Part 10

19. _Joule's Determinations of the Mechanical Equivalent._--The honour of placing the mechanical theory of heat on a sound _experimental_ basis belongs almost exclusively to J. P. Joule, who showed by direct experiment that in all the most important cases in which heat was generated by the expenditure of mechanical work, or mechanical work was produced at the expense of heat, there was a constant ratio of equivalence between the heat generated and the work expended and vice versa. His first experiments were on the relation of the chemical and electric energy expended to the heat produced in metallic conductors and voltaic and electrolytic cells; these experiments were described in a series of papers published in the _Phil. Mag._, 1840-1843. He first proved the relation, known as Joule's law, that the heat produced in a conductor of resistance R by a current C is proportional to C^2R per second. He went on to show that the total heat produced in any voltaic circuit was proportional to the electromotive force E of the battery and to the number of equivalents electrolysed in it. Faraday had shown that electromotive force depends on chemical affinity. Joule measured the corresponding heats of combustion, and showed that the electromotive force corresponding to a chemical reaction is proportional to the heat of combustion of the electrochemical equivalent. He also measured the E.M.F. required to decompose water, and showed that when part of the electric energy EC is thus expended in a voltameter, the heat generated is less than the heat of combustion corresponding to EC by a quantity representing the heat of combustion of the decomposed gases. His papers so far had been concerned with the relations between electrical energy, chemical energy and heat which he showed to be mutually equivalent. The first paper in which he discussed the relation of heat to mechanical power was entitled "On the Calorific Effects of Magneto-Electricity, and on the Mechanical Value of Heat" (_Brit. Assoc._, 1843; _Phil. Mag._, 23, p. 263). In this paper he showed that the heat produced by currents generated by magneto-electric induction followed the same law as voltaic currents. By a simple and ingenious arrangement he succeeded in measuring the mechanical power expended in producing the currents, and deduced the mechanical equivalent of heat and of electrical energy. The amount of mechanical work required to raise 1 lb. of water 1 deg. F. (1 B.Th.U.), as found by this method, was 838 foot-pounds. In a note added to the paper he states that he found the value 770 foot-pounds by the more direct method of forcing water through fine tubes. In a paper "On the Changes of Temperature produced by the Rarefaction and Condensation of Air" (_Phil. Mag._, May 1845), he made the first direct measurements of the quantity of heat disengaged by compressing air, and also of the heat absorbed when the air was allowed to expand against atmospheric pressure; as the result he deduced the value 798 foot-pounds for the mechanical equivalent of 1 B.Th.U. He also showed that there was no appreciable absorption of heat when air was allowed to expand in such a manner as not to develop mechanical power, and he pointed out that the mechanical equivalent of heat could not be satisfactorily deduced from the relations of the specific heats, because the knowledge of the specific heats of gases at that time was of so uncertain a character. He attributed most weight to his later determinations of the mechanical equivalent made by the direct method of friction of liquids. He showed that the results obtained with different liquids, water, mercury and sperm oil, were the same, namely, 782 foot-pounds; and finally repeating the method with water, using all the precautions and improvements which his experience had suggested, he obtained the value 772 foot-pounds, which was accepted universally for many years, and has only recently required alteration on account of the more exact definition of the heat unit, and the standard scale of temperature (see CALORIMETRY). The great value of Joule's work for the general establishment of the principle of the conservation of energy lay in the variety and completeness of the experimental evidence he adduced. It was not sufficient to find the relation between heat and mechanical work or other forms of energy in one particular case. It was necessary to show that the same relation held in all cases which could be examined experimentally, and that the ratio of equivalence of the different forms of energy, measured in different ways, was independent of the manner in which the conversion was effected and of the material or working substance employed.

As the result of Joule's experiments, we are justified in concluding that heat is a form of energy, and that all its transformations are subject to the general principle of the conservation of energy. As applied to heat, the principle is called the first law of thermodynamics, and may be stated as follows: _When heat is transformed into any other kind of energy, or vice versa, the total quantity of energy remains invariable; that is to say, the quantity of heat which disappears is equivalent to the quantity of the other kind of energy produced and vice versa._

The number of units of mechanical work equivalent to one unit of heat is generally called the mechanical equivalent of heat, or Joule's equivalent, and is denoted by the letter J. Its numerical value depends on the units employed for heat and mechanical energy respectively. The values of the equivalent in terms of the units most commonly employed at the present time are as follows:--

777 foot-pounds (Lat. 45 deg.) are equivalent to 1 B.Th.U. (lb. deg. Fahr.) 1399 foot-pounds " " " 1 lb. deg. C. 426.3 kilogrammetres " " 1 kilogram-deg. C. or kilo-calorie. 426.3 grammetres " " 1 gram-deg. C. or calorie. 4.180 joules " " 1 gram-deg. C. or calorie.

The water for the heat units is supposed to be taken at 20 deg. C. or 68 deg. F., and the degree of temperature is supposed to be measured by the hydrogen thermometer. The acceleration of gravity in latitude 45 deg. is taken as 980.7 C.G.S. For details of more recent and accurate methods of determination, the reader should refer to the article CALORIMETRY, where tables of the variation of the specific heat of water with temperature are also given.

The second law of thermodynamics is a title often used to denote Carnot's principle or some equivalent mathematical expression. In some cases this title is not conferred on Carnot's principle itself, but on some axiom from which the principle may be indirectly deduced. These axioms, however, cannot as a rule be directly applied, so that it would appear preferable to take Carnot's principle itself as the second law. It may be observed that, as a matter of history, Carnot's principle was established and generally admitted before the principle of the conservation of energy as applied to heat, and that from this point of view the titles, first and second laws, are not particularly appropriate.

20. _Combination of Carnot's Principle with the Mechanical Theory._--A very instructive paper, as showing the state of the science of heat about this time, is that of C. H. A. Holtzmann, "On the Heat and Elasticity of Gases and Vapours" (Mannheim, 1845; Taylor's _Scientific Memoirs_, iv. 189). He points out that the theory of Laplace and Poisson does not agree with facts when applied to vapours, and that Clapeyron's formulae, though probably correct, contain an undetermined function (Carnot's F't, Clapeyron's 1/C) of the temperature. He determines the value of this function to be J/T by assuming, with Seguin and Mayer, that the work done in the isothermal expansion of a gas is a measure of the heat absorbed. From the then accepted value .078 of the difference of the specific heats of air, he finds the numerical value of J to be 374 kilogrammetres per kilo-calorie. _Assuming the heat equivalent of the work to remain in the gas_, he obtains expressions similar to Clapeyron's for the total heat and the specific heats. In consequence of this assumption, the formulae he obtained for adiabatic expansion were necessarily wrong, but no data existed at that time for testing them. In applying his formulae to vapours, he obtained an expression for the saturation-pressure of steam, which agreed with the empirical formula of Roche, and satisfied other experimental data on the supposition that the coefficient of expansion of steam was .00423, and its specific heat 1.69--values which are now known to be impossible, but which appeared at the time to give a very satisfactory explanation of the phenomena.

The essay of Hermann Helmholtz, _On the Conservation of Force_ (Berlin, 1847), discusses all the known cases of the transformation of energy, and is justly regarded as one of the chief landmarks in the establishment of the energy-principle. Helmholtz gives an admirable statement of the fundamental principle as applied to heat, but makes no attempt to formulate the correct equations of thermodynamics on the mechanical theory. He points out the fallacy of Holtzmann's (and Mayer's) calculation of the equivalent, but admits that it is supported by Joule's experiments, though he does not seem to appreciate the true value of Joule's work. He considers that Holtzmann's formulae are well supported by experiment, and are much preferable to Clapeyron's, because the value of the undetermined function F't is found. But he fails to notice that Holtzmann's equations are fundamentally inconsistent with the conservation of energy, because the heat equivalent of the external work done is supposed to remain in the gas.

That a quantity of heat equivalent to the work performed actually disappears when a gas does work in expansion, was first shown by Joule in the paper on condensation and rarefaction of air (1845) already referred to. At the conclusion of this paper he felt justified by direct experimental evidence in reasserting definitely the hypothesis of Seguin (_loc. cit._ p. 383) that "the steam while expanding in the cylinder loses heat in quantity exactly proportional to the mechanical force developed, and that on the condensation of the steam the heat thus converted into power is not given back." He did not see his way to reconcile this conclusion with Clapeyron's description of Carnot's cycle. At a later date, in a letter to Professor W. Thomson (Lord Kelvin) (1848), he pointed out that, since, according to his own experiments, the work done in the expansion of a gas at constant temperature is equivalent to the heat absorbed, by equating Carnot's expressions (given in S 17) for the work done and the heat absorbed, the value of Carnot's function F't must be equal to J/T, in order to reconcile his principle with the mechanical theory.

Professor W. Thomson gave an account of Carnot's theory (_Trans. Roy. Soc. Edin._, Jan. 1849), in which he recognized the discrepancy between Clapeyron's statement and Joule's experiments, but did not see his way out of the difficulty. He therefore adopted Carnot's principle provisionally, and proceeded to calculate a table of values of Carnot's function F't, from the values of the total-heat and vapour-pressure of steam-then recently determined by Regnault (_Memoires de l'Institut de Paris_, 1847). In making the calculation, he assumed that the specific volume v of saturated steam at any temperature T and pressure p is that given by the gaseous laws, pv = RT. The results are otherwise correct so far as Regnault's data are accurate, because the values of the efficiency per degree F't are not affected by any assumption with regard to the nature of heat. He obtained the values of the efficiency F't over a finite range from t to 0 deg. C., by adding up the values of F't for the separate degrees. This latter proceeding is inconsistent with the mechanical theory, but is the correct method on the assumption that the heat given up to the condenser is equal to that taken from the source. The values he obtained for F't agreed very well with those previously given by Carnot and Clapeyron, and showed that this function diminishes with rise of temperature roughly in the inverse ratio of T, as suggested by Joule.

R. J. E. Clausius (_Pogg. Ann._, 1850, 79, p. 369) and W. J. M. Rankine (_Trans. Roy. Soc. Edin._, 1850) were the first to develop the correct equations of thermodynamics on the mechanical theory. When heat was supplied to a body to change its temperature or state, part remained in the body as intrinsic heat energy E, but part was converted into external work of expansion W and ceased to exist as heat. The part remaining in the body was always the same for the same change of state, however performed, as required by Carnot's fundamental axiom, but the

## part corresponding to the external work was necessarily different for

different values of the work done. Thus in any cycle in which the body was exactly restored to its initial state, the heat remaining in the body would always be the same, or as Carnot puts it, the quantities of heat absorbed and given out in its diverse transformations are exactly "compensated," so far as the body is concerned. But the quantities of heat absorbed and given out are not necessarily equal. On the contrary, they differ by the equivalent of the external work done in the cycle. Applying this principle to the case of steam, Clausius deduced a fact previously unknown, that the specific heat of steam maintained in a state of saturation is negative, which was also deduced by Rankine (loc. cit.) about the same time. In applying the principle to gases Clausius assumes (with Mayer and Holtzmann) that the heat absorbed by a gas in isothermal expansion is equivalent to the work done, but he does not appear to be acquainted with Joule's experiment, and the reasons he adduces in support of this assumption are not conclusive. This being admitted, he deduces from the energy principle alone the propositions already given by Carnot with reference to gases, and shows in addition that the specific heat of a perfect gas must be independent of the density. In the second part of his paper he introduces Carnot's principle, which he quotes as follows: "The performance of work is equivalent to a transference of heat from a hot to a cold body without the quantity of heat being thereby diminished." This is not Carnot's way of stating his principle (see S 15), but has the effect of exaggerating the importance of Clapeyron's unnecessary assumption. By equating the expressions given by Carnot for the work done and the heat absorbed in the expansion of a gas, he deduces (following Holtzmann) the value J/T for Carnot's function F't (which Clapeyron denotes by 1/C). He shows that this assumption gives values of Carnot's function which agree fairly well with those calculated by Clapeyron and Thomson, and that it leads to values of the mechanical equivalent not differing greatly from those of Joule. Substituting the value J/T for C in the analytical expressions given by Clapeyron for the latent heat of expansion and vaporization, these relations are immediately reduced to their modern form (see THERMODYNAMICS, S 4). Being unacquainted with Carnot's original work, but recognizing the invalidity of Clapeyron's description of Carnot's cycle, Clausius substituted a proof consistent with the mechanical theory, which he based on the axiom that "heat cannot of itself pass from cold to hot." The proof on this basis involves the application of the energy principle, which does not appear to be necessary, and the axiom to which final appeal is made does not appear more convincing than Carnot's. Strange to say, Clausius did not in this paper give the expression for the efficiency in a Carnot cycle of finite range (Carnot's Ft) which follows immediately from the value J/T assumed for the efficiency F't of a cycle of infinitesimal range at the temperature t C or T Abs.

Rankine did not make the same assumption as Clausius explicitly, but applied the mechanical theory of heat to the development of his hypothesis of molecular vortices, and deduced from it a number of results similar to those obtained by Clausius. Unfortunately the paper (loc. cit.) was not published till some time later, but in a summary given in the _Phil. Mag._ (July 1851) the principal results were detailed. Assuming the value of Joule's equivalent, Rankine deduced the value 0.2404 for the specific heat of air at constant pressure, in place of 0.267 as found by Delaroche and Berard. The subsequent verification of this value by Regnault (_Comptes rendus_, 1853) afforded strong confirmation of the accuracy of Joule's work. In a note appended to the abstract in the _Phil. Mag._ Rankine states that he has succeeded in proving that the maximum efficiency of an engine working in a Carnot cycle of finite range t1 to t0 is of the form (t1 - t0)/(t1 - k), where k is a constant, the same for all substances. This is correct if t represents temperature Centigrade, and k = -273.

Professor W. Thomson (Lord Kelvin) in a paper "On the Dynamical Theory of Heat" (_Trans. Roy. Soc. Edin._, 1851, first published in the _Phil. Mag._, 1852) gave a very clear statement of the position of the theory at that time. He showed that the value F't = J/T, assumed for Carnot's function by Clausius without any experimental justification, rested solely on the evidence of Joule's experiment, and might possibly not be true at all temperatures. Assuming the value J/T with this reservation, he gave as the expression for the efficiency over a finite range t1 to t0 C., or T1 to T0 Abs., the result,

W/H = (t1 - t0)/(t1 + 273) = (T1 - T0)/T1 (4)

which, he observed, agrees in form with that found by Rankine.

21. _The Absolute Scale of Temperature._--Since Carnot's function is the same for all substances at the same temperature, and is a function of the temperature only, it supplies a means of measuring temperature independently of the properties of any particular substance. This proposal was first made by Lord Kelvin (_Phil. Mag._, 1848), who suggested that the degree of temperature should be chosen so that the efficiency of a perfect engine at any point of the scale should be the same, or that Carnot's function F't should be constant. This would give the simplest expression for the efficiency on the caloric theory, but the scale so obtained, when the values of Carnot's function were calculated from Regnault's observations on steam, was found to differ considerably from the scale of the mercury or air-thermometer. At a later date, when it became clear that the value of Carnot's function was very nearly proportional to the reciprocal of the temperature T measured from the absolute zero of the gas thermometer, he proposed a simpler method (_Phil. Trans._, 1854), namely, to define absolute temperature [theta] as proportional to the reciprocal of Carnot's function. On this definition of absolute temperature, the expression ([theta]1 - [theta]0)/[theta]1 for the efficiency of a Carnot cycle with limits [theta]1 and [theta]0 would be exact, and it became a most important problem to determine how far the temperature T by gas thermometer differed from the absolute temperature [theta]. With this object he devised a very delicate method, known as the "porous plug experiment" (see THERMODYNAMICS) of testing the deviation of the gas thermometer from the absolute scale. The experiments were carried out in conjunction with Joule, and finally resulted in showing (_Phil. Trans._, 1862, "On the Thermal Effects of Fluids in Motion") that the deviations of the air thermometer from the absolute scale as above defined are almost negligible, and that in the case of the gas hydrogen the deviations are so small that a thermometer containing this gas may be taken for all practical purposes as agreeing exactly with the absolute scale at all ordinary temperatures. For this reason the hydrogen thermometer has since been generally adopted as the standard.

22. _Availability of Heat of Combustion._--Taking the value 1.13 kilogrammetres per kilo-calorie for 1 deg. C. fall of temperature at 100 deg. C., Carnot attempted to estimate the possible performance of a steam-engine receiving heat at 160 deg. C. and rejecting it at 40 deg. C. Assuming the performance to be simply proportional to the temperature fall, the work done for 120 deg. fall would be 134 kilogrammetres per kilo-calorie. To make an accurate calculation required a knowledge of the variation of the function F't with temperature. Taking the accurate formula of S 20, the work obtainable is 118 kilogrammetres per kilo-calorie, which is 28% of 426, the mechanical equivalent of the kilo-calorie in kilogrammetres. Carnot pointed out that the fall of 120 deg. C. utilized in the steam-engine was only a small fraction of the whole temperature fall obtainable by combustion, and made an estimate of the total power available if the whole fall could be utilized, allowing for the probable diminution of the function F't with rise of temperature. His estimate was 3.9 million kilogrammetres per kilogramme of coal. This was certainly an over-estimate, but was surprisingly close, considering the scanty data at his disposal.