by Hasok Chang
The actual operationalization of absolute temperature was effected by a slightly different strategy, in which the concrete image used was not the steam-water system, but the gas-only system. As explained in "Using Gas Thermometers to Approximate Absolute Temperature," the new thermodynamic theory yielded the
56. Thomson 1880, 46, §44. See also Thomson [1879-80a] 1911 and Thomson [1879-80b] 1911.
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proposition that an ideal gas would expand uniformly with increasing absolute temperature. By taking the expansion of the ideal gas as the theoretical framework for the definition of absolute temperature, Thomson freed himself from the need to look for a plausible concrete image of the Carnot cycle. The expansion of an ideal gas had a straightforward concrete image (an actual body of gas expanding against some external pressure), and it was certainly possible to perform actual operations matching that image: heat a gas in a cylinder with a piston pressing down on it; the temperature is indicated by the product of the pressure and the volume of the gas. (We have seen in "The Achievement of Observability, by Stages" in chapter 2 that high-quality gas thermometers were much more complicated, but we can ignore such complications for the moment.)
As there were no insurmountable obstacles in making and using gas thermometers, it was easy to proceed with the second step of this operationalization. The interesting problems in this case emerged in the process of examining the exact match between the image and the actual operations. Regnault had demonstrated very clearly that thermometers made with different gases disagreed with each other except at the fixed points. Therefore it was impossible to get an exact match between the image of the ideal gas thermometer and the collection of all actual gas thermometers, because the image stipulated only one temperature value in each situation, and the collection of the gas thermometers returned multiple values. There are various options one could take in order to get an exact match. (1) One option, which I have never seen advocated, is to modify the image by saying that temperature can have multiple values in a given situation, to negate what I have called the principle of single value in "Comparability and the Ontological Principle of Single Value" in chapter 2. (2) Another is to declare that one of the actual thermometers is the correct measurer of absolute temperature, and the rest must be corrected to conform to it. There is no logical problem in making such a conventionalist declaration, but it would tie the absolute temperature concept to the properties of one particular substance, for no convincing reason. As I have already emphasized, that was something that Thomson explicitly set out to avoid. (3) That leaves us with the option that Thomson (and Joule) actually took, which is to say that most likely none of the actual gas thermometers are exactly accurate indicators of absolute temperature, and they all have to be corrected so that they agree with each other. This scheme for operationalizing absolute temperature is summarized graphically in figure 4.16.
The challenge in that last option was to come up with some unified rationale according to which all the thermometers could be corrected so that they gave converging values of absolute temperature. The reasoning behind the Joule-Thomson experiment provided such a rationale, as explained in "Using Gas Thermometers to Approximate Absolute Temperature." It makes sense within energy-based thermodynamics that a gas which expands regularly with increasing temperature would also remain at the same temperature when it expands without doing work. Both types of behavior indicate that the internal energy of such a gas does not change merely by virtue of a change in its volume, and both can be considered different manifestations of Mayer's hypothesis (see Hutchison 1976a, 279-280). When thermodynamic theory is supplemented by the kinetic theory of gases, this condition can be understood
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Figure 4.16. Joule and Thomson's operationalization of absolute temperature, by means of the gas-only system.
as the stipulated absence of intermolecular forces in an ideal gas. But in actual gases one would expect some degree of intermolecular interactions, therefore some irregularity in thermal expansion as well as some cooling (or heating) in Joule-Thomson expansion. As all actual gas thermometers could be corrected on this same basis, the temperature concept itself would maintain its generality (unlike in option 2). If these coordinated corrections did result in bringing the measured values closer together, then such convergence would be a step toward perfecting the match between the actual gas thermometers and the image of the ideal gas thermometer. The corrections carried out by Joule and Thomson, and by later investigators, seem to have produced just such a move toward convergence. (However, it would be worthwhile to compile the data more carefully in order to show the exact degree and shape of the convergence achieved.)
Accuracy through Iteration
One major puzzle still remains. I began my discussion in the last section by saying that we should avoid thinking about operationalization in terms of correctness, but ended
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by discussing corrections made to actual measuring instruments. If we are not concerned with correctness, what can the "corrections" possibly be about? The logic of this situation needs a more careful examination. The point of rejecting the question of correctness was to acknowledge that there are no pre-existing values of abstract concepts that we can try to approach; as mentioned earlier, the "real" values only come into being as a result of a successful operationalization. But I need to clarify and articulate the sense of validity that is hiding behind the word "successful" there.
Thomson himself did not quite give a clear explanation of the epistemic character of his maneuvers even in his 1880 article, as discussed in "Dealing with Abstractions." Fortunately, a much more satisfying understanding of the problem of operationalizing absolute temperature was to emerge within a decade, apparently starting with the work of Hugh Longbourne Callendar (1863-1930), English physicist and engineer who made important observations on the properties of steam and crucial contributions to electric-resistance thermometry. My discussion of Callendar's work will rely on the exposition given by Henri Louis Le Chatelier (who was introduced in the discussion of pyrometry in chapter 3), which is much more helpful than Callendar's own presentation.57 The Callendar-Le Chatelier operationalization of absolute temperature can be understood as an instance of the process of epistemic iteration, first introduced in "The Iterative Improvement of Standards" in chapter 1.
The starting point of epistemic iteration is the affirmation of a certain system of knowledge, which does not have an ultimate justification and may need to be changed later for various reasons. The initial assumption for Callendar was that air-thermometer temperature and absolute temperature values were very close to each other. We start by writing the law governing the thermal behavior of actual gases as follows:
where R is a constant, T is absolute temperature, and φ is an as-yet unknown function of T and p. The factor φ is what makes equation (20) different from the ideal gas law, and it is a different function for each type of gas; it is presumed to be small in magnitude, which amounts to an assumption that actual gases roughly obey the ideal gas law. Such an assumption is not testable (or even fully meaningful) at that stage, since T is not operationalized yet; however, it may be vindicated if the correction process is in the end successful or discarded as implausible if the correction process cannot be made to work.
The next step is to estimate φ, which is done by means of the results of the Joule-Thomson experiment, following Thomson's method. Le Chatelier gives the following empirical result, calculated from the data obtained in experiments with atmospheric air:
57. Callendar 1887, 179, gives the starting point of his analysis and the final results, but does not give the details of the reasoning; my discussion follows Le Chatelier and Boudouard 1901, 23-26.
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where p 0is the standard atmospheric pressure and T 0is the absolute temperature of melting ice. I have presented the derivation of this result in some detail elsewhere, but one important point can be gathered from merely inspecting the fin
al outcome. Equation (21) is supposed to be an empirical result, but it expresses φ as a function of absolute temperature T, not as a function of t a , (Amontons) temperature measured by an ordinary thermometer in the Joule-Thomson experiment. What happens in this derivation is a deliberate conflation of absolute temperature and air temperature (or mercury temperature), as Callendar and Le Chatelier take the empirical Joule-Thomson formula expressed in t aand simply substitutes it into theoretical formulas expressed in T, letting t astand in for T. This is allowed, as an approximation, on the assumption that T and t aare roughly equal because φ is very small.58
Unlike Thomson, Le Chatelier was very clear that equation (21) did not give the final correction (Le Chatelier and Boudouard 1901, 25): "This is still an approximate result, for we have depended upon the experiments of Joule and Thomson and on the law of adiabatic expansion." Here Le Chatelier was also acknowledging the fact that in the derivation of (21) he had helped himself to the adiabatic gas law, knowing that it was not known to be exactly true but assuming that it was approximately true. A further round of corrections could be made with the help of the correction indicated in (21). This would involve recalibrating the air thermometer, according to the law of expansion that is obtained by inserting (21) into (20); recall that the air thermometer was initially calibrated on the basis of the assumption that the expansion of air was exactly regular (φ = 0). With the recalibration of the air thermometer, one could either do the Joule-Thomson experiments again or just reanalyze the old data. Either way, the refined version of the Joule-Thomson experiment would yield a more refined estimate of φ, giving an updated version of (21). This process could be repeated as often as desired. A similar assessment of the situation was given twenty years later by A. L. Day and R. B. Sosman, with the most succinct conceptual clarity on the matter that I have seen: It is important at this point to recall that our initial measurements with the gas-thermometer tell us nothing about whether the gas in question obeys the law pv = kθ or not. Only measurements of the energy-relations of the gas can give us that information. But since such measurements involve the measurement of temperature, it is evident that the realisation of the temperature scale is logically a process of successive approximations. (Day and Sosman 1922, 837; emphasis added)
However, it seems that in practice no one was worried enough to enter into second-round corrections or beyond. Callendar calculated the first-round corrections on air up to 1000°C; although the corrections got larger with increasing temperature, they turned out to be only 0.62° at 1000°C for the constant-volume air thermometer, and 1.19° for the constant-pressure air thermometer. It was seen that the corrections would grow rapidly beyond that point, but that was not so
58. See Chang and Yi (forthcoming) for details.
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much of a practical concern since 1000°C was about the very limit at which any gas thermometers could be made to function at all in any case.59 Le Chatelier was happy to declare: The deviations of the air-thermometer at high temperatures are thus very slight if concordance is established at 0° and 100°; we shall not have to occupy ourselves further with the differences between the indications of the thermodynamic thermometer and those of the gas-thermometer. (Le Chatelier and Boudouard 1901, 26)
One only needed to avoid gases like carbon dioxide, for which the corrections were significantly larger. Day and Sosman gave a similar view (1922, 837): "Practically, the first approximation is sufficient, so nearly do the gases commonly used in gas-thermometers conform to the 'ideal' behaviour expressed in the law pv = kθ."
This is a pleasing result, but we must also keep in mind that the smallness of the first-round correction is hardly the end of the story, for two reasons. First of all, for each gas we would need to see whether the corrections actually continue to get smaller in such a way as to result in convergence. In mathematics conditions of iterative convergence can be discerned easily enough because the true function we are trying to approximate is already known or at least knowable. In epistemic iteration the true function is not known, and in cases like the present one, not even defined. So the only thing we can do is to carry on with the iteration until we are pragmatically satisfied that a convergence seems destined to happen. In the case of absolute temperature one iterative correction seemed to be enough for all practical purposes, for several gases usable in thermometers. However, to the best of my knowledge no one gave a conclusive demonstration that there would be convergence if further rounds of corrections were to be carried out.
The second point of caution emerges when we recall the discussion at the end of the last section. If we are to respect Thomson's original aim of taking the definition of temperature away from particular substances, it is not good enough to have convergence in the corrections of one type of gas thermometer. Various gas thermometers need to converge not only each in itself but all of them with each other. Only then could we have a perfect match between the single-valued image of absolute temperature and the operational absolute temperature measured by a collection of gas thermometers. It is perhaps plausible to reject some particular gases as legitimate thermometric fluids if there are particular reasons that should disqualify them, but at least some degree of generality would need to be preserved if we are to remain faithful to the spirit of Thomson's enterprise.
Seeing the "correction" of actual thermometers as an iterative process clarifies some issues that have been left obscure in my analysis so far. The clarification stems from the realization that in an iterative process, point-by-point justification of each and every step is neither possible nor necessary; what matters is that each stage leads on to the next one with some improvement. The point here is not just that
59. See Callendar 1887, 179. According to Day and Sosman (1922, 859), up to that time only four attempts had been made to reach 1000°C with gas thermometers. See also the discussion about the high-temperature limits of the air thermometer in "Ganging Up on Wedgwood" in chapter 3.
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slightly incorrect information fed into an iterative process may well be corrected (as Peirce pointed out in the example shown in "The Iterative Improvement of Standards" in chapter 1). As already noted, the question of correctness does not even apply, unless and until the iterative process produces a successful outcome. Therefore it makes sense to relax the sort of demand for justification that can lead us to seek illusory rigor.
There are several aspects of this relaxation. (1) First of all, one's exact starting point may not be important. In the case of absolute temperature, assuming that the concrete image of the ideal gas law was approximately true of actual gases happened to hit the nail nearly on the head, but the iterative correction process could also have started from other initial approximations and reached similar final results. (2) Some looseness can also be allowed in the process of reasoning adopted beyond the initial starting point. Thomson was able to make certain shortcuts and apparently unwarranted approximations in his various derivations without much of a tangible consequence.60 Similarly, Le Chatelier helped himself to the adiabatic gas law, knowing full well that it was not guaranteed to be exactly right. (3) Empirical data that may not be exactly right can also be used legitimately. Therefore Thomson's defense of the use of Joule's mercury thermometer in the Joule-Thomson experiment was not only invalid (as I have argued in "Dealing with Abstractions") but also unnecessary. A recognition of the nature of the iterative process would have spared Thomson from an illusory problem and a pseudo-solution to it. (4) Just as different starting points may lead toward the same conclusion, different paths of reasoning may do so as well. Thomson himself proposed various methods of operationalizing absolute temperature, though only one was pursued sufficiently so it is difficult to know whether the same outcome would have been reached through his other strategies. But I think it is safe to say that the Joule-Thomson experiment was not the only possible way to obtain the desired results. In fact Joule and Thomson themselves noted in 1862, with evident pleasure, that Rankine h
ad used Regnault's data to obtain a formula for the law of expansion of actual gases that was basically the same as their own result based on the Joule-Thomson experiment (equation (18)).61
One more important issue remains to be clarified. In the process of operationalizing an abstract concept, what exactly do we aim for, and what exactly do we get? The hoped-for outcome is an agreement between the concrete image of the abstract concept and the actual operations that we adopt for an empirical engagement with the concept (including its measurement). That is the correspondence that makes the most sense to consider, not the complacently imagined correspondence between theory and experience, or theory and "reality." With an iterative process we do not expect ever to have an exact agreement between the
60. For a more detailed sense of this aspect of Thomson's work, see Chang and Yi (forthcoming).
61. Joule and Thomson [1862] 1882, 430. They refer to Rankine's article in the Philosophical Transactions, 1854, part II, p. 336. In an earlier installment of their article Joule and Thomson ([1854] 1882, 375-377) had already reproduced a private communication from Rankine on the subject, with commentary.