Treatise on Thermodynamics (1903) — part 4 of 14

are generally measured in calories (the heat equivalent of the external effects). The external work, W, is small com- pared with Q. Furthermore, most chemical processes are accompanied by a rise in temperature, or, if the initial temperature be re-established, by an external yield of heat (exothermal processes). Therefore, in thermochemistry, the heat given out to the surroundings in order to restore the initial temperature is denoted as the " positive heat effect " of the process. In our equations we shall therefore use Q (the heat absorbed) with the negative sign, in processes with positive heat effect ie g. combustion) ; with the positive sign, in those with negative heat effect {e.g. evaporation, fusion, dissociation).

§ 94. To make equation (45) suitable for thermochemistry it is expedient to denote the internal energy U of a system in a given state, by a symbol denoting its chemical con- stitution. J. Thomsen introduced a symbol of this kind. He denoted by the formulae for the atomic or molecular weight of the substances enclosed in brackets, the internal energy of a corresponding weight referred to an arbitrary zero of energy. Thus [Pb], [S], [PbS] denote the energies of an atom of lead, an atom of sulphur, and a molecule of lead sulphide respectively. In order to express the fact that the formation of a molecule of lead sulphide from its atoms is accompanied by a heat effect of 18,400 cal., the external work of the process being negligible, we put

Ui = [Pb]-f [S]; U, = [PbS]; W = 0; Q= - l'8,400cal.,

APPLICA TIONS TO NON-HOMOGENEOUS SYSTEMS. 69

and equation (45) becomes

• 18,400 cal. = [PbS] - [Pb] - [S],

or, as usually written,

[Pb] + [S] - [PbS] = 18,400 cal.

This means that the internal energy of lead and sulphur, when separate, is 18,400 calories greater than that of their combination at the same temperature. That the internal energies compared actually refer to the same material system, can be checked by the use of the molecular formulae. The equation could be simplified by selecting the uncombined state of the elements Pb and S as the zero of energy. Then (§ 64), [Pb] + [S] = 0, and

[PbS] = - 18,400 cal.

§ 95. To define accurately the state of a substance, and thereby its energy, besides its chemical nature and mass, its temperature and pressure must be given. If no special statement is made, as in the above example, mean laboratory temperature, i.e. about 18° C, is generally assumed, and the pressure is supposed to be atmospheric pressure. The pressure has, however, very little influence on the internal energy; in fact, none at alKin the case of perfect gases [equation (35)].

The state of aggregation should also be indicated. This may be done, where necessary, by using brackets for the solids, parentheses for liquids, and braces for gases. Thus [H2O], (H2O), {H2O} denote the energies of a molecule of ice, water, and water vapour respectively. Hence, for the fusion of ice at 0° C,

(H2O) - [H2O] = 80 X 18 = 1440 cal.

It is often desirable, as in the case of solid carbon, sulphur, arsenic, or isomeric compounds, to denote by some means the special modification of the substance.

These symbols may be treated like algebraic quantities, whereby considerations, which would otherwise present

70 THERMODYNAMICS.

considerable complications, 'may be materially shortened. Examples of this are given below,

§ 96. To denote the energy of a solution or mixture of several compounds, we may write the formulae for the mole- cular weights with the requisite number of molecules. Thus,

(H2SO4) + 5(H20) - (H2SO4.5H2O) = 13,100 cal.

means that the solution of 1 molecule of sulphuric acid in 5 molecules of water gives out 13,100 calories of heat. Similarly, the equation

(H2SO4) + lOCHaO) - (H2SO, . IOH2O) = 15,100 cal.

gives the heat effect on dissolving the same in ten molecules of ^\ ater. By subtracting the first equation from the second, we get

(H2SO1.5H2O) + 5(H20) - (H2SOi.l0H2O) = 2000 cal.,

i.e. on diluting a solution of 1 molecule of sulphuric acid dissolved in 5 molecules of water, by the addition of another 5 molecules of water, 2000 calories are given out.

§ 97. As a matter of experience, in very dilute solutions further dilution no longer yields any appreciable amount of heat. Thus, in indicating the internal energy of a dilute solution it is often unnecessary to give the number of mole- cules of the solvent. We write briefly

(H2SO4) + (aq.) - (H2S04aq.) = 17,900 cal.

to express the heat effect of infinite dilution of a molecule of sulphuric acid. Here (aq.) denotes any amount of water sufficient for the practical production of an infinitely dilute solution.

§ 98. Volumetric changes being very slight in chemical processes which involve only solids and liquids, the heat equivalent of the external work W (§ 93) is a negligible

APPLICATIONS TO NON-HOMOGENEOUS SYSTEMS. 71

quantity compared with the heat effect. The latter alone, then, represents the change of energy of the system —

U2 - Ui = Q.

It, therefore, depends on the initial and final states only, and not on the intermediate steps of the process. These considerations do not apply, in general, when gaseous sub- stances enter into the reaction. It is only in the combustions in the " calorimetric bomb," extensively used by Berthelot and Stohmann in their investigations, that the volume remains constant and the external work is zero. In these reactions the heat effect observed represents the total change of energy. In other cases, however, the amount of external work W may assume a considerable value, and it is materially iafluenced by the process itself. Thus, a gas may be allowed to expand, at the same time performing work, which may have any value within certain limits, from zero upwards. But since its change of energy U2 — Ui depends on the initial and final states only, a greater amount of work done against the external forces necessitates a smaller heat effect for the process, and vice verm. To find the latter, not only the change of the internal energy, but also the amount of the external work must be known. This renders necessary an account of the external conditions under which the process takes place.

§ 99. Of all the external conditions that may accompany a chemical process, constant (atmospheric) pressure is the one which is of the most practical importance : p = j)q. The external work is then, according to equation (20),

W

= -J'i;oc?V=MVi-V2); . . (46)

that is, equal to the product of the pressure and the decrease of volume. This, according to (45), gives

U2-Ui = Q+MVi-V2). . . (47)

Now, the total decrease of volume, Vi — V2, may generally

72 THERMODYNAMICS.

be put equal to the decrease of volume of the gaseous portions of the system, neglecting that of the solids and liquids. Since, by (16),

Vi - V.2 = E-(>M - n^^,

where Ui, W2 are the number of gas molecules present before and after the reaction, the heat equivalent of the external work at constant pressure is, by (46) and (34),

y- = -^— ^^ — y = j6{ni — n.2) = l-9/0(»i — n-i) cal.

The heat effect of a process at constant pressure is therefore

• Q = Ui - U2 + l-970(«i - n^2) cal. . (48)

If, for instance, one gram molecule of hydrogen and half a gram molecule of oxygen, both at 18^ C, combine at constant pressure to form water at 18' C, we put

Ui = {H2} + UO.2} ; U2 = (H2O) ; n, = % ; ih = 0; 6 = 291.

The heat of combustion is, therefore, by (48),

• Q = {H2} + ^{Os} - (H2O) + 860 cal.,

i.e. 860 cal. more than would correspond to the decrease of the internal energy, or to the combustion without the simultaneous performance of external work.

§ 100. If we write equation (47) in the form

(U + poV)-! - (U + po)i = Q, . . (49)

it will be seen that, in processes under constant pressure 2>o, the heat effect depends only on the initial and final states, just as in the case when there is no external work. The heat effect, however, is not equal to the difference of the internal energies U, but to the difference of the values of the quantity (U + j^oV) at the beginning and end of the process. This quantity is Gibbs's " heat function at constant

APPLICA TIONS To non-homogeneous systems. 73

pressure." If, then, only processes at constant pressure be considered, it will be expedient to regard the symbols {Hq}, {H2O}, etc., as representing the above function (U + |)oV), instead of simply the energy U. Thus the difference in the two values of the function will, in all cases, directly represent the heat effect. This notation is therefore adopted in the following.

§ 101. To determine the heat effect of a chemical reaction at constant pressure, the initial and final values of the heat function, U -f i^oV, of the system suffice. The general solution of this problem, therefore, amounts to finding the heat functions of all imaginable material systems in all possible states. Frequently, different ways of transition from one state of a system to another may be devised, which may serve either as a test of the theory, or as a check upon the accuracy of the observations. Thus J. Thomsen found the heat of neutralization of a solution of sodium bicarbonate with caustic soda to be :

(NaHCOg aq.) + (NaHO aq.) - (NaaCOg aq.) = 9200 cal.

He also found the heat of neutralization of carbon dioxide to be :

(CO2 aq.) + 2(NaH0 aq.) - (NaaCOa aq.) = 20,200 cal.

By subtraction

(CO2 aq.) + (NaHO aq.) - (NaHCOa aq.) = 11,000 cal.

This is the heat effect corresponding to the direct com- bination of carbon dioxide and caustic soda to form sodium bicarbonate. Berthelot verified this by direct measurement.

§ 102. Frequently, of two ways of transition, one is better adapted for calorimetric measurements than the other. Thus, the heat effect of the decomposition of hydrogen peroxide into water and oxygen cannot readily be measured directly. Thomsen therefore oxidized a solution

74 THERMODYNAMICS.

of stannous chloride in hydrochloric acid first by means of hydrogen peroxide :

(SnCla. 2HC1 aq.) + (H.2O2 aq.) - (SnCl^ aq.) = 88,800 cal.,

then by means of oxygen gas :

(SnCls . 2HC1 aq,) + MO2) - (SnCl^ aq.) = 65,700 cal.

Subtraction gives

(H2O2 aq.) - ilO^} - (aq.) = 23,100 cal.

for the heat effect of the decomposition of dissolved hydro- gen peroxide into oxygen and water.

§ 103. The lieat of formation of carbon monoxide from solid carbon and oxygen cannot be directly determined, because carbon never burns completely to carbon monoxide, but always, in part, to carbon dioxide as well. Therefore Favre and Silbermann determined the heat effect of the complete combustion of carbon to carbon dioxide :

[C] + (O.2} - {CO.2} = 97,000 cal.,

and then determined the heat effect of the combustion of carbon monoxide to carbon dioxide :

{CO} + ^{Oa} - {CO2} = 68,000 cal.

By subtraction we get

[C] + ilOa) - {CO} = 29,000 cal.,

the required heat of formation of carbon monoxide.

§ 104. According to the above, theory enables us to calculate the heat effect of processes which cannot be directly realized, for as soon as the heat function of a system has been found in any way, it may be compared with other heat functions.

Let the problem be, e.g., to find the heat of formation of liquid carbon bisulphide from solid carbon and solid sulphur, which do not combine directly. The following represent the reactions : —

APPLICA TIONS TO NON-HOMOGENEOUS SYSTEMS. 75

The combustion of solid sulphur to sulphur dioxide gas :

[S] + {O2} - {SO2} = 71,100 cal.

The combustion of solid carbon to carbon dioxide :

[C] + {O2} - {CO2} = 97,000 cal.

The combustion of carbon bisulphide vapour to carbon dioxide and sulphur dioxide :

{CSo} + 3(02} - {CO2} - 2{S02} = 265,100 cal.

The condensation of carbon bisulphide vapour :

{CS2} - (CS2) = 6400 cal.

Elimination by purely mathematical processes furnishes the required heat of formation :

[C] + 2[S] - (CS2) = - 19,500 cal.,

hence negative.

In organic thermochemistry the most important method of determining the heat of formation of a compound consists in determining the heat of combustion, first of the compound, and then of its constituents.

Methane (marsh gas) gives by the complete combustion to carbon dioxide and water (liquid) :

{CH4} + 2(02} - {CO2} - 2(H20) = 211,900 cal., but {H2} + ^{02} - (H2O) = 68,400 cal., (50)

and [C] + {O2} - {CO2} = 97,000 cal.;

therefore, by elimination, we obtain the heat of formation of methane from solid carbon and hydrogen gas :

[C] + 2{H2} - (CHJ = 21,900 cal.

§ 105. The external heat, Q, of a given change at constant pressure will depend on the temperature at which the pro- cess is carried out. In this respect the first law of thermo- dynamics leads to the following relation : —

76 THERMODYNAMICS.

From equation (49) it follows that, for any two given temperatures, and 0',

(Ua H-iJoVa)^ - (Ui + i^oVi)«* = ^ and (Ua + i>oV2)e' - (Ui + ^oVi)<y' = Q«"

Hence, by subtraction,

Q.' -%= [(U, + i^oVa).' - (Ua 4- i^oVa).]

?.e. the difference in the heat effects (Q^ — Q./) resulting from performing the process at different temperatures, is equal to the difference in the quantities of heat which, before and after the reaction, would be required to raise the temperature of the system from to 0'.

Thus the influence of the temperature on the combustion of hydrogen to water (liquid) may be found by comparing the heat capacity of the mixture (H2 + \0.^ with that of the water (H2O). The former is equal to the molecular heat of hydrogen plus half the molecular heat of oxygen. According to the table in § 87, this is

6-82 + 3-47 = 10-29. The latter is 1 x 18 = 18.

The difference between these values is — 7*71, and, therefore, the heat of combustion of a gram molecule of hydrogen decreases with rising temperature by 7'7 cal. per degree Centigrade.

PART III.

The Second Fundamental PRiNcirLE OF Thermodynamics.

CHAPTER I.

INTRODUC TIO N.

§ 106. The second law of thermodynamics is essentially different from the first law, since it deals with a question in no way touched upon by the first law, viz. the direction in which a process takes place in nature. Not every change which is consistent with the principle of the conservation of energy satisfies also the additional conditions which the second law imposes upon the processes, which actually take place in nature. In other words, the principle of the con- servation of energy does not suffice for a unique determi- nation of natural processes.

If, for instance, an exchange of heat by conduction takes place between two bodies of different temperature, the first law, or the principle of the conservation of energy, merely demands that the quantity of heat given out by the one body shall be equal to that taken up by the other. Whether the flow of heat, however, takes place from the colder to the hotter body, or vice versa, cannot be answered by the energy principle alone. The very notion of temperature is alien to that principle, as can be seen from the fact that it yields no exact definition of temperature. Neither does the general equation (17) of the first law contain any statement with

78 THERMODYNAMICS.

regard to the direction of the particular process. The special equation (50), for instance,

IH2} + HO2} - (H2O) = 68,400 cal,

means only that, if hydrogen and oxygen combine under constant pressure to form water, the restablishment of the initial temperature requires a certain amount of heat to be given up to surrounding bodies ; and vice versa, that this amount of heat is absorbed when water is decomposed into hydrogen and oxygen. It offers no information, however, as to whether hydrogen and oxygen actually combine to form water, or water decomposes into hydrogen and oxygen, or whether such a process can take place at all in either direction. From the point of view of the first law, the initial and final stages of any process are completely eqnivalent.

§ 107. In one particular case, however, does the principle of the conservation of energy prescribe a certain direction to a process. This occurs when, in a system, one of the various forms of energy is at an absolute maximum (or minimum). It is evident that, in this case, the direction of the change must be such that the particular form of energy will decrease (or increase). This particular case is realized in mechanics by a system of particles at rest. Here the kinetic energy is at an absolute minimum, and, therefore, any change of the system is accompanied by an increase of the kinetic energy, and, if it be an isolated system, by a decrease of the potential energy. This gives rise to an important proposition in mechanics, which characterizes the direction of possible motion, and lays down, in consequence, the general condition of mechanical equilibrium. It is evident that, if both the kinetic and potential energies be at a minimum, no change can possibly take place, since none of these can increase at the expense of the other. The system must, therefore, remain at rest.

If a heavy liquid be initially at rest at different levels in two communicating tubes, then motion will set in, so as to

INTRODUCTION. 79

equalize the levels, for the centre of gravity of the system is thereby lowered, and the potential energy diminished. Equilibrium exists when the centre of gravity is at its lowest, and therefore the potential energy at a minimum, i.e. when the liquid stands at the same level in both tubes. If no special assumption be made with regard to the initial velocity of the liquid, the above proposition no longer holds. The potential energy need not decrease, and the higher level might rise or sink according to circumstances.

If our knowledge of thermal phenomena led us to recognize a state of minimum energy, a similar proposition would hold for this, but only for this, particular state. In reality no such minimum has been detected. It is, there- fore, hopeless to seek to reduce the general laws regarding the direction of thermodynamical changes, as well as those of thermodynamical equilibrium, to the corresponding pro- positions in mechanics which hold good only for systems at rest.

§ 108. Although these considerations make it evident that the principle of the conservation of energy cannot serve to determine the direction of a thermodynamical process, and therewith the conditions of thermodynamical equilibrium, unceasing attempts have been made to make the principle of the conservation of energy in some way or other serve this purpose. These attempts have, in many cases, stood in the way of a clear presentation of the second law. Occasionally we still find the endeavour made to' represent this law as contained in the energy principle, in that the doubtless too restricted term of " energetics " is applied to all investigations on these questions. The con- ception of energy is not sufficient for the second law. It cannot be exhaustively treated by breaking up a natural process into a series of changes of energy, and then in- vestigating the direction of each change. We can always tell, it is true, what are the different kinds of energy exchanged for one another ; for there is no doubt that the principle of energy must be fulfilled, but the expression of

8o THERMOD YNA MICS.

the conditions of these changes remains arbitrary, and this ambiguity cannot be completely removed by any general assumption.

We often find the second law stated as follows : The change of mechanical work into heat may be complete, but, on the contrary, that of heat into work must needs be incomplete, since, whenever a certain quantity of heat is transformed into work, another quantity of heat must undergo a corresponding and compensating change ; e.g. transference from higher to lower temperature. This is quite correct in certain very special cases, but it by no means expresses the essential feature of the process, as a simple example will show. An achievement which is closely associated with the discovery of the principle of energy, and which is one of the most important for the theory of heat, is the proposition expressed in equation (19), § 70, that the total internal energy of a gas depends only on the temperature, and not on the volume. If a perfect gas be allowed to expand, doing external work, and be pre- vented from cooling by connecting it with a heat-reservoir of higher temperature, the temperature of the gas, and at the same time its internal energy, remains unchanged, and it may be said that the amount of heat given out by the reservoir is completely changed into work without an ex- change of energy taking place anywhere. Not the least objection can be made to this. The proposition of the " incomplete transformability of heat into work " cannot be applied to this case, except by a different way of viewing the process, which, however, changes nothing in the physical facts, and cannot, therefore, be confirmed or refuted by them, namely, by the introduction of new kinds of energy, only invented ad hoc. This consists in dividing the energy of the gas into several parts, which may then individually depend also on the volume. This division has, however, to be carried out differently for different cases {e.g. in one way for isothermal, in another for adiabatic pro- cesses), and necessitates complicated considerations even in cases of physical simplicity. But when we pass from the

INTRODUCTION, 8i

consideration of the first law of thermodynamics to that of the second, we have to deal with a new fact, and it is evident that no definition, however ingenious, although it contain no contradiction in itself, will ever permit of the deduction of a new fact.

§ 109. There is but one way of clearly showing the signi- ficance of the second law, and that is to base it on facts by formulating propositions which may be proved or dis- proved by experiment. The following proposition is of this character : It is in no way possible to completely reverse any process in which heat has been produced by friction. For the sake of example we shall refer to Joule's experi- ments on friction, described in § 60, for the determination of the mechanical equivalent of heat. Applied to these, our proposition says that, when the falling weights have generated heat in water or mercury by the friction of the paddles, no process can be invented which will completely restore everywhere the initial state of that experiment, i.e. which will raise the weights to their original height, cool the liquid, and otherwise leave no change. The appliances used may be of any kind whatsoever, mechanical, thermal, chemical, electrical, etc., but the condition of complete restoration of the initial state renders it necessary that all materials and machines used must ultimately be left exactly in the condition in which they were before their application. Such a proposition cannot be proved a priori^ neither does it amount to a definition, but it contains a definite asser- tion, to be stated precisely in each case, which may be verified by actual experiment. The proposition is there- fore correct or incorrect.

§ 110. Another proposition of this kind, and closely connected with the former, is the following : It is in no way possible to completely reverse any process in which a gas expands without performing work or absorbing heat, i.e. with constant total energy (as described in § 68). The word " completely " again refers to the accurate reproduc- tion of the initial conditions. To test this, the gas, after

G

82 THERMOD YNA MICS.

it had assumed its new state of equilibrium, might first be compressed to its former volume by a weight falling to a lower level. External work is done on the gas, and it is thereby heated. The problem is now to bring the gas to its initial condition, and to raise the weight. The gas might be reduced to its original temperature by conducting the heat of compression into a colder heat-reservoir. In order that the process may be completely reversed, the reservoir must be deprived of the heat gained thereby, and the weight raised to its original position. This is, however, exactly what was asserted in the preceding paragraph to be im- practicable.

§ 111. A third proposition in point refers to the con- duction of heat. Supposing that a body receives a certain quantity of heat from another of higher temperature, the problem is to completely reverse this process, i.e. to convey back the heat without leaving any change whatsoever. In the description of Camot's reversible cycle it has been pointed out, that heat can at any time be drawn from a heat-reservoir and transferred to a hotter reservoir without leaving any change except the expenditure of a certain amount of work, and the transference of an equivalent amount of heat from one reservoir to the other. If this heat could be removed, and the corresponding work re- covered without other changes, the process of heat-conduc- tion would be completely reversed. Here, again, we have the problem which was declared in § 109 to be impracticable.

Further examples of processes to which the same con- siderations apply are, diifusion, the freezing of an overcooled liquid, the condensation of a supersaturated vapour, all explosive reactions, and, in fact, every transformation of a system into a state of greater stability.

§ 112. A process which can in no way be completely reversed is termed irreversible, all other processes re- versible. That a process may be irreversible, it is not sufficient that it cannot be directly reversed. This is the

INTRODUCTION. 83

case with many mechanical processes which are not irre- versible (c/. § 113). The full requirement is, that it be impossible, even with the assistance of all agents in nature, to restore everywhere the exact initial state when the process has once taken place. The propositions of the three preceding paragraphs, therefore, declare, that the generation of heat by friction, the expansion of a gas without the per- formance of external work and the absorption of external heat, the conduction of heat, etc., are irreversible processes.

§ 113. We now turn to the question of the actual existence of reversible and irreversible processes. Numerous reversible processes can at least be imagined, as, for instance, those consisting of a succession of states of equilibrium, as fully explained in § 71, and, therefore, directly reversible in all their parts. Further, all perfectly periodic processes, e.g. an ideal pendulum or planetary motion, are reversible, for, at the end of every period, the initial state is completely restored. Also, all mechanical processes with absolutely rigid bodies and absolutely incompressible liquids, as far as friction can be avoided, are reversible. By the introduction of suitable machines with absolutely unyielding connecting rods, frictionless joints and bearings, inextensible belts, etc., it is always possible to work the machines in such a way as to bring the system completely into its initial state without leaving any change in the machines, for the machines of themselves do not perform work.

If, for instance, a heavy liquid, originally at rest at different levels in two communicating tubes (§ 107), be set in motion by gravity, it will, in consequence of its kinetic energy, go beyond its position of equilibrium, and, since the tubes are supposed frictionless, again swing back to its exact original position. The process at this point has been completely reversed, and therefore belongs to the class of reversible processes. As soon as friction is admitted, how- ever, its reversibility is at least questionable. Whether reversible processes exist in nature or not, is not a priori evident or demonstrable. There is, however, no purely

84 • THERMODYNAMICS.

logical objection to imagining that a means may some day be found of completely reversing some process hitherto con- sidered irreversible : one, for example, in which friction or heat-conduction plays a part. But it can be demonstrated — and this will be done in the following chapter — that if, in a single instance, one of the processes declared to be irre- versible in §§ 109, etc., should be found to be reversible, then all of these processes must be reversible in all cases. Consequently, either all or none of these processes are irreversible. There is no third possibility. If those pro- cesses are not irreversible, the entire edifice of the second law will crumble. None of the numerous relations deduced from it, however many may have been verified by experience, could then be considered as universally proved, and theo- retical work would have to start from the beginning. (The so-called proofs of " energetics " are not a substitute, for a closer test shows all of them to be more or less imperfect paraphrases of the propositions to be proved. This is not the place, however, to demonstrate this point.) It is this foundation on the physical fact of irreversibility which forms the strength of the second law. If, therefore, it must be admitted that a single experience contradicting that fact would render the law untenable, on the other hand, any confirmation of part supports the whole structure, and gives to deductions, even in seemingly remote regions, the full significance possessed by the law itself.

§ 114. Since the decision as to whether a particular process is irreversible or reversible depends only on whether the process can in any manner whatsoever be completely reversed or not, the nature of the initial and final states, and not the intermediate steps of the process, entirely settle it. The question is, whether or not it is possible, starting from the final state, to reach the initial one in any way with- out any other change. The second law, therefore, furnishes a relation between the quantities connected with the initial and final states of any natural process. The final state of an irreversible process is evidently in some way discriminate

INTRODUCTION, 85

from the initial state, while in reversible processes the two states are in certain respects equivalent. The second law points out this characteristic property of both states, and also shows, when the two states are given, whether a trans- formation is possible in nature from the first to the second, or from the second to the first, without leaving changes in other bodies. For this purpose, of course, the two states must be fully characterized. Besides the chemical consti- tution of the systems in question, the physical conditions — viz. the state of aggregation, temperature, and pressure in both states — must be known, as is necessary for the applica- tion of the first law.

The relation furnished by the second law will evidently be simpler the nearer the two states are to one another. On this depends the great fertility of the second law in its treatment of cyclic processes, which, however complicated they may be, give rise to a final state only slightly different from the initial state (§ 91).

§ 115. Since there exists in nature no process entirely free from friction or heat-conduction, all processes which actually take place in nature, if the second law be correct, are in reality irreversible ; reversible processes form only an ideal limiting case. They are, however, of considerable importance for theoretical demonstration and for application to states of equilibrium.

CHAPTER II.

PROOF.

§ 116. The second fundamental principle of thermo- dynamics being, like the first, an empirical law, we can speak of its proof only in so far as its total purport may be deduced from a single self-evident proposition. We, there- fore, put forward the following proposition as being given directly by experience : It is imjjossible to construct an engine which will work in a complete cynle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir. Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made. It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost ; fur the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody's immediate disposal. For this reason we take the above proposition as our starting point. Since we are to deduce the second law from it, we expect, at the same time, to make a most serviceable appli- cation of any natural phenomenon which may be discovered to deviate from the second law. As soon as a phenomenon is found to contradict any legitimate conclusions from the second law, this contradiction must arise from an inaccuracy in our first assumption, and the phenomenon could be used

PROOF. 87

for the construction of the above-described engine. We shall in the following, according to the proposal of Ostwald, speak of perpetual motion of the second kind, since it stands in the same relation to the second law as perpetual motion of the first kind does to the first law. In connection with all objections to the second law, it must be borne in mind that, if no errors are to be found in the line of proof, they are ultimately directed against the impossibility of perpetual motion of the second kind (§ 136).*

§ 117. From the impossibility of perpetual motion of the second kind, it follows, in the first place, that the generation of heat by friction is irreversible (cf. def. § 112). For supposing it were not so, i.e. supposing a method could be found by which a process involving generation of heat by friction could be completely reversed, this very method would produce what is identically perpetual motion of the second kind : viz. a change which consists of nothing but the production of work, and the absorption of an equivalent amount of heat.

§ 118. It follows, further, that the expansion of a gas *^ without the performance of external ^Iteat^ or the absorption -^ of heat, is irreversible. For, suppose a method were known of completely reversing this process, i.e. of reducing the volume of a gas, without leaving any other change what- soever, this method could be utilized for the production of perpetual motion of the second kind in the following manner. Allow the gas to do work by expansion, supplying the energy

• I desire to emphasize here, that the starting point selected by me for the proof of the second law coincides fundamentally with that which R. Clausius, or which Sir W. Thomson, or which J. Clerk Maxwell used for the same purpose. The fundamental proposition which each of these investi- gators placed at the beginning of his deductions asserts each time, only in diflferent form, the impossibility of the realization of perpetual motion of the second kind. I have selected the above form of expression, because of its apparent technical significance. Not a single really rational proof of the second law has thus far been advanced which does not require this fun- damental principle, however numerous the attempts in this direction may have been in recent times, nor do I believe that such an attempt will ever meet with success.

88 THERMODYNAMICS.

lost thereby by the conduction of heat from a reservoir at the same or higher temperature, and then, by the assumed method, reduce the volume of the gas to its initial value without leaving any other change. This process might be repeated as often as we please, and would therefore represent an engine working in a complete cycle, and producing no eifect except the performance of work, and the withdrawal of heat from a reservoir, i.e. perpetual motion of the second kind.

On the basis of the proposition we have just proved, that the expansion of a gas without the performance of work and the absorption of heat is irreversible, we shall now carry through the proof of the second law for those bodies whose thermodynamical properties are most completely known, viz. for perfect gases.

§ 119. If a perfect gas be subjected to infinitely slow compression or expansion, and if, at the same time, heat be applied or withdrawn, we have, by equation (22), in each infinitely small portion of the process, per unit mass,

gr = (III _j- j>f/l'

or, since for a perfect gas,

du = c^dd,

, BO

and « = — . -,

■* m V

q = cdO ■] dv.

If the process be adiabatic, then g' = 0, and the inte- gration of the above equation gives (as in § 88) the function

■p

c„ log B ■{ — log v

equal to a constant. We shall now put

(p == c^ log H — log V + const., . . (51)

PROOF. 89

and call this function, after Clausius, the entropij of unit mass of the gas. The constant, which has to be added, can be determined by arbitrarily fixing the zero state. Accord- ingly

• = M^ = WU log 61 + - log V + const.] . (52)

is the entropy of mass M of the gas. The entropy of the gas, therefore, remains constant during the described adiabatic change of state.

§ 120. On the application of heat, the entropy of the gas changes, in the case considered, by

It increases or decreases according as heat is absorbed or

evolved.

, The absorbed heat Q has here been broken up into two

A'^^actors, B and cl<^. According to a view which has recently

^^ been brought forward, this breaking up of heat into factors is

regarded as a general property of heat. It should, however,

be emphasized that equation (53) is by no means generally

true. It holds only in the particular case where the external

work performed by the gas is expressed by j;fZV. The

relation

d^ = M(c^^ + ) = ^^ —

holds, quite generally, for any process in which the tempera- ture of the gas is increased by dO, and the volume by ^V. It is, in fact, only a different mathematical form for the definition of the entropy given in (52). On the other hand, the equation

Q = dV +pdY .

holds by no means in all cases, but should, in general, be replaced by

Q + W = (ZU,

90 THERMOD YNA MICS.

where W, the work done on the substance, may have any value within certain limits. For instance, W = 0, if the gas be conveyed into its new state of equilibrium without performing external work (as described in § 68). In this case, Q = d], and the equation Q = M^ no longer holds.

§ 121. We shall now consider two gases which can com- municate heat to one another by conduction, but may, in general, be under different pressures. If the volume of one, or both, of the gases be changed by some reversible process, care being taken that the temperatures of the gases equalize at each moment, and that no exchange of heat takes place with surrounding bodies, we have, according to equation (53), during any element of time, for the first gas,

''*' = |'

and, for the second gas.

Q2

d<^» =

According to the conditions of the process. Ox = e, and Qi + Q2 = 0, whence, cZ4>i + fZt|>2 =

or, for a finite change,

<I>i + $2 = const (54)

The sum of the entropies of the two gases remains constant during the described j)rocess.