But if heat can be generated by mechanical power, so inversely must heat be competent to produce mechanical effects, and, in fact, according to the mechanical theory of heat, one heat unit must be regarded as capable of performing the work of 424 metre-kilograms, or, in other words, for each metre-kilogram of work performed 0.002358 units of heat must be expended.

This equivalent results, in the first place, theoretically, if we assume as known the ratio of the specific heat of gases, under constant pressure and in constant volume, as well as the coefficients of the expansion of the gases. One cubic metre of air at 0°, under ordinary atmospheric pressure, weighing 1.293 kilogram must be heated to 273° C., if, with unaltered volume, its elasticity be augmented to two atmospheres. But for this are required,

273 × 1.293 × 0.1686=59 units of heat,

since 0.1686 is the specific heat of air with constant volume. But if one cubic metre of air at 0°, with atmospheric pressure, be raised to the temperature of 273° C., while, with constant pressure, it is free to expand, its volume will be increased to two cubic metres, and the quantity of heat necessary, therefore, is

273 × 1.293 × 0.2377=83 units of heat,

since 0.2377 is the specific heat of air with constant pressure. The difference, 83-59-24 units of heat, is thus necessary, over and above the increase of temperature, to expand the gas, under constant pressure, to double the volume.

Let us now inquire into the quantity of mechanical work thereby performed. Let us conceive the above-mentioned quantity of air enclosed in a hollow cylinder, having a transverse section of one square metre, and confined above by a moving piston, which, at its starting point, is elevated one metre above the immovable floor. On this piston the atmosphere bears with a pressure of 10333 kilograms. If the enclosed air, with unaltered pressure, be now expanded to a double volume, it must necessarily push the piston one metre, which corresponds to a mechanical work of 10,333 kilogram-metres. Thus to execute a mechanical work of 10333 metre-kilograms, 24 units of heat are necessary; hence one unit of heat corresponds to a mechanical work of =430 kilogram-metres, a result which so nearly coincides with that obtained in the inverse way, namely, by conversion of mechanical work into heat, that no doubt of the complete reciprocity between mechanical work and heat can longer exist.

10333
24

The experiments and observations above recited have served to establish the proposition that "in all cases in which work is produced by heat, a quantity of heat proportional to the work produced disappears or is consumed, and that inversely the same quantity of heat may be generated by the expenditure of an equal amount of work," a proposition which is usually received as the first law of the mechanical theory of heat, and which, with this degree of precision and generality, was first enunciated by Clausius. It is the proposition which forms the starting point of the mathematical development of the mechanical theory of heat, and in regard to it the learned have furnished us with a series of articles in Poggendorff's Annalen. These articles, accompanied by annotations by Friedrich Vieweg and son, have recently (1865) appeared at Brunswick in a single volume.

Besides Clausius; Holzmann, Clapeyron, W. Thomson, Rankine, and others have occupied themselves with the mathematical development of the mechanical theory of heat, while Zeuner may claim the merit of having collected in a clear and comprehensive form the leading characteristics of the theory and of having illustrated it by manifold applications, (Grundzüge der mechanischen Wärmetheorie, 1st edition, 1860, 2d, 1866, Leipzig.) Another highly acceptable work on this important subject is the Théorie mecanique de la chaleur, by Hirn, (2d edition, Paris, 1865,) in which, together with the analytic development, the experimental part is very thoroughly treated.

In all these writings the mechanical theory of heat is developed, as it needs must be when a general and complete solution of the problem is contemplated, through a certain amount of the higher mathematical analysis. In what follows, however, I shall endeavor to set forth at least the most important principles thereof in an elementary form, and, by the application of these principles to saturated vapors, to show some material consequences of their action. I shall hope, thereby, not only to communicate a right idea of the nature and signification of the theory in question to those who are deficient in a knowledge of the differential and integral calculus, but also to supply to those who may be provided with that learning, a sort of introduction or preparation for the more thorough study of those doctrinal difficulties which are apt to oppose themselves at the entrance upon such an investigation. But before proceeding to a nearer consideration of the grounds of the mechanical theory of heat, we will examine more definitely the loss of heat, corresponding, in some particular cases, to the performance of work.

V.-DISAPPEARANCE OF HEAT THROUGH THE PERFORMANCE OF WORK.

By a large number of experiments directed to the subject under consideration, Hirn has shown that in steam-engines a quantity of heat disappears directly corresponding to the work executed, he having employed with that view engines of from 90 to 150 horse-power. The machines with which he experimented were of the expansion order, in which the steam, after it had operated, passed off into a condenser. In order to avoid errors which might arise from water being mechanically carried over by the engine from the boiler, or steam already condensed in the expansion, Hirn caused the machinery to work with overheated steam, which he procured by means of an appropriate apparatus, whereby the vapors proceeding from the boiler were heated, before their entrance into the cylinder, to a definite temperature ascertained by a thermometer.

The quantity of water p which entered, in a second, the vessel for evaporation, and arrived through the machine in the condenser, was ascertained by exact measurement of the quantity of water which, during the space of a whole day, was conveyed by the feed-pump into the boiler, under a uniform working of the machine, and with an unchanged height of water in the boiler. In like manner the quantity of water P immitted each second into the condenser was ascertained by the determination of the quantity of water of condensation discharged during a whole day under a uniform influx.

The quantity of heat given up by the steam condensed during each second, was found in the following manner: If t be the temperature at which the steam is formed in the boiler, then according to Regnault's experiments, the whole quantity of heat which is contained in one kilogram of steam at this temperature more than that contained in one kilogram of water at 0°, is q1=606.5+0.305t units of heat. But this steam, before its entrance into the cylinder of the steam engine, is heated to the degree of T, whence there is further necessary for each kilogram of steam q2=0.5 (T-t) units of heat, if we take, as may be done with approximative correctness, 0.5 as the specific heat of the steam.

The quantity of heat which this kilogram of steam loses, until it is condensed and cooled to the temperature of ƒ degrees, with which the water of condensation leaves the condenser, is thus 1+92-f, and hence the whole quantity of heat, which the quantity of steam p traversing the engine every second gives up, is

Q1=P (1+92-),

=p [606.5+0.305t+0.5 (T—t)—ƒ ].

If, now, no heat were consumed by the performance of work in the cylinder the whole quantity of heat Q1 must be carried over to the condenser, and her serve to raise the temperature of the condensation-water. If i be the temperature at which that water enters the condenser, but ƒ the temperature at which it issues therefrom, then fi is the quantity of heat which each kilogram of condensation

=

water takes up in the condenser, whence the quantity of heat Q2 which is taken up, in each second, by the condensation-water is Q=P (fi,) and thus must Q1 Q2 if no heat be expended for the work. But experiment shows in fact that Q-Q2 is in nowise equivalent to nothing. In such an experiment, for example, in which the ratio of expansion was 1 : 2, the tension of the steam in the boiler gave a result equal to 4.5 atmospheres, and thus t=148°.3. Also,

There had, consequently, been 40.34 units of heat expended for the work done, while the net effect of the machine was found, by Prony's dynamometer equal to 11250 metre-kilograms, and hence there results for one unit of heat consumed

11250 =278 metre-kilograms,

40.34

as the practical or net effect.

By another experiment with the same machine, in which the ratio of expansion was 1:6, the following values were given:

t=152°.2

p=0.23548 kilograms.

f=2°.605

T-t=93°
P=5.8718 kilograms.
i=3°.2

whence there results

Q1-Q2=158.81-123.3=30.51 units of heat;

while the net effect of the machine was found equal to 8700 metre-kilograms, and thus, for one unit of heat consumed, there results

Thus it will be seen from these premises that not only is there really a consumption of heat for mechanical work, but also that the practical effect of steamengines is very nearly proportional to the loss of heat.

We might, from these investigations alone, calculate the mechanical equivalent of heat, if the practical effect measured were equal to the whole work done by the steam. Let us suppose now that the net effect of the machine amounted to about 70 per cent. of the entire work done by the steam, and we shall have, as the mean of the two above experiments, the mechanical equivalent of heat equal to 400 metre-kilograms.

In like manner Clausius derived from a great number of experiments, which were conducted with steam-engines by Hirn, the number 413 as the mean value for the mechanical equivalent of heat. Now, if we assume the mechanical equivalent of heat equal to 424, it would result from the two experiments, whose details have been given above, that the practical effect of these machines amounts to about 66 per cent. of the whole work done by the steam.

Every process which is of a nature to produce heat can also perform mechanical work; but such work is always attended by a corresponding consumption of heat, or, in other words, the quantity of heat produced by a definite process suffers a corresponding diminution, if together with the generation of heat mechanical effect is produced. This proposition is well illustrated by the electro-magnetic motor. When an electric current traverses the metallic coils of a magnetizing spiral, the wire is heated, and the heat produced, in a given time, in the whole circuit is expressed by the equation

w='k s2l.

where s indicates the quantity of the current, the total resistance of the circuit, and k a constant factor. The quantity of heat produced, however, is, under like conditions, always proportional to the consumption of zinc in the battery, (local action being of course disregarded.) Were the conducting wire, for instance, so much lengthened that the whole resistance of the circuit would be doubled, and thus raised to 27, the quantity of the current and consumption of zinc would be thereby reduced to half, but the heat produced by the current

Thus with the quantity of current and consumption of zinc, the production of heat also would be reduced to half.

Quite otherwise is the result when the diminution of the quantity of the current is produced, not by the augmentation of the resistance to conduction, but by the expenditure of power.

In a previous section of this work it has been seen that the strength of the current, which traverses any electro-magnetic motor in a state of repose, is instantly reduced when the motor begins to rotate, and that the current becomes weaker as the rotation is more rapid. Let us suppose that the burden of the machine be so regulated that the strength of the current of the rotating machine be just half as great as in that at rest, then, with the quantity of current reduced to half, the consumption of zinc will also be reduced to half; but the production of heat will have decreased in a quite different proportion. Since now the strength of the current iss, but the resistance the same as in the machine at rest, namely, 7, we shall have as the quantity of heat produced

Thus the zinc consumption reduced to half produces only a quantity of heat reduced to one-fourth; a part of the zinc-consumption, therefore, is not employed in the production of heat, but in the performance of mechanical work, or, in other words, for the quantity of heat w, an equivalent mechanical work has been performed.

We observe a similar state of things if we investigate the performance of labor by human or animal forces. Animal heat, we know, is generated by a slow combustion, kept up through the process of breathing. For the oxygen which we inspire, carbonic acid and vapor are exhaled; with every breath, therefore, a definite quantity of carbon and hydrogen leaves the body, and the corporeal mass must necessarily undergo a corresponding diminution; a diminution which, if not determinable by weight for every breath, is readily so for an interval of a few hours. This loss of material in the process of breathing is replaced through the reception of food.

But the proportion between the production of heat and the consumption of corporeal matter is quite different, according as the person remains perfectly at rest, or is engaged in the performance of some more or less considerable labor. The production of heat and consumption of oxygen, and, consequently, the bodily diminution of weight, are at a minimum, if the individual continues for some time sitting or lying in complete inactivity. If he perform, on the contrary, some strenuous labor, both the consumption of oxygen and the reduction of weight will be found in the same space of time to have been much more sensible. Through the accelerated breathing and more rapid pulsation the production of heat in the body is undoubtedly augmented, but it results from the principles of the mechanical theory of heat that the development of warmth cannot be taken as directly proportional to the consumption of oxygen, but that the increased interchange of matter in the body serves only in part for the production of heat, while the rest has been spent in the producing mechanical effect.

The correctness of this proposition has been verified by Hirn in a series of

carefully conducted experiments. With this view he occupied a small hermetically-closed chamber, constructed of deal-boards, and lighted by a glass window, the contents of which chamber measured about four cubic metres. At one end of this structure was a chair on which he sat when occupied with the development of heat under conditions of repose. At the other end was a tread-wheel, the axis of which passed, but so as to be air-tight, through the wall, and was connected on the outside with such an apparatus that by the turning of the wheel a mechanical work was executed. The quantity of this labor performed at each revolution of the wheel is manifestly equal to the lifting of the bodily weight of the experimenter to a height represented by the circumference of the wheel. By means of a counter adapted to the axis of the tread-wheel, the number of revolutions in a given time could be counted, while the quantity of external mechanical work done in an hour could be determined with great accuracy.

Before the mouth of the experimenter a valve apparatus was attached, from which a caoutchouc tube was carried to a gasometer which furnished the air required for breathing, while a second tube of like material passed to another gasometer which received the exhaled gases; these, as well as the air inhaled, were carefully analyzed. The chamber was placed in the midst of a larger apartment, the temperature of which varied but little and slowly. Sensitive thermometers gave the temperature of the air both without and within the chamber. If, during repose or labor, the interior thermometer had become stationary, its indication was noted, and the valve apparatus placed before the mouth, so that the consumption of oxygen during an unaltered condition of the experimenter might be ascertained.

It is clear that if the interior thermometer ceased to rise, the loss of heat in the chamber through its walls had become equal to the quantity of heat which the experimenter developed. By a series of preliminary experiments, Hirn had determined what quantity of heat must be developed within the chamber, in order to maintain within and without definite differences of temperature. With this view a flame of hydrogen gas, supplied by a constant stream, was suffered to burn in the interior of the chamber. For a definite magnitude of the flame, when the condition of equilibrium is attained, a determinate difference of temperature within and without the chamber is established; and when the quantity of hydrogen consumed in a given time is ascertained, we can calculate what quantity of heat has been developed in that time, since we know how many units of heat are developed by the burning of one gram of hydrogen, (§ 277.) From the repetition of these experiments for different sizes of flame, Hirn obtained the empirical law on which depends the excess of temperature in the interior from the quantity of heat there developed, and he could thus deduce, in later experiments, from the observed difference of temperature the quantity of heat developed by the experimenter.

When, during such an experiment, Hirn occupied the chamber, he found that, with absolute rest of his person, 29.65 grams of oxygen were consumed in an hour, while the development of heat during that time amounted to 155 units of 155 heat, (calories,) being or 5.22 units of heat to one gram of oxygen. 29.65' When, on the other hand, the experimenter labored on the tread-wheel, so that the work done in an hour amounted to 27448 metre-kilograms, the consumption of oxygen in that space of time was 131.74 grams, while the quantity of heat developed, as indicated by the thermometer, amounted to 251 units.

In the state of rest, however, the 131.74 grams of oxygen consumed would be 131.74 × 5.22=687.68 units of heat, and thus 436.68 would be exhibited more than had in fact been developed; but instead of the vanished 436.68 units of heat, work had been done, partly without, on the tread-wheel, and partly within, in the organism itself.