from which, ifCp−Cvis known, the value ofJcan be found.
In Mayer's time the difference of the specific heats of air was imperfectly known, and soJcould not be found with anything like accuracy. From Regnault's experiments on the specific heat at constant pressure, and from the known ratio of the specific heats as deduced from the velocity of sound combined with Regnault's result, the value ofCp−Cvmay be taken as .0686. ThusJworks out to 42.2 × 106, in ergs per calorie, which is not far from the true value. Mayer obtained a result equivalent to 36.5 × 106ergs per calorie.
The assumption on which this calculation is founded is that there is no alteration of the internal energy of the gas in consequence of expansion. If the air when raised in temperature, and at the same time increased in volume, contained less internal energy than when simply heated without alteration of volume, the energy evolved would be available to aid the performance of the work done against external forces, and less heat would be required, or, in the contrary case, more heat would be required, than would be necessary if the internal energy remained unaltered. Thus puttingdWforpdv, the work done,efor the internal energy before expansion, anddHfor the heat given to the gas, we have obviously the equation
JdH=de+dW
wheredeis the change of internal energy due to the alteration of volume, together with the alteration of temperature. If now the temperature be altered without expansion, no external work is done anddWfor that case is zero. Let∂eand∂Hbe the energy change and the heat supplied, then in this case
J∂H=∂e+O
Thus
J(dH−∂H) =de−∂e+dW
and the assumption is thatde=∂e, so thatdW=J(dH−∂H); that is,dW=J(Cp−Cv), when the rise of temperature is 1° C. and the mass of air is one gramme. This assumption requires justification, and by an experiment of Joule's, which was repeated in a more sensitive form devised by Thomson, it was shown to be a very close approximation to thetruth. Joule's experiment is well known: the explanation given above may serve to make clear the nature of the research undertaken later by Thomson and Joule conjointly.
The inverse process, the conversion of heat into work, required investigation, and it is this that constitutes the science of thermodynamics. It was the subject of the celebratedRéflexions sur la Puissance Motrice du Feu, et sur les Machines Propres à Développer cette Puissance, published in 1824 by Sadi Carnot, an uncle of the late President of the French Republic. Only a few copies of this essay were issued, and its text was known to very few persons twenty-four years later, when it was reprinted by the Academy of Sciences. Its methods and conclusions were set forth by Thomson in 1849 in a memoir which he entitled, "An Account of Carnot's Theory of the Motive Power of Heat." Numerical results deduced from Regnault's experiments on steam were included; and the memoir as a whole led naturally in Thomson's hands to a corrected theory of heat engines, which he published in 1852. Carnot's view of the working of a heat engine was founded on the analogy of the performance of work by a stream of water descending from a higher level to a lower. The same quantity of water flows away in a given time from a water wheel in the tail-race as is received in that time by the wheel from the supply stream. Now a heat engine receives heat from a supplying body, or source, at one temperature and parts with heat to another body (for example, the condenser of a steam engine) at a lower temperature. This body is usually called the refrigerator. According to Carnotthese temperatures corresponded to the two levels in the case of the water wheel; the heat was what flowed through the engine. Thus in his theory as much heat was given up by a heat engine to the body at the lower temperature as was received by it from the source. The heat was simply transferred from the body at the higher temperature to the body at the lower; and this transference was supposed to be the source of the work.17
The first law of thermodynamics based on Joule's proportionality of heat produced to work expended, and the converse assumed and verifieda posteriori, showed that this view is erroneous, and that the heat delivered to the refrigerator must be less in amount than that received from the source, by exactly the amount which is converted into work, together with the heat which, in an imperfect engine, is lost by conduction, etc., from the cylinder or other working chamber. This change was made by Thomson in his second paper: but he found the ideas of Carnot of direct and fruitful application in the new theory. These were the cycle of operations and the ideal reversible engine.
In the Carnot cycle the working substance—which might be a gas or a vapour, or a liquid, or a vapour and its liquid in contact: it did not matter what for the result—was supposed to be put through a succession of changes in which the final state coincided with the initial. Thus the substance having been broughtback to the same physical condition as it had when the cycle began, has the same internal energy as it had at the beginning, and in the reckoning of the work done by or against external forces, nothing requires to be set to the account of the working substance. This is the first great advantage of the method of reasoning which Carnot introduced.
The ideal engine was a very simple affair: but the notion of reversibility is difficult to express in a form sufficiently definite and precise. Carnot does not attempt this; he merely contents himself with describing certain cycles of operations which obviously can be carried through in the reverse order. Nor does Thomson go further in his "Account of Carnot's Theory," though he states the criterion of a perfect engine in the words, "A perfect thermodynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency, if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced." This proposition was proved by Carnot: and the following formal statement in the essay is made: "La puissance motrice de la chaleur est independante des agents mis en œuvre pour la réaliser: sa quantité est fixée uniquement par les temperatures des corps entre lesquels se fait, en dernier résultat, le transport du calorique." The result involved in each, that the work done in a cycle by an ideal engine depends on the temperatures between which it works and not at all on the working substance, is, as we shall see, of the greatest importance. The proof of the proposition, by supposing a more efficient engine than the ideal one to exist, and to be coupled with the latter, so that the moreefficient would perform the cycle forwards and the ideal engine the same cycle backwards, is well known. In Carnot's view the former would do more work by letting down a given quantity of heat from the higher to the lower temperature than was spent on the latter in transferring the same quantity of heat from the lower to the higher temperature, so that no heat would be taken from or given to source or refrigerator, while there would be a gain of work on the whole. This would be equivalent to admitting that useful work could be continually performed without any resulting thermal or other change in the agents performing the work. Even at that time this could not be admitted as possible, and hence the supposition that a more efficient engine than the reversible one could exist was untenable.
Carnot showed that the work done by an ideal engine, in transferring heat from one temperature to another, was to be found by means of a certain function of the temperature, hence called "Carnot's function." The corresponding function in the true dynamical theory is always called Carnot's. A certain assignment of value to it gave, as we shall see, Thomson's famous absolute thermodynamic scale of temperature.
In the light of the facts and theories which now exist, and are almost the commonplaces of physical text books, it is very interesting to review the ideas and difficulties which occurred to the founders of the science of heat sixty years ago. For example, Thomson asks, in his "Account of Carnot's Theory," what becomes of the mechanical effect which might be produced by heat which is transferred from one body to another by conduction. The heat leaves one bodyand enters another and no mechanical effect results: if it passed from one to the other through a heat engine, mechanical effect would be produced: what is produced in place of the mechanical effect which is lost? This he calls a very "perplexing question," and hopes that it will, before long, be cleared up. He states, further, that the difficulty would be entirely avoided by abandoning Carnot's principle that mechanical effect is obtained by "the transference of heat from one body to another at a lower temperate." Joule urges precisely this solution of the difficulty in his paper, "On the Changes of Temperature produced by the Rarefaction and Condensation of Air" (Phil. Mag., May 1845). Thomson notes this, but adds, "If we do so, however, we meet with innumerable other difficulties—insuperable without further experimental investigation, and an entire reconstruction of the theory of heat from its foundation. It is in reality to experiment that we must look, either for a verification of Carnot's axiom, and an explanation of the difficulty we have been considering, or for an entirely new basis of the Theory of Heat."
The experiments here asked for had already, as was soon after perceived by Thomson, been made by Joule, not merely in his determinations of the dynamical equivalent of heat, but in his exceedingly important investigation of the energy changes in the circuit of a voltaic cell, or of a magneto-electric machine. Moreover, the answer to this "very perplexing question" was afterwards to be given by Thomson himself in his paper, "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy," published in the EdinburghProceedingsin 1852.
Again, we find, a page or two earlier in the "Account of Carnot's Theory," the question asked with respect to the heat evolved in the circuit of a magneto-electric machine, "Is the heat which is evolved in one part of the closed conductor merely transferred from those parts which are subject to the inducing influence?" and the statement made that Joule had examined this question, and decided that it must be answered in the negative. But Thomson goes on to say, "Before we can finally conclude that heat is absolutely generated in such operations, it would be necessary to prove that the inducing magnet does not become lower in temperature and thus compensate for the heat evolved in the conductor."
Here, apparently, the idea of work done in moving the magnet, or the conductor in the magnetic field, is not present to Thomson's mind; for if it had been, the idea that the work thus spent might have its equivalent, in part, at least, in heat generated in the circuit, would no doubt have occurred to him and been stated. This idea had been used just a year before by Helmholtz, in his essay "Die Erhaltung der Kraft," to account for the heat produced in the circuit by the induced current, that is, to answer the first question put above in the sense in which Joule answered it. The subject, however, was fully worked out by Thomson in a paper published in thePhilosophical Magazinefor December 1851, to which we shall refer later.
Tables of the work performed by various steam engines working between different stated temperatures were given at the close of the "Account," and compared with the theoretical "duty" as calculated forCarnot's ideal perfect engine. Of course the theoretical duty was calculated from the temperatures of the boiler and condenser; the much greater fall of temperature from the furnace to the boiler was neglected as inevitable, so that the loss involved in that fall is not taken account of. Carnot's theory gave for the theoretical duty of one heat unit (equivalent to 1390 foot-pounds of work) 440 foot-pounds for boiler at 140° C. and condenser at 30° C.; and the best performance recorded was 253 foot-pounds, giving a percentage of 57.5 per cent. The worst was that of common engines consuming 12 lb. of coal per horse-power per hour, and gave 38.1 foot-pounds, or a percentage of 8.6 per cent. These percentages become on the dynamical theory 68 and 10.3, since the true theoretical duty for the heat unit is only 371 foot-pounds.
It is worthy of notice that the indicator-diagram method of graphically representing the changes in a cycle of operations is adopted in Thomson's "Account," but does not occur in Carnot's essay. The cycles consist of two isothermal changes and two adiabatic changes; that is, two changes at the temperatures of the source and refrigerator respectively, and two changes—from the higher to the lower temperature, and from the lower to the higher. These changes are made subject to the condition in each case that the substance neither gains nor loses energy in the form of heat, but is cooled in the one case by expansion and heated in the other by compression. The indicator diagram was due not to Thomson but to Clapeyron (see p.99above), who used it to illustrate an account of Carnot's theory.
There appeared in the issue of the EdinburghPhilosophical Transactionsfor January 2, 1849, along with the "Account of Carnot's Theory," a paper by James Thomson, entitled, "Theoretical Considerations on the Effect of Pressure in Lowering the Freezing Point of Water." The author predicted that, unless the principle of conservation of energy was at fault, the effect of increase of pressure on water in the act of freezing would be to lower the freezing point; and he calculated from Carnot's theory the amount of lowering which would be produced by a given increment of pressure. The prediction thus made was tested by experiments carried out in the Physical Laboratory by Thomson, and the results obtained completely confirmed the conclusions arrived at by theory. This prediction and its verification have been justly regarded as of great importance in the history of the dynamical theory of heat; and they afford an excellent example of the predictive character of a true scientific theory. The theory of the matter will be referred to in the next chapter.
Thefirst statement of the true dynamical theory of heat, based on the fundamental idea that the work done in a Carnot cycle is to be accounted for by an excess of the heat received from the source over the heat delivered to the refrigerator, was given by Clausius in a paper which appeared inPoggendorff's Annalenin March and April 1850, and in thePhilosophical Magazinefor July 1850, under a title which is a German translation of that of Carnot's essay. In that paper the First Law of Thermodynamics is explicitly stated as follows: "In all cases in which work is produced by the agency of heat, a quantity of heat proportional to the amount of work produced is expended, and, inversely, by the expenditure of that amount of work exactly the same amount of heat is generated." Modern thermodynamics is based on this principle and on the so-called Second Law of Thermodynamics; which is, however, variously stated by different authors. According to Clausius, who used in his paper an argument like that of Carnot based on the transference of heat from the source to the refrigerator, the foundation of the second law was the fact that heat tends to pass from hotter to colder bodies. In 1854 (Pogg. Ann., Dec. 1854) he stated his fundamental principle explicitly in the form: "Heat can neverpass from a colder to a hotter body, unless some other change, connected therewith, take place at the same time," and gives in a note the shorter statement, which he regards as equivalent: "Heat cannot of itself pass from a colder to a hotter body."
We shall not here discuss the manner in which Clausius applied this principle: but he arrived at and described in his paper many important results, of which he must therefore be regarded as the primary discoverer. His theory as originally set forth was lacking in clearness and simplicity, and was much improved by additions made to it on its republication, in 1864, with other memoirs on the Theory of Heat.
In theTransactions R.S.E., for March 1851, Thomson published his great paper, "On the Dynamical Theory of Heat." The object of the paper was stated to be threefold: (1) To show what modifications of Carnot's conclusions are required, when the dynamical theory is adopted: (2) To indicate the significance in this theory of the numerical results deduced from Regnault's observations on steam: (3) To point out certain remarkable relations connecting the physical properties of all substances established by reasoning analogous to that of Carnot, but founded on the dynamical theory.
This paper, though subsequent to that of Clausius, is very different in character. Many of the results are identical with those previously obtained by Clausius, but they are reached by a process which is preceded by a clear statement of fundamental principles. These principles have since been the subject of discussion, and are not free from difficulty even now; but a great step in advance was made by their careful formulation inThomson's paper, as a preliminary to the erection of the theory and the deduction of its consequences. Two propositions are stated which may be taken as the First and Second Laws of Thermodynamics. One is equivalent to the First Law as stated in p.116, the other enunciates the principle of Reversibility as a criterion of "perfection" of a heat engine. We quote these propositions.
"Prop. I (Joule).—When equal quantities of mechanical effect are produced by any means whatever from purely thermal sources, or lost in purely thermal effects, equal quantities of heat are put out of existence or are generated."
"Prop. II (Carnot and Clausius).—If an engine be such that when worked backwards, the physical and mechanical agencies in every part of its motions are all reversed, it produces as much mechanical effect as can be produced by any thermodynamic engine, with the same temperatures of source and refrigerator, from a given quantity of heat."
Prop. I was proved by assuming that heat is a form of energy and considering always the work effected by causing a working substance to pass through a closed cycle of changes, so that there was no change of internal energy to be reckoned with.
Prop. II was proved by the following "axiom": "It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects." This is rather a postulate than an axiom; for it can hardly be contended that it commands assent as soon as it is stated, even from a mind which is conversant with thermal phenomena.It sets forth clearly, however, and with sufficient guardedness of statement, a principle which, when the process by which work is done is always a cyclical one, is not found contradicted by experience, and one, moreover, which can be at once explicitly applied to demonstrate that no engine can be more efficient than a reversible one, and that therefore the efficiency of a reversible engine is independent of the nature of the working substance.
It has been suggested by Clerk Maxwell that this "axiom" is contradicted by the behaviour of a gas. According to the kinetic theory of gases an elevation of temperature consists in an increase of the kinetic energy of the translatory motion of the gaseous particles; and no doubt there actually is, from time to time, a passage of some more quickly moving particles from a portion of a gas in which the average kinetic energy is low, to a region in which the average kinetic energy is high, and thus a transference of heat from a region of low temperature to one of higher temperature. Maxwell imagined a space filled with gas to be divided into two compartments A and B by a partition in which were small massless trapdoors, to open and shut which required no expenditure of energy. At each of these doors was stationed a "sorting demon," whose duty it was to allow every particle having a velocity greater than the average to pass through from A to B, and to stop all those of smaller velocity than the average. Similarly, the demons were to prevent all quickly moving particles from going across from B to A, and to pass all slowly moving particles. In this way, without the expenditure of work, all the quickly moving particles could be assembled in onecompartment, and all the slowly moving particles in the other; and thus a difference of temperatures between the two compartments could be brought about, or a previously existing one increased by transference of heat from a colder to a hotter mass of gas.
Contrary to a not uncommon belief, this process does not invalidate Thomson's axiom as he intended it to be understood. For the gas referred to here is what he would have regarded as the working substance of the engine, by the cycles of which all the mechanical effect was derived; and it is not, at the end of the process, in the state as regards average kinetic energy of the particles in which it was at first. That this was his answer to the implied criticism of his axiom contained in Maxwell's illustration, those who have heard him refer to the matter in his lectures are well aware. But of course it is to be understood that the substance returns to the same state only in a statistical sense.
Thomson's demonstration that a reversible engine is the most efficient is well known, and need not here be repeated in detail. The reversible engine may be worked backwards, and the working substance will take in heat where in the direct action it gave it out, andvice versa: the substance will do work against external forces where in the direct action it had work done upon it, andvice versa: in short, all the physical and mechanical changes will be of the same amount, but merely reversed, at every stage of the backward process. Thus if an engine A be more efficient than a reversible one B, it will convert a larger percentage of an amount of heatHtaken in at the source into work than would the reversible one working between the same temperatures. Thus ifhbe the heat givento the refrigerator by A, andh'that given by B when both work directly and take inH;hmust be less thanh'. Then couple the engines together so that B works backwards while A works directly. A will take inHand deliverh, and do work equivalent toH-h. B will takeh'from the refrigerator and deliverHto the source, and have work equivalent toH-h'spent upon it. There will be no heat on the whole given to or taken from the source; but heath'-hwill be taken from the refrigerator, and work equivalent to this will be done. Thusby a cyclical process, which leaves the working substance as it was, work is done at the expense of heat taken from the refrigerator, which Thomson's postulate affirms to be impossible. Therefore the assumption that an engine more efficient than the reversible engine exists must be abandoned; and we have the conclusion that all reversible engines are equally efficient.
Thomson acknowledged in his paper the priority of Clausius in his proof of this proposition, but stated that this demonstration had occurred to him before he was aware that Clausius had dealt with the matter. He now cited, as examples of the First Law of Thermodynamics, the results of Joule's experiments regarding the heat produced in the circuits of magneto-electric machines, and the fact that when an electric current produced by a thermal agency or by a battery drives a motor, the heat evolved in the circuit by the passage of the current is lessened by the equivalent of the work done on the motor.
Fig. 12.Fig. 12.
Fig. 12.
In the Carnot cycle, the first operation is an isothermal expansion (ABin Fig.12), in which the substance increases in volume bydv, and takes in fromthe source heat of amountMdv. The second operation is an adiabatic expansion,BC, in which the volume is further increased and the temperature sinks bydtto the temperature of the refrigerator. The third operation is an isothermal compression,CD, until the volume and pressure are such that an adiabatic compressionDAwill just bring the substance back to the original state. If ∂p⁄ ∂tbe the rate of increase of pressure with temperature when the volume is constant, the step of pressure from one isothermal to the other is∂p ⁄ ∂t . dt; and thus the area of the closed cycle in the diagram which measures the external work done in the succession of changes is∂p ⁄ ∂t . dtdv. Now, by the second law, the work done must be a certain fraction of the work-equivalent of the heat,Mdv, taken in from the source. This fraction is independent of the nature of the working substance, but varies with the temperature, and istherefore a function of the temperature. Its ratio to the difference of temperaturedtbetween source and refrigerator was called "Carnot's function," and the determination of this function by experiment was at first perhaps the most important problem of thermodynamics. Denoting it byμ, we have the equation
which may be taken as expressing in mathematical language the second law of thermodynamics.Mis here so chosen thatMdvis the heat expressed in units of work, so that μ does not involve Joule's equivalent of heat. This equation was given by Carnot: it is here obtained by the dynamical theory which regards the work done as accounted for by disappearance, not transference merely, of heat.
The work done in the cycle becomes nowμMdtdv, or ifHdenoteMdv, it isμHdt. The fraction of the heat utilised is thusμdt. This is called theefficiencyof the engine for the cycle.
From the first law Thomson obtained another fundamental equation. For every substance there is a relation connecting the pressurep(or more generally the stress of some type), the volumev(or the configuration according to the specified stress), and the temperature. We may therefore take arbitrary changes of any two of these quantities: the relation referred to will give the corresponding change of the third. Thomson chosevandtas the quantities to be varied, and supposed them to sustain arbitrary small changesdvanddtin consequence of the passage of heat to thesubstance from without. The amount of heat taken in isMdv+Ndt, whereMdvandNdtare heats required for the changes taken separately. But the substance expanding throughdvdoes external work pdv. Thus the net amount of energy given to the substance from without isMdv+Ndt−pdvor (M−p)dv+Ndt; and if the substance is made to pass through a cycle of changes so that it returns to the physical state from which it started, the whole energy received in the cycle must be zero. From this it follows that the rate of variation ofM−pwhen the temperature but not the volume varies, is equal to the rate of variation ofNwhen the volume but not the temperature varies. To see that this relation holds, the reader unacquainted with the properties of perfect differentials may proceed thus. Let the substance be subjected to the infinitesimal closed cycle of changes defined by (1) a variation consisting of the simultaneous changesdv,dtof volume and temperature, (2) a variation −dvof volume only, (3) a variation −dtof temperature only.M−pandNvary so as to have definite values for the beginning and end of each step, and the proper mean values can be written down for each step at once, and therefore the value of (M−p)dv+Ndtobtained. Adding together these values for the three steps we get the integral for the cycle. The condition that this should vanish is at once seen to be the relation stated above.
This result combined with the equationAderived from the second law, gives an important expression for Carnot's function.
We shall not pursue this discussion further: so much is given to make clear how certain results as tothe physical properties of substances were obtained, and to explain Thomson's scale of absolute thermodynamic temperature, which is by far the most important discovery within the range of theoretical thermodynamics.
There are several scales of temperature: in point of fact the scale of a mercury-in-glass thermometer is defined by the process of graduation, and therefore there are as many such scales as there are thermometers, since no two specimens of glass expand in precisely the same way. Equal differences of temperature do not correspond to equal increments of volume of the mercury: for the glass envelope expands also and in its own way. On the scale of a constant pressure gas thermometer changes of temperature are measured by variations of volume of the gas, while the pressure is maintained constant; on a constant volume gas thermometer changes of temperature are measured by alterations of pressure while the volume of the gas is kept constant. Each scale has its own independent definition, thus if the pressure of the gas be kept constant, and the volume at temperature 0° C. bev0and that at any other temperature bev1we define the numerical valuet, this latter temperature, by the equationv=v0(1 +Et), whereEis 1 ⁄ 100 of the increase of volume sustained by the gas in being raised from 0° C. to 100° C. These are the temperatures of reference on an ordinary centigrade thermometer, that is, the temperature of melting ice and of saturated steam under standard atmospheric pressure, respectively. Thusthas the value (v⁄v0− 1) ⁄E, and is the temperature (on the constant pressure scale of the gas thermometer) corresponding to the volumev. Equaldifferences of temperature are such as correspond to equal increments of the volume at 0° C.
Similarly, on the constant volume scale we obtain a definition of temperature from the pressurep, by the equationt= (p⁄p0− 1) ⁄E', wherep0is the pressure at 0° C., andE'is 1 ⁄ 100 of the change of pressure produced by raising the temperature from 0° C. to 100° C.
For airEis approximately 1 ⁄ 273, and thust= 273 (v−v0) ⁄v0. If we take the case ofv= 0, we gett= − 273. Now, although this temperature may be inaccessible, we may take it as zero, and the temperature denoted bytis, when reckoned from this zero, 273 +t. This zero is called the absolute zero on the constant pressure air thermometer. The value ofE'is very nearly the same as that ofE; and we get in a similar manner an absolute zero for the constant volume scale. If the gas obeyed Boyle's law exactly at all temperatures,Ewould not differ fromE'.
It was suggested to Thomson by Joule, in a letter dated December 9, 1848, that the value ofμmight be given by the equationμ=JE⁄ (1 +Et). Here we take heat in dynamical units, and therefore the factorJis not required. With these units Joule's suggestion is thatμ=E⁄ (1 +Et), or withE= 1 ⁄ 273μ= 1 ⁄ (273 +t), that is,μ= 1 ⁄TwhereTis the temperature reckoned in centigrade degrees from the absolute zero of the constant pressure air thermometer.
The possibility of adopting this value of μ was shown by Thomson to depend on whether or not the heat absorbed by a given mass of gas in expanding without alteration of temperature is the equivalent of the work done by the expanding gas against external pressure.The heatHabsorbed by the air in expanding from volumeVto another volumeV'at constant temperature is the integral ofMdvtaken from the former volume to the latter. But by the value ofMgiven on p.121, ifWbe the integral ofpdv, that is the work done by the air in the expansion, ∂W⁄ ∂t= μH. The equation fulfilled by the gas at constant pressure (the defining equation fort),v=v0(1 +Et), gives for the integral ofpdv, that isW, the equationW=pv0(1 +Et) log (V'⁄V), so that ∂W⁄ ∂t=EW⁄ (1 +Et). Thus μH=EW⁄ (1 +Et).
Hence it follows that if μ =E⁄ (1 +Et), the value ofHwill be simplyW. Thus Joule's suggested value of μ is only admissible if the work done by the gas in expanding from a given volume to any other is the equivalent of the heat absorbed; or, which is the same thing, if the external work done in compressing the gas from one volume to another is the equivalent of the heat developed.
This result naturally suggests the formation of a new scale of thermometry by the adoption of the defining relationT= 1 ⁄ μ, whereTdenotes temperature. A scale of temperature thus defined is proposed in the paper by Joule and Thomson, "On the Thermal Effects of Fluids in Motion," Part II, which was published in thePhilosophical Transactionsfor June 1854, and is what is now universally known as Thomson's scale of absolute thermodynamic temperature. It can, of course, be made to give 100 as the numerical value of the temperature difference between 0° C. and 100° C. by properly fixing the unit ofT. This scale was the natural successor, in the dynamical theory, of one which Thomson had suggested in 1848, and whichwas founded, according to Carnot's idea, on the condition that a unit of heat should do the same amount of work in descending through each degree. This, as he pointed out, might justly be called anabsolutescale, since it would be independent of the physical properties of any substance. In the same sense the scale defined byT= 1 ⁄ μ is truly an absolute scale.
The new scale gives a simple expression for the efficiency of a perfect engine working between two physically given temperatures, and assigns the numerical values of these temperatures; for the heatHtaken in from the source in the isothermal expansion which forms the first operation of the cycle (p.120) isMdv, and, as we have seen, the work done in the cycle is∂p⁄∂t.dtdv, or μHdt. If we adopt the expression 1 ⁄Tfor μ, we may putdTfordt; and we obtain for the work done the expressionHdT⁄T. The work done is thus the fractiondT⁄Tof the heat taken in, and this is what is properly called the efficiency of the engine for the cycle.
If we suppose the difference of temperatures between source and refrigerator to be finite,T−T', say, then sinceTis the temperature of the source, we have for the efficiency (T−T') ⁄T. If the heat taken in beH, the heat rejected isHT'⁄T, so that the heat received by the engine is to the heat rejected by it in the ratio ofT'toT. Thus, as was done by Thomson, we may define the temperatures of the source and refrigerator as proportional to the heat taken in from the source and the heat rejected to the refrigerator by a perfect engine, working between those temperatures. The scale may be made to have 100 degrees between the temperature of melting ice and the boiling point,as already explained. We shall return to the comparison of this scale with that of the air thermometer. At present we consider some of the thermodynamic relations of the properties of bodies arrived at by Thomson.
First we take the working substance of the engine as consisting of matter in two states or phases; for example, ice and water, or water and saturated steam. Let us apply equation (A) to this case. Ifv1,v2be the volume of unit of mass in the first and second states respectively, the isothermal expansion of the first part of the cycle will take place in consequence of the conversion of a massdmfrom the first state to the second. Thusdv, the change of volume, isdm(v2−v1). Also ifLbe the latent heat of the substance in the second state,e.g.the latent heat of water,Mdv=Ldm; so thatM(v2−v1) =L. Ifdpbe the step of pressure corresponding to the stepdTof temperature, equation (A) becomes
In the case of coexistence of the liquid and solid phases, this gives us the very remarkable result that a change of pressuredpwill raise or lower the temperature of coexistence of the two phases, that is, the melting point of the solid, by the difference of temperature,dT, according asv2is greater or less thanv1Thus a substance like water, which expands in freezing, so thatv2−v1is negative, has its freezing point lowered by increase of pressure and raised by diminution of pressure. This is the result predicted by Professor James Thomson and verified experimentally by his brother (p.113above). On the other hand, a substance like paraffin wax,which contracts in solidifying, would have its melting point raised by increase of pressure and lowered by a diminution of pressure.
The same conclusions would be applicable when the phases are liquid and vapour of the same substance, if there were any case in whichv2−v1is negative. As it is we see, what is well known to be the case, that the temperature of equilibrium of a liquid with its vapour is raised by increase of pressure.
Another important result of equation (B), as applied to the liquid and vapour phases of a substance, is the information which it gives as to the density of the saturated vapour. When the two phases coexist the pressure is a function of the temperature only. Hence if the relation of pressure to temperature is known,dp⁄dTcan be calculated, or obtained graphically from a curve; and the volumev2per unit mass of the vapour will be given in terms ofdp⁄dT, the temperatureT, and the volumevper unit mass of the liquid. The density of saturated steam at different temperatures is very difficult to measure experimentally with any approach of accuracy: but so far as experiment goes equation (B) is confirmed. The theory here given is fully confirmed by other results, and equation (B) is available for the calculation ofv2for any substance for which the relation betweenpandTis known. It is thus that the density of saturated steam can best be found.
We can obtain another important result for the case of the working substance in two phases from equation (B). The relation is
wherecandhare the specific heats of the substance in the two phases respectively, andLis the latent heat of the second phase at absolute temperatureT.
We shall obtain the relation in another way, which will illustrate another mode of dealing with a cycle of operations which Thomson employed. Any small step of change of a substance may be regarded as made up of a step of volume, say, followed by a step of temperature, that is, by an isothermal step followed by an adiabatic step. In this way any cycle of operations whatever may be regarded as made up of a series of Carnot cycles. But without regarding any cycle of a more general kind than Carnot's as thus compounded, we can draw conclusions from it by the dynamical theory provided only it is reversible. Suppose a gramme, say, of the substance to be taken at a specified temperatureTin the lower phase, and to be changed to the other phase at that temperature. The heat taken in will beLand the expansion will bev2−v1. Next, keeping the substance in the second phase, and in equilibrium with the first phase (that is, for example, if the second phase is saturated vapour, the saturation is to continue in the further change), let the substance be lowered in temperature bydT. The heat given out by the substance will behdT, wherehis the specific heat of the substance in the second phase. Now at the new temperatureT−dTlet the substance be wholly brought back to the second phase; the heat given out will beL−∂L⁄∂T.dT. Finally, let the substance, now again all in the first phase, be brought to the original temperature: the heat taken in will becdt, wherecis the specific heat in the first phase. Thus the net excess of heat taken in over heat givenout in the cycle is (∂L⁄∂T+c−h)dT. This must, in the indicator diagram for the changes specified, be the area of the cycle or (v2−v1)∂p⁄∂T.dT. But by equation (B)L⁄T(v2−v1) =∂p⁄∂T, and the area of the cycle is (L⁄T)dT. Equating the two expressions thus found for the area we get equation (C).
This relation was arrived at by Clausius in his paper referred to above, and the priority of publication is his: it is here given in the form which it takes when Thomson's scale of absolute temperature is used.
Regnault's experimental results for the heat required to raise unit mass of water from the temperature of melting ice to any higher temperature and evaporate it at that temperature enable the values ofL⁄Tand∂L⁄∂Tto be calculated, and therefore that ofhto be found. It appears thathis negative for all the temperatures to which Regnault's experimental results can be held to apply. This, as was pointed out by Thomson, means that if a mass of saturated vapour is made to expand so as at the same time to fall in temperature, it must have heat given to it, otherwise it will be partly condensed into liquid; and, on the other hand, if the vapour be compressed and made to rise in temperature while at the same time it is kept saturated, heat must be taken from it, otherwise the vapour will become superheated and so cease to be saturated.
It is convenient to notice here the article onHeatwhich Thomson wrote for the ninth edition of theEncyclopædia Britannica. In that article he gave a valuable discussion of ordinary thermometry, of thermometry by means of the pressures of saturated vapour of different substances—steam-pressure thermometers, he called them—of absolute thermodynamic thermometry,all enriched with new experimental and theoretical investigations, and appended to the whole a valuable synopsis, with additions of his own, of the Fourier mathematics of heat conduction.
First dealing with temperature as measured by the expansion of a liquid in a less expansible vessel, he showed how it is in reality numerically reckoned. This amounted to a discussion of the scale of an ordinary mercury-in-glass thermometer, a subject concerning which erroneous statements are not infrequently made in text-books. A sketch of Thomson's treatment of it is given here.
Considering this thermometer as a vessel consisting of a glass bulb and a long glass stem of fine and uniform bore, hermetically sealed and containing only mercury and mercury vapour, he explained the numerical relation between the temperature as shown by the instrument and the volumes of the mercury and vessel. The scale is really defined by the method of graduation adopted. Two points of reference are marked on the stem at which the top of the mercury stands when the vessel is immersed (1) in melting ice, (2) in saturated steam under standard atmospheric pressure. The stem is divided into parts of equal volume of bore between these two points and beyond each of them. For a centigrade thermometer the bore-space between the two points is divided into 100 equal parts, and the lower point of reference is marked 0 and the upper 100, and the other dividing marks are numbered in accordance with this along the stem. Each of these parts of the bore may be called a degree-space.
Now let the instrument contain in its bulb and stem, up to the mark 0,Ndegree-spaces, and letvbethe volume of a degree-space at that temperature. The volume up to the mark 0 will beNv, at that temperature; and if the substance of the vessel be quite uniform in quality and free from stress,Nwill be the same for all temperatures. Ifv0be the volume of a degree-space at the temperature of melting ice the volume of the mercury at that temperature will beNv0. IfGbe the expansion of the glass when the volume of a degree-space is increased fromv0tovby the rise of temperature, thenv=v0(1 +G). The volume of the mercury has been increased therefore to (N+n)v0(1 +G) by the same rise of temperature, if the top of the column is thereby made to rise from the mark 0 so as to occupyndegree-spaces more than before. But ifEbe the expansion of the mercury between the temperature of melting ice and that which has now been attained, the volume of the mercury is alsoNv0(1 +E). HenceN(1 +E) = (N+n) (1 +G). This givesn=N(E−G) ⁄ (1 +G).
If we take, as is usual,nas measuring the temperature, and substitute for it the symbolt, we have, sinceN= 100 (1 +G100) ⁄ (E100−G100),