by Hasok Chang
So we are faced with the following puzzle: how were Joule and Thomson able to judge how much actual gases deviated from the ideal, when they only had ordinary thermometers and would not have known how an ideal gas should behave when monitored by ordinary thermometers? Their experiments employed Joule's mercury thermometers for the estimation of three crucial parameters: not only the size of the cooling effect but also the specific heat of the gases (K) and the mechanical equivalent of heat (J), all of which entered into the correction factor in equation (18) in "Using Gas Thermometers to Approximate Absolute Temperature." In that situation, how can we be assured that the correction of thermometers on the basis of the Joule-Thomson measurements was correct? (Note that the situation here is fundamentally different from that of Thomson's scheme for operationalizing his first absolute temperature. In that case, the use of empirical data taken with the air thermometer did not pose a problem because Thomson was seeking an explicit correlation of absolute temperature and air-thermometer temperature.)
Thomson recognized this problem clearly in his entry on "heat" in the ninth edition of the Encyclopaedia Britannica (1880, 49, §55), where he confessed with great insight: "[W]e have no right to measure these [Joule-Thomson] heating and cooling effects on any scale of temperature, as we have not yet formed a thermometric scale." Thomson indicated how the problem could be avoided, in principle: "Now, instead of reckoning on any thermometric scale the cooling effect or the heating effect of passage through the plug, we have to measure the quantity of work (δw) required to annul it." But he admitted that "the experiments as actually made by Joule and Thomson simply gave the cooling effects and heating effects shown by mercury thermometers." The justification that Thomson produced at the end of this remarkable discourse is disappointing: The very thermometers that were used [in the Joule-Thomson experiment] had been used by Joule in his original experiments determining the dynamical equivalent of heat [J], and again in his later experiments by which for the first time the specific heat of air at constant pressure [K] was measured with sufficient accuracy for our present purpose. Hence by putting together different experiments which had actually been made with those thermometers of Joule's, the operation of measuring δw, at all events for the case of air, was virtually completed. Thus according to our present view the mercury thermometers are merely used as a step in aid of the measurement of δw, and their scales may be utterly arbitrary.
What Thomson claims here is that the temperature measurements are merely ways of getting at the value of the quantity δw, and the final result is independent of
end p.204
the particular method by which it is obtained. Thomson's claim is hollow, unless it happens to be the case that the resulting empirical formula for δw is not a function of mercury temperature at all. But δw is a function of mercury temperature (t) in general, according to Joule and Thomson's own results. The empirical formula derived from their experiments is the following:
where is the temperature change in the gas, and A is a constant whose value depends on the nature of the gas (A was determined to be 0.92 air, and 4.64 for carbon dioxide); δw is given by multiplying by K (the specific heat of the gas) and J (the mechanical equivalent of heat).49 I do not see how it can be argued that δw would in general have no dependence on t. The same point can be seen even more clearly if we take the view that certain errors are introduced into the measured values of , K, and J, if those values are obtained on the basis of the assumption that the mercury thermometer readings indicate the absolute temperature. Thomson's claim amounts to insisting a priori that all such errors cancel each other out when the three quantities are worked into a formula to produce the final result. That is possible in particular cases, but by no means guaranteed.
This, however, seems to be where Thomson left the problem. In the corpus of his work after the Britannica article and a couple of related articles published around the same time, I have not found any further contributions to the measurement of absolute temperature. Thomson was apparently quite satisfied with the theoretical understanding of absolute temperature that he had been able to secure in the framework of a fully developed theory of thermodynamics, and in practical terms he was happy with the old Joule-Thomson empirical results that seemed to give a sufficient indication that the deviations of gas-thermometer temperature from his second absolute temperature were quite small.
Let us take stock. I have noted a circularity: the operationalization of temperature seems to require the use of an already operationalized concept of temperature. Thomson and Joule broke this circularity by using an existing operationalization of temperature that was fully admitted to be unjustified—namely, a mercury thermometer, which was by no means guaranteed to indicate absolute temperature. Was it possible to guard against the errors that might have been introduced in using a thermometer that was admittedly open to corrections? Thomson does not seem to have given a very convincing answer to that question. Can we go beyond Thomson here?
Operationalization and Its Validity
The discussion in the last section brought us to a dead end of justification. I believe that the problem arises from applying an inappropriate notion of justification to operationalization, as I will try to show in this section and the next. It is helpful
49. Joule and Thomson [1862] 1882, 428-429; for further exposition on how this result was obtained, see Chang and Yi (forthcoming). In the case of hydrogen, dφ/dp was apparently not a function of temperature; see Thomson 1880, 49.
end p.205
here to start with Nancy Cartwright's (1999, 180) unequivocal statement of the relation between theories and the world:50 "theories in physics do not generally represent what happens in the world; only models represent in this way, and the models that do so are not already part of any theory." As Cartwright explains, this amounts to an important revision of her earlier view that was famously expressed in How the Laws of Physics Lie. I take her current view to mean that the laws of physics are not even capable of lying when it comes to states of affairs that are amenable to actual physical operations, because the theoretical laws cannot say anything about the actual situations—unless and until the concepts occurring in the laws have been operationalized.
To get a clearer view on operationalization, it is helpful to see it as a two-step process, as represented in figure 4.14. A theoretical concept receives its abstract characterization through its various relations with other theoretical concepts; those relations constitute an abstract system. The first step in operationalization is imaging: to find a concrete image of the abstract system that defines the abstract concept. I call this an "image" rather than a "model," in order to avoid any possible suggestion that it might be an actual physical system; the concrete image is still an imagined system, a conception, consisting of conceivable physical entities and operations, but not actual ones. The concrete image is not a physical embodiment of the abstract system.51 Finding the concrete image is a creative process, since the abstract system itself will not dictate how it should be concretized. When we take the Carnot cycle and say it could be realized by a frictionless cylinder-and-piston system filled with a water-steam mixture that is heated and cooled by heat reservoirs of infinite capacities, we are proposing a concrete image of an abstract system. And there are many other possible concrete images of the Carnot cycle.
After finding a concrete image of the abstract system, the second step in operationalization is matching: to find an actual physical system of entities and operations that matches up with the image. Here it cannot be taken for granted that we will be able to find such a matching system. If it seems clear that there cannot be any matching actual systems, then we pronounce the image to be idealized.52 We can expect that the concrete images we need to deal with in many cases will be idealized ones. In the context of thermodynamics and statistical mechanics, perfect reversibility is an idealization in the macroscopic realm. But it is possible to estimate how closely actual systems approximate idealized ones.
&nb
sp; With the help of this two-step view of operationalization, let us return to the question of the justification or validation of operationalization. The first thing we need to do is lose the habit of thinking in terms of simple correctness. It is very tempting to think that the ultimate basis on which to judge the validity of
50. She attributes the inspiration for this view partly to Margaret Morrison (1998).
51. Although I do not use the term "model," some of the recent literature on the nature and function of models is clearly relevant to the discussion here. See especially Morrison and Morgan's view (1999, 21-23) that theoretical models can function as measuring instruments.
52. Strictly speaking, this notion of idealization does not apply to abstract systems; however, if an abstract system only seems to have idealized concrete images, then we could say that the abstract system itself is idealized.
end p.206
Figure 4.14. The two-step view of operationalization.
an operationalization should be whether measurements made on its basis yield values that correspond to the real values. But what are "the real values"? Why do we assume that unoperationalized abstract concepts, in themselves, possess any concrete values at all? For instance, we are apt to think that each phenomenon must possess a definite value of absolute temperature, which we can in principle find out by making the correct measurement. It is very difficult to get away from this sort of intuition, but that is what we must to do in order to avoid behaving like the fly that keeps flying into the windowpane in trying to get out of the room. An unoperationalized abstract concept does not correspond to anything definite in the realm of physical operations, which is where values of physical quantities belong. To put it somewhat metaphorically: equations in the most abstract theories of physics contain symbols (and universal constants), but no numerical values of any concrete physical property; the latter only appear when the equations are applied to concrete physical situations, be they actual or imagined.53 Once an operationalization is made, the abstract concept possesses values in concrete
53. I am not advocating an ontological nihilism regarding the reality of possessed values of properties, only a caution about how meaningful our concepts are at various levels. A concept that is fully meaningful at the theoretical level may be devoid of meaning at the operational level or vice versa. Differentiating the levels of meaning should also make it clear that I am not advocating an extreme operationalism of the kind that I rejected in "Beyond Bridgman" in chapter 3. An abstract concept within a coherent theory is not meaningless even if it is not operationalized, but it must be operationalized in order to have empirical significance.
end p.207
situations. But we need to keep in mind that those values are products of the operationalization in question, not independent standards against which we can judge the correctness of the operationalization itself. That is the root of the circularity that we have encountered time and again in the attempt to justify measurement methods.
If we come away from the idea of correspondence to real values, how can the question of validity enter the process of operationalization at all? That is a difficult question, to which I am not able to offer a complete answer. But it seems to me that the answer has to lie in looking at the correspondence between systems, not between the real and measured values of a particular quantity. A valid operationalization consists in a good correspondence between the abstract system and its concrete image, and between the concrete image and some system of actual objects and operations. The correspondence between systems is not something that can be evaluated on a one-dimensional scale, so the judgment on the validity of a proposed operationalization is bound to be a complex one, not a yes-no verdict.
Let us now return to the case of Thomson's absolute temperature and review the strategies of operationalization that he attempted, which were initially described in "Semi-Concrete Models of the Carnot Cycle" and "Using Gas Thermometers to Approximate Absolute Temperature" in the narrative. Logically the simplest method would have been to make a concrete image of the Carnot cycle and to find actual operations to approximate it; however, any concrete images of the Carnot cycle were bound to be so highly idealized that Thomson never made a serious attempt in that direction. The most formidable problem is the requirement of perfect reversibility, which any concrete image of the Carnot cycle would have to satisfy. As noted earlier, this is not merely a matter of requiring frictionless pistons and perfect insulation. Reversibility also means that all heat transfer should occur between objects with equal temperatures (since there would be an increase of entropy in any heat transfer across a temperature difference). But Thomson himself stated as a matter of principle, or even definition, that no net movement of heat can occur between objects that are at the same temperature.54 The only way to make any sense of the reversibility requirement, then, is to see it as a limiting case, a clearly unattainable ideal that can only be approximated if the temperature differences approach zero, which would mean that the amount of time required for the transfer of any finite amount of heat will approach infinity. The fact that the Carnot cycle is a cycle exacerbates the difficulty because it requires that not just one, but a few different physical processes satisfying the reversibility requirement need to be found and put together.
Therefore Thomson's preference was not to deal with a Carnot cycle directly, but to deduce the relevant features of the abstract Carnot cycle from simpler abstract
54. See the definition from 1854 quoted in note 38.
end p.208
Figure 4.15. Thomson's operationalization of absolute temperature, by means of the water-steam system.
processes and to operationalize the simpler processes. Thomson's most sophisticated attempt in this direction came in 1880, in a renewed attempt to operationalize his second absolute temperature. He was assisted in this task by the principle of energy conservation, which allowed him to deduce theoretically the work-heat ratio for the cycle (and hence the heat input-output ratio needed for the definition of temperature) merely by considering isothermal expansion.55 Then he only had to find the concrete image of the first stroke of the cycle rather than the whole cycle. That stroke was still an ideal reversible process, but Thomson thought that its concrete image could be an irreversible process, if it could be demonstrated to be equivalent in the relevant sense to the reversible process. This scheme of operationalization is summarized graphically in figure 4.15.
Thomson's favorite concrete image of the first stroke of the Carnot cycle was the production of steam from water. Thomson was able to allow the image to be a nonreversible process because the amount of mechanical work generated in this process, for a given amount of heat input, was not thought to be affected by
55. For further details on this maneuver, see Chang and Yi (forthcoming).
end p.209
whether the process was reversible or not. (That would be on the assumption that the internal energy of the steam-water system is a state function, so it has fixed values for the same initial and final states of the system regardless of how the system gets from one to the other.) Now, as described earlier, the particularly nice thing about the steam-water system is that the generation of steam from liquid water is a process that takes place isothermally under constant pressure, which makes it easy to compute the mechanical work generated (simply the constant pressure multiplied by the volume increment). That is not the case for gas-only systems, for instance, because an isothermal expansion of a gas cannot take place under constant pressure, so the computation of mechanical work produced in that expansion requires an exact knowledge of the pressure-volume relation. In addition, if the pressure varies during the heat intake process, computing the amount of heat entering the gas requires an exact knowledge of the variation of specific heat with pressure. All in all, the work-heat ratio pertaining to the isothermal expansion of a simple gas is surrounded with uncertainty, and it is not obvious whether and how much the work-heat ratio would depend on whether the expansion is c
arried out reversibly or not. There is no such problem in the steam-water case.
At last, Thomson had identified clearly a simple concrete image of an abstract thermodynamic system embodying the absolute temperature concept, an image that also had obvious potential for correspondence to actual systems. This explains his elation about the steam-water system, expressed in his 1880 article in the Encyclopaedia Britannica: We have given the steam thermometer as our first example of thermodynamic thermometry because intelligence in thermodynamics has been hitherto much retarded, and the student unnecessarily perplexed, and a mere quicksand has been given as a foundation for thermometry, by building from the beginning on an ideal substance called 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. (Thomson 1880, 47, §46; emphases added)
Unfortunately, Thomson was never able to bring his proposed steam-water thermometer into credible practical use. For one thing, no one, not even Regnault, had produced the necessary data giving the density of steam (both in itself and in relation to the density of water) as a function of temperature in the whole range of temperatures. Lacking those data, Thomson could not graduate a steam-water thermometer on the absolute scale without relying on unwarranted theoretical assumptions. Despite his attraction to the virtues of the steam-water system, Thomson was forced to calibrate his steam-water thermometers against gas thermometers.56