Inventing Temperature Page 13
19. For a general discussion of Dalton's caloric theory, see Fox 1968. Going beyond mere criticism of De Luc, Dalton (1808, 9ff.) advanced a complex theoretical and experimental argument that the expansion of mercury was quadratic rather than linear with temperature. He even devised a new temperature scale on the basis of this belief, the correctness of which seemed to him confirmed beyond doubt by the way it simplified several empirical laws governing thermal phenomena. See also Cardwell 1971, 124-126.
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amount that went into the combined state. Without knowing what the latter amount was, one could hardly say anything theoretically about specific heat.
A most instructive case of the effect of this theoretical worry is the authoritative Elementary Treatise on Natural Philosophy (1803) by the renowned mineralogist and divine René-Just Haüy (1743-1794), one of the founders of modern crystallography. This textbook was personally commissioned by Napoleon for use in the newly established lycées, and it promptly became a recommended text for the École Polytechnique as well. Thus, it was with Napoleon's authority as well as his own that Haüy had asserted: The experiments of De Luc have served … to render evident the advantage possessed by mercury, of being amongst all known liquids,20 that which approaches the most to the state of undergoing dilatations exactly proportional to the augmentations of heat, at least between zero and the degrees of boiling water. (Haüy [1803] 1807, 1:142)
However, within just three years Haüy withdrew his advocacy of De Luc in the second edition of his textbook, where he gave more highly theoretical treatments. Haüy now emphasized that the expansion of a body and the raising of its temperature were two distinct effects of the caloric that entered the body. He attributed the distinction between these two effects to Laplace (quite significantly, as we shall see in the next section), referring to Lavoisier and Laplace's famous 1783 memoir on heat.
Haüy (1806, 1:86) traced expansion to the part of added caloric that became latent, and the raising of temperature to the part that remained sensible. Then the crucial question in thermometry was the relation between those two amounts: "[I]f the amount of dilatation is to give the measure of the increase in tension,21 the amount of the caloric that works to dilate the body must be proportional to the amount that elevates the temperature" (1:160). According to Haüy's new way of thinking, De Luc's reasoning was at least slightly negligent. The crucial complication noted by Haüy was that the expansion of water would require more caloric at lower temperatures, since there was stronger intermolecular attraction due to the intermolecular distances being smaller. For that reason, he argued that the real temperature of a mixture would always be lower than the value given by De Luc's simple-minded calculations.22
20. As for the air thermometer, Haüy ([1803] 1807, 1:259-260) discussed its disadvantages in a similar vein to De Luc, referring to Amontons's instrument.
21. Haüy (1806, 1:82) defined temperature as the "tension" of sensible caloric, a notion advanced by Marc-Auguste Pictet in conscious analogy to Volta's concept of electric tension; see Pictet 1791, 9.
22. Haüy's reasoning is worth following in some detail. Consider the mixing of two equal portions of hot and cold water. In reaching equilibrium, the hot water gives out some heat, which is absorbed by the cold water. One part of the heat given up serves to contract the hot water (call that amount of caloric C1), and the rest (C2) serves to cool it; likewise, one part of the caloric absorbed by the cold water (C3) serves to expand it, and the rest (C4) serves to warm it. In order to know the resulting temperature of the mixture, it is necessary to know the quantities C2 and C4. Since they are not necessarily equal to each other, the temperature of the mixture is not necessarily the arithmetic average of the starting temperatures. Haüy starts with the assumption that the total amount of caloric given out by the hot water should be equal to the amount absorbed by the cold water (C1 + C2 = C3 + C4). Then he reasons that C3 is greater than C1, so C2 must be greater than C4. This is because the thermal expansion of water must require more caloric at lower temperatures, since the molecules of matter would be closer together and therefore offer stronger resistance to the expansive action of caloric; at higher temperatures the intermolecular attraction would be weaker because of the larger distances involved. Therefore the contraction of the hot water would cause less caloric to be given out than the amount taken up by the expansion of the cold water by the same amount (C1 < C3), which means that there is more caloric taken away to cool the hot water than that added to heat the cold water (C2 > C4). Here Haüy seems to be assuming that the volume of the mixture would be the same as the sum of the initial volumes; this assumption was disputed by Dalton, as we have seen. Apparently also assuming that the specific heat of water is constant if we only consider the part of the caloric that is actually used for raising temperature, Haüy concluded that the temperature of a mixture would always be below the value calculated by De Luc. See Haüy 1806, 1:166-167.
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In summary, it seems that mature theoretical reflections tended to do irrevocable damage to the method of mixtures, by rendering the constancy or variation of the specific heat of water an entirely open question. There is evidence that even De Luc himself recognized this point of uncertainty, actually before Dalton's and Haüy's criticisms were published. Crawford, who was cited by Dalton as the authority who taught him about the method of mixtures, noted in the second edition of his book on animal heat: Mr De Luc has, however, himself observed, in a paper, with which he some time ago favoured me on this subject,23 that we cannot determine with certainty from those experiments, the relation which the expansion of mercury bears to the increments of heat. For when we infer the agreement between the dilatations of mercury and the increments of heat from such experiments, we take it for granted, that the capacity of water for receiving heat, continues permanent at all temperatures between the freezing and boiling points. This, however, should not be admitted without proof. (Crawford 1788, 32-33)
Although Crawford still maintained his belief in the real correctness of mercury thermometers,24 the statement shows that even the two most important advocates of the method of mixtures came to doubt its theoretical cogency.
23. I have not been able to ascertain which paper of De Luc's Crawford is referring to here.
24. This was on the basis of an additional test of thermometers that Crawford devised. In this experiment he contrived two open metal cylinders containing air at the temperatures of boiling water and melting ice. These two cylinders were put in communication with each other at their open faces, and a mercury thermometer was inserted at that point of contact. Crawford believed that the real temperature of air at that boundary was the arithmetic mean of the two extreme temperatures, and that is what his mercury thermometer indicated. From this result he inferred that the mercury thermometer was indeed correct, and consequently that the method of mixtures must have been quite correct after all. (Incidentally, Crawford believed that the mercury thermometer was almost exactly accurate and disputed De Luc's result that the mercury temperature was appreciably below the real temperature in the middle of the range between the boiling and freezing points of water; cf. the data presented in table 2.2.) See Crawford 1788, 34-54. It is doubtful that anyone would have been persuaded that Crawford's setup was reliably producing the exact mean temperature as intended; I have not come across any discussion of this experiment by anyone else.
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The Calorist Mirage of Gaseous Linearity
If the caloric theories rendered the method of mixtures groundless, what alternative did they present in making the choice of thermometric fluids? The answer was not immediately clear. Haüy and Dalton, whom I have discussed as two chief critics of De Luc, only agreed that De Luc was wrong. As for the true temperature of a mixture, Dalton thought it should be higher than De Luc's value, and Haüy thought it should be lower than De Luc's. Was that disagreement ever resolved? I have not seen any evidence of a serious debate on that issue. In fact, Haüy and D
alton were exceptional among calorists in making an attempt at all to theorize quantitatively about the thermal expansion of liquids. The problem was too difficult, and microscopic reasoning rapidly became groundless once it got involved in figuring out the amount of caloric required for effecting the expansion of liquids in opposition to the unspecified forces which material particles exerted on each other.25 Instead, most caloric theorists were seduced by an apparently easier way out. Caloric theory taught that the action of heat was most purely manifested in gases, rather than liquids or solids. In gases the tiny material particles would be separated too far from each other to exert any nonnegligible forces on each other; therefore, all significant action in gases would be due to the caloric that fills the space between the material particles. Then the theorist could avoid dealing with the uncertainties of the interparticle forces altogether.26
Faith in the simplicity of the thermal behavior of gases was strengthened enormously by the observation announced by Joseph-Louis Gay-Lussac (1802) and independently by Dalton (1802a) that all gases expanded by an equal fraction of the initial volume when their temperature was increased by the same amount. This seemed to provide striking confirmation that the thermal behavior of gases had remarkable simplicity and uniformity, and led many calorists to assume that gases expanded uniformly with temperature. A typical instance was the treatment by Louis-Jacques Thenard (1777-1857) in his highly regarded textbook of chemistry, which he dedicated to Gay-Lussac: "[A]ll gases, in contrast [to liquids and solids], expand equally, and their expansion is uniform and equal for each degree—1/266.67 of their volume at 0°, under atmospheric pressure. The discovery of this law must be attributed to Dalton and Gay-Lussac" (Thenard 1813, 1:37). There was, however, a logical gap in that reasoning, as recognized very clearly by both Dalton and Gay-Lussac themselves despite their works always being cited as evidence for this common opinion. Even if we grant that the thermal expansion of gases is a phenomenon determined exclusively by the effect of temperature, it still does not follow that the volume of a gas should be a linear function of temperature.
25. In fact, given the lack of a quantitative estimate, Haüy's reasoning could even have been construed as a further vindication of the mercury thermometer, since in De Luc's mixing experiments mercury did give readings somewhat lower than the calculated real temperatures! In any case Haüy must have thought that the errors in De Luc's calculations were small enough, since he still endorsed De Luc's test when it came to the verdict that mercury was better than alcohol. See Haüy 1806, 1:165.
26. For more details on this widespread notion, see Fox 1971, ch. 3, "The Special Status of Gases."
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Not all functions of one variable are linear!27 An additional argument was needed for getting linearity, and it was Laplace who took up the challenge most seriously.
The revival of interest in gas thermometers occurred in the context of the ascendancy of what Robert Fox (1974) has famously termed "Laplacian physics," the most dominant trend in French physical science in the years roughly from 1800 to 1815. The mathematician, astronomer, and physicist Pierre-Simon Laplace (1749-1827) worked in close association with the chemist Claude-Louis Berthollet (1748-1822), after each collaborating with Lavoisier. Together they set out a new program for the physical sciences and fostered the next generation of scientists who would carry out the program.28 Berthollet and Laplace subscribed to a "Newtonian" research program seeking to explain all phenomena by the action of central forces operating between pointlike particles. Renowned for his mathematical refinement of Newtonian celestial mechanics, Laplace aspired to bring its rigor and exactitude to the rest of physics: "[W]e shall be able to raise the physics of terrestrial bodies to the state of perfection to which celestial physics has been brought by the discovery of universal gravitation" (Laplace 1796, 2:198). In the first decade of the nineteenth century Laplace and his followers won wide acclaim by creating new theories of optical refraction, capillary action, and acoustics, based on short-range forces.29 Heat theory was an obvious next target, since it was already an essential part of Laplace's treatment of the speed of sound, and one of his long-standing interests dating back to his early collaboration with Lavoisier. Besides, with the gradual demise of Irvinism the theoretical lead in heat theory fell to the Lavoisierian chemical tradition, which Laplace transformed in interesting ways as we shall see.
Laplace's early attempt at an argument for the air thermometer, included in the fourth volume of his classic Treatise of Celestial Mechanics, was brief and loose (1805, xxii and 270). Laplace said it was "at least very probable" that an air thermometer indicated accurately "the real degrees of heat," but his entire argument consisted in this: "[I]f we imagine the temperature of the air to increase while its volume remains the same, it is very natural to suppose that its elastic force, which is caused by heat, will increase in the same ratio." Then he imagined a relaxation of the external pressure confining the heated gas; if the pressure were brought back to the initial value, the volume of the gas would increase in the same ratio as the pressure had done under constant volume. This last step just follows from assuming
27. Dalton (1808, 9) wrote: "Since the publication of my experiments on the expansion of elastic fluids by heat and those of Gay Lussac, immediately succeeding them … it has been imagined by some that gases expand equally; but this is not corroborated by experience from other sources." Rather, he thought that gases expanded "in geometric progression to equal increments of temperature" (11). See also Gay-Lussac 1802, 208-209, including the passage cited as the epigraph to this chapter. Haüy (1806, 1:263-264), who was clearly aware of this point before he was distracted by Laplacian theorizing, even reported that Gay-Lussac had found the coefficient of thermal expansion of air to vary as a function of temperature.
28. For a detailed treatment of Laplace and Berthollet's circle, see Crosland 1967.
29. On the details of these theories, see Gillispie 1997, and also Heilbron 1993, 166-184.
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Mariotte's (Boyle's) law.30 This non-argument, only buttressed by the word "natural," seems to have convinced many people, even the judicious Haüy (1806, 1:167-168). Calorist plausibility combined with Laplacian authority catapulted the air thermometer into the position of the "true thermometer" in the eyes of many active researchers. Thomas Thomson (1773-1852), Regius Professor of Chemistry at the University of Glasgow, granted that "it is at present the opinion of chemists, that … the expansion of all gases is equable," reversing his own earlier view that "none of the gaseous bodies expand equably."31 It became a general view that the only consolation for the mercury thermometer was that it was practically more convenient to use than the air thermometer, and that its readings agreed closely enough with those of the air thermometer between the freezing and boiling points of water, as shown most clearly by Gay-Lussac (1807).
Meanwhile Laplace himself was not quite satisfied with his 1805 argument for the air thermometer and went on to develop a more detailed and quantitative argument.32 To make the concept of temperature more precise, he adopted the approach of the Genevan physicist and classicist Pierre Prevost (1751-1839), who had defined temperature through the equilibrium of radiant caloric, conceiving caloric as a "discrete fluid."33 Extending that kind of view to the molecular level of description, Laplace defined temperature as the density of intermolecular caloric, produced by a continual process of simultaneous emission and absorption between molecules.34 But why should the caloric contained in molecules be radiated away at all? There would have to be some force that pushes the caloric away from the material core of the molecule that attracts and normally holds it. This force, according to Laplace, was the repulsion exerted by the caloric contained in other molecules nearby. Laplace's model might seem to fit well with the old distinction between free and latent caloric: some of the latent caloric, contained in molecules, would be disengaged by caloric-caloric repulsion and become free caloric. However, that would have conflicted with the Lavoisierian
conception that latent caloric was chemically
30. The second step would have been unnecessary for a constant-volume air thermometer, which indicates temperature by pressure, but Laplace was obliged to add it because he was considering a constant-pressure air thermometer.
31. For the earlier view, see T. Thomson 1802, 1:273; for the later view, T. Thomson 1830, 9-10. In his advocacy of the equable expansion of air, Thomson admitted that "it is scarcely possible to demonstrate the truth of this opinion experimentally, because we have no means of measuring temperature, except by expansion." But he added that "the opinion is founded on very plausible reasons," without actually giving those reasons.
32. Laplace presented some important derivations on the behavior of gases from general calorist principles to the Paris Academy of Sciences on 10 September 1821. The mathematical article was published promptly (Laplace 1821), and a verbal summary was printed with some delay (Laplace [1821] 1826). Updated versions were included in the fifth volume of Traité de mécanique céleste (Laplace [1823] 1825).
33. This view was initially proposed in Prevost 1791 and elaborated in several subsequent publications.
34. Here we must note that Laplace's molecules did not touch each other in a gas, unlike Dalton's atoms (each of which consisted of a dense core surrounded by an "atmosphere of caloric") that filled up space even in a gas.