CLAUSIUS extended the operation of the Second Law or, what is the same thing, the scope of entropy, to all bodies. See RÜHLMANN'S "Handbuch d. mech. Wärmtheorie," Vol. I, pp. 395-405.
BOLTZMANN says in this connection: "As regards entropy, solid and liquid bodies do not differ qualitatively from perfect gases; the discussion of the entropy of the former, however, presents greater mathematical difficulties."
Certain features of the entropy of solid and liquid bodies have, however, been derived with the help of ideal gases as temporary auxiliaries. We consider this argument the simplest and therefore now give an outline of PLANCK'S presentation of the matter.[25]
[25]Thermodynamik, 2d Ed., pp. 87-100.
PLANCK'S PROOF THAT ALSO FOR ANY OTHER BODIES THAN GASES THERE REALLY EXISTS A FUNCTION WHICH POSSESSES THE CHARACTERISTICS OF ENTROPY; THE MAIN STEPS ARE NUMBERED
(1) Expression of entropy for an ideal gas and properties of entropy.
where the elastic forces do a ; strictly speaking, is not differential of the heat supply.
(3) Two gases (1) and (2) thermally connected, are maintained at same temperature but different pressure and change adiabatically while experiencing change of volumes; then it can be shown that for this finite change, , that is for the two gases the sum of the final No other change is effected in any other bodies but in these two gases; here emphasis is laid on preposition in; for the work done may be the lifting or lowering of a load and such change of location in rigid bodies involves no change of inner energy. Changes of density in external bodies can be also avoided by having the two gas tanks located in a vacuum.
(4) A similar proposition can be established for a system of any number of gases by successively treating the gases in pairs as above. The theorem then reads: "If the gas system as a whole possesses the same entropy in two different states then the system can be brought from one state to the other in a reversible manner without changes remaining in other bodies."
(5) We know that the expansion of an ideal gas without doing external work and receiving any heat supply is an irreversible process. The consequence is that the entropy of this gas increases. It follows at once that "it is impossible to diminish the entropy of an ideal gas without changes remaining in other bodies."
(6) The same result obtains for a system of any number of ideal gases. Consequently "there exists in the whole of Nature no means (be they of the mechanical, thermal, chemical or electrical sort) of diminishing the entropy of a system of ideal gases, without changes remaining in other bodies."
(7) "If a system of ideal gases has changed to another state (possibly in an entirely unknown way) without changes remaining in other bodies, then the final entropy can certainly not be smaller, it can only be greater than or equal to the initial condition. In the former case this process is an irreversible one, in the latter case a reversible one.
"Equality of entropy in the two states therefore constitutes a sufficient and at the same time a necessary condition for the complete reversibility of the passage from one state to the other, provided no changes are to remain behind in other bodies."
(8) "This proposition has a very considerable range of validity; for there was expressly no limiting assumption made concerning the way in which the gas system reached its final condition; the proposition is therefore valid not only for slowly and simply changing processes but also for any physical and chemical processes provided at the end no changes remained in any body outside of the gas system. Nor need we believe that entropy of a gas has significance only for states of equilibrium, provided we can suppose the gas mass (moving in any way) to consist of sufficiently small parts each so homogeneous that it possesses entropy."[26]
Then the summation must extend over all these gas parts. "The velocity has no influence on the entropy, just as little as the height of the heavy gas parts above a particular horizontal plane."
(9) "The laws thus far deduced for ideal gases can in the same way be transferred to any other bodies, the main difference in general being that the expression for the entropy of any body cannot be written in finite magnitudes because the equation of condition is not generally known. But it can always be shown—and this is the decisive point—that for any other body there really exists a function possessing the characteristic properties of entropy."
Now let us assume any physically "homogeneous body, by which is meant that the smallest visible space parts of the system are completely alike. Here it does not matter whether or no the substance is chemically homogeneous, i.e., whether it consists of entirely like molecules, and consequently it also does not here matter whether in the course of the prospective changes of state it experiences chemical transformation.... When the substance is stationary the whole energy of the system will consist of the so-called 'inner' energy , which depends only on the mass and inner constitution of the substance, which constitution is conditioned by the temperature and density."
(10) Let us suppose that with such a homogeneous body there is conducted a certain reversible or irreversible cycle process which therefore brings the body exactly back again to its initial condition. Let the external influences on the body consist in the performance of work and in heat supply or withdrawal, which heat exchange is to be effected by any number of suitable heat reservoirs. At the end of the process no changes remain in the body itself, only the heat reservoirs have altered their state. Now let us suppose the heat carriers in the reservoirs to be composed of purely ideal gases, which may be kept at constant volume or under constant pressure, at any rate only be subject to reversible changes of volume. According to the last proved proposition, the sum of the entropies of all the gases cannot have become smaller, for at the end of the process no changes remain in any other body, not even in the body which completed the cycle process.
(11) Let be the heat gained by the body from some reservoir in an element of time and the temperature of the reservoir[27] at the same moment, then the change of entropy experienced by the reservoir at this instant will be and in the whole course of time all the reservoirs together will experience the entropy change and then we know that there must be satisfied the condition which is the form in which CLAUSIUS first enunciated the Second Law.
(12) Another condition for the process considered is furnished by the First Law. For each element of time , where is the inner energy of the body and the work expended upon it in an element of time by external means.
Now let us consider a more special case in which the external pressure at each instant is equal to the pressure p of the supposedly stationary body. Then the external work will be represented by and then it follows that
(13) Furthermore let the temperature of each heat reservoir, at the instant when it comes into use, be equal to the simultaneous temperature of the body; then the cycle process becomes a reversible one and the inequality of the second law becomes an equality, and substitution of above value for gives
In this equation there occur only quantities referring to the state of the body itself and therefore it can be interpreted without any reference to the heat reservoirs. It contains the following proposition:
"If a homogeneous body by suitable treatment is allowed to pass through a series of continuous states of equilibrium and thus finally to come back to its initial condition, the summation of the differential, for all the changes of state will be equal to zero. From this follows at once that if the change of state is not allowed to continue to the restoration of the initial condition (1), but is stopped at any state (2), the value of the sum will depend solely on the final state (2) and on the initial state (1), and not on the course of the passage from 1 to 2."[28]
"The last expression is 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 particular state is, therefore, like energy, completely determined down to an additive constant depending on the choice of the zero state."
(14) "Let us again designate the entropy by , then or, what amounts to the same thing, by which reduced to the unit of mass becomes
"This is evidently identical with the value found for an ideal gas. But it is equally applicable to every body when its energy and volume are known as functions, say, of and , for the expression for entropy can then be directly determined by integration. But since these functions are not completely known for any other substance we must in general rest content with the differential equation. For the present proof, however, and for many applications of the Second Law it suffices to know that this differential equation really contains a unique definition of entropy."
As with an ideal gas, we can now always speak of the entropy of any substance as a certain finite magnitude determined by the values of the temperature and volume at the instant, and can so speak even when the substance experiences any reversible or irreversible change. Moreover, the differential equation (43) is applicable to any change of state, even an irreversible one.
In thus applying the idea of entropy there is no conflict with its derivation. The entropy of a state is measured by a reversible process which conducts the body from its present state to the zero state, but this ideal process has nothing to do with the changes of state that the body has experienced or is going to experience.
"On the other hand, we must emphasize that differential equation (43) for is valid only for changes of temperature and volume and is not so for changes of mass or of chemical composition. For changes of the latter sort were never considered in defining entropy."
(15) "Finally, we may designate the sum of the entropies of several bodies as the entropy of the system of all the bodies, provided the system can be subdivided into infinitesimal elements for which uniform density and temperature can be assumed; but velocity and force of gravitation do not at all enter into the expression for entropy."
[26]If the motion of the gas is so turbulent that temperature and density cannot be defined, then we must have recourse to BOLTZMANN'S broader definition of entropy.
[27]It does not here matter what the temperature of the body is at this instant.
[28]This is evident from the fact that the quantities , , , and , under the integral are each a function of the state only and do not depend on its past history. This falls far short of being true for turbulent states, for which it is difficult to get and . PLANCK does not make the preceding statement, but gives instead a rigorous proof based on cyclical considerations.