[105] "Philosophical Transactions," 1798.

Rumford, after showing that this heat could not have been derived from any of the surrounding objects, or by compression of the materials employed or acted upon, says: "It appears to me extremely difficult, if not impossible, to form any distinct idea of anything capable of being excited and communicated in the manner that heat was excited and communicated in these experiments, except it be motion."[106] He then goes on to urge a zealous and persistent investigation of the laws which govern this motion. He estimates the heat produced by a power which he states could easily be exerted by one horse, and makes it equal to the "combustion of nine wax candles, each three-quarters of an inch in diameter," and equivalent to the elevation of "25.68 pounds of ice-cold water" to the boiling-point, or 4,784.4 heat-units.[107]

The time was stated at "150 minutes." Taking the actual power of Rumford's Bavarian "one horse" as the most probable figure, 25,000 pounds raised one foot high per minute,[108] this gives the "mechanical equivalent" of the foot-pound as 783.8 heat-units, differing but 1.5 per cent. from the now accepted value.

[106] This idea was not by any means original with Rumford. Bacon seems to have had the same idea; and Locke says, explicitly enough: "Heat is a very brisk agitation of the insensible parts of the object ... so that what in our sensation is heat, in the object is nothing but motion."

[107] The British heat-unit is the quant.i.ty of heat required to heat one pound of water 1 Fahr. from the temperature of maximum density.

[108] Rankine gives 25,920 foot-pounds per minute--or 432 per second--for the average draught-horse in Great Britain, which is probably too high for Bavaria. The engineer's "horse-power"--33,000 foot-pounds per minute--is far in excess of the average power of even a good draught-horse, which latter is sometimes taken as two-thirds the former.

Had Rumford been able to eliminate all losses of heat by evaporation, radiation, and conduction, to which losses he refers, and to measure the power exerted with accuracy, the approximation would have been still closer. Rumford thus made the experimental discovery of the real nature of heat, proving it to be a form of energy, and, publishing the fact a half-century before the now standard determinations were made, gave us a very close approximation to the value of the heat-equivalent. Rumford also observed that the heat generated was "exactly proportional to the force with which the two surfaces are pressed together, and to the rapidity of the friction," which is a simple statement of equivalence between the quant.i.ty of work done, or energy expended, and the quant.i.ty of heat produced. This was the first great step toward the formation of a Science of Thermo-dynamics.

Rumford's work was the corner-stone of the science.

Sir Humphry Davy, a little later (1799), published the details of an experiment which conclusively confirmed these deductions from Rumford's work. He rubbed two pieces of ice together, and found that they were melted by the friction so produced. He thereupon concluded: "It is evident that ice by friction is converted into water....

Friction, consequently, does not diminish the capacity of bodies for heat."

Bacon and Newton, and Hooke and Boyle, seem to have antic.i.p.ated--long before Rumford's time--all later philosophers, in admitting the probable correctness of that modern dynamical, or vibratory, theory of heat which considers it a mode of motion; but Davy, in 1812, for the first time, stated plainly and precisely the real nature of heat, saying: "The immediate cause of the phenomenon of heat, then, is motion, and the laws of its communication are precisely the same as the laws of the communication of motion." The basis of this opinion was the same that had previously been noted by Rumford.

So much having been determined, it became at once evident that the determination of the exact value of the mechanical equivalent of heat was simply a matter of experiment; and during the succeeding generation this determination was made, with greater or less exactness, by several distinguished men. It was also equally evident that the laws governing the new science of thermo-dynamics could be mathematically expressed.

Fourier had, before the date last given, applied mathematical a.n.a.lysis in the solution of problems relating to the transfer of heat without transformation, and his "Theorie de la Chaleur" contained an exceedingly beautiful treatment of the subject. Sadi Carnot, twelve years later (1824), published his "Reflexions sur la Puissance Motrice du Feu," in which he made a first attempt to express the principles involved in the application of heat to the production of mechanical effect. Starting with the axiom that a body which, having pa.s.sed through a series of conditions modifying its temperature, is returned to "its primitive physical state as to density, temperature, and molecular const.i.tution," must contain the same quant.i.ty of heat which it had contained originally, he shows that the efficiency of heat-engines is to be determined by carrying the working fluid through a complete cycle, beginning and ending with the same set of conditions. Carnot had not then accepted the vibratory theory of heat, and consequently was led into some errors; but, as will be seen hereafter, the idea just expressed is one of the most important details of a theory of the steam-engine.

Seguin, who has already been mentioned as one of the first to use the fire-tubular boiler for locomotive engines, published in 1839 a work, "Sur l'Influence des Chemins de Fer," in which he gave the requisite data for a rough determination of the value of the mechanical equivalent of heat, although he does not himself deduce that value.

Dr. Julius R. Mayer, three years later (1842), published the results of a very ingenious and quite closely approximate calculation of the heat-equivalent, basing his estimate upon the work necessary to compress air, and on the specific heats of the gas, the idea being that the work of compression is the equivalent of the heat generated.

Seguin had taken the converse operation, taking the loss of heat of expanding steam as the equivalent of the work done by the steam while expanding. The latter also was the first to point out the fact, afterward experimentally proved by Hirn, that the fluid exhausted from an engine should heat the water of condensation less than would the same fluid when originally taken into the engine.

A Danish engineer, Colding, at about the same time (1843), published the results of experiments made to determine the same quant.i.ty; but the best and most extended work, and that which is now almost universally accepted as standard, was done by a British investigator.

James Prescott Joule commenced the experimental investigations which have made him famous at some time previous to 1843, at which date he published, in the _Philosophical Magazine_, his earliest method. His first determination gave 770 foot-pounds. During the succeeding five or six years Joule repeated his work, adopting a considerable variety of methods, and obtaining very variable results. One method was to determine the heat produced by forcing air through tubes; another, and his usual plan, was to turn a paddle-wheel by a definite power in a known weight of water. He finally, in 1849, concluded these researches.

[Ill.u.s.tration: James Prescott Joule.]

The method of calculating the mechanical equivalent of heat which was adopted by Dr. Mayer, of Heilbronn, is as beautiful as it is ingenious: Conceive two equal portions of atmospheric air to be inclosed, at the same temperature--as at the freezing-point--in vessels each capable of containing one cubic foot; communicate heat to both, retaining the one portion at the original volume, and permitting the other to expand under a constant pressure equal to that of the atmosphere. In each vessel there will be inclosed 0.08073 pound, or 1.29 ounce, of air. When, at the same temperature, the one has doubled its pressure and the other has doubled its volume, each will be at a temperature of 525.2 Fahr., or 274 C, and each will have double the original temperature, as measured on the absolute scale from the zero of heat-motion. But the one will have absorbed but 6-3/4 British thermal units, while the other will have absorbed 9-1/2. In the first case, all of this heat will have been employed in simply increasing the temperature of the air; in the second case, the temperature of the air will have been equally increased, and, besides, a certain amount of work--2,116.3 foot-pounds--must have been done in overcoming the resistance of the air; it is to this latter action that we must debit the additional heat which has disappeared. Now, 2,116.3/(2-3/4) = 770 foot-pounds per heat-unit--almost precisely the value derived from Joule's experiments. Had Mayer's measurement been absolutely accurate, the result of his calculation would have been an exact determination of the heat-equivalent, provided no heat is, in this case, lost by internal work.

Joule's most probably accurate measure was obtained by the use of a paddle-wheel revolving in water or other fluid. A copper vessel contained a carefully weighed portion of the fluid, and at the bottom was a step, on which stood a vertical spindle carrying the paddle-wheel. This wheel was turned by cords pa.s.sing over nicely-balanced grooved wheels, the axles of which were carried on friction-rollers. Weights hung at the ends of these cords were the moving forces. Falling to the ground, they exerted an easily and accurately determinable amount of work, _W_ _H_, which turned the paddle-wheel a definite number of revolutions, warming the water by the production of an amount of heat exactly equivalent to the amount of work done. After the weight had been raised and this operation repeated a sufficient number of times, the quant.i.ty of heat communicated to the water was carefully determined and compared with the amount of work expended in its development. Joule also used a pair of disks of iron rubbing against each other in a vessel of mercury, and measured the heat thus developed by friction, comparing it with the work done. The average of forty experiments with water gave the equivalent 772.692 foot-pounds; fifty with mercury gave 774.083; twenty with cast-iron gave 774.987--the temperature of the apparatus being from 55 to 60 Fahr.

Joule also determined, by experiment, the fact that the expansion of air or other gas without doing work produces no change of temperature, which fact is predicable from the now known principles of thermo-dynamics. He stated the results of his researches relating to the mechanical equivalent of heat as follows:

1. The heat produced by the friction of bodies, whether solid or liquid, is always proportional to the quant.i.ty of work expended.

2. The quant.i.ty required to increase the temperature of a pound of water (weighed _in vacuo_ at 55 to 60 Fahr.) by one degree requires for its production the expenditure of a force measured by the fall of 772 pounds from a height of one foot. This quant.i.ty is now generally called "Joule's equivalent."

During this series of experiments, Joule also deduced the position of the "absolute zero," the point at which heat-motion ceases, and stated it to be about 480 Fahr. below the freezing-point of water, which is not very far from the probably true value,-493.2 Fahr. (-273 C.), as deduced afterward from more precise data.

The result of these, and of the later experiments of Hirn and others, has been the admission of the following principle:

Heat-energy and mechanical energy are mutually convertible and have a definite equivalence, the British thermal unit being equivalent to 772 foot-pounds of work, and the metric _calorie_ to 423.55, or, as usually taken, 424 kilogrammetres. The exact measure is not fully determined, however.

It has now become generally admitted that all forms of energy due to physical forces are mutually convertible with a definite quantivalence; and it is not yet determined that even vital and mental energy do not fall within the same great generalization. This quantivalence is the sole basis of the science of Energetics.

The study of this science has been, up to the present time, princ.i.p.ally confined to that portion which comprehends the relations of heat and mechanical energy. In the study of this department of the science, thermo-dynamics, Rankine, Clausius, Thompson, Hirn, and others have acquired great distinction. In the investigations which have been made by these authorities, the methods of transfer of heat and of modification of physical state in gases and vapors, when a change occurs in the form of the energy considered, have been the subjects of especial study.

According to the law of Boyle and Marriotte, the expansion of such fluids follows a law expressed graphically by the hyperbola, and algebraically by the expression PV^{_x_} = A, in which, with unchanging temperature, _x_ is equal to 1. One of the first and most evident deductions from the principles of the equivalence of the several forms of energy is that the value of x must increase as the energy expended in expansion increases. This change is very marked with a vapor like steam--which, expanded without doing work, has an exponent less than unity, and which, when doing work by expanding behind a piston, partially condenses, the value of _x_ increases to, in the case of steam, 1.111 according to Rankine, or, probably more correctly, to 1.135 or more, according to Zeuner and Grashof. This fact has an important bearing upon the theory of the steam-engine, and we are indebted to Rankine for the first complete treatise on that theory as thus modified.

Prof. Rankine began his investigations as early as 1849, at which time he proposed his theory of the molecular const.i.tution of matter, now well known as the theory of molecular vortices. He supposes a system of whirling rings or vortices of heat-motion, and bases his philosophy upon that hypothesis, supposing sensible heat to be employed in changing the velocity of the particles, latent heat to be the work of altering the dimensions of the orbits, and considering the effort of each vortex to enlarge its boundaries to be due to centrifugal force. He distinguished between real and apparent specific heat, and showed that the two methods of absorption of heat, in the case of the heating of a fluid, that due to simple increase of temperature and that due to increase of volume, should be distinguished; he proposed, for the latter quant.i.ty, the term heat-potential, and for the sum of the two, the name of thermo-dynamic function.

[Ill.u.s.tration: Prof. W. J. M. Rankine.]

Carnot had stated, a quarter of a century earlier, that the efficiency of a heat-engine is a function of the two limits of temperature between which the machine is worked, and not of the nature of the working substance--an a.s.sertion which is quite true where the material does not change its physical state while working. Rankine now deduced that "general equation of thermo-dynamics" which expresses algebraically the relations between heat and mechanical energy, when energy is changing from the one state to the other, in which equation is given, for any a.s.sumed change of the fluids, the quant.i.ty of heat transformed. He showed that steam in the engine must be partially liquefied by the process of expanding against a resistance, and proved that the total heat of a perfect gas must increase with rise of temperature at a rate proportional to its specific heat under constant pressure.

Rankine, in 1850, showed the inaccuracy of the then accepted value, 0.2669, of the specific heat of air under constant pressure, and calculated its value as 0.24. Three years later, the experiments of Regnault gave the value 0.2379, and Rankine, recalculating it, made it 0.2377. In 1851, Rankine continued his discussion of the subject, and, by his own theory, corroborated Thompson's law giving the efficiency of a perfect heat-engine as the quotient of the range of working temperature to the temperature of the upper limit, measured from the absolute zero.

During this period, Clausius, the German physicist, was working on the same subject, taking quite a different method, studying the mechanical effects of heat in gases, and deducing, almost simultaneously with Rankine (1850), the general equation which lies at the beginning of the theory of the equivalence of heat and mechanical energy. He found that the probable zero of heat-motion is at such a point that the Carnot function must be approximately the reciprocal of the "absolute"

temperature, as measured with the air thermometer, or, stated exactly, that quant.i.ty as determined by a perfect gas thermometer. He confirmed Rankine's conclusion relative to the liquefaction of saturated vapors when expanding against resistance, and, in 1854, adapted Carnot's principle to the new theory, and showed that his idea of the reversible engine and of the performance of a cycle in testing the changes produced still held good, notwithstanding Carnot's ignorance of the true nature of heat. Clausius also gave us the extremely important principle: It is impossible for a self-acting machine, unaided, to transfer heat from one body at a low temperature to another having a higher temperature.

Simultaneously with Rankine and Clausius, Prof. William Thomson was engaged in researches in thermo-dynamics (1850). He was the first to express the principle of Carnot as adapted to the modern theory by Clausius in the now generally quoted propositions:[109]

[109] _Vide_ Tait's admirable "Sketch of Thermodynamics," second edition, Edinburgh, 1877.

1. When equal mechanical effects are produced by purely thermal action, equal quant.i.ties of heat are produced or disappear by transformation of energy.

2. If, in any engine, a reversal effects complete inversion of all the physical and mechanical details of its operation, it is a perfect engine, and produces maximum effect with any given quant.i.ty of heat and with any fixed limits of range of temperature.

William Thomson and James Thompson showed, among the earliest of their deductions from these principles, the fact, afterward confirmed by experiment, that the melting-point of ice should be lowered by pressure 0.0135 Fahr, for each atmosphere, and that a body which contracts while being heated will always have its temperature decreased by sudden compression. Thomson applied the principles of energetics in extended investigations in the department of electricity, while Helmholtz carried some of the same methods into his favorite study of acoustics.

The application of now well-settled principles to the physics of gases led to many interesting and important deductions: Clausius explained the relations between the volume, density, temperature, and pressure of gases, and their modifications; Maxwell reestablished the experimentally determined law of Dalton and Charles, known also as that of Gay-Lussac (1801), which a.s.serts that all ma.s.ses of equal pressure, volume, and temperature, contain equal numbers of molecules.

On the Continent of Europe, also, Hirn, Zeuner, Grashof, Tresca, Laboulaye, and others have, during the same period and since, continued and greatly extended these theoretical researches.

During all this time, a vast amount of experimental work has also been done, resulting in the determination of important data without which all the preceding labor would have been fruitless. Of those who have engaged in such work, Cagniard de la Tour, Andrews, Regnault, Hirn, Fairbairn and Tate, Laboulaye, Tresca, and a few others have directed their researches in this most important direction with the special object of aiding in the advancement of the new-born sciences. By the middle of the present century, the time which we are now studying, this set of data was tolerably complete. Boyle had, two hundred years before, discovered and published the law, which is now known by his name[110] and by that of Marriotte,[111] that the pressure of a gas varies inversely as its volume and directly as its density; Dr. Black and James Watt discovered, a hundred years later (1760), the latent heat of vapors, and Watt determined the method of expansion of steam; Dalton, in England, and Gay-Lussac, in France, showed, at the beginning of the nineteenth century, that all gaseous fluids are expanded by equal fractions of their volume by equal increments of temperature; Watt and Robison had given tables of the elastic force of steam, and Gren had shown that, at the temperature of boiling water, the pressure of steam was equal to that of the atmosphere; Dalton, Ure, and others proved (1800-1818) that the law connecting temperatures and pressures of steam was expressed by a geometrical ratio; and Biot had already given an approximate formula, when Southern introduced another, which is still in use.

[110] "New Experiments, Physico-Mechanical, etc., touching the Spring of Air," 1662.

[111] "De la Nature de l'Air," 1676.

The French Government established a commission in 1823 to experiment with a view to the inst.i.tution of legislation regulating the working of steam-engines and boilers; and this commission, MM. de p.r.o.ny, Arago, Girard, and Dulong, determined quite accurately the temperatures of steam under pressures running up to twenty-four atmospheres, giving a formula for the calculation of the one quant.i.ty, the other being known. Ten years later, the Government of the United States inst.i.tuted similar experiments under the direction of the Franklin Inst.i.tute.

The marked distinction between gases, like oxygen and hydrogen, and condensible vapors, like steam and carbonic acid, had been, at this time, shown by Cagniard de la Tour, who, in 1822, studied their behavior at high temperatures and under very great pressures. He found that, when a vapor was confined in a gla.s.s tube in presence of the same substance in the liquid state, as where steam and water were confined together, if the temperature was increased to a certain definite point, the whole ma.s.s suddenly became of uniform character, and the previously existing line of demarkation vanished, the whole ma.s.s of fluid becoming, as he inferred, gaseous. It was at about this time that Faraday made known his then novel experiments, in which gases which had been before supposed permanent were liquefied, simply by subjecting them to enormous pressures. He then also first stated that, above certain temperatures, liquefaction of vapors was impossible, however great the pressure.

Faraday's conclusion was justified by the researches of Dr. Andrews, who has since most successfully extended the investigation commenced by Cagniard de la Tour, and who has shown that, at a certain point, which he calls the "critical point," the properties of the two states of the fluid fade into each other, and that, at that point, the two become continuous. With carbonic acid, this occurs at 75 atmospheres, about 1,125 pounds per square inch, a pressure which would counterbalance a column of mercury 60 yards, or nearly as many metres, high. The temperature at this point is about 90 Fahr., or 31 Cent.

For ether, the temperature is 370 Fahr., and the pressure 38 atmospheres; for alcohol, they are 498 Fahr., and 120 atmospheres; and for bisulphide of carbon, 505 Fahr., and 67 atmospheres. For water, the pressure is too high to be determined; but the temperature is about 775 Fahr., or 413 Cent.

Donny and Dufour have shown that these normal properties of vapors and liquids are subject to modification by certain conditions, as previously (1818) noted by Gay-Lussac, and have pointed out the bearing of this fact upon the safety of steam-boilers. It was discovered that the boiling-point of water could be elevated far above its ordinary temperature of ebullition by expedients which deprive the liquid of the air usually condensed within its ma.s.s, and which prevent contact with rough or metallic surfaces. By suspension in a mixture of oils which is of nearly the same density, Dufour raised drops of water under atmospheric pressure to a temperature of 356 Fahr.--180 Cent.--the temperature of steam of about 150 pounds per square inch.

Prof. James Thompson has, on theoretical grounds, indicated that a somewhat similar action may enable vapor, under some conditions, to be cooled below the normal temperature of condensation, without liquefaction.

Fairbairn and Tate repeated the attempt to determine the volume and temperature of water at pressures extending beyond those in use in the steam-engine, and incomplete determinations have also been made by others.

Regnault is the standard authority on these data. His experiments (1847) were made at the expense of the French Government, and under the direction of the French Academy. They were wonderfully accurate, and extended through a very wide range of temperatures and pressures.

The results remain standard after the lapse of a quarter of a century, and are regarded as models of precise physical work.[112]