The Absolute and Its Measurement: William Thomson on Temperature

Abstract

In this paper we give a full account of the work of William Thomson (Lord Kelvin) on absolute temperature, which to this day provides the theoretical underpinnings for the most rigorous measurements of temperature. When Thomson fashioned his concepts of ‘absolute’ temperature, his main concern was to make the definition of temperature independent of the properties of particular thermometric substances (rather than to count temperature from an absolute zero). He tried out a succession of definitions based on the thermodynamics of ideal heat engines; most notably, in 1854 he gave the ratio of two temperatures as the ratio of quantities of heat taken in and given out at those temperatures in a Carnot cycle. But there were difficulties with using such definitions for experimental work, since it was not possible even to approximate an ideal Carnot engine in reality. More generally, it is not trivial to connect an abstract concept with concrete operations in order to make physical measurements possible. In the end, Thomson argued that an ideal gas thermometer would indicate his absolute temperature, and that the deviation of actual gas thermometers from the ideal could be estimated by means of the Joule‐Thomson effect. However, the measurement of the Joule‐Thomson effect itself required measurements of temperature, so there was a problem of circularity.

Acknowledgements

We would like to thank Olivier Darrigol and Keith Hutchison for their helpful and detailed comments on earlier versions of this paper. For help with earlier phases of work leading up to this paper, we would like to thank Jed Buchwald, Sam Schweber, and the members of the LSE/Amsterdam research group on Measurement in Economics and Physics. We thank Cambridge University for permission to cite from the Kelvin papers.

Notes

1. Silvanus P. Thompson, The Life of William Thomson, Baron Kelvin of Largs, 2 vols. (London: Macmillan, 1910), 267–69, 274, 293, 689; Crosbie Smith, ‘William Thomson and the Creation of Thermodynamics: 1840–1855’, Archive for History of Exact Sciences, 16 (1977), 231–88 (240–41); Crosbie Smith and M. Norton Wise, Energy and Empire: A Biographical Study of Lord Kelvin (Cambridge: Cambridge University Press, 1989), 249–50, 300–2, 324–25; Harold I. Sharlin, Lord Kelvin: The Dynamic Victorian, in collaboration with Tiby Sharlin (University Park and London: The Pennsylvania State University Press, 1979), 97–98, 118, 190.

2. D. S. L. Cardwell, From Watt to Clausius (Ithaca: Cornell University Press, 1971), esp. pp. 239–40, 258–60; D. S. L. Cardwell, James Joule: A Biography (Manchester: Manchester University Press, 1989); Keith Hutchison, ‘Mayer's Hypothesis: A Study of the Early Years of Thermodynamics’, Centaurus, 20 (1976), 279–304.

3. M. Norton Wise and Crosbie Smith, ‘Measurement, Work and Industry in Lord Kelvin's Britain’, Historical Studies in the Physical Sciences, 17 (1986), 147–73.

4. Andrew Gray, Lord Kelvin: An Account of His Scientific Life and Work (London: J. M. Dent & Co., 1908), chapter 8, esp. 123–27, 135–138.

5. Clifford Truesdell, The Tragicomical History of Thermodynamics 1822–1854 (New York: Springer‐Verlag, 1980), esp. sections 11B, 11H, and 9D (including an appendix by C. S. Man).

6. We focus on Thomson's work in detail, at the expense of discussions of broader historical contexts and general epistemological issues. For a less technical treatment of the topic in a broader setting, see Hasok Chang, Inventing Temperature: Measurement and Scientific Progress (New York: Oxford University Press, 2004), chapter 4.

7. William Thomson, ‘On an Absolute Thermometric Scale founded on Carnot's Theory of the Motive Power of Heat, and calculated from Regnault's Observations’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 100–06 (100); emphasis added.

8. Henri Victor Regnault, ‘Relations des expériences entreprises par ordre de Monsieur le Ministre des Travaux Publics, et sur la proposition de la Commission Centrale des Machines à Vapeur, pour déterminer les principales lois et les données numériques qui entrent dans le calcul des machines à vapeur’, Mémoires de l'Académie Royale des Sciences de l'Institut de France, 21 (1847), 1–748. For further discussion of Regnault's minimalist metrology, see Hasok Chang, ‘Spirit, air and quicksilver: The search for the ‘real’ scale of temperature’, Historical Studies in the Physical and the Biological Sciences, 31:2 (2001), 249–84.

9. Thomson (note 7), 102; emphases original; see also James Prescott Joule and William Thomson, ‘On the Thermal Effects of Fluids in Motion, Part 2’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 357–400 (393).

10. Thomson (note 7), 100.

11. For a history of this problem, see Chang (note 8), 253–63.

12. Thomson (note 7), 101.

13. Ibid., p. 102; emphasis original.

14. See Wise and Smith (note 3), esp. pp. 150, 158–59. Interestingly, they also point out that Thomson's initial ideas about absolute electrical measurements were inspired by certain aspects of Regnault's work.

15. See Simon Schaffer, ‘Late Victorian Metrology and its Instrumentation: A Manufactory of Ohms’, in R. Bud and S. E. Cozzens, eds., Invisible Connections: Instruments, Institutions, and Science (Bellingham, Washington: Spie Optical Engineering Press, 1992), 23–56; Paul Tunbridge, Lord Kelvin, His Influence on Electrical Measurements and Units (London: P. Peregrinus on behalf of the Institution of Electrical Engineers, 1992). The most convenient summary is provided in Dong‐Won Kim, Leadership and Creativity: A History of the Cavendish Laboratory, 1871–1919 (Dordrecht: Kluwer, 2002), 38–43. I thank Professor Kim for alerting me to this parallel between electrical and thermal measurements in Thomson's work.

16. Thomson (note 7), 102; emphases original.

17. Ibid., 104; emphasis added.

18. This influence commonly attributed to Amontons is actually not very easy to document. Cardwell, From Watt to Clausius (note 2), 31, says that George Martine took this idea from Amontons's work. Jacques Payen, in his entry on Amontons in the Dictionary of Scientific Biography, vol. 1, 138, states that J. H. Lambert's ideas were stimulated in a similar way.

19. Thomson, quoted in Schaffer (note 15), 42.

20. For the treatment of the air engine, see William Thomson, ‘An Account of Carnot's Theory of the Motive Power of Heat; with Numerical Results deduced from Regnault's Experiments on Steam’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 113–55 (127–33). This paper was originally published in 1849 in the Transactions of the Royal Society of Edinburgh, vol. 16, 541–74.

21. This is adapted from the figure in Thomson, ibid., 124.

22. See Ibid., 125–26, for this passage and the following reasoning.

23. This table, promised in Thomson (note 7), 105, can be found in Thomson (note 20), 139, 141, in slightly different forms than originally announced.

24. Thomson (note 7), 104–5. Thomson had to use Boyle's and Gay‐Lussac's laws to get anywhere in reasoning out the air‐engine case, too; see Thomson (note 20), 129, 131.

25. Thomson (note 7), 106.

26. See Cardwell, James Joule (note 2), and Crosbie Smith, The Science of Energy: A Cultural History of Energy Physics in Victorian Britain (London and Chicago: The Athlone Press and the University of Chicago Press, 1988).

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

28. Thomson to Joule, 27 October 1848, Kelvin Papers, Add. 7342, J62, University Library, Cambridge.

29. Thomson (note 20), 116–17.

30. See the retrospective note attached to Thomson (note 7), 106; emphasis added.

31. Hutchison (note 2), 299; Truesdell (note 5), 321.

32. We have extracted this relation from formula (7) given in Thomson (note 20), 134, §31, by assuming that μ is a constant, which is a consequence of Thomson's first definition of absolute temperature.

33. The following account of Carnot's function is taken from William Thomson, ‘On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule’s equivalent of a Thermal Unit, and M. Regnault's Observations on Steam’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 174–210 (187–88, §21). This article was originally published in 1851 in the Transactions of the Royal Society of Edinburgh. Here and elsewhere, we have changed Thomson's notation slightly for improved clarity in relation with other formulae.

34. James Prescott Joule and William Thomson, ‘On the Thermal Effects of Fluids in Motion, Part 2’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 357–400 (393); emphasis added. This paper was originally published in 1854 in the Philosophical Transactions of the Royal Society, vol. 144. Joule and Thomson stated that this liberalized conception was already expressed in Thomson's 1848 paper (note 7), but that does not seem quite correct.

35. William Thomson, ‘Thermo‐electric Currents’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 232–91 (233, footnote). This article was originally published in 1854 in the Transactions of the Royal Society of Edinburgh, vol. 21.

36. Thomson (note 33), 199. Here Thomson cited Joule's letter of 9 December 1848.

37. Thomson (note 20), 131.

38. We have taken the value 273.7 from Joule and Thomson (note 34), 394.

39. See the 6th form of Mayer's hypothesis listed in Hutchison (note 2), 279.

40. Joule and Thomson (note 34), 393–94; emphasis added.

41. Ibid., 394. Essentially the same definition was also attached to Thomson's paper 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.” See Thomson (note 35), 235.

42. Truesdell (note 5), 310.

43. Joule to Thomson, 9 December 1848, Kelvin Papers, Add. 7342, J64, University Library, Cambridge; see also the account of this letter in Cardwell, James Joule (note 2), 99–100.

44. See the same frustration noted by Cardwell, ibid., 98, and also Sharlin (note 1), 249, note 35 to chapter 6.

45. Thomson to Joule, 27 October 1848, Kelvin Papers, Add. 7342, J62, University Library, Cambridge. For the published version, see Thomson (note 20), 146–48 (read 30 April 1849).

46. Joule assumed what Hutchison identifies as the second form of Mayer's hypothesis: ‘The heat absorbed (or released) by a gas during [reversible] isothermal expansion (or compression, respectively) is the exact equivalent of the external work done.’ See Hutchison (note 2), 279.

47. We are helped here by the exposition in Gray (note 4), esp. 125. The argument reconstructed here is not so far from the modern ones, for example given in M. W. Zemansky and R. H. Dittman, Heat and Thermodynamics, 6th ed. (New York: McGraw‐Hill, 1981), 175–77.

48. The same by what standard of temperature? That is actually not very clear. But it is probably an innocuous enough assumption that a phenomenon occurring at a constant temperature by any good thermometer will also be occurring at a constant absolute temperature.

49. See William Thomson, ‘On a Method of discovering experimentally the Relation between the Mechanical Work spent, and the Heat produced by the Compression of a Gaseous Fluid’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 210–22 (esp., 211). This article was originally published in 1851 in the Transactions of the Royal Society of Edinburgh, vol. 20.

50. See James Prescott Joule and William Thomson, ‘On the Thermal Effects of Fluids in Motion, Part 4’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 415–31 (427–31); what we reproduce here is the equation from p. 430. This article was originally published in 1862 in the Philosophical Transactions of the Royal Society, vol. 152.

51. Cf. the harshly critical discussion in Truesdell (note 5), section 9D, including an appendix by C. S. Man.

52. Thomson (note 49), 218–19.

53. Thomson's reasoning from 1851 on this point is quite murky, but a corresponding passage in Joule and Thomson (note 34), 379, is clearer.

54. This is the most sensible rendition we can make of Joule and Thomson's reasoning on this point, which Hutchison (note 2), 297, finds ‘strange’ and ‘unsatisfying’, a judgement with which we have sympathy. Not much more helpful is the brief summary given in William Thomson, ‘On the Quantities of Mechanical Energy contained in a Fluid in Different States, as to Temperature and Density’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 222–32 (230–32). This article was originally published in 1851 in the Transactions of the Royal Society of Edinburgh, vol. 20.

55. Thomson (note 49), 219.

56. This relation is taken from James Prescott Joule and William Thomson, ‘On the Thermal Effects of Fluids in Motion’, in W. Thomson, Mathematical and Physical Papers, vol. 1 (Cambridge: Cambridge University Press, 1882), 346–56 (347). This article was originally published in 1853 in the Philosophical Transactions of the Royal Society. They claim that this relation is equivalent to ‘equation (f), §74, or equation (17), §95, and (8), §88’, from Thomson (note 49) and Thomson (note 54).

57. This is listed as equation (6) in Joule and Thomson (note 34), 397, and explicated on pp. 393–96.

58. Ibid., 398–400; our equation (28) is Joule and Thomson's equation (15) on p. 398.

59. Joule and Thomson (note 50), 428. They also stated there that ‘a direct demonstration’ of this equation could be ‘readily obtained’ from ‘elementary thermodynamic principles’, but did not actually give such a demonstration. Olivier Darrigol (personal communication, February 2004) gives a modern derivation of equation (29) as follows. The Joule‐Thomson process conserves the enthalpy H = U+PV because there is no heat communicated to the gas from outside. In standard modern notation (with θ for temperature on an arbitrary scale), this implies the relation δθ/δP = ϑθ/ϑP|_H = −(h+V)/C_p, where C _p and h are defined through the relation δQ = C_pdθ+hdP. Carnot's theorem further implies the relation ϑV/ϑθ|_p = −hμ, if the ideal gas law holds in terms of temperature θ. Hence we have δθ/δP = C_p ⁻¹(μ⁻¹ϑV/ϑθ|_p−V), which is the modern rendering of equation (29).

60. Ibid., 428–29. The data summarized in this formula were gathered and processed from various tables earlier in that paper, most conveniently summarized in two tables on p. 429. These tables also include theoretical values of the cooling effects, agreeing quite well with the actual values; it is not very clear how the theoretical values were obtained. A quasi‐empirical derivation of a similar result, building on Rankine's empirical formula for the pressure of carbonic acid, seems to be given in Joule and Thomson (note 34), 375–77; see also §1 of ‘Theoretical Deductions’ in that paper, 377–85.

61. Joule and Thomson (note 50), 429–30. Unfortunately Thomson left this derivation in a very precarious state, since in the reprint of this paper for his collected papers he added a note (dated Aug. 1879) that the assumption of the approach to ‘the rigorous fulfilment of Boyle's law at very high temperatures’ was ‘certainly false’ (430).

62. Joule and Thomson (note 34), 395–96.

63. For further discussion of ‘epistemic iteration’ in general terms, see Chang (note 6), 226–31.

64. The first formulation, given in William Thomson, Elasticity and Heat (Being articles contributed to the Encyclopaedia Britannica) (Edinburgh: Adam and Charles Black, 1880), 43, §35, is as follows: ‘(1) Alter the bulk or shape of the thermometric substance till it becomes warmer to any desired degree. (2) Keeping it now at this higher temperature, alter bulk or shape farther, and generate the heat which the substance takes to keep its temperature constant, by stirring water, or a portion of the substance itself, if it is partly fluid, and measure the quantity of work spent in this stirring. (3) Bring it back towards its original bulk and shape till it becomes cooled to its original temperature. (4) Keeping it at this temperature, reduce it to its original bulk and shape, carrying off, by a large quantity of water, the heat which it must part with to prevent it from becoming warmed. Find by a special experiment how much work must be done to give an equal amount of heat to an equal amount of water by stirring. Then the ratio of the first measured quantity of work to the second is the ratio of the higher temperature to the lower on the absolute thermodynamic scale.’ There are significant difficulties in interpreting the process described here as a Carnot cycle, especially regarding the directions of expansion or contraction of the working substance.

65. Ibid., 43–44, §37; emphasis original.

66. See Gray (note 4), chapter 8.

67. The chief difficulty is the fact that Thomson described this non‐adiabatic expansion as one in which the working substance is ‘infinitesimally warmed’. Therefore it would seem that what he had in mind was not an isothermal expansion, which means that it cannot be considered as the first stroke of a Carnot cycle. Worse yet, such a process would not fit into his second definition of absolute temperature at all, since that definition only applies to a cycle in which heat is taken in or given out only at fixed temperatures. We see no choice but to interpret being ‘warmed’ in the 1880 formulation as meaning ‘taking in heat’, rather than ‘increasing in temperature’ (heated but not warmed up, as it were).

68. Thomson (note 64), 44, §38.

69. Ibid., 47, §46.

70. Ibid., 46, §44. See also William Thomson, ‘On Steam‐Pressure Thermometers of Sulphurous Acid, Water, and Mercury’, in W. Thomson, Mathematical and Physical Papers, vol. 5 (Cambridge: Cambridge University Press, 1911), 77–87; William Thomson, ‘On a Realised Sulphurous Acid Steam‐Pressure Differential Thermometer; Also a Note on Steam‐Pressure Thermometers’, in W. Thomson, Mathematical and Physical Papers, vol. 5 (Cambridge: Cambridge University Press, 1911), 90–95. These articles were originally published in the Proceedings of the Royal Society of Edinburgh, vol. 10 (1879–80).

71. In the following exposition we take our cue from the account given in Gray (note 4), 136–137. For Thomson's own derivations, see Thomson (note 64), 47–50, §§47–58. The notation we use is basically Thomson's.

72. See Thomson (note 64), 49, §55, for the discussion of this matter and for all of the quoted passages below.

73. Even in the case of hydrogen, where Thomson seems to regard δw as a constant, the experimental estimates that he cites actually do show a variation with temperature. See ibid., 49, §56.

74. See, for example, H. L. Callendar, ‘On the Practical Measurement of Temperature: Experiments Made at the Cavendish Laboratory, Cambridge’, Philosophical Transactions of the Royal Society of London, A178 (1887), 161–230.

75. See Chang (note 6), 212–17.