§ 122. Any such process with two gases is evidently reversible in all its parts, for it may be directly reversed without leaving changes in the surroundings. From this follows the proposition that it is always possible to bring two gases, by a reversible process, without leaving changes

PRQOF. 91

in * other bodies, from any given state to any other given state, if the sum of the entropies in the two states be equal. Let an initial state of the gases be given by the tempe- ratures 01, 02> and the specific volumes v, V2 ; a second state by the corresponding values 61, 6.2 ; v(, v^. We now suppose that

$1 + $2 = <I>i' + a>2' . . . . (55)

Bring the first gas to the temperature 02 by a reversible adiabatic compression or expansion ; then place the two gases in thermal contact with one another, and continue to com- press or expand the first infinitely slowly. Heat will now pass between the two gases, and the entropy of the first one will change, and it will be possible to make this entropy assume the value <l>i'. But, according to (54), during the above process the sum of the two entropies remains constant, and = <I>i -{- <p2; therefore the entropy of the second gas is (<I>i -t- ^2) — $1', which is, according to (55), equal to fp^'. If we now separate the two gases, and compress or expand each one adiabatically and reversibly until they have the required temperatures 0/ and d^, the specific volumes must then be Vi and v-J, and the required final state has been reached.

This process is reversible in all its parts, and no changes remain in other bodies ; in particular, the surroundings have neither gained nor lost heat. The Conditions of the problem have therefore been fulfilled, and the proposition proved.

§ 123. A similar proposition can readily be proved for any number of gases. It is always possible to bring a system of n gases from any one state to any other by a reversible process without leaving changes in other bodies, if the sum of the entropies of all the gases is the same in both states, i.e. if

<I>i + a>2 +...+<!>„ = «I>i' + $2' + ... + ^n' (56)

• The emphasis is to be put on the word " in." Changes of position of ponderable bodies (for example, the raising or lowering of weights) are not internal changes ; but, of course, temperature and density changes are.

92 THERMOD YNA MICS.

By the process described in the preceding paragraph we may, by the successive combination of pairs of gases of the system, bring the first, then the second, then the third, and 80 on to the (n — l)th gas, to the required entropy. Now, in each of the successive processes the sum of the entropies of all the gases remains constant, and, since the entropies of the first {n — 1) gases are <I>i', ^-2 - • - <t>'„ _ „ the entropy of the nth gas is necessarily

(*i + <^2 + . . . + 4>„) - (<l>i' + <I>2' + . . . + 4>',.-i)-

This is, according to (56), the required value ^'. Each gas can now be brought by an adiabatic reversible process into the required state, and the problem is solved.

If we call the sum of the entropies of all the gases the entropy of the whole system, we may then say : If a system of gases has the same entropy in tu'O different states, it may he transformed from the one to the other hy a reversible process, without leaving changes in other bodies.

§ 124. We now introduce the proposition proved in § 118, that the expansion of a perfect gas, without per- forming external work or absorbing heat, is irreversible ; or, what is the same thing, that the transition of a perfect gas to a state of greater volume and equal temperature, without external effects, as described in § 68, is irreversible. Such a process corresponds to an increase of the entropy, accord- ing to the definition (52). It immediately follows that it is altogether impossible to decrease the entropy of a gas without producing a change in surrounding objects. If this were possible, the irreversible expansion of a gas could be completely reversed. After the gas had expanded without external effects, and had assumed its new state of equili- brium, the entropy of the gas could be reduced to its initial value, without leaving changes in other bodies, by the supposed method, and then, by an adiabatic reversible process, brought to its initial temperature, and thereby also to its original volume. This would completely reverse the

PROOF. 93

first expansion, and furnish, according to § 118, perpetual motion of the second kind.

§ 125. A system of two or more gases behaves in the same way. There exists, in nature, no means of diminishing the entropy of a system of perfect gases, without leaving changes in bodies outside the system. A contrivance which would accomplish this, be it mechanical, thermal, chemical, or electrical in nature, might be used to reduce the entropy of a single gas without leaving changes in other bodies.

Suppose a system of gases to have passed in any manner from one state in which their entropies are <[>i, <I>2 • • • ^n, to a state where they are 4>i', ^^ . . . $„', and that no change has been produced in any body outside the system, and let

<l»i' + 4>2' + . . . + <!>,; < <I>i + <I>2 + . . . + ^n, (57)

then it is possible, according to the proposition proved in § 123, to bring the system by a reversible process, without leaving changes in other bodies, into any other state in which the sum of the entropies is

4>i' + <i>2' + • . . 4- $;,

and accordingly into a state in which the first gas has the entropy <I>i, the second the entropy <I>2 . . ., the (w — i)th the entropy $„.i, and the nih. in consequence the entropy

(a>i' + $2' + . . . + <!>„') - <^i - <I)2 - . . . - <I>«-i (58)

The first (« — 1) gases may now be reduced to their original state by reversible adiabatic processes. The nih. gas possesses the entropy (08), which is, according to the supposi- tion (57), smaller than the original entropy tpn. The entropy of the nih. gas has, therefore, been diminished without leaving changes in other bodies. This we have already proved in the preceding paragraph to be impossible.

94 THERMODYNAMICS.

Tlie general proposition has, therefore, been proved, and we may immediately add the following.

§ 126. If a system of 'perfect gases pass in any ivay from one state to another, and no changes remain in surrounding bodies, the entropy of the system is certainly not smaller, hut either greater than, or, in the limit, equal to that of the initial state; in other words, the total change of the entropy

0. The sign of inequality corresponds to an irreversible process, the sign of equality to a reversible one. The equality of the entropies in both states is, therefore, not only a sufficient, as described in § 123, but also a necessary condition of the complete reversibility of the transformation from the one state to the other, provided no changes are to remain in other bodies.

§ 127. The scope of this proposition is considerable, since there have designedly been imposed no restrictions regarding the way in which the system passes from its initial to its final state. The proposition, therefore, holds not only for slow and simple processes, but also for physical and chemical ones of any degree of complication, provided that at the end of the process no changes remain in any body outside the system. It must not be supposed that the entropy of a gas has a meaning only for states of equilibrium. We may assume each sufficiently small particle, even of a gas in turmoil, to be homogeneous and at a definite temperature, and must, therefore, according to (52), assign to it a definite value of the entropy. M, v, and B are then the mass, specific volume, and temperature of the particle under consideration. A summation extending over all the particles of the mass — within which the values of V and d may vary from particle to particle — gives the entropy of the whole mass of the gas in the particular state. The proposition still holds, that the entropy of the whole gas must continually increase during any process which does not give rise to changes in other bodies, e.g. when a gas flows from a vessel into a vacuum (§ 68). It will be seen that the velocity of the gas particles does not influence

PROOF.

95

the value of the entrojjy ; neither does their height above a certain horizontal plane, although they are considered to have weight.

§ 128. The laws which we have deduced for perfect gases may be transferred to any substance in exactly the same way. The main difference is, that the expression for the entropy of any body cannot, in general, be written down in finite quantities, since the characteristic equation is not generally known. But it can be demonstrated — and this is the deciding point — that, for any other body, there exists a function with the characteristic properties of the entropy.

Imagine any homogeneous body to pass through a cer- tain reversible or irreversible cycle and to be brought back to its exact original state, and let the external effects of this process consist in the performance of work and in the addition or withdrawal of heat. The latter may be brought about by means of any required number of suitable heat- reservoirs. After the process, no changes remain in the substance itself; the heat-reservoirs alone have suffered change. We shall now assume all the heat-reservoirs to be perfect gases, kept either at constant volume or at constant pressure, but, at any rate, subject only to reversible changes of volume. According to our last proposition, the sum of the entropies of all these gases cannot have decreased, since after the process no change remains in any other body.

If Q denote the amount of heat given to the substance during an infinitely small element of time by one of the reservoirs; d, the temperature of the reservoir at that moment ; then, according to equation (53), the reservoir's change of entropy during that element of time is

d'

The change of the entropy of all the reservoirs, during all the elements of time considered, is

96 THERMODYNAMICS.

Now, according to § 126, we have the following condition : —

• v5>o 2f ^ ^-

This is the form in which the second law was first enunciated by Clausius.

A further condition is given by the first law ; for, ac- cording to (17) in § 63, we have, during every element of time of the process,

Q + W = fTT,

U being the initial energy of the body, and W the work done on 'the body during the element of time.

§ 129. If we now make the special assumption that the external pressure is, at any moment, equal to the pressure p of the substance, the work of compression becomes, according to (20),

whence Q = cfU + 'pdN.

If, further, each heat-reservoir be exactly at the tempe- rature of the substance at the moment when brought into operation, the cyclic process is reversible, and the inequality of the second law becomes an equality —

^5 =

or, on substituting the value of Q,

^ dX^ -f- i^dN _ ^

All the quantities in this equation refer to the state of the substance itself. It admits of interpretation without reference to the heat-reservoirs, and amounts to the folloAv- ing proposition.

PROOF.

97

§ 130. If a homogeneous hody he taken through a series of states of equilibrium (§ 71), that follow continuously from one another, hack to its initial state, then the summation of the differential

d] + pdV

e

extending over all the states of that process gives the value zero. It follows that, if the process be not continued until the initial state, 1, is again reached, but be stopped at a certain state, 2, the value of the summation

/:

'^^ (59)

depends only on the states 1 and 2, not on the manner of the transformation from state 1 to state 2. If two series of changes leading from 1 to 2 be considered (e.g. curves a and j3 in Fig. 2, § 75), these can be com- bined into an infinitely slow cyclic process. We may, for example, go from 1 to 2 along a, and return to 1 along j3.

It has been demonstrated that over the entire cycle —

r d\J + pdY f

d\J-{-pdY , f dUj-pdV_

whence

f dJJ +pdY _ f J.) » ~ Jo

dJJ +pdY _ [ dV + pdY

The integral (59) with the above-proved properties has been called by Clausius the entropy of the body in state 2, referred to state 1 as the zero state. The entropy of a body in a given state, like the internal energy, is completely determined up to an additive constant, whose value depends on the zero state.

H

93 THERMOD YNA MICS.

Denoting the entropy, as formerly, by $, we have :

and ,7^=^^ + i>^ (60)

or per unit mass :

(h = 0— (61)

This, again, leads to the value (51) for a perfect gas. The expression for the entropy of any body may be found by immediate integration (§ 254), provided its energy, U = Mt«, and its volume, V = Mr, are known as functions, say, of and j^- Since, however, these are not completely known except for perfect gases, we have to content ourselves in general with^ the differential equation. For the proof, and for many applications of the second law, it is, however, sufficient to know that this differential equation contains in reality a unique definition of the entropy.

§ 131. We may, therefore, just as in the case of perfect gases, speak of the entropy of any substance as of a finite quantity determined by the momentary values of tempe- rature and volume, even when the substance undergoes reversible or irreversible changes. The differential equation (61) holds, as was stated in § 120 in the case of perfect gases, for any change of state, including irreversible changes. This more general application of the conception of the entropy in no wise contradicts the manner of its deduction. The entropy of a body in any given state is measured by means of a reversible process which brings the body from ^that state to the zero state. This ideal process, however, has nothing to do with any actual reversible or irreversible changes which the body may have undergone or be about to undergo.

On the other hand, it should be stated that the differ- ential equation (60), while it holds for changes of volume and temperature, does not apply to changes of mass, for

PROOF.

99

this kind of change was in no way referred to in the definition of the entropy.

Finally, we shall call the sum of the entropies of a number of bodies briefly the entropy of the system composed of those bodies. Thus the entropy of a body whose particles are not at uniform temperature, and have different velocities, may be found, as in the case of gases (§ 127), by a sum- mation extending over all its elements of mass, provided the temperature and density within each infinitely small element of mass may be considered uniform. Neither the velocity nor the weight of the particles enter into the expression for the entropy.

§ 132. The existence and the value of the entropy having been established for all states of a body, there is no difficulty in transferring the proof, which was given for perfect gases (beginning in § 119), to any system of bodies. Just as in § 119 we find that, during reversible adiabatic expansion or compression of a body, its entropy remains constant, while by the absorption of heat the change of the entropy is

^Z^=? (62)

This relation holds only for reversible changes of volume, as was shown for perfect gases in § 120. Besides, it is found, as in § 121, that during reversible expansion or compression of two bodies at a common temperature, if they be allowed to exchange heat by conduction with one another, but not with surrounding bodies, the sum of their entropies remains constant. A line of argument corresponding fully to that advanced for perfect gases then leads to the following general result : * It is impossible in any way to diminish the

• With regard to the generalization of the theorem which was proved for a perfect gas in § 124, it may be stated that a certain difficulty arises in the special case of an incompressible body. In this case the body cannot be expanded. Professor Krigar-Menzel, who drew my attention to this, sent me at the same time the following proof. Proposition : It is impossible to diminish the entropy of an incompressible body without leaving changes in other bodies. Proof: Bring the body into thermal contact with a perfect

TOO THERMODYNAMICS.

entrojyy of a system of bodies without thereby leaving behind changes in other bodies. If, therefore, a system of bodies has changed its state in a physical or chemical way, without leaving any change in bodies not belonging to the system, then the entropy in the final state is greater than, or, in the limit, equal to the entropy in the initial state. The limit- ing case corresponds to reversible, all others to irreversible, processes.

§ 133. The restriction, hitherto indispensable, that no changes must remain in bodies outside the system is easily dispensed with by including in the system all bodies that may be aifected in any way by the process considered. The proposition then becomes : Every 'physical or chemical process in nature takes place in such a way as to increase the sum of the entropies of all the bodies taking any part in the process. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged. This is the most general statement of the second law of Thermodynamics.

§ 134. As the impossibility of perpetual motion of the first kind leads to the first law of Thermodynamics, or the principle of the conservation of energy ; so the impossibility of perpetual motion of the second kind has led to the second law, properly designated as the principle of the increase of the entropy. This principle may be presented under other forms, which possess certain practical advantages, especially for isothermal or isopiestic processes. They will be men- tioned in our next chapter. It should be emphasized, how- ever, that the form here given is the only one of unrestricted

gas, isolate the system adiabatically, and diminish the volume of the gas by reversible compression. Heat thereby passes from the gas into the body, and the entropy of the gas diminishes in consequence, while that of the body increases by an equal amount. Now separate the body from the gas. If the proposition were false, and there existed an uncompensated entropy diminish- ing process, we could by means of it bring the body back to its original smaller entropy, and therewith to its initial state. The only outstanding change of the whole process would be the diminution of the entropy of the perfect gas. But this contradicts § 118. The proposition is therefore not false, but true.

PROOF. 10 1

applicability to any finite process, and that no other universal measure of the irreversibility of processes exists than the amount of the increase of the entropy to which they lead. All other forms of the second law are either applicable to infinitesimal changes only, or presuppose, when extended to finite changes, the existence of some special condition im- posed upon the process (§§ 140, etc.). The real meaning of the second law has frequently been looked for in a " dissi- pation of energy." This view, proceeding, as it does, from the irreversible phenomena of conduction and radiation of heat, presents only one side of the question. There are irreversible processes in which the final and initial states show exactly the same form of energy, e.g. the diffusion of two perfect gases (§ 238), or further dilution of a dilute solution. Such processes are accompanied by no perceptible transference of heat, nor by external work, nor by any notice- able transformation of energy.* They occur only for the reason that they lead to an appreciable increase of the entropy. The amount of "lost work" yields a no more definite general measure of irreversibility than does that of "dissipated energy." This is possible only in the case of isothermal processes (§ 143). An exhaustive general state- ment of the second law can be made only by means of the conception of the entropy.

§ 135. Clausius summed up the first law by saying that the energy of the world remains constant; the second by saying that the entropy of the world tends towards a maximum. Objection has justly been raised to this form of expression. The energy and the entropy of the world have no meaning, because such quantities admit of no accurate definition. Nevertheless, it is not difficult to

• In reply to a criticism of this statement, I have simply to refer to § 108, wherein the remark is made, that, to be sure, by the introduction of new kinds of energy, conceived ad hoc, it is possible to speak of an energy transformation even for the cases now under discussion. There is, however, nothing arbitrary in the statement made in the text, where the energy appears as completely defined by § 56, but rather in the introduction of the new kinds of energy.

I02 THERMODYNAMICS.

express the characteristic feature of those propositions of Clausius in such a way as to give them a meaning, and to bring out more clearly what Clausius evidently wished to express by them.

The energy of any system of bodies changes according to the measure of the effects produced by external agents. It remains constant, only, if the system be isolated. Since, strictly speaking, every system is acted on by external agents — for complete isolation cannot be realized in nature — ^the energy of a finite system may be approximately, but never absolutely, constant. Nevertheless, the more ex- tended the system, the more negligible, in general, will the external effects become, in comparison with the magnitude of the energy of the system, and the changes of energy of its parts (§ 55) ; for, while the external effects are of the order of magnitude of the surface of the system, the internal energy is of the order of magnitude of the volume. In very small systems (elements of volume) the opposite is the case for the same reason, since here the energy of the system may be neglected in comparison with any one of the external effects. Frequent use is made of this proposition, e.g. in establishing the limiting conditions in the theory of the con- duction of heat. In the case here considered, it may, there- fore, be said that the more widely extended a system we assume, the more approximately, in general, will its energy remain constant. A comparatively small error will be com- mitted in assuming the energy of our solar system to be constant, a proportionately smaller one if the system of all known fixed stars be included. In this sense an actual significance belongs to the ^proposition, that the energy of an infinite system, or the energy of the world, remains constant.

The proposition regarding the increase of the entropy should be similarly understood. If we say that the en- tropy of a system increases quite regardless of all outside changes, an error will, in general, be committed, but the more comprehensive the system, the smaller does the pro- portional error become.

PROOF. 103

§ 136. In conclusion, we shall briefly discuss the question of the possible limitations to the second law. If there exist any such limitations — a view still held by many scientists and philosophers — this much may be asserted, that their existence presupposes an error in our starting-point, viz. the impossibility of perpetual motion of the second kind, or a fault in our method of proof. From the beginning we have recognized the legitimacy of the first of these objections, and it cannot be removed by any line of argument. The second objection generally amounts to the following. The impracticability of perpetual motion of the second kind is granted, yet its absolute impossibility is contested, since our limited experimental appliances, supposing it were possible, would be insufficient for the realization of the ideal processes which the line of proof presupposes. This position, however, proves untenable. It would be absurd to assume that the validity of the second law depends in any way on the skill of the physicist or chemist in observing or experimenting. The gist of the second law has nothing to do with experi- ment ; the law asserts briefly that there exists in nature a quantity which changes always in the same sense in all natural 'processes. The proposition stated in this general form may be correct or incorrect ; but whichever it may be, it will remain so, irrespective of whether thinking and measuring beings exist on the earth or not, and whether or not, assuming they do exist, they are able to measure the details of physical- or chemical processes more accurately by one, two, or a hundred decimal places than we can.* The limitations to the law, if any, must lie in the same province as its essential idea, in the observed Nature, and not in the Observer. That man's experience is called upon in the de- duction of the law is of no consequence ; for that is, in fact,

• We do not say that the second law is applicable to every single detail of a process. Upon closer examination the matter appears to be thus. The entropy, like temperature, pressure, and density, cannot be defined as an absolute, continuous quantity, but as a certain average value of a large number of single values. As long, therefore, as we regard simply one or more single values, the entropy cannot be defined any more than the temperature or pressure, and the second law neither applied nor proved.

104 THERMODYNAMICS.

our only way of arriving at a knowledge of natural law. But the law once discovered must receive recognition of its independence, at least in so far as Natural Law can be said to exist independent of Mind. Should any one deny this, he would have to deny the possibility of natural science.

The case of the first law is quite similar. To most unprejudiced scientists the impossibility of perpetual motion of the first kind is certainly the most direct of the general proofs of the principle of energy. Nevertheless, hardly any one would now think of making the validity of that principle depend on the degree of accuracy of the experi- mental proof of that general empirical proposition. Pre- sumably the time will come when the principle of the increase of the entropy will be presented without any connection with experiment. Some metaphysicists may even put it forward as being a 'priori valid. In the mean time, no more effective weapon can be used by both champions and opponents of the second law than inde- fatigable endeavour to follow the real purport of this law to the utmost consequences, taking the latter one by one to the highest court of appeal — experience. Whatever the decision may be, lasting gain will accrue to us from such a proceeding, since thereby we serve the chief end of natural science — the enlargement of our stock of knowledge.

CHAPTER III.

GENERAL DEDUCTIONS.

§ 137. Our first application of the principle of the entropy which was expressed in its most general form in the pre- ceding chapter, will be to Garnet's cycle, described in detail for perfect gases in § 90. This time, the system operated upon may be of any character whatsoever, and chemical re- actions, too, may take place, provided they are reversible. Resuming the notation used in § 90, we may at once state the result.

In a cyclic process, according to the first law, the heat, Q2, given out by the hotter reservoir is equivalent to the sum of the work done by the system, W = — W, and the heat received by the colder reservoir, Qi' = — Qi :

Q2 = W + Qi' or Qi + Q2 + W = . . . . (63)

According to the second law, since the process is re- versible, all bodies which show any change of state after the process, i.e. the two heat-reservoirs only, possess the same total entropy as before the process. The change of the entropy of the two reservoirs is, according to (62) :

§^'= _ ^ for the first, and - §? for the second, (64)

their sum: §^ + ^' = (65)

whence, by (63),

Qi:Q2:W = (-0i):02:(«i-02)

. to6 THE R MOD YNA MICS.

as in (44), but without any assumption as to the nature of the substance passing through the cycle of operations.

In order, therefore, to gain the mechanical work, W, by means of a reversible Carnot cycle of operations with any substance between two heat-reservoirs at the tempera- tures By and d^ {62 > 61), the quantity of heat

must pass from the hotter to the colder reservoir. In other words, the passage of the quantity of heat Q/ from O2 to Oi may be taken advantage of to gain the mechanical work

W' = ^^Q/ (66)

§ 138. For an irreversible cycle, i.e. one involving any irreversible physical or chemical changes of the substance operated upon, the equation of energy (63) still holds, but the equation for the change of the entropy (Q5) is replaced by the inequality :

_ Qi _ Q2 /^

Observe, however, that the expressions (64) for the change of the entropy of the reservoirs are still correct, provided we assume that any changes of volume of the substances used as reservoirs are reversible. Thus,

sr + e.^" ("')

or Qa < ^^Q,',

hence, from (67) and (63),

GENERAL DEDUCTIONS. 107

This means that the amount of work, W, to be gained by means of a cyclic process from the transference of the heat, Qi', from a hotter to a colder reservoir, is always smaller for an irreversible process than for a reversible one. Conse- quently the equation (66) represents the maximum amount of work to be gained from any cyclic process between heat- reservoirs at the temperatures d^ and Bi.

In particular, if W' = 0, it follows from the equation of energy (63) that

Q2 = — Qi = Qi

and the inequality (67) becomes

' <0.

<.rl)

In this case the cyclic process results in the transference of heat (Q2) from the reservoir of temperature O.2 to that of temperature Qi, and the inequality means that this flow of heat is always directed from the hotter to the colder reservoir.

Again, a special case of this type of process is the direct passage of heat by conduction between heat-rjeservoirs, with- out any actual participation of the system supposed to pass through the cycle of operations. It is seen to be an irre- versible change, since it brings about an increase of the sum of the entropies of the two heat-reservoirs.

§ 139. We shall now apply the principle of the entropy to any reversible or irreversible cycle with any system of bodies, in the course of which only one heat-reservoir of constant temperature Q is used. Whatever may be the nature of the process in detail, there remains at its close no change of the entropy except that undergone by the heat- reservoir. According to the first law, we have'

W -f Q = 0.

W is the work done on the system, and Q the heat absorbed by the system from the reservoir.

According to the second law, the change of the entropy

1 08 THERMOD YNAMICS,

of the reservoir, within which only reversible changes of volume are supposed to take place, is

-Q>0

or Q < 0,

whence W > 0.

Work has been expended on the system, and heat added to the reservoir. If, in the limit, the process be reversible, the signs of inequality disappear, and both the work W and the heat Q are zero. On this proposition rests the great fertility of the second law in its application to isothermal reversible cycles.

§ 140. We shall no longer deal with cycles, but shall consider the general question of the direction in which a change will set in, when any system in nature is given. For chemical reactions in particular is this question of importance. It is completely answered by the second law in conjunction with the first, for the second law con- tains a condition necessary for all natural processes. Let us imagine any homogeneous or heterogeneous system of bodies at the common temperature 0, and investigate the conditions for the starting of any physical or chemical change. According to the first law, we have for any infinitesimal change :

cZU = Q + W, (68) .

where U is the total internal energy of the system, Q the heat absorbed by the system during the process, and W the work done on the system.

According to the second law, the change of the total entropy of all the bodies taking part in the process is

(^4> + cZ4>o g

where <l> is the entropy of the system, <!>„ the entropy of the surrounding medium (air, calorimetric liquid, walls of

GENERAL DEDUCTIONS. icg

vessels, etc.). Here the sign of equality holds for reversible cases, which, it is true, should be considered as an ideal limiting case of actual processes (§ 115).

If we assume that all changes of volume in the surround- ing medium are reversible, we have, according to (62),

^^ Q

or, by (68), ^<I>o = - ^ '"^ ,

On substituting the value of d^Q, we have

d^ ^ > (69)

or d\J -Bd^<V^ (70)

All conclusions with regard to thermodynamic chemical changes, hitherto drawn by different authors in different ways, culminate in this relation (70). It cannot in general be integrated, since the left-hand side is not, in general, a perfect differential. The second law, then, does not lead to a general statement with regard to finite changes of a system taken by itself unless something be known of the external conditions to which it is subject. This was to be expected, and holds for the first law as well. To arrive at a law governing finite changes of the system, the knowledge of such external conditions as will permit the integration of the differential is indispensable. Among these the following are singled out as worthy of note.

§ 141. Case I. Adiabatic Process.— No exchange of heat with the surroundings being permitted, we have Q = 0, and, by (68),

dV = W. Consequently, by (70), d^ > 0.

The entropy of the system increases or remains constant, a case which has already been sufficiently discussed.

no THERMODYNAMICS.

§ 142. Case II. Isothermal Process.— The temperature beiug kept constant, (70) passes into

fZ(U - 0<I>) < AV

i.e. the increment of the quantity (U - flfp) is smaller than, or, in the limit, equal to, the work done on the system. This theorem is well adapted for application to chemical processes, since isothermal changes play an important part in nature.

Putting U - 04> = F, (71)

we have, for reversible isothermal changes :

^F = W and, on integrating,

F, - Fi = 2W (72)

For finite reversible isothermal changes the total work done on the system is equal to the increase of F ; or, the entire work performed by the system is equal to the decrease of F, and, therefore, depends only on the initial and final states of the system. Where Fi = F2, as in cyclic processes, the external work is zero.

The function F, thus bearing the same relation to the external work that the energy U does to the sum of the external heat and work, has been called by H. v. Helmholtz the free energy (freie Energie) of the system. (It should rather be called "free energy for isothermal processes.") Corresponding to this, he calls U the total energy (Gesammt- energie), and the difference U — F = B<^, the latent energy (gebundene Energie) of the system. The change of the latter in reversible isothermal processes gives the amount of the external heat absorbed. This sjjlitting up of total energy into free and latent energy is applicable to isothermal processes only.

In irreversible processes, on the other hand, clY < W, and on integrating we have

F, - Fi < ^W (73)

GENERAL DEDUCTIONS. m

The free energy increases by a less amonnt than that which corresponds to the work done on the system. The results for reversible and irreversible processes may be stated thus. In irreversible isothermal processes the work done on the system is more, or the work done by the system is less, than it would be if the same change were brought about by a reversible process, for in that case it would be the differ- ence of the free energies at the beginning and end of the process (72).

Hence, any reversible transformation of the system from one state to another yields the maximum amount of work that can be gained by any isothermal process between those two states. In all irreversible processes a certain amount of work is lost, viz. the difference between the maximum work to be gained (the decrease of the free energy) and the work actually gained.

The fact that, in the above, irreversible as well as reversible processes between the same initial and final states were considered, does not contradict the proposition that between two states of a system either only reversible or only irreversible processes are possible, if no external changes are to remain in other bodies. In fact, the pro- cess here discussed involves such changes in the surrounding medium ; for, in order to keep the system at constant temperature, an exchange of heat between it and the sur- rounding medium must take place in one direction or the other.

§ 143. If the work done during an isothermal process

vanish, as is practically the case in most chemical reactions,

we have

SW = 0,

and, by (73), Fa - Fi < 0,

i.e. the free energy decreases. The amount of this decrease may be used as a measure of the work done by the forces (chemical affinity) causing the process, for the same is not available for external work.

For instance, let an aqueous solution of some non-volatile

1 1 2 THERMOD YNAMICS.

salt be diluted isothermally, the heat of dilution being furnished or received by a heat-reservoir according as the energy, U2, of the diluted solution (final state) is greater or less than the sum, Ui, of the energies of the undiluted solution and the water added (initial state). The free energy, F2, of the diluted solution, on the other hand, is necessarily smaller than the sum, Fi, of the free energies of the undiluted solution and the water added. The amount of the decrease of the free energy, or the work done by the " affinity of the solution for water " during the process of dilution may be measured. For this purpose, the dilution should be performed in some reversible isothermal manner, when, according to (72), the quantity to be measured is actually gained in the form of external work. For instance, evaporate the water, which is to be added, infinitely slowly under the pressure of its saturated vapour. When it has all been changed to water vapour, allow the latter to expand isothermally and reversibly until its density equals that which saturated water vapour would possess at that tempera- ture when in contact with the solution. Now establish lasting contact between the water vapour and the solution, \ whereby the equilibrium will not be disturbed. Finally, by ^^ isothermal compression, condense the water vapour infinitely ' slowly when in direct contact with the solution. It will then be uniformly distributed throughout the latter. Such a process, as here described, is composed only of states of equilibrium. Hence it is reversible, and the external work thereby gained represents at the same time the decrease of the free energy, F2 — Fi, which takes place on directly mixing the solution and the water.