation, which would prove to be much more usable and fruitful: If any substance whatever, subjected to a perfectly reversible cycle of operations, takes in heat only in a locality kept at a uniform temperature, and emits heat only in another locality kept at a uniform temperature, the temperatures of these localities are proportional to the quantities of heat taken in or emitted at them in a complete cycle of operations.37

  In mathematical form, we may write this as follows:

  36. For this expression, see Thomson [1851a] 1882, 199; here Thomson cited Joule's letter to him of 9 December 1848. I have taken the value 273.7 from Joule and Thomson [1854] 1882, 394.

  37. Joule and Thomson [1854] 1882, 394. Essentially the same definition was also attached to Thomson's article on thermo-electricity published in the same year: "Definition of temperature and general thermometric assumption.—If two bodies be put in contact, and neither gives heat to the other, their temperatures are said to be the same; but if one gives heat to the other, its temperature is said to be higher. The temperatures of two bodies are proportional to the quantities of heat respectively taken in and given out in localities at one temperature and at the other, respectively, by a material system subjected to a complete cycle of perfectly reversible thermodynamic operations, and not allowed to part with or take in heat at any other temperature: or, the absolute values of two temperatures are to one another in the proportion of the heat taken in to the heat rejected in a perfect thermo-dynamic engine working with a source and refrigerator at the higher and lower of the temperatures respectively" (Thomson [1854] 1882, 235).

  end p.185

  where the T's indicate the absolute temperatures of the isothermal processes (strokes 1 and 3 of the Carnot cycle) and the Q's indicate the amounts of heat absorbed or emitted in the respective processes.

  How is this alternate formulation justified? The Joule-Thomson article itself is not very clear on that point, but it is possible to show, as follows, that definition (4) follows as a consequence of definition (5), which means that we can take (5) as the primary definition. Take a Carnot cycle operating between absolute temperatures T and T′ (where T > T′), in which the engine absorbs heat Q in the first stroke and releases Q′ in the third stroke (Q > Q′). Energy conservation dictates that the net mechanical work produced in that cycle is J(Q − Q′), where J is the constant giving the mechanical equivalent of heat, and (Q − Q′) gives the amount of heat destroyed (converted in
to mechanical work). Now, using the definition of absolute temperature given in (5), we can express the work as follows:

  If we consider a cycle in which the temperature difference is infinitesimal, we may write equation (6) as follows:

  Now recall the definition of Carnot's function given in equation (2), W = QμdT. Equating that with (7) gives μ = J/T, which is the definition expressed in equation (4), so we have the desired result.

  Definition (5) marked a point of closure in Thomson's theoretical work on thermometry, although he would return to the subject many years later. In subsequent discussions I will refer to definitions (4) and (5) together as Thomson's "second absolute temperature." This closes my discussion of the theoretical development of the temperature concept. But how was this abstract concept to be made susceptible to measurement? That is the subject of the next two sections.

  Semi-Concrete Models of the Carnot Cycle

  In a way, creating a theoretical definition of temperature was the easy part of Thomson's task. Anyone can make up a theoretical definition, but the definition will not be useful for empirical science, unless it can be connected to the realm of physical operations. Linking up an abstract theoretical concept with concrete physical operations is a challenging task in general, as I will discuss more carefully in the analysis part of this chapter, but it was made starkly difficult in Thomson's case, since he had deliberately fashioned the absolute temperature concept to make sure that any connections whatsoever to any particular objects or materials were severed. How was Thomson going to turn around later and say, excuse me, but now I would like to have those connections back? The operationalization of Thomson's absolute temperature was a problem that remained nontrivial well into the twentieth century. Herbert Arthur Klein reports: The situation is well summarized by R. D. Huntoon, former director of the Institute for Basic Standards of the U.S. National Bureau of Standards. Surveying the status

  end p.186

  of standards for physical measurement in the mid-1960s, he noted that the actual relationships between the universally used IPTS [International Practical Temperature Scale] scales and the corresponding thermodynamic scale are "not precisely known." (Klein 1988, 333)

  In order to have a true appreciation of the problem that Thomson faced, we need to pause for a moment to reflect on the nature of Carnot's theory. As noted earlier, Carnot wanted a completely general theory of heat engines, which meant that the "working substance" in his theoretical engine was conceived as an abstract body possessing only the properties of pressure, volume, temperature, and heat content. How was Carnot able to deduce anything useful at all about the behavior of a substance of such skeletal description? He made use of some general assumptions, such as the conservation of heat and some propositions found in the latest physics of gases, but they were still not sufficient to allow the deduction of anything definite about engine efficiency. Quite plainly, in actual situations the efficiency will depend on the particular design of the engine and the nature of the particular working substance.

  Carnot made headway by treating only a very restricted class of heat engines, though he still avoided invoking particular properties of the working substance in the general theory. The following were the most important restrictions. (1) Carnot only treated engines that worked through a cyclical process, in which the working substance returned, at the end of a cycle, exactly to the state in which it started. (2) The Carnot engine was not merely cyclical, but cyclical in a very specific way, with definite strokes constituting the cycle. (3) Finally, and most crucially, the Carnot cycle was also perfectly reversible; reversibility implied not only the absence of friction and other forms of dissipation of heat and work within the engine but also no transfer of heat across any temperature differences, as I will discuss further shortly. Those restrictions allowed Carnot to prove some important results about his abstract heat engine.

  Still, Thomson faced a great difficulty as long as the object of Carnot's theory remained so removed from any actually constructable heat engine. Let us consider how Thomson endeavored to deal with that difficulty, first going back to his original definition of absolute temperature in 1848. The conceptually straightforward scheme for measuring Thomson's first absolute temperature would have been the following: take an object whose temperature we would like to measure; use it as a Carnot heat reservoir and run a Carnot engine between that and another reservoir whose temperature is previously known; and measure the amount of mechanical work that is produced, which gives the difference between the two temperatures. The difficulty of realizing that procedure can only be imagined, because there is no record of anyone ever who was crazy enough to attempt it. In order to meet the standard of precision in thermometry established by Regnault, the instrument used would have needed to be frighteningly close to the theoretical Carnot engine. That route to the operationalization of absolute temperature was a nonstarter.

  So Thomson took a conceptual detour. Instead of attempting to measure temperature directly with a thermometer constructed out of an actual, fully concrete Carnot engine, he theorized about versions of the Carnot engine that were concrete

  end p.187

  enough to allow the use of certain empirical data in the description of its workings. The key to making a reliable connection between the abstract definition of absolute temperature and actual empirical data was to use a model that was sufficiently concrete, but still ideal in the sense of satisfying Carnot's propositions about engine efficiency. Thomson worked out two such quasi-concrete models, following some moves made by Carnot himself: a system made of water and steam, and a system with only air in it. Here I will only give the details of Thomson's water-steam system.38 The important advantage of this system is that the pressure of "saturated" steam is a function of temperature only (see "The Understanding of Boiling" in chapter 1), which simplifies the reasoning a great deal as we shall see later. This quasi-concrete model allowed Thomson to compute the heat-work relation from empirical data. As we shall see, the relevant empirical data were certain parameters measured as functions of temperature measured by an air thermometer. Putting such data into the definition of absolute temperature yielded a relation between absolute temperature and air-thermometer temperature, with which he could convert air-thermometer temperature into absolute temperature. Let us now see how this calculation was made.

  Thomson was still working with the original version of the Carnot theory before energy conservation when he was trying to measure his first absolute temperature. To recap the relevant parts of that theory briefly: a Carnot engine produces a certain amount of work, W, when a certain amount of heat, H, is passed through it; we need to evaluate W, which is visually represented by the area enclosed by the four-sided figure AA 1 A 2 A 3in figure 4.8, representing the amount of work done by the steam-water mixture in strokes 1 and 2, minus the amount of work done to the steam-water mixture in strokes 3 and 4. Thomson estimated the area in question actually by performing the integration along the pressure axis, as follows:

  where p 1and p 2are the pressures in strokes 3 and 1, which are constant because the temperature is constant in each stroke; ξ is the length of the line spanning the curved sides of the figure (AA 3and A 1 A 2 ).

  Now, what does ξ represent physically? That is the crucial question. Thomson answered it as follows: We see that ξ is the difference of the volumes below the piston at corresponding instants of the second and fourth operations, or instants at which the saturated steam and the water in the cylinder have the same pressure p, and, consequently, the same temperature which we may denote by t. Again, throughout the second operation [curve A 1 A 2in the figure] the entire contents of the cylinder possess a greater amount of heat by H units than during the fourth [curve A 3 A]; and, therefore, at any instant of the second operation there is as much more steam as

  38. For the treatment of the air engine, see Thomson [1849] 1882, 127-133.

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  Figure 4.8. The indicator-diagram representation of the workin
g of the ideal steam-water cycle, adapted from Thomson [1849] 1882, 124.

  contains H units of latent heat, than at the corresponding instants of the fourth operation. (Thomson [1849] 1882, 125-126)

  The crucial assumption here is that there is a strict correlation between the temperature and pressure of saturated steam; by the time Thomson was writing, this was generally accepted as an empirical law (see "The Understanding of Boiling" in chapter 1). Now we must ask how much increase of volume results from the production of the amount of steam embodying latent heat H. That volume increment is given as follows:

  where k denotes the latent heat per unit volume of steam at a given temperature, and σ is the ratio of the density of steam to the density of water. The formula makes sense as follows: the input of heat H produces H/k liters of steam, for which σH/k liters of water needs to be vaporized; the net increase of volume is given by subtracting that original water volume from the volume of steam produced.

  Substituting that expression into equation (8), we have: