THERMODYNAMICS, or THE ME CHANICAL THEORY OF HEAT, that branch of p'hysical science which treats of the relation of heat energy to energy of other kinds, and particularly of the convertibility of heat energy into mechanical energy, and the converse. In order to discuss, quantitatively, the conversion of one Icind of energy into another lcind, we must first have a definite method of measuring each of them. Mechanical energy (see ENERGETICS) is measured by. de termining the amount of work that a given quantity of it can perform; the customary unit employed for this purpose being the °foot pound* or the "metre-kilogram') in engineering practice and the "erg° in scientific work; the °ere being defined as the quantity of work done in overcoming a resistance of one dyne, through a distance of one centimeter. The unit employed in the measurement of heat is.almost universally the quantity of heat reqtured to raise. the temperature of some definite mass of water through one degree, on some stated part of the thermometric scale. The ordinary "British thermal unit,)) whict is used in engi neering practice in English-spealcing countries, is the quantity of heat required to raise the temperature •of one pound of water by one Fahrenheit degree; and in countries that use the metric system, the engineering unit is the quantity of heat required to raise the tem perature of one lcilogram of water through one centigrade degree. As the specific heat (q.v.) of water varies slightly at different tempera tures, these definitions are not absolutely defi nite, unless the part of the thermometric scale at which the experiment is to be performed is specified. Unfortunately there is no general agreement among engineers on this point; and for most purposes in practical engineering it is customary to ignore the slight variation in the specific heat of water and to consider the foregoing definitions to be sufficiently precise as they stand. For scientific purposes, where the greatest possible accuracy is required, this course is not permissible, and it becomes neces sary to specify the particular degree through which the temperature of the water is to be raised. Even here there is no definitely estab lished convention; but there appears to be a growing tendency to adopt the degree that ex tends from 14.5° C. to 15.5° C. In scientific work, too, it is customary to define the thermal unit in terms of a gram of water, instead of a kilogram; and the scientific heat unit (which is called the "small calorie? to distinguish it from the "greater calorie') that is used in engineer ing) may be defined as the quantity of heat re quired to raise the temperature of one gram of water from 14.5° C. to 15.5° C. The quan tity of heat required to raise the temperature of one gram of water from 3.5° C. to 4.5° C.— 4.0° being the temperature at which water has its maximum' density—is also known as the atherm.)) The science of thermodynamics is founded upon two general, fundamental laws, which, so far as we are aware, are absolutely rigorous and which are respectively known as the "first') and °second" laws. These we shall consider in order.

The °first law of thermodynamics° is noth ing but a special application of the general principle of the conservation of energy. (See ENERGETICS). It states that whenever heat energy is converted into mechanical energy (or the reverse), then for each unit of one kind of energy that disappears there is always a per fectly definite and constant quantity of energy of the other kind which appears. Mayer and Joule discovered this fact independently, about the year 1840. There has been in the past some considerable controversy as to the credit that should be assigned to these respective investi gators. We cannot enter into this discussion, but the reader who desires to follow it up will find an admirable and very fair statement of the facts of the case in two papers on the Copley medalists of 1870 and 1871, in Tyndall's 'Fragments of Science.> Joule did a vast amount of experimental work for the purpose of determining the exact value of the °me chanical equivalent of heat," as the constant is called, which expresses the number of units of mechanical energy that are equivalent to one unit of heat; and in the course of his labors he tried many different experimental methods. (Consult The Scientific Papers of James Pres cott Joule' ). His best known method consisted in stirring a known mass of water and measur ing the rise in temperature so produced as well as the quantity of mechanical work expended in the stirring. He concluded that the tempera ture of one pound of water is raised by one Fahrenheit degree by the expenditure of 772 foot-pounds of mechanical energy. This con stant, which is known as °Joule's equivalent" and is denoted by the symbol J, has played an all-important part in engineering and scientific work for more than half a century. A better value of it was obtained by Rowland in 1879 (consult 'The Physical Papers of Henry Augustus Rowland') ; but the prestige of Joule was so great that the superiority of Rowland's work was not generally recognized for many years. Rowland's method was similar to that of Joule, but he worked with far better appara tus, and took advantage of the advances that had been made since Joule's work was done, both in calorimetry and in thermometry. It was in the course of this work that Rowland made the discovery that when temperature is defined in accordance with the scale of the normal, constant-volume air thermometer, the specific heat of water has a minimum value at a little above 30° C.,— a discovery which im plies a high degree of precision in the ex-. perimental methods employed, and which has been abundantly verified by later investigators. Rowland's value of the mechanical equivalent of heat may be stated as follows: Taking as the unit of heat the quantity of heat required to raise the temperature of one kilogram of water from 14.5° C. to 15.5° C., the mechanical equivalent is 427.4 kilogram-metres at sea-level in the latitude of Baltimore. If the unit of heat is the quantity of heat required to raise the temperature of a pound of water from 59° F. to 60° F., then the mechanical equivalent is 779.0 foot-pounds. If the unit of heat is the quantity of heat required to raise the tem perature of one gram of water from 14.5° C. to 15.5° C., then the mechanical equivalent is 41,890,000 ergs. Numerous other experimenters have made determinations of the mechanical equivalent, both by the method followed by Joule and Rowland, and by other methods. Prominent among these is Griffiths, who heated the water in his calorimeter mainly by means of a known electrical current, traversing a known resistance, and hence giving out a known quantity of heat. Taking as a unit of heat the quantity of heat required to raise the tempera ture of a kilogram of water from 14.5° C. to 15.5° C., Griffiths found the mechanical equiva lent to be 427.45 kilogram-metres at sea-level in the latitude of Greenwich. Rowland's value, when expressed in these same units and cor rected to the latitude of Greenwieth, is 427. .0. For further details concerning the experimental determination of the mechanical equivalent. consult Preston, 'Theory of and for numerous interesting illustrations of the first law of thermodynamics, consult Tyndall, 'Heat a Mode of Motion.' The °second law of thermodynamics" is hard to explain in a limited space, or without the use of higher mathematics; and, as Ran kine remarked, its exposition has been much neglected by the writers of popular works, so that °the consequence is that most of those who depend altogether on such works for their scientific information remain in ignorance, not only of the second law, but of the fact that there is a second law; and knowing the first law only, imagine that they know the whole principles of thermodynamics." In its simplest form, the °second law" merely states that heat always tends to pass from a hotter body to a colder one. This fact is obvious enough in its simpler manifestations; for every housewife knows that to make the kettle boil she must put it on the stove and not in the refrigerator. It is not so evident, however, that there are no conditions whatever under which heat will pass of its own natural tendency from a lower tem perature to a higher one. It is not evident at first thought, for example, that we cannot make a burning glass big enough to give a tempera ture, at its focus, which shall be higher than the temperature of the sun; yet we cannot do so, if the second law of thermodynamics is true, for the heat at the focus of the glass cer tainly comes from the sun, and if that focus were hotter than the sun, we should have a case in which heat is passing by its own natural radiative tendency from a cooler body (the sun) to a hotter one (the focus of the glass). The second law was first proposed, as a broad principle of nature, by Clausius; and although numerous distinguished mathematicians and physicists have questioned its validity from time to time, it is now recognized as a great, universal truth, applicable to all classes of phenomena, without exception, so long as we are dealing with large masses of matter. We cannot undertake, to discuss the seeming exceptions that occur in connection with par ticles microscopic or less in size. (See BROWN IAN MOVEMENT). It is indeed true that heat can be abstracted from a body and made to pass into a warmer one, and this is actually done on a commercial scale in cold storage plants and in the manufacture of artificial ice; but the point is that this feat cannot be accom plished without the expenditure of energy. We

are to think of heat, in its tendency to pass from a 'higher temperature to a lower one, in much the same way as we think of water tend ing to run down hill. Water will not run up hill of its own accord, but it may be forced to pass from a lower level to a higher one by the expenditure of energy upon a pump or other equivalent device. The correctness of Clausius' hypothesis with regard to heat is substantiated by the fact that no case has yet been discovered in which it is demonstrably violated. On the other hand, many previously unknown phe nomena of nature have been predicted by its aid, and in every instance subsequent experiment has borne out the prediction in every respect. For a short account of some of the better known objections that have been urged against the soundness of the °second law,' consult the latter portion of Browne's translation of Clau sius's Mechanical Theory of Heat.' In studying the transformation of heat ener into mechanical energy (or the re , it is customary to think of the con version as being performed by a suitable type of heat-engine; for this conception helps to make the problem so that the mind can readily grasp the principles involved. The imaginary engine is usually conceived to be perfect in construction, so as to run without friction and without losses by radiation or con duction. In fact, the material of which the engine is composed is assumed to be incapable of absorbing any heat at all. Some of its parts may, however, be assumed to be perfectly transparent to heat, and others to be absolutely opaque to it; and we may make such other extravagant assumptions as may be convenient for the discussion of the problem in hand, the only office of the imaginary engine being to assist the mind in the presentation and discussion of the es sential facts, whatever those may be. These fictive engines are usually assumed, further more, to be °perfectly reversible,' so that when, by the expenditure of mechanical power, they are forced to run backwards, all of the normal operations of the engine take place precisely as before, but in a contrary sense. If, for ex ample, the engine, at some instant in its for ward motion, absorbed a quantity Q of heat from an outside body whose temperature was T, then when the engine reaches the corre sponding state in its reversed motion, it must give out this same quantity, Q, of heat, and must give it out again to the same body from which it originally abstracted it, and at the same temperature, T. An engine which ful fils all of these various conditions is called a °perfectly reversible engine'; or, more briefly, an °ideal engine.' Carnot's Theorem, In 1824 Carnot gave a remarkable theorem (consult his 'Reflections on the Motive Power of Heat,' Thurston's translation), which may be stated in the fol lowing language: Of all the possible kinds of heat engine, which run by converting heat energy into mechanical energy, and which take in their heat all at one given temperature and give out all that they do give out (if any) at another given temperature, there is none that is more efficient than the ideal, reversible engine; °efficiency" being defined as the fraction of the absorbed heat-energy that is converted into mechanical work. This theorem is of exceeding importance, as it holds true not only for the untold thousands of kinds of ideal engines that we might be able to think of at the present time, but also for any others that may depend upon principles of nature as yet undiscovered; always supposing that the two fundamental laws of thermodynamics, as stated above, are true. In Carnot's time, heat was believed to be a substance; and Carnot's proof of his theorem is based upon this view. After the newer con ception of heat had been attained, however, Clausius proved that Carnot's theorem is cap able of equally sound demonstration in accord ance with the two thermodynamical laws now admitted. The proof is as follows: Let us assume that the theorem is false, and that there is some other engine, which we will designate as B, which is more efficient than some par ticular ideal reversible engine, A, which runs between the same two temperature limits. Let 7; be the temperature at which both engines take in their heat, and let T, be the temperature at which each rejects such heat (if any) as it does not transform into work. Let H, and HI, respectively, be the quantities of heat taken in and rejected, during a given time, by the reversible engine, A, and let H't and H's be the quantities taken in and rejected, respectively, by the other engine B. The quantities of heat that are transformed into work by A and B, respectively, • are then (H,—H,) and (11',—H's); and the efficiencies are respectively (Hi— Hi)/Hi and (H's—H'i)/H'i. The condition that we are assuming, in violation of the theorem, is that the efficiency of the engine B is greater than that of A; that is, (H',— H's)/ift> (H1— Hs)/Ht; or, what is the same thing, Than > H', (Ht— H,). Now suppose that the two engines are coupled together so that the engine B runs Ard and drives the re versible engine, A, backward. Then A, owing to its reversibility, for every Hs) units of mechanical work that it absorbs, takes in H2 units of heat at the temperature T,, and re jects H, units of heat at the higher tempera ture, T.; while the other engine B, for every (H'i— H',) units of mechanical work that it performs, takes in H', units of heat at the higher temperature T,, and rejects In at the lower temperature Ts. Now in the case supposed, where one of the engines drives the other one backward, the mechanical energy devoleped by the engine B is entirely absorbed by the reversed engine, A. Hence we have (H', — H's)= (H1— Ht); and this equation, taken in connection with the fore going inequality, gives H, That is, the heat delivered by the doubled engine to the source whose temperature is T1, is greater than the heat that is being withdrawn from that source; so that if we regard the doubled engine as a single machine, we have a case in which heat is passing, by its own natural tendency and without external compulsion, from a tempera ture T, to a higher temperature, Tt. But this is contrary to the second law of thermo dynamics; and hence if that law is sound, it must be that no such engine as B exists. In other words, there is no engine which takes its heat all at a temperature T,, and rejects what it does reject at a lower temperature T. which has a higher efficiency than the ideal re versible engine running between these same temperature limits. It will be observed that in case both of the engines are reversible, the foregoing proof can easily be made to show that neither one is more efficient than the other one. It follows, therefore, that all ideal re versible engines which take in no heat except at T., and reject none except at T,, have the same identical efficiency; and this efficiency can, therefore, depend upon nothing but the two temperatures T, and T,. In the language of mathematics, the efficiency of an ideal reversible engine which runs as here described is a °func tion° of the temperatures at which heat is absorbed and rejected, and of nothing else. In the foregoing demonstration it was as sumed that all of the heat taken in by the engine B is either transformed into mechanical energy or rejected at the temperature T,. If the engine B is of such a kind that this con dition is not fulfilled, by reason of the engine losing some of its heat at temperatures inter mediate to 7', and T, (or by reason of any other imperfaction in design or construction), then the theorem is still true; for the assump tion that we have made above is the one that is least favorable to the demonstration.

Absolute Let us consider an ideal, reversible engine, which in each unit of time takes in H. units of heat at the tem perature T,, and rejects units of heat at the temperature T,. Then the efficiency of the en gine is (Hi—HO/Hi; and this (as we have seen) must be equal to some function of T1 and T,. It will he more convenient, however, to write the efficiency in the form 1— (WHO ; which is obviously permissible. Since this is a function of the two temperatures, so also is fh/112• and we may write 112/1/2 =-1 (T2, T2).

I Now being the heat rejected by the given engine at the temperature T2, may be used again in a second ideal reversible engine, which we may assume to take its heat at T2, and to reject what it does reject (if any) at some still lower temperature, T. The second en gine, considered separately, would give a second equation entirely analogous to the one already written ; and we should have H2/H2= f Tg). But we might consider the two engines. coupled together, to con stitute a single ideal reversible engine, taking in a quantity H, of heat at T., and rejecting a quantity 111 at the temperature L. From this point of view we could write H,/H, = f Ts). But if we multiply H,/H, by we obtain H./112; and hence we see that the function f must be of such a nature that we have the identical relation f (7's, 7'3). f (Ti, 7'1) T.), whatever the values of T,, T, and T,. Exam ination of this equation will show that the dis appearance of T, by the multiplication of the two terms in the first member involves that the function f shall be of the form f T2).=F(Ti)/F(Ti).

Hence we have the general relation