CHAPTER VIII

THERMODYNAMICS AND ABSOLUTE THERMOMETRY

The first 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 in _Poggendorff's Annalen_ in March and April 1850, and in the _Philosophical Magazine_ for July 1850, under a t.i.tle 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 quant.i.ty 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 pa.s.s from hotter to colder bodies. In 1854 (_Pogg. Ann._, Dec. 1854) he stated his fundamental principle explicitly in the form: "Heat can never pa.s.s 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 pa.s.s 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 the _Transactions 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 a.n.a.logous 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 in Thomson'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 quant.i.ties of mechanical effect are produced by any means whatever from purely thermal sources, or lost in purely thermal effects, equal quant.i.ties 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 quant.i.ty of heat."

Prop. I was proved by a.s.suming that heat is a form of energy and considering always the work effected by causing a working substance to pa.s.s 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 a.s.sent 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 pa.s.sage 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 s.p.a.ce filled with gas to be divided into two compartments A and B by a part.i.tion in which were small ma.s.sless 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 pa.s.s 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 pa.s.s all slowly moving particles. In this way, without the expenditure of work, all the quickly moving particles could be a.s.sembled in one compartment, 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 ma.s.s 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 ill.u.s.tration, 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, and _vice versa_: the substance will do work against external forces where in the direct action it had work done upon it, and _vice 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 heat H taken in at the source into work than would the reversible one working between the same temperatures. Thus if h be the heat given to the refrigerator by A, and h' that given by B when both work directly and take in H; h must be less than h'. Then couple the engines together so that B works backwards while A works directly. A will take in H and deliver h, and do work equivalent to H - h. B will take h' from the refrigerator and deliver H to the source, and have work equivalent to H - h' spent upon it. There will be no heat on the whole given to or taken from the source; but heat h' - h will be taken from the refrigerator, and work equivalent to this will be done.

Thus _by 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 a.s.sumption 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 pa.s.sage of the current is lessened by the equivalent of the work done on the motor.

[Ill.u.s.tration: FIG. 12.]

In the Carnot cycle, the first operation is an isothermal expansion (AB in Fig. 12), in which the substance increases in volume by dv, and takes in from the source heat of amount Mdv. The second operation is an adiabatic expansion, BC, in which the volume is further increased and the temperature sinks by dt to the temperature of the refrigerator. The third operation is an isothermal compression, CD, until the volume and pressure are such that an adiabatic compression DA will just bring the substance back to the original state. If ?p??t be 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 is therefore a function of the temperature. Its ratio to the difference of temperature dt between 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 by , we have the equation

?p??t = M ... (A)

which may be taken as expressing in mathematical language the second law of thermodynamics. M is here so chosen that Mdv is the heat expressed in units of work, so that 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 now Mdtdv, or if H denote Mdv, it is Hdt. The fraction of the heat utilised is thus dt. This is called the efficiency of the engine for the cycle.

From the first law Thomson obtained another fundamental equation. For every substance there is a relation connecting the pressure p (or more generally the stress of some type), the volume v (or the configuration according to the specified stress), and the temperature. We may therefore take arbitrary changes of any two of these quant.i.ties: the relation referred to will give the corresponding change of the third.

Thomson chose v and t as the quant.i.ties to be varied, and supposed them to sustain arbitrary small changes dv and dt in consequence of the pa.s.sage of heat to the substance from without. The amount of heat taken in is Mdv + Ndt, where Mdv and Ndt are heats required for the changes taken separately. But the substance expanding through dv does external work pdv. Thus the net amount of energy given to the substance from without is Mdv + Ndt - pdv or (M - p)dv + Ndt; and if the substance is made to pa.s.s 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 of M - p when the temperature but not the volume varies, is equal to the rate of variation of N when 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 changes dv, dt of volume and temperature, (2) a variation -dv of volume only, (3) a variation -dt of temperature only. M - p and N vary 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 + Ndt obtained. 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 equation A derived 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 to the 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-gla.s.s 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 gla.s.s expand in precisely the same way. Equal differences of temperature do not correspond to equal increments of volume of the mercury: for the gla.s.s 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. be v0 and that at any other temperature be v1 we define the numerical value t, this latter temperature, by the equation v = v0(1 + Et), where E is 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. Thus t has the value (v?v0 - 1)?E, and is the temperature (on the constant pressure scale of the gas thermometer) corresponding to the volume v.

Equal differences 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 pressure p, by the equation t = (p?p0 - 1)?E', where p0 is the pressure at 0 C., and E' is 1?100 of the change of pressure produced by raising the temperature from 0 C. to 100 C.

For air E is approximately 1?273, and thus t = 273(v - v0)?v0.

If we take the case of v = 0, we get t = -273. Now, although this temperature may be inaccessible, we may take it as zero, and the temperature denoted by t is, when reckoned from this zero, 273 + t.

This zero is called the absolute zero on the constant pressure air thermometer. The value of E' is very nearly the same as that of E; 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, E would not differ from E'.

It was suggested to Thomson by Joule, in a letter dated December 9, 1848, that the value of might be given by the equation = JE?(1 + Et). Here we take heat in dynamical units, and therefore the factor J is not required. With these units Joule's suggestion is that = E?(1 + Et), or with E = 1?273 = 1?(273 + t), that is, = 1?T where T is the temperature reckoned in centigrade degrees from the absolute zero of the constant pressure air thermometer.

The possibility of adopting this value of was shown by Thomson to depend on whether or not the heat absorbed by a given ma.s.s of gas in expanding without alteration of temperature is the equivalent of the work done by the expanding gas against external pressure. The heat H absorbed by the air in expanding from volume V to another volume V' at constant temperature is the integral of Mdv taken from the former volume to the latter. But by the value of M given on p. 121, if W be the integral of pdv, that is the work done by the air in the expansion, ?W??t = H. The equation fulfilled by the gas at constant pressure (the defining equation for t), v = v0(1 + Et), gives for the integral of pdv, that is W, the equation W = pv0(1 + Et)log(V'?V), so that ?W??t = EW?(1 + Et). Thus H = EW?(1 + Et).

Hence it follows that if = E?(1 + Et), the value of H will be simply W. Thus Joule's suggested value of 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 relation T = 1?, where T denotes 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 the _Philosophical Transactions_ for 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 of T.

This scale was the natural successor, in the dynamical theory, of one which Thomson had suggested in 1848, and which was 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 an absolute scale, since it would be independent of the physical properties of any substance. In the same sense the scale defined by T = 1? 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 a.s.signs the numerical values of these temperatures; for the heat H taken in from the source in the isothermal expansion which forms the first operation of the cycle (p. 120) is Mdv, and, as we have seen, the work done in the cycle is ?p??t.dtdv, or Hdt. If we adopt the expression 1?T for , we may put dT for dt; and we obtain for the work done the expression HdT?T. The work done is thus the fraction dT?T of 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 since T is the temperature of the source, we have for the efficiency (T - T')?T. If the heat taken in be H, the heat rejected is HT'?T, so that the heat received by the engine is to the heat rejected by it in the ratio of T' to T. 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. If v1, v2 be the volume of unit of ma.s.s 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 ma.s.s dm from the first state to the second. Thus dv, the change of volume, is dm(v2 - v1). Also if L be the latent heat of the substance in the second state, _e.g._ the latent heat of water, Mdv = Ldm; so that M(v2 - v1) = L. If dp be the step of pressure corresponding to the step dT of temperature, equation (A) becomes

dT?T = dp(v2 - v1)?L ... (B)

In the case of coexistence of the liquid and solid phases, this gives us the very remarkable result that a change of pressure dp will 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 as v2 is greater or less than v1 Thus a substance like water, which expands in freezing, so that v2 - v1 is 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. 113 above).

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 which v2 - v1 is 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?dT can be calculated, or obtained graphically from a curve; and the volume v2 per unit ma.s.s of the vapour will be given in terms of dp?dT, the temperature T, and the volume v per unit ma.s.s 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 of v2 for any substance for which the relation between p and T is 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

?L??T + c - h = L?T ... (C)

where c and h are the specific heats of the substance in the two phases respectively, and L is the latent heat of the second phase at absolute temperature T.

We shall obtain the relation in another way, which will ill.u.s.trate 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 temperature T in the lower phase, and to be changed to the other phase at that temperature. The heat taken in will be L and the expansion will be v2 - 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 by dT. The heat given out by the substance will be hdT, where h is the specific heat of the substance in the second phase. Now at the new temperature T - dT let the substance be wholly brought back to the second phase; the heat given out will be L - ?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 be cdt, where c is the specific heat in the first phase.

Thus the net excess of heat taken in over heat given out 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 ma.s.s of water from the temperature of melting ice to any higher temperature and evaporate it at that temperature enable the values of L?T and ?L??T to be calculated, and therefore that of h to be found. It appears that h is 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 ma.s.s 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 on Heat which Thomson wrote for the ninth edition of the _Encyclopaedia 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-gla.s.s 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.