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this important subject was resumed, and who again recalled it to the attention of physicists in a treatise published, in 1842, in the Annals of Chemistry and Pharmacy, under the title of Observations on the forces of inanimate nature, was Mayer, a practising physician of Heilbronn.
III. –THE MECHANICAL EQUIVALENT OF HEAT. Mayer first enunciated the idea that, as a definite proportion universally exists between cause and effect, there must always, in the production of heat by mechanical means, be an invariable proportion between the heat generated and the mechanical power consumed for that purpose; and, in fact, he thus early established, with closely approximate exactness, the mechanical equivalent of heat. This was, at a later period, still more accurately determined through the researches of Joule and Hirn.
In 1843 the observation was made by Joule (Phil. Mag., vol. xxiii) that, in the passage of water through a narrow tube, heat is generated, and that a mechanical power of 770 foot-pounds* is consumed in raising the temperature of one pound of water to 1° F., a result which, as we shall see,
differrent from that obtained by compression of the air.
Joule sought, also, to ascertain by other methods the proportion of the heat generated by friction to the mechanical power thereby expended. In a copper vessel (A, Fig. 3,) a paddle wheel, whose construction is represented
in Fig 4, was so disposed as to be capable of revolving around a vertical axis. Eight paddles of thin plate, at an angle of 45° with one another, are placed at the height h, eight others at the height
9. They move between metallic plates, which are fixed to the wall of the vessel; four of Fig. 4. these plates standing at right angles to one another, being at the height f, and four others at the height c.
The vessel A, which stands on a wooden pedestal, is filled with water, and the revolution of the paddle-wheel is effected, in the manner represented in the figure, by means of the weights B and D, which by their descent co-operate in communicating motion to the axis of the wheel, and have a fall of about 63 inches. After these weights had reached the floor, by withdrawing the peg s,
1 the connection of the cylinder V with the axis of rotation of the paddle-wheel was severed, the weights B and D were again wound up,
A foot-pound is the power developed in the fall of a weight of one pound through a height of one foot. An English unit of heat is the quantity required to raise a pound of water one degree Fahrenheit. A kilogram-metre is the power developed in the fall of a kilogram (2.204 pounds) through one metre. A French unit of heat is the quantity required to raise one kilogram of water one degree centigrade.
and the same operation was repeated. After this had been done 20 times the elevation of the temperature produced by the above means in the water of the reservoir A was measured, and was found to amount to nearly 0°.6 F. The mechanical power expended in the production of this effect is obtained by multiplying the weights by the total distance through which they have fallen, with allowance, however, for the acceleration with which, each time, they have descended to the floor.
In the mode here described Joule has conducted a long series of researches, and calculates, as a mean after the application of the necessary corrections, that an expenditure of power equivalent to 773.64 foot-pounds develops under the above circumstances as much heat as is required to raise the temperature of one pound of water 1° F., or, in other terms, that a unit of heat (French) is the thermal equivalent of a mechanical expenditure of power of 425 kilogram-metres.
The friction of an iron paddle-wheel in quicksilver gave 776.3 foot-pounds, and the friction of cast-iron plates with one another 774.88 foot-pounds as the expendture of power which is necessary to raise the temperature of one pound of water 1° F.
A not very different result was obtained by Joule when he compared the quantity of heat set free in the coils of an electro-magnet rotating between strong magnetic poles, with the mechanical power necessary to produce this rotation, (Phil. Mag. vol. xxiji.) For determining the heat developed in the coils of the rotating electro-magnet, the latter was introduced into a glass tube in such a manner that the interval between the magnet and the glass wall formed a vessel closed on all sides, which was filled with water. Through the heat devel. oped by the rotation of the electro-magnet, the temperature of this water was raised, and the increase of temperature carefully ascertained. In order to determine the mechanical power required to produce the rotation, a string was wound around the prolongation of the axis of rotation, and the revolution of the magnet effected by a weight suspended to the string. From this experiment Joule computed that, for the production of an amount of heat capable of raising one pound of water 1° F. a mechanical power of 838 foot-pounds is requisite, and thus the unit of heat corresponds to an expenditure of power of 460 kilogram-metres.
To the same physicist we owe an experiment for determining the mechanical equivalent of heat through that which is liberated by compression of the air, (Krönig's Journal, iii). Into a copper reservoir A, 12 inches in length, 1361 cubic contents, 4-inch thickness of wall, by means of a compression pump screwed to it, air was pressed, as into the bulb of an air-gun, until an elastic force of nearly 22 atmospheres was attained. During this operation the copper reservoir, together with the pump, was immersed in a vessel which held 45 pounds 3 ounces of water. By 300 strokes of the piston the air in the vessel was condensed from 1 to 21.654 atmospheres, and so much heat was thereby developed that the temperature of cool water rose 0°:645 F. This increase of temperature, however, did not arise alone from the compression of the air, but also from the friction of the piston. To eliminate this last, the tube through which the air had been introduced was closed, and it was found that, through 300 strokes of the piston, which now were not attended by a compression of the air in the reservoir, the temperature of the cool water was raised 09.297 F. By this first experiment, therefore, on computing the results of compression of the air, an elevation of temperature of 0°.348 F. is given.
After making the necessary reductions and corrections, it now resulted that through the compression of 2956 cubic inches of dry air, of atmospheric density, into a space of 136.5 cubic inches, such a quantity of heat was developed as was necessary to raise the temperature of one pound of water 130.628 F. This is equivalent to the quantity of heat required to raise the temperature of 3437 grams of water 1° Û.
Let us now seek to determine the mechanical power which is required in order to compress 2956 cubic inches of air, of atinospheric density, to a 21.654-fold density. For this purpose we will conceive the quantity of air just mentioned to be contained in a tube a b, (Fig. 5,) which is supposed to be 21.65.1 feet long, and to havo such a transverse section (11.376 square inches) that the contents of a portion of this tube one foot long shall be equal to the contents of the vessel A, (136.5 cubic inches); thus the above compression will plainly ensue if we force a piston from the upper end a down to the point c, which is situated one foot above the bottom of the tube. If now lines be drawn at different points of the tube per
h pendicular to its axis, and of a length always proportional to the pressure under which the included air stands, and if the piston be driven to the point c, so that the line cg shall be 21.654 times as great as a f, the curve f hig, which connects the terminal points of the above lines, will be a portion of an equilateral hyperbola, and the hyperbolic surface a c ghy will represent the power which must have been employed in pressing the piston down from a to c. Let us denote gc by y, b c by x, and ba by x'; thus the
ax contents of the surface in question will be H=x• y: log nat" and if x=1 Hry log nat x', or H=2.3026 · Y · log a'
(1) if log represent the common logarithm referable to 10 as a base. By the test of experiment the barometer stood at 30.2 English inches, which makes on the transverse section of our tube 168.5 pounds. The line fa thus represents to us the pressure 168.5, but gc the pressure 21.654 x 168.5=3648.7. Let us now put in equation (1) X=21.654 and y=3648.7; we shall thus have
H=2.3026 X 3648.7 log 21.654=11220 foot-pounds, as the expenditure of power which is required to compress 2956 cubic inches of air of atmospheric density into a space 21.654 times smaller, whereby, as has been seen 3.437 units of heat are developed.
Hence, according to these experiments, 3437 units of heat are the thermic equivalent for an expenditure of power of 11220 foot-pounds, or 1552 kilogram-metres. In order, therefore, to produce one unit of heat through compression of the air, an expenditure of power of 451 kilogram-metres is needed.
For the purpose of measuring the absorption of heat which results from the discharge of compressed air, the vessel A, after the air had been compressed in it to 22 atmospheres, was placed in a reservoir containing 21 pounds of water. As the compressed air was now allowed to escape from the vessel A through a leaden pipe, the temperature of the surrounding water was found to be lowered 4.°1 F. With due regard to all necessary corrections, it may be hence computed that the quantity of heat which disappears by the discharge of the air from
the vessel is perhaps just as great as that which is developed in the vessel through the compression of the air. In this experiment the escaping air has to overcome the resistance of the atmosphere, and thus to perform a mechanical work.
In another series of experiments, to the vessel A, in which air had been compressed to 22 atmospheres, was screwed, by means of a short metallic pipe, an equally large vessel B, exhausted of air, and after both vessels, A and B, had been placed in the same reservoir, holding 164 pounds of water, a suitably adapted cock was opened, so that half of the air compressed in A could flow over into B. Through this process no observable change of temperature was produced in the water surrounding the vessels A and B, whence Joule draws the conclusion that no change of temperature occurs when air expands in such a way as to create no mechanical power.
When the two receivers A and B were placed in separate vessels of water, a lowering of temperature of 2°.36 F. was observed in the vessel which contained the receiver A, out of which flowed the compressed air, while the water which surrounded the receiver B, into which the air flowed, acquired a nearly equal elevation of temperature.
Hirn, also, (Théorie méchanique de la chaleur, Paris, 1865,) has made a series of experiments for the determination of the mechanical equivalent of heat, among which we adduce that on the development of heat through the compression of lead as being distinguished at once for its simplicity and conclusiveness.
A cylinder A, of wrought iron, 350 kilograms in weight, which we will call the hammer, is suspended by two pairs of strings about three metres in length, as is shown by Fig. 6. Opposite to this cylinder is suspended in like manner
a prismatic block of sandstone, of the weight of 941 kilograms, which we will term the anvil, and which is furnished on the side opposed to the hammer with an iron plate C. Between the hammer and anvil is placed a cylindrical piece of lead P, having a weight of 2.948 kilograms, and supported by a light wooden holder, (Holzgabel.) This piece of lead is in part hollowed out in the direction of its axis. Its temperature before the experiment was ascertained by means of a thermometer temporarily introduced into the cavity to be 79.87.
The hammer was now drawn back by a pulley until it reached the position A', and then again released. In recovering its position of equilibrium, it delivered a strong blow upon the lead, which compressed and heated. Yet was not tho entire living force of the falling hammer spent in the compression of the lead; for, after impact, the stone block and iron cylinder were again driven somewhat apart. According to an experiment of this kind, the height of fall of the hammer was 1.166 metre; the recoil of the same after impact 0.087 metre; the recoil of the anvil after impact 0.103 metre. Hence the living force which the iron hammer had attained in falling was,
L=350 · 1.166=408.100 metre-kilograms; but the living force with which hammer and anvil after the blow recoiled from one
I=0.103 (941+2.95)+0.087.350=127.677 metre kilograms; the living force, therefore, expended in the compression of the lead is,
L-I=408.100-127.677=280.423 metre-kilograms. In order to determine the quantity of heat which was developed through the compression of the lead, the latter, after receiving
Fig. 8. the blow, was quickly withdrawn from between the hammer and anvil, and by means of two threads, which had been already attached to it, was suspended in the manner shown at Fig. 8. Into the cavity of the compressed piece of lead, 18.5 grams of water at 0° C. wero poured, and the temperature thereof, which very quickly became the same with that of the lead, was ascertained by means of an immersed thermometer. This temperature was: 4 minutes after the impact
129.10 8 minutes after the impact
119.75 Thus in four minutes, from the end of the fourth minute to the end of the eighth, the cooling amounted to 09.35. If we assume, now, what may at least be accepted as an approximation, that the rate of cooling, during the first four minutes after the blow, was maintained during the following four minutes, we have 11.75 : 0.35=12.1 : 2; whence results x=0.36. Since, therefore, the temperature of the lead had, at the moment of compression, been 12°.10+0.36=120.46, the calefaction from the blow would be 120.46—70.87= 4°.59; consequently, the quantity of heat developed through the collision is, 49.59 · 2.948 · 0.03145+12.46 • 0.0185=0.656 units of heat, since 0.03145 is the specific heat of lead.
If we divide the work spent in the compression of the lead, 280.423 metrekilograms, by the corresponding quantity of heat, 0.656 thermic units, we obtain the work necessary for producing one thermic unit,
0.656 Instead of this number, however, 425.2 is the result, if the cooling of the lead is not calculated approximately, as above, but by exact formulas.
IV.-EQUIVALENCE OF HEAT AND WORK. As a mean, there results, from the best experiments which have been made on this subject, 424 metre-kilograms as the mechanical equivalent of heat, or, to use a more accurate expression, the work equivalent of the unit of heat; and the quantity of heat A, which corresponds to the unit of work, is
424 that is to say, the caloric equivalent of the work unit is 0.002358 units of heat; by the expenditure of one metro-kilogram, therefore, 0.002358 units of heat may be generated.