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 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 never pass 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 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 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 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 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 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 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, 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 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.

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 μ, we have the equation

∂p⧸∂t = μM ... (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 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 now μMdtdv, or if H denote Mdv, it is μHdt. The fraction of the heat utilised is thus μdt. 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 quantities: the relation referred to will give the corresponding change of the third. Thomson chose v and t as the quantities to be varied, and supposed them to sustain arbitrary small changes dv and dt in consequence of the passage 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 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 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-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. 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 μ might be given by the equation μ = 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 μ = E⧸(1 + Et), or with E = 1⧸273 μ = 1⧸(273 + t), that is, μ = 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 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 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 = μH. 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 μH = EW⧸(1 + Et).

Hence it follows that if μ = E⧸(1 + Et), the value of H will be simply W. 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 relation T = 1⧸μ, 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⧸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 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 μHdt. If we adopt the expression 1⧸T for μ, 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 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 mass 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 mass of the vapour will be given in terms of dp⧸dT, the temperature T, and the volume v per 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 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 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 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 mass 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 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 on Heat which Thomson wrote for the ninth edition of the _Encyclopædia 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, N degree-spaces, and let v be the volume of a degree-space at that temperature. The volume up to the mark 0 will be Nv, at that temperature; and if the substance of the vessel be quite uniform in quality and free from stress, N will be the same for all temperatures. If v₀ be the volume of a degree-space at the temperature of melting ice the volume of the mercury at that temperature will be Nv₀. If G be the expansion of the glass when the volume of a degree-space is increased from v₀ to v by the rise of temperature, then v = 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 occupy n degree-spaces more than before. But if E be the expansion of the mercury between the temperature of melting ice and that which has now been attained, the volume of the mercury is also Nv₀(1 + E). Hence N(1 + E) = (N + n)(1 + G). This gives n = N(E － G)⧸(1 + G).

If we take, as is usual, n as measuring the temperature, and substitute for it the symbol t, we have, since N = 100(1 + G₁₀₀)⧸(E₁₀₀ － G₁₀₀),

t = 100 {(1 + G₁₀₀)⧸(1 + G)} {(E － G)⧸(E₁₀₀ － G₁₀₀)} ... (D)

In this reckoning the definition of any temperature, let us say 37° C., is the temperature of the vessel and its contents when the top of the mercury column stands at the mark 37 above 0, on the scale defined by the graduation of the instrument; but the numerical signification with relation to the volumes is given by equation (D). This shows that the numerical measure of any temperature involves both the expansion of the vessel and that of the glass vessel between the temperature of melting ice and the temperature in question. This result may be contrasted with the erroneous statement frequently made that equal increments of temperature correspond to equal increments of the volume of the thermometric substance. It also shows that different mercury-in-glass thermometers, however accurately made and graduated, need not agree when placed in a bath at any other temperature than 0° C. or 100° C. This fact, and the results of the comparison of thermometers made with different kinds of glass with the normal air thermometer, which was carried out by Regnault, were always insisted on by Thomson in his teaching when he dealt with the subject of heat. The scale of a mercury-in-glass thermometer is too often in text-books, and even in Acts of Parliament regarded as a perfectly definite thing, and the expansion of a gas is not infrequently defined by this indefinite scale, instead of being used as it ought to be, as the basis of definition of the scale of the gas thermometer. The whole treatment of the so-called gaseous laws is too often, from a logical point of view, a mass of confusion.

In his article on Heat Thomson gave two definitions of the scale of absolute temperature. One is that stated on p. 126 above, namely, that the temperature of the source and refrigerator are in the ratio of the heat taken in from the source to the heat given to the refrigerator, when the engine describes a Carnot cycle consisting of two isothermal and two adiabatic changes.

The other definition is better adapted for general use, as it applies to any cycle whatever which is reversible. Let the working substance expand under constant pressure by an amount dv (AB' in Fig. 12), and let heat H be given to the substance at the same time. The external work done is pdv. Thomson called pdv⧸H the work ratio. Now let the temperature be raised by dT without giving heat to the substance or taking heat from it, and let the corresponding pressure rise be dp; and call dp⧸p the pressure ratio. The temperature ratio dT⧸T is equal to the product of the work ratio and the pressure ratio, that is,

dT⧸T = dvdp⧸H

This is clearly true; for dvdp is the area of a cycle like AB'C'D, represented in Fig. 12, for which an amount of heat H is taken in, though not in this case strictly at one temperature. And clearly, since in Fig. 12 the change from B' to B is adiabatic, H is the heat which would have to be taken in for the isothermal change AB in the Carnot cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of the cycle is dvdp⧸H, and this by the former definition is dT⧸T.

Or we may regard the matter thus:--The amount of heat H which corresponds to an infinitesimal expansion dv may be used in equation (A) whether the expansion is isothermal or not, if we take T as the average temperature of the expansion. Hence we have dp⧸dT = H⧸(dv.T), that is, dT⧸T = dpdv⧸H. The theorem on p. 128 is obtained by what is virtually this process.

COMPARISON OF ABSOLUTE SCALE WITH SCALE OF AIR THERMOMETER

The comparison which Joule and Thomson carried out of the absolute thermodynamic scale with the scale of the constant pressure gas thermometer has already been referred to, and it has been shown that the two scales would exactly agree, that is, absolute temperature would be simply proportional to the volume of the gas in a gas thermometer kept at the temperature to be measured, if the internal energy of the gas were not altered by an alteration of volume without alteration of temperature, that is, if the de － ∂e of p. 107 above were zero. Joule tested whether this was the case by immersing two vessels, connected by a tube which could be opened or closed by a stopcock, in the water of a calorimeter, ascertaining the temperature with a very sensitive thermometer, and then allowing air which had already been compressed into one of the vessels to flow into the other, which was initially empty. It was found that no alteration of temperature of the water of the calorimeter that could be observed was produced. But the volume of the air had been doubled by the process, and if any sensible alteration of internal energy had taken place it would have shown itself by an elevation or a lowering of the temperature of the water, according as the energy had been diminished or increased.

Thomson suggested that the gas to be examined should be forced through a pipe ending in a fine nozzle, or, preferably, through a plug of porous material placed in a pipe along which the gas was forced by a pump, and observations made of the temperature in the steady stream on both sides of the plug. The experiments were carried out with a plug of compressed cotton-wool held between two metal disks pierced with holes, in a tube of boxwood surrounded also by cotton-wool, and placed in a bath of water closely surrounding the supply pipe. This was of metal, and formed the end of a long spiral all immersed in the bath. Thus the temperature of the gas approaching the plug was kept at a uniform temperature determined by a delicate thermometer; another thermometer gave the temperature in the steady stream beyond the plug.

In the case of hydrogen the experiments showed a slight heating effect of passage through the plug; air, oxygen, nitrogen and carbonic acid were cooled by the passage.

The theory of the matter is set forth in the original papers, and in a very elegant manner in the article on Heat. The result of the analysis shows that if ∂w be the positive or negative work-value of the heat which will convert one gramme of the gas after passage to its original temperature; and T be absolute temperature, and v volume of a gramme of the gas at pressure p, and the difference of pressure on the two sides of the plug be dp, the equation which holds is

(1⧸T) (∂T⧸∂v) = 1⧸{v + (∂w⧸dp)} ... (E)[18]

It was found by Joule and Thomson that ∂w was proportional to dp for values of dp up to five or six atmospheres. At different temperatures, however, in the case of hydrogen the heating effect was found to diminish with rise of temperature, being .100 of a degree centigrade at 4° or 5° centigrade, and .155 at temperatures of from 89° to 93° centigrade for a difference of pressure due to 100 inches of mercury.

If there is neither heating nor cooling ∂w = 0, and we obtain by integration T = Cv, where C is a constant.

Elaborate discussions of the theory of this experiment will be found in modern treatises on thermodynamics, and in various recent memoirs, and the differential equation has been modified in various ways, and integrated on various suppositions, which it would be out of place to discuss here.

The cooling effect of passing a gas such as air or oxygen through a narrow orifice has been used to liquefy the gas. The stream of gas is pumped along a pipe towards the opening, and that which has passed the orifice and been slightly cooled is led on its way back to the pump along the outside of the pipe by which more gas is approaching the orifice, and so cools slightly the advancing current. The gas which emerges later is thus cooler than that which emerged before, and the process goes on until the issuing gas is liquefied and falls down into the lower part of the pipe surrounding the orifice, whence it can be drawn off into vessels constructed to receive and preserve it.

It is possible thus to liquefy hydrogen, which shows that at the low temperature at which the process is usually started (an initial cooling is applied) the passage through the orifice has a cooling effect as in the other cases.

Another idea, that of _thermodynamic motivity_, on which Thomson suggested might be founded a fruitful presentation of the subject of thermodynamics, may be mentioned here. It was set forth in a letter written to Professor Tait in May 1879. If a system of bodies be given, all at different temperatures, it is possible to reduce them to a common temperature, and by doing so to extract a certain amount of mechanical energy from them. The temperatures must for this purpose be equalised by perfect thermodynamic engines working between the final temperature T₀, say, and the temperatures of the different parts of the system. This process is one of the levelling up and the levelling down of temperature; and the temperature T₀ is such that exactly the heat given out at T₀ by certain engines, receiving heat from bodies of higher temperature than T₀, is supplied to the engines which work between T₀ and bodies at lower temperatures. The whole useful work obtained in this way was called by Thomson the motivity of the system. Of course equalisation of temperature may be obtained by conduction, and in this case the energy which might be utilised is lost. With two equal and similar bodies at absolute temperatures T, T' the temperature to which they are reduced when their motivity is extracted is √(TT'). If the temperatures are equalised by conduction the resulting temperature is higher, being ½(T + T'). Thus, if only the two bodies are available for engines to work between, the motivity is the measure of the energy lost when conduction brings about equalisation of temperature.

A very suggestive paper on the subject was published by Lord Kelvin in the _Trans. R.S.E._, vol. 28, 1877-8.

DISSIPATION OF ENERGY

In connection with the theory of heat must be mentioned Thomson's great generalisation, the theory of the dissipation of energy.[19] Most people have some notion of the meaning of the physical doctrine of conservation of energy, though in popular discourses it is usually misstated. What is meant is that in a finite material system, which is isolated in the sense that it is not acted on by force from without, the total amount of energy--that is, energy of motion and energy of relative position (including energy of chemical affinity) of the parts--remains constant. The usual misstatement is that the energy of the universe is constant. This may be true if the universe is finite; if the universe is infinite in extent the statement has no meaning. In any case, we know nothing about the universe as a whole, and therefore make no statements regarding it.

But while there is thus conservation or constancy of amount of energy in an isolated and finite material system, this energy may to residents on the system become unavailable. For useful work within such a system is done by conversion of energy from one form to another and the total amount remains unchanged. But if this conversion is prevented all processes which involve such conversion must cease, and among these are vital processes.

The unavailable form which the energy of the system with which we are directly and at present concerned, whatever may become of us ultimately, is taking, according to Thomson's theory, is universally diffused heat. How this comes about may be seen as follows. Even a perfect engine, if the refrigerator be at the lowest available temperature, rejects a quantity of heat which cannot be utilised for the performance of the work. This heat is diffused by conduction and radiation to surrounding bodies, and so to bodies more remote, and the general temperature of the system is raised. Moreover, as heat engines are imperfect there is heat rejected to the surroundings by conduction, and produced by work done against friction, so that the heat thrown on the unavailable or waste heap is still further increased.

Conduction of heat is the great agency by which energy is more and more dispersed in this unavailable form throughout the totality of material bodies. As has been seen, available motivity is continually wasted through its agency; and in the flow of heat in the earth and in the sun and other unequally heated bodies of our system the waste of energy is prodigious. Aided by convection currents in the air and in the ocean it continually equalises temperatures, but does so at an immense cost of useful energy.

Then in our insanely wasteful methods of heating our houses by open fires, of half burning the coal used in boiler furnaces, and allowing unconsumed carbon to escape into the atmosphere in enormous quantities, while a very large portion of the heat actually generated is allowed to escape up chimneys with heated gases, the store of unavailable heat is being added to at a rate which will entail great distress, if not ruin, on humanity at no indefinitely distant future. It will be the height of imprudence to trust to the prospect, not infrequently referred to at the present time, of drawing on the energy locked up in the atomic structure of matter. He would be a foolish man who would wastefully squander the wealth he possesses, in the belief that he can recoup himself from mines which all experience so far shows require an expenditure to work them far beyond any return that has as yet been obtained.

It is not apart from our present theme to urge that it is high time the question of the national economy of fuel, and the desirability of utilising by afforestation the solar energy continually going to waste on the surface of the earth, were dealt with by statesmen. If statesmen would but make themselves acquainted with the results of physical science in this magnificent region of cosmic economics there would be some hope, but, alas! as a rule their education is one which inevitably leads to neglect, if not to disdain of physical teaching.

From the causes which have been referred to, energy is continually being dissipated, not destroyed, but locked up in greater and greater quantity in the general heat of bodies. There is always friction, always heat conduction and convection, so that as our stores of motional or positional energy, whether of chemical substances uncombined, the earth's motion, or what not, are drawn upon, the inevitable fraction, too often a large proportion, is shed off and the general temperature raised. After a large part of the whole existent energy has gone thus to raise the dead level of things, no difference of temperature adequate for heat engines to work between will be possible, and the inevitable death of all things will approach with headlong rapidity.

THERMOELASTICITY AND THERMOELECTRICITY

In the second definition of the scale of absolute temperature just discussed, stress of any type may be substituted for pressure, and the corresponding displacement s for the change of volume. Thus for a piece of elastic material put through a cycle of changes we may substitute dS for dp and Ads for dv; where A is such a factor that AdSds is the work done in the displacement ds by the stress dS. As an example consider a wire subjected to simple longitudinal stress S. Longitudinal extension is produced, but this is not the only change; there is at the same time lateral contraction. However, s within certain limits is proportional to S.

Let heat dH in dynamical measure be given to the wire while the stress S is maintained constant, and let the extension increase from s to s + ds. The stress S will do work ASds _on the wire_, and the work ratio will be -ASds⧸dH. Now let the stress be increased to S + dS while the extension is kept constant, and the absolute temperature raised from T to T + dT. The stress ratio (as we may call it) is dS⧸S and the temperature ratio dT⧸T. Thus we obtain (p. 134 above)

－(dS⧸dT) = (1⧸TA) (dH⧸ds)

In his Heat article Thomson used the alteration e of strain under constant stress (that is ds⧸l, where l is the length of the wire) corresponding to an amount of heat sufficient to raise the temperature under constant stress by 1°. Hence if K be the specific heat under constant stress, and le be put for ds in the sense just stated, we have

dT = －(TedS⧸Kρ) ... (F)

where ρ is the density, since dH = KρlA.

The ratio of dH to the increase ds of the extension is positive or negative, that is, the substance absorbs or evolves heat, when strained under the condition of constant stress, according as dS⧸dT is negative or positive. Or we may put the same thing in another way which is frequently useful. If a wire subjected to constant stress has heat given to it, ds is negative or positive, in other words the wire shortens or lengthens, according as dS⧸dT is positive or negative, that is, according as the stress for a given strain is increased or diminished by increase of temperature.

It is known from experiment that a metal wire expands under constant stress when heat is given to it, and thus we learn from the equation (F) that the stress required for a given strain is diminished when the temperature of the wire is raised. Again, a strip of india-rubber stretched by a weight is shortened if its temperature is raised, consequently the stress required for a given strain is increased by rise of temperature.

These results, from a qualitative point of view, are self-evident. But from what has been set forth it will be obvious that an equation exactly similar to (F) holds whether the change ds of s is taken as before under constant stress, or at uniform temperature, or whether the change dS of S is effected adiabatically or at constant strain.

In all these cases the same equation

dT = －T (edS⧸Kρ) ... (G)

applies, with the change of meaning of dT involved.

This equation differs from that of Thomson as given in various places (_e.g._ in the _Encyclopædia Britannica_ article on Elasticity which he also wrote) in the negative sign on the right-hand side, but the difference is only apparent. According to his specification a pressure would be a positive stress, and an expansion a positive displacement, and in applying the equation to numerical examples this must be borne in mind so that the proper signs may be given to each numerical magnitude. As an example of adiabatic change, a sudden extension of the wire already referred to by an increase of stress dS may be considered. If there is not time for the passage of heat from or to the surroundings of the wire, the change of temperature will be given by equation (G).

This equation was applied by Thomson (article Elasticity) to find the relation between what he called the kinetic modulus of elasticity and the static modulus, that is, between the modulus for adiabatic strain and the modulus for isothermal strain.

The augmentation of the strain produced by raising the temperature 1° is e, and therefore edT, that is, －Te²dS⧸Kρ, is the increase of strain due to the sudden rise of temperature dT. This added to the isothermal strain produced by dS will give the whole adiabatic strain. Thus if M be the static or isothermal modulus, the adiabatic strain is dS⧸M － Te²dS⧸Kρ. If M' denote the kinetic or adiabatic modulus its value is dS divided by the whole adiabatic strain, that is, M' = M⧸(1 － MTe²⧸Kρ) and the ratio M'⧸M = 1⧸(1 － MTe²⧸Kρ).

It is well known and easy to prove, without the use of any theorem which can be properly called thermodynamic, that this ratio of moduli is equal to the ratio of the specific heat K of the substance, under the condition of constant stress, to the specific heat N under the condition of constant strain of the corresponding type. This, indeed, is self-evident if two changes of stress, one isothermal the other adiabatic, _which produce the same steps of displacement ds_, be considered, and it be remembered that the step ∂T of temperature which accompanies the adiabatic change may be regarded as made up of a step －dT of temperature, accompanying a displacement ds effected at constant stress, and then two successive steps dT and ∂T effected, at constant strain, along with the steps of stress dS. The ratio M'⧸M is easily seen to have the value (∂T + dT)⧸dt, and since －KdT + N(∂T + dT) = 0, by the adiabatic condition, the theorem is proved.

Laplace's celebrated result for air, according to which the adiabatic bulk-modulus is equal to the static bulk-modulus multiplied by the ratio of the specific heat of air pressure constant to the specific heat of air volume constant, is a particular example of this theory.

Thomson showed in the Elasticity article how, by the value of M'⧸M, derived as above from thermodynamic theory, the value of K⧸N could be obtained for different substances and for different types of stress, and gave very interesting tables of results for solids, liquids, and gases subjected to pressure-stress (bulk-modulus) and for solids subjected to longitudinal stress (Young's modulus).

The discussion as to the relation of the adiabatic and isothermal moduli of elasticity is part of a very important paper on "Thermoelastic, Thermomagnetic, and Thermoelectric Properties of Matter," which he published in the _Philosophical Magazine_ for January 1878. This was in the main a reprint of an article entitled, "On the Thermoelastic and Thermomagnetic Properties of Matter, Part I," which appeared in April 1855 in the first number of the _Quarterly Journal of Mathematics_. Only thermoelasticity was considered in this article; the thermomagnetic results had, however, been indicated in an article on "Thermomagnetism" in the second edition of the _Cyclopædia of Physical Science_, edited and in great part written by Professor J. P. Nichol, and published in 1860. For the same Cyclopædia Thomson also wrote an article entitled, "Thermo-electric, Division I.--Pyro-Electricity, or Thermo-Electricity of Non-conducting Crystals," and the enlarged _Phil. Mag._ article also contained the application of thermodynamics to this kind of thermoelectric action.

This great paper cannot be described without a good deal of mathematical analysis; but the student who has read the earlier thermodynamical papers of Thomson will have little difficulty in mastering it. It must suffice to say here that it may be regarded as giving the keynote of much of the general thermodynamic treatment of physical phenomena, which forms so large a part of the physical mathematics of the present day, and which we owe to Willard Gibbs Duhem, and other contemporary writers.

Thomson had, however, previous to the publication of this paper, applied thermodynamic theory to thermoelectric phenomena. A long series of papers containing experimental investigations, and entitled, "Electrodynamic Qualities of Metals," are placed in the second volume of his _Mathematical and Physical Papers_. This series begins with the Bakerian Lecture (published in the _Transactions of the Royal Society_ for 1856) which includes an account of the remarkable experimental work accomplished during the preceding four or five years by the volunteer laboratory corps in the newly-established physical laboratory in the old College. The subjects dealt with are the Electric Convection of Heat, Thermoelectric Inversions, the Effects of Mechanical Strain and of Magnetisation on the Thermoelectric Qualities of Metals, and the Effects of Tension and Magnetisation on the Electric Conductivity of Metals. It is only possible to give here a very short indication of the thermodynamic treatment, and of the nature of Thomson's remarkable discovery of the electric convection of heat.

It was found by Seebeck in 1822 that when a circuit is formed of two different metals (without any cell or battery) a current flows round the circuit if the two junctions are not at the same temperature. For example, if the two metals be rods of antimony and bismuth, joined at their extremities so as to form a complete circuit, and one junction be warmed while the other is kept at the ordinary temperature, a current flows across the hot junction in the direction from bismuth to antimony. Similarly, if a circuit be made of a copper wire and an iron wire, a current passes across the warmer junction from copper to iron. The current strength--other things being the same--depends on the metals used; for example, bismuth and antimony are more effective than other metals.

It was found by Peltier that when a current, say from a battery, is sent round such a circuit, that junction is cooled and that junction is heated by the passage of the current, which, being respectively heated and cooled, would without the cell have caused a current to flow in the same direction. Thus the current produced by the difference of temperature of the junctions causes an absorption of heat from the warmer junction, and an evolution of heat at the colder junction.

This naturally suggested to Thomson the consideration of a circuit of two metals, with the junctions at different temperatures, as a heat engine, of which the hot junction was the source and the cold junction the refrigerator, while the heat generated in the circuit by the current and other work performed, if there was any, was the equivalent of the difference between the heat absorbed and the heat evolved. Of course in such an arrangement there is always irreversible loss of heat by conduction; but when such losses are properly allowed for the circuit is capable of being correctly regarded as a reversible engine.

Shortly after Seebeck's discovery it was found by Cumming that when the hot junction was increased in temperature the electromotive force increased more and more slowly, at a certain temperature of the hot junction took its maximum value, and then as the temperature of the hot junction was further increased began to diminish, and ultimately, at a sufficiently high temperature, in most instances changed sign. The temperature of maximum electromotive force was found to be independent of the temperature of the colder junction. It is called the temperature of the neutral point, from the fact that if the two junctions of a thermoelectric circuit be kept at a constant small difference of temperature, and be both raised in temperature until one is at a higher temperature than the neutral point, and the other is at a lower, the electromotive force will fall off, until finally, when this point is reached, it has become zero.

Thus it was found that for every pair of metals there was at least one such temperature of the hot junction, and it was assumed, with consequences in agreement with experimental results, that when the temperature was the neutral temperature there was neither absorption nor evolution of heat at the junction. But then the source provided by the thermodynamic view just stated had ceased to exist. The current still flowed, there was evolution of heat at the cold junction, and likewise Joulean evolution of heat in the wires of the circuit in consequence of their resistance. Hence it was clear that energy must be obtained elsewhere than at the junctions. Thomson solved the problem by showing that (besides the Joulean evolution of heat) there is absorption (or evolution) of heat when a current flows in a conductor along which there is a gradient of temperature. For example, when an electric current flows along an unequally heated copper wire, heat is evolved where the current flows from the hot parts to the cold, and heat is absorbed where the flow is from cold to hot. When the hot junction is at the temperature of zero absorption or evolution of heat--the so-called neutral temperature--the heat absorbed in the flow of the circuit along the unequally heated conductors is greater than that evolved on the whole, by an amount which is the equivalent of the energy electrically expended in the circuit in the same time.

It was found, moreover, that the amount of heat absorbed by a given current in ascending or descending through a given difference of temperature is different in different metals. When the current was unit current and the temperature difference also unity, Thomson called the heat absorbed or evolved in a metal the specific heat of electricity in the metal, a name which is convenient in some ways, but misleading in others. The term rather conveys the notion that electricity has a material existence. A substance such as copper, lead, water, or mercury has a specific heat in a perfectly understood sense; electricity is not a substance, hence there cannot be in the same proper sense a specific heat of electricity.

However, this absorption and evolution of heat was investigated experimentally and mathematically by Thomson, and is generally now referred to in thermoelectric discussions as the "Thomson effect."