Up to the present time the changes caused in the temperature of bodies by motion have been very little studied. This class of phenomena merits, however, the attention of observers. When bodies are in motion, especially when that motion disappears, or when it produces motive power, remarkable changes take place in the distribution of heat, and perhaps in its quantity.
We will collect a few facts which exhibit this phenomenon most clearly.
1. _The Collision of Bodies._—We know that in the collision of bodies there is always expenditure of motive power. Perfectly elastic bodies only form an exception, and none such are found in nature.
We also know that always in the collision of bodies there occurs a change of temperature, an elevation of temperature. We cannot, as did M. Berthollet, attribute the heat set free in this case to the reduction of the volume of the body; for when this reduction has reached its limit the liberation of heat would cease. Now this does not occur.
It is sufficient that the body change form by percussion, without change of volume, to produce disengagement of heat.
If, for example, we take a cube of lead and strike it successively on each of its faces, there will always be heat liberated, without sensible diminution in this disengagement, so long as the blows are continued with equal force. This does not occur when medals are struck. In this case the metal cannot change form after the first blows of the die, and the effect of the collision is not conveyed to the medal, but to the threads of the screw which are strained, and to its supports.
It would seem, then, that heat set free should be attributed to the friction of the molecules of the metal, which change place relatively to each other, that is, the heat is set free just where the moving force is expended.
A similar remark will apply in regard to the collision of two bodies of differing hardness—lead and iron for instance. The first of these metals becomes very hot, while the second does not vary sensibly in temperature. But the motive power is almost wholly exhausted in changing the form of the first of these metals. We may also cite, as a fact of the same nature, the heat produced by the extension of a metallic rod just before it breaks. Experiment has proved that, other things being equal, the greater the elongation before rupture, the more considerable is the elevation of temperature.
(2) [The remainder is blank.]
When a hypothesis no longer suffices to explain phenomena, it should be abandoned.
This is the case with the hypothesis which regards caloric as matter, as a subtile fluid.
The experimental facts tending to destroy this theory are as follows:
(1) The development of heat by percussion or the friction of bodies (experiments of Rumford, friction of wheels on their spindles, on the axles, experiments to be made). Here the elevation of temperature takes place at the same time in the body rubbing and the body rubbed. Moreover, they do not change perceptibly in form or nature (to be proved). Thus heat is produced by motion. If it is matter, it must be admitted that the matter is created by motion.
(2) When an air-pump is worked, and at the same time air is admitted into the receiver, the temperature remains constant in the receiver. It remains constant on the outside. Consequently, the air compressed by the pumps must rise in temperature above the air outside, and it is expelled at a higher temperature. The air enters then at a temperature of 10°, for instance, and leaves at another, 10° + 90° or 100°, for example. Thus heat has been created by motion.
(3) If the air in a reservoir is compressed, and at the same time allowed to escape through a little opening, there is by the compression elevation of temperature, by the escape lowering of temperature (according to Gay-Lussac and Welter). The air then enters at one side at one temperature and escapes at the other side at a higher temperature, from which follows the same conclusion as in the preceding case.
(Experiment to be made: To fit to a high-pressure boiler a cock and a tube leading to it and emptying into the atmosphere; to open the cock a little way, and present a thermometer to the outlet of the steam; to see if it remains at 100° or more; to see if steam is liquefied in the pipe; to see whether it comes out cloudy or transparent.)
(4) The elevation of temperature which takes place at the time of the entrance of the air into the vacuum, an elevation that cannot be attributed to the compression of the air remaining (air which may be replaced by steam), can therefore be attributed only to the friction of the air against the walls of the opening, or against the interior of the receiver, or against itself.
(5) M. Gay-Lussac showed (it is said) that if two receivers were put in communication with each other, the one a vacuum, the other full of air, the temperature would rise in one as much as it would fall in the other. If, then, both be compressed one half, the first would return to its previous temperature and the second to a much higher one. Mixing them, the whole mass would be heated.
When the air enters a vacuum, its passage through one small opening and the motion imparted to it in the interior appear to produce elevation of temperature.
We may be allowed to express here an hypothesis in regard to the nature of heat.
At present, light is generally regarded as the result of a vibratory movement of the ethereal fluid. Light produces heat, or at least accompanies the radiating heat, and moves with the same velocity as heat. Radiating heat is then a vibratory movement. It would be ridiculous to suppose that it is an emission of matter while the light which accompanies it could be only a movement.
Could a motion (that of radiating heat) produce matter (caloric)?
No, undoubtedly; it can only produce a motion. Heat is then the result of a motion.
Then it is plain that it could be produced by the consumption of motive power, and that it could produce this power.
All the other phenomena—composition and decomposition of bodies, passage to the gaseous state, specific heat, equilibrium of heat, its more or less easy transmission, its constancy in experiments with the calorimeter—could be explained by this hypothesis. But it would be difficult to explain why, in the development of motive power by heat, a cold body is necessary; why, in consuming the heat of a warm body, motion cannot be produced.
It appears very difficult to penetrate into the real essence of bodies. To avoid erroneous reasoning, it would be necessary to investigate carefully the source of our knowledge in regard to the nature of bodies, their form, their forces; to see what the primitive notions are, to see from what impressions they are derived; to see how one is raised successively to the different degrees of abstraction.
Is heat the result of a vibratory motion of molecules? If this is so, quantity of heat is simply quantity of motive power. As long as motive power is employed to produce vibratory movements, the quantity of heat must be unchangeable; which seems to follow from experiments with the calorimeter; but when it passes into movements of sensible extent, the quantity of heat can no longer remain constant.
Can examples be found of the production of motive power with actual consumption of heat? It seems that we may find production of heat with consumption of motive power (re-entrance of the air into a vacuum, for example).
What is the cause of the production of heat in combinations of substances? What is radiant caloric?
Liquefaction of bodies, solidification of liquids, crystallization—are they not forms of combinations of integrant molecules?
Supposing heat due to a vibratory movement, how can the passage from the solid or the liquid to the gaseous state be explained?
When motive power is produced by the passage of heat from the body _A_ to the body _B_, is the quantity of this heat which arrives at _B_ (if it is not the same as that which has been taken from _A_, if a portion has really been consumed to produce motive power) the same whatever may be the substance employed to realize the motive power?
Is there any way of using more heat in the production of motive power, and of causing less to reach the body _B_? Could we even utilize it entirely, allowing none to go to the body _B_? If this were possible, motive power could be created without consumption of combustible, and by mere destruction of the heat of bodies.
Is it absolutely certain that steam after having operated an engine and produced motive power can raise the temperature of the water of condensation as if it had been conducted directly into it?
Reasoning shows us that there cannot be loss of living force, or, which is the same thing, of motive power, if the bodies act upon each other without directly touching each other, without actual collision. Now everything leads us to believe that the molecules of bodies are always separated from each other by some space, that they are never actually in contact. If they touched each other, they would remain united, and consequently change form.
If the molecules of bodies are never in close contact with each other whatever may be the forces which separate or attract them, there can never be either production or loss of motive power in nature. This power must be as unchangeable in quantity as matter. Then the direct re-establishment of equilibrium of the caloric, and its re-establishment with production of motive power, would be essentially different from each other.
Heat is simply motive power, or rather motion which has changed form. It is a movement among the particles of bodies. Wherever there is destruction of motive power there is, at the same time, production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power.
We can then establish the general proposition that motive power is, in quantity, invariable in nature; that it is, correctly speaking, never either produced or destroyed. It is true that it changes form, that is, it produces sometimes one sort of motion, sometimes another, but it is never annihilated.
According to some ideas that I have formed on the theory of heat, the production of a unit of motive power necessitates the destruction of 2.70 units of heats.
A machine which would produce 20 units of motive power per kilogram of coal ought to destroy (20 × 2.70)/(7000) of the heat developed by the combustion. (20 × 2.70)/(7000) = (8)/(1000) about; that is, less than (1)/(100).
(Each unit of motive power, or dyname, representing the weight of one cubic metre of water raised to the height of one metre.)
_Experiments to be made on Heat and Motive Power._
To repeat Rumford’s experiments in the drilling of a metal in water, but to measure the motive power consumed at the same time as the heat produced; same experiments on several metals and on wood.
To strike a piece of lead in various ways, to measure the motive power consumed and the heat produced. Same experiments on other metals.
To strongly agitate water in a small cask or in a double-acting pump having a piston pierced with a small opening.
Experiment of the same sort on the agitation of mercury, alcohol, air and other gases. To measure the motive power consumed and heat produced.
To admit air into a vacuum or into air more or less rarefied; _id._ for other gases or vapors. To examine the elevation of temperature by means of the manometer and the thermometer of Bréguet. Estimation of the error of the thermometer in the time required for the air to vary a certain number of degrees. These experiments would serve to measure the changes which take place in the temperature of the gas during its changes of volume. They would also furnish means of comparing these changes with the quantities of motive power produced or consumed.
Expel the air from a large reservoir in which it is compressed, and check its velocity in a large pipe in which solid bodies have been placed; measure the temperature when it has become uniform. See if it is the same as in the reservoir. Same experiments with other gases and with vapor formed under different pressures.
To repeat Dalton’s experiments and carry them on to pressures of thirty or forty atmospheres. To measure the constituent heat of the vapor within these limits.
_Id._ on the vapor of alcohol, of ether, of essence of turpentine, of mercury, to prove whether the agent employed makes any difference in the production of motive power.
_Id._ on water charged with a deliquescent salt, the calcium chloride, for instance.
Is the law of tensions always the same? To measure the specific heat of vapor.
_Experiments to be made on the Tension of Vapors._
A graduated capillary tube filled with water, mercury, or with oil and air. Plunge this tube into a bath of oil, of mercury, or of melted lead. To measure the temperature by an air-thermometer.
Same experiments with alcohol, ether, sulphide of carbon, muriatic ether, essence of turpentine, sulphur, phosphorus.
Experiments on the tension of steam with a boiler, and a thermometric tube full of air. A thermometer will be placed in a tube immersed in the boiler, open outwards and filled with oil or mercury.
Experiments by means of a simple capillary tube filled with three successive parts—first of air, second of mercury, third of water or other liquid of which the tension can be measured (of alcohol, of ether, of essence of turpentine, of lavender, of sulphide of carbon, of muriatic ether, etc.). One end of the tube may be immersed in a bath of mercury or oil, the temperature of which is to be measured. The column of mercury can be made long enough to allow of the air being previously compressed or rarefied.
[Illustration:
FIG. 6. ]
The tube will be bent into a spiral at one end, the straight part being graduated (thus permitting the tension of mercurial vapor to be measured).
[Illustration:
FIG. 7. ]
Experiments on the tension of vapors at low temperature, with a thermometric tube bent round, and filled partly with mercury, partly with water or alcohol. The mercury will operate by its weight. The upper part of the tube will be empty and sealed, or fully open to the atmosphere.
The bulb will be immersed in water the temperature of which is to be measured. If the tube is sealed, the upper part must be cooled.
The bulb might contain water, ether, or essence of turpentine.
If the tube is sealed, the tension of mercurial vapor could be measured.
Experiments on the constituent heat of vapors by means of a barometric tube having two enlarged bulbs. One of the bulbs may be immersed in cold water, and the elevation of temperature of this water will indicate the constituent heat of the vapor.
[Illustration:
FIG. 8. ]
The other bulb may be warmed either by boiling liquid or by fire.
Water, alcohol, steam, ether, mercury, acetic acid, sulphide of carbon.
The operation may be repeated and add the results.
_Experiments to be made on Gases and Vapors._
To measure the temperature acquired by the air introduced into a vacuum or space containing previously rarefied air.
[Illustration:
FIG. 9. ]
If the vacuum is made under the glass receiver of an air-pump, and the cock admitting the outer air be suddenly opened, the introduction of this air will cause a Bréguet thermometer to rise to 50° or 60°. To examine the movement of this thermometer when the reintroduction takes place only by degrees, to compare it with the movement of the manometer.
Construction of a manometer which may give the pressure almost instantaneously.
Imagine a capillary tube bent into a spiral at one end, and having one extremity closed, the other open. This tube will be perfectly dry and a small index of mercury may be introduced into it. The diameter of the tube will be small enough for the air enclosed in it to take almost instantly the temperature of the glass. We shall try to ascertain the time necessary for the establishment of this equilibrium of temperature by placing the tube under the receiver of the air-pump, making a partial vacuum, and admitting the air. We shall see whether, some seconds after the introduction, the index perceptibly moves. The index must be of very light weight to avoid oscillation as much as possible.
For the same reason, the capillary tube should be also as narrow as possible. If the straight part of the tube is equal to the bent part and the index be placed at the beginning of the bent part, for a pressure equal to atmospheric pressure, it would not be necessary to subject the instrument to a less pressure than ½ atmosphere. It is between these two limits that it would serve as a measure.
It might end in an open enlargement to prevent the projection of the mercury outside the tube. Disposed in this way, it could be used as a general measure for pressures between _p_ and (½)_p_; _p_ being anything whatever. The apparatus will be fastened to a board bearing a graduated scale placed against the straight tube. The scale will be, for instance, numbered by fives or tens. A corresponding table denoting pressures would be required.
Placing the instrument under the receiver and forming a partial vacuum, the index will rise into the enlargement. Then, admitting the air by degrees and very slowly, we may note the correspondence between the heights of the ordinary mercury manometer and the point which will be reached by the lower face of the index of the instrument. This will answer to form a comparative table of the pressures and the numbers of the scale. The pressures would be represented by their relations to the observed pressure at the moment of the passage of the index over zero, for any other fixed number of the scale.
Thus, for example, suppose that we observed on the manometer 400 or _n_ millimetres of mercury when the index is on _o_, then _n′_ when the index is on 1, _n″_ when on 2, and so on. This will give the ratios _n′_/_n_, _n″_/_n_, ... which must be inscribed in the table. Then _n_ could be varied at pleasure, and the table could still be used.
In fact, according to the law of Mariotte, volumes preserving the same ratios, pressures should also preserve the same ratios to each other.
Let _p_ be the pressure when the index is on _o_, _v_ the volume of air at the same moment, _p′_ and _v′_ the same pressures and volume at the moment when the index is on 1. Whether the air be expelled or admitted the pressures would be instead of _p_ and _p′_, _q_ and _q′_. But there would follow
_p_ : _p′_ :: _v′_ : _v_ and _q_ : _q′_ :: _v′_ : _v_; then _p_ : _p′_ :: _q_ : _q′_.
We should moreover work at a uniform temperature and note the variations.
If the straight part of the tube were perfectly calibrated, the volumes, and consequently the pressures, would form a geometrical progression, when the figures of the scale would be found to be in arithmetical progression, and a table of logarithms would enable one to be found from the other.
In order to increase as required the mass of air enclosed in the tube the instrument must be placed on its side or flat, in the air-pump receivers. The mercury index would be placed in the lateral part of the enlargement of the tube, and the atmospheric air would enter. The instrument might also be heated in this position.
Care must be taken to admit only very dry air, which could be obtained by placing under the receiver calcium chloride or any other substance which absorbs moisture greedily.
Instead of bending the tube into a spiral, it might be bent in the middle in the form of a ᑌ, or it might be better to form three, four or more parallel branches. Making the tube very long, the index would have a larger range for the same changes of pressure, and the results produced could then be measured by a slight variation in density in the air of the receiver.
_Comparison of the Rapidity with which the Air cools in the Receiver and in the Tube._
Let us suppose, what I believe to be very near the truth, that the heat absorbed is proportional to the surface of the bodies in contact. From this we can infer without difficulty, that the rapidity of the cooling of the air in two cylindrical tubes would be inversely as their diameters.
If the receiver is considered as a tube of two decimetres in diameter, and the manometer as a tube of one millimetre diameter, the rapidity of the cooling of the air would be in the ratio, very nearly, of 1 to 200.
_Extent of the Movement of the Index._
Suppose the tube turned up on itself five times and having a total length of 1 metre; a variation of density equal to ⅒ in the air will give a movement of 1 decimetre; a variation of heat of 1 degree supposed to be equivalent to a variation of density of ¹⁄₂₆₆ will give ¹⁄₂₆₆ of a metre, or about 3^{mm}.70, quite an appreciable quantity. As to the time required to move the mercury index, regard being had to its mass, if we suppose it 1 centimetre long, and the variation of pressure ¹⁄₁₀₀ of an atmosphere, it would require about ⅙ of a second to make it pass over one decimetre.
_Use of the Instrument in Measuring the Variations of the Tensions of the Air under a Pneumatic Receiver._
At each stroke of the piston which expands the air under the pneumatic receiver when a vacuum is to be created, a lowering of pressure is produced, and undoubtedly a change of temperature. It can be determined approximately, at least, by observing the position of the manometer at the instant after the dilatation has taken place, and again after a time long enough for the temperature to have returned to its original point, that of the surrounding bodies. Comparison of the elastic force in the two cases will lead to comparison of the temperatures.
The temperature having returned to its original point, we will give a second stroke of the piston which will rarefy the air more than the former, and thus we will make two observations of the manometer, before and after the return to the former temperature. And so on.
APPENDIX B. CARNOT’S FOOT-NOTES.
NOTE A.—The objection may perhaps be raised here, that perpetual motion, demonstrated to be impossible by mechanical action alone, may possibly not be so if the power either of heat or electricity be exerted; but is it possible to conceive the phenomena of heat and electricity as due to anything else than some kind of motion of the body, and as such should they not be subjected to the general laws of mechanics? Do we not know besides, _à posteriori_, that all the attempts made to produce perpetual motion by any means whatever have been fruitless?—that we have never succeeded in producing a motion veritably perpetual, that is, a motion which will continue forever without alteration in the bodies set to work to accomplish it? The electromotor apparatus (the pile of Volta) has sometimes been regarded as capable of producing perpetual motion; attempts have been made to realize this idea by constructing dry piles said to be unchangeable; but however it has been done, the apparatus has always exhibited sensible deteriorations when its action has been sustained for a time with any energy.
The general and philosophic acceptation of the words _perpetual motion_ should include not only a motion susceptible of indefinitely continuing itself after a first impulse received, but the action of an apparatus, of any construction whatever, capable of creating motive power in unlimited quantity, capable of starting from rest all the bodies of nature if they should be found in that condition, of overcoming their inertia; capable, finally, of finding in itself the forces necessary to move the whole universe, to prolong, to accelerate incessantly, its motion. Such would be a veritable creation of motive power. If this were a possibility, it would be useless to seek in currents of air and water or in combustibles this motive power. We should have at our disposal an inexhaustible source upon which we could draw at will.
NOTE B.—The experimental facts which best prove the change of temperature of gases by compression or dilatation are the following:
(1) The fall of the thermometer placed under the receiver of a pneumatic machine in which a vacuum has been produced. This fall is very sensible on the Bréguet thermometer: it may exceed 40° or 50°. The mist which forms in this case seems to be due to the condensation of the watery vapor caused by the cooling of the air.
(2) The inflammation of German tinder in the so-called pneumatic tinder-boxes; which are, as we know, little pump-chambers in which the air is rapidly compressed.
(3) The fall of a thermometer placed in a space where the air has been first compressed and then allowed to escape by the opening of a cock.
(4) The results of experiments on the velocity of sound. M. de Laplace has shown that, in order to secure results accurately by theory and computation, it is necessary to assume the heating of the air by sudden compression.
The only fact which may be adduced in opposition to the above is an experiment of MM. Gay-Lussac and Welter, described in the _Annales de Chimie et de Physique_. A small opening having been made in a large reservoir of compressed air, and the ball of a thermometer having been introduced into the current of air which passes out through this opening, no sensible fall of the temperature denoted by the thermometer has been observed.
Two explanations of this fact may be given: (1) The striking of the air against the walls of the opening by which it escapes may develop heat in observable quantity. (2) The air which has just touched the bowl of the thermometer possibly takes again by its collision with this bowl, or rather by the effect of the _détour_ which it is forced to make by its rencounter, a density equal to that which it had in the receiver,—much as the water of a current rises against a fixed obstacle, above its level.
The change of temperature occasioned in the gas by the change of volume may be regarded as one of the most important facts of Physics, because of the numerous consequences which it entails, and at the same time as one of the most difficult to illustrate, and to measure by decisive experiments. It seems to present in some respects singular anomalies.
Is it not to the cooling of the air by dilatation that the cold of the higher regions of the atmosphere must be attributed? The reasons given heretofore as an explanation of this cold are entirely insufficient; it has been said that the air of the elevated regions receiving little reflected heat from the earth, and radiating towards celestial space, would lose caloric, and that this is the cause of its cooling; but this explanation is refuted by the fact that, at an equal height, cold reigns with equal and even more intensity on the elevated plains than on the summit of the mountains, or in those portions of the atmosphere distant from the sun.
NOTE C.—We see no reason for admitting, _à priori_, the constancy of the specific heat of bodies at different temperatures, that is, to admit that equal quantities of heat will produce equal increments of temperature, when this body changes neither its state nor its density; when, for example, it is an elastic fluid enclosed in a fixed space. Direct experiments on solid and liquid bodies have proved that between zero and 100°, equal increments in the quantities of heat would produce nearly equal increments of temperature. But the more recent experiments of MM. Dulong and Petit (see _Annales de Chimie et de Physique_, February, March, and April, 1818) have shown that this correspondence no longer continues at temperatures much above 100°, whether these temperatures be measured on the mercury thermometer or on the air-thermometer.
Not only do the specific heats not remain the same at different temperatures, but, also, they do not preserve the same ratios among themselves, so that no thermometric scale could establish the constancy of all the specific heats. It would have been interesting to prove whether the same irregularities exist for gaseous substances, but such experiments presented almost insurmountable difficulties.
The irregularities of specific heats of solid bodies might have been attributed, it would seem, to the latent heat employed to produce a beginning of fusion—a softening which occurs in most bodies long before complete fusion. We might support this opinion by the following statement: According to the experiments of MM. Dulong and Petit, the increase of specific heat with the temperature is more rapid in solids than in liquids, although the latter possess considerably more dilatability. The cause of irregularity just referred to, if it is real, would disappear entirely in gases.
NOTE D.—In order to determine the arbitrary constants _A_, _B_, _A′_, _B′_, in accordance with the results in M. Dalton’s table, we must begin by computing the volume of the vapor as determined by its pressure and temperature,—a result which is easily accomplished by reference to the laws of Mariotte and Gay-Lussac, the weight of the vapor being fixed.
The volume will be given by the equation
_v_ = _c_ (267 + _t_)/(_p_),
in which _v_ is this volume, _t_ the temperature, _p_ the pressure, and _c_ a constant quantity depending on the weight of the vapor and on the units chosen. We give here the table of the volumes occupied by a gramme of vapor formed at different temperatures, and consequently under different pressures.
───────────────────────┬───────────────────────┬─────────────────────── _t_ │ _p_ │ _v_ or degrees Centigrade. │or tension of the vapor│ or volume of a gramme │ expressed in │ of vapor expressed in │millimetres of mercury.│ litres. ───────────────────────┼───────────────────────┼─────────────────────── ° │ mm. │ lit. 0│ 5.060 │ 185.0 20│ 17.32 │ 58.2 40│ 53.00 │ 20.4 60│ 144.6 │ 7.96 80│ 352.1 │ 3.47 100│ 760.0 │ 1.70 ───────────────────────┴───────────────────────┴───────────────────────
The first two columns of this table are taken from the _Traité de Physique_ of M. Biot (vol. i., p. 272 and 531). The third is calculated by means of the above formula, and in accordance with the result of experiment, indicating that water vaporized under atmospheric pressure occupies a space 1700 times as great as in the liquid state.
By using three numbers of the first column and three corresponding numbers of the third column, we can easily determine the constants of our equation
_t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_).
We will not enter into the details of the calculation necessary to determine these quantities. It is sufficient to say that the following values,
_A_ = 2268, _A′_ = 19.64, _B_ = −1000, _B′_ = 3.30,
satisfy fairly well the prescribed conditions, so that the equation
_t_ = (2268 − 1000 log _v_)/(19.64 + 3.30 log _v_)
expresses very nearly the relation which exists between the volume of the vapor and its temperature. We may remark here that the quantity _B′_ is positive and very small, which tends to confirm this proposition—that the specific heat of an elastic fluid increases with the volume, but follows a slow progression.
NOTE E.—Were we to admit the constancy of the specific heat of a gas when its volume does not change, but when its temperature varies, analysis would show a relation between the motive power and the thermometric degree. We will show how this is, and this will also give us occasion to show how some of the propositions established above should be expressed in algebraic language.
Let _r_ be the quantity of motive power produced by the expansion of a given quantity of air passing from the volume of one litre to the volume of _v_ litres under constant temperature. If _v_ increases by the infinitely small quantity _dv_, _r_ will increase by the quantity _dr_, which, according to the nature of motive power, will be equal to the increase _dv_ of volume multiplied by the expansive force which the elastic fluid then possesses; let _p_ be this expansive force. We should have the equation
_dr_ = _pdv_. (1)
Let us suppose the constant temperature under which the dilatation takes place equal to _t_ degrees Centigrade. If we call _q_ the elastic force of the air occupying the volume 1 litre at the same temperature _t_, we shall have, according to the law of Mariotte,
(_v_)/(1) = (_q_)/(_p_), whence _p_ = (_q_)/(_v_).
If now _P_ is the elastic force of this same air at the constant volume 1, but at the temperature zero, we shall have, according to the rule of M. Gay-Lussac,
_q_ = _P_ + _P_ (_t_)/(267) = (_P_)/(267)(267 + _t_);
whence
_q_ = _p_ = (_P_)/(267) (267 + _t_)/(_v_).
If, to abridge, we call _N_ the quantity (_P_)/(267), the equation would become
_p_ = _N_ (_t_ + 267)/(_v_);
whence we deduce, according to equation (1),
_dr_ = _N_ (_t_ + 267)/(_v_)_dv_.
Regarding _t_ as constant, and taking the integral of the two numbers, we shall have
_r_ = _N_(_t_ + 267) log _v_ + _C_.
If we suppose _r_ = 0 when _v_ = 1, we shall have _C_ = 0; whence
_r_ = _N_(_t_ + 267) log _v_. (2)
This is the motive power produced by the expansion of the air which, under the temperature _t_, has passed from the volume 1 to the volume _v_. If instead of working at the temperature _t_ we work in precisely the same manner at the temperature _t_ + _dt_, the power developed will be
_r_ + δ_r_ = _N_(_t_ + _dt_ + 267) log _v_.
Subtracting equation (2), we have
δ_r_ = _N_ log _vdt_. (3)
Let _e_ be the quantity of heat employed to maintain the temperature of the gas constant during its dilatation. According to the reasoning of page 69, δ_r_ will be the power developed by the fall of the quantity _e_ of heat from the degree _t_ + _td_ to the degree _t_. If we call _u_ the motive power developed by the fall of unity of heat from the degree _t_ to the degree zero, as, according to the general principle established page 68, this quantity _u_ ought to depend solely on _t_, it could be represented by the function _Ft_, whence _u_ = _Ft_.
When _t_ is increased it becomes _t_ + _td_, _u_ becomes _u_ + _du_; whence
_u_ + _du_ = _F_(_t_ + _dt_).
Subtracting the preceding equation, we have
_du_ = _F_(_t_ + _dt_) − _Ft_ = _F′tdt_.
This is evidently the quantity of motive power produced by the fall of unity of heat from the temperature _t_ + _dt_ to the temperature _t_.
If the quantity of heat instead of being a unit had been _e_, its motive power produced would have had for its value
_edu_ = _eF′tdt_. (4)
But _edu_ is the same thing as δ_r_; both are the power developed by the fall of the quantity _e_ of heat from the temperature _t_ + _dt_ to the temperature _t_; consequently,
_edu_ = δ_r_,
and from equations (3), (4),
_eF′tdt_ = _N_ log _vdt_;
or, dividing by _F′tdt_,
_e_ = (_N_)/(_F′t_) log _v_ = _T_ log _v_.
Calling _T_ the fraction (_N_)/(_F′t_) which is a function of _t_ only, the equation
_e_ = _T_ log _v_
is the analytical expression of the law stated pp. 80, 81. It is common to all gases, since the laws of which we have made use are common to all.
If we call _s_ the quantity of heat necessary to change the air that we have employed from the volume 1 and from the temperature zero to the volume _v_ and to the temperature _t_, the difference between _s_ and _e_ will be the quantity of heat required to bring the air at the volume 1 from zero to _t_. This quantity depends on _t_ alone; we will call it _U_. It will be any function whatever of _t_. We shall have
_s_ = _e_ + _U_ = _T_ log _v_ + _U_.
If we differentiate this equation with relation to _t_ alone, and if we represent it by _T′_ and _U′_, the differential coefficients of _T_ and _U_, we shall get
(_ds_)/(_dt_) = _T′_ log _v_ + _U′_; (5)
_ds_/_dt_ is simply the specific heat of the gas under constant volume, and our equation (1) is the analytical expression of the law stated on page 86.
If we suppose the specific heat constant at all temperatures (hypothesis discussed above, page 92), the quantity _ds_/_dt_ will be independent of _t_; and in order to satisfy equation (5) for two particular values of _v_, it will be necessary that _T′_ and _U′_ be independent of _t_; we shall then have _T′_ = _C_, a constant quantity. Multiplying _T′_ and _C_ by _dt_, and taking the integral of both, we find
_T_ = _Ct_ + _C_{1}_;
but as _T_ = _N_/_F′t_, we have
_F′t_ = (_N_)/(_T_) = (_N_)/(_Ct_ + _C_{1}_).
Multiplying both by _dt_ and integrating, we have
_Ft_ = (_N_)/(_C_) log (_Ct_ + _C_{1}_) + _C_{2}_;
or changing arbitrary constants, and remarking further that _Ft_ is 0 when _t_ = 0°,
_Ft_ = _A_ log (1 + (_t_)/(_B_)). (6)
The nature of the function _Ft_ would be thus determined, and we would thus be able to estimate the motive power developed by any fall of heat. But this latter conclusion is founded on the hypothesis of the constancy of the specific heat of a gas which does not change in volume—an hypothesis which has not yet been sufficiently verified by experiment. Until there is fresh proof, our equation (6) can be admitted only throughout a limited portion of the thermometric scale.
In equation (5), the first member represents, as we have remarked, the specific heat of the air occupying the volume _v_. Experiment having shown that this heat varies little in spite of the quite considerable changes of volume, it is necessary that the coefficient _T′_ of log _v_ should be a very small quantity. If we consider it nothing, and, after having multiplied by _dt_ the equation
_T′_ = 0,
we take the integral of it, we find
_T_ = _C_, constant quantity;
but
_T_ = _N_/_F′t_,
whence
_F′t_ = _N_/_T_ = _N_/_C_ = _A_;
whence we deduce finally, by a second integration,
_Ft_ = _At_ + _B_.
As _Ft_ = 0 when _t_ = 0, _B_ is 0; thus
_Ft_ = _At_;
that is, the motive power produced would be found to be exactly proportional to the fall of the caloric. This is the analytical translation of what was stated on page 98.
NOTE F.—M. Dalton believed that he had discovered that the vapors of different liquids at equal thermometric distances from the boiling-point possess equal tensions; but this law is not precisely exact; it is only approximate. It is the same with the law of the proportionality of the latent heat of vapors with their densities (see Extracts from a Mémoire of M. C. Despretz, _Annales de Chimie et de Physique_, t. xvi. p. 105, and t. xxiv. p. 323). Questions of this nature are closely connected with those of the motive power of heat. Quite recently MM. H. Davy and Faraday, after having conducted a series of elegant experiments on the liquefaction of gases by means of considerable pressure, have tried to observe the changes of tension of these liquefied gases on account of slight changes of temperature. They have in view the application of the new liquids to the production of motive power (see _Annales de Chimie et de Physique_, January, 1824, p. 80).
According to the above-mentioned theory, we can foresee that the use of these liquids would present no advantages relatively to the economy of heat. The advantages would be found only in the lower temperature at which it would be possible to work, and in the sources whence, for this reason, it would become possible to obtain caloric.
NOTE G.—This principle, the real foundation of the theory of steam-engines, was very clearly developed by M. Clement in a memoir presented to the Academy of Sciences several years ago. This Memoir has never been printed, and I owe the knowledge of it to the kindness of the author. Not only is the principle established therein, but it is applied to the different systems of steam-engines actually in use. The motive power of each of them is estimated therein by the aid of the law cited page 92, and compared with the results of experiment.
The principle in question is so little known or so poorly appreciated, that recently Mr. Perkins, a celebrated mechanician of London, constructed a machine in which steam produced under the pressure of 35 atmospheres—a pressure never before used—is subjected to very little expansion of volume, as any one with the least knowledge of this machine can understand. It consists of a single cylinder of very small dimensions, which at each stroke is entirely filled with steam, formed under the pressure of 35 atmospheres. The steam produces no effect by the expansion of its volume, for no space is provided in which the expansion can take place. It is condensed as soon as it leaves the small cylinder. It works therefore only under a pressure of 35 atmospheres, and not, as its useful employment would require, under progressively decreasing pressures. The machine of Mr. Perkins seems not to realize the hopes which it at first awakened. It has been asserted that the economy of coal in this engine was ⁹⁄₁₀ above the best engines of Watt, and that it possessed still other advantages (see _Annales de Chimie et de Physique_, April, 1823, p. 429). These assertions have not been verified. The engine of Mr. Perkins is nevertheless a valuable invention, in that it has proved the possibility of making use of steam under much higher pressure than previously, and because, being easily modified, it may lead to very useful results.
Watt, to whom we owe almost all the great improvements in steam-engines, and who brought these engines to a state of perfection difficult even now to surpass, was also the first who employed steam under progressively decreasing pressures. In many cases he suppressed the introduction of the steam into the cylinder at a half, a third, or a quarter of the stroke. The piston completes its stroke, therefore, under a constantly diminishing pressure. The first engines working on this principle date from 1778. Watt conceived the idea of them in 1769, and took out a patent in 1782.
We give here the Table appended to Watt’s patent. He supposed the steam introduced into the cylinder during the first quarter of the stroke of the piston; then, dividing this stroke into twenty parts, he calculated the mean pressure as follows:
Portions of the descent from the top of the Decreasing pressure of the cylinder. steam, the entire pressure being 1. Steam arriving 0.05 freely from the 1.000 Total pressure. boiler. 0.10 „ 1.000 „ 0.15 „ 1.000 „ 0.20 „ 1.000 „ Quarter 0.25 „ 1.000 „ The steam being cut off and the 0.30 descent taking 0.830 place only by expansion. 0.35 „ 0.714 0.40 „ 0.625 0.45 „ 0.555 Half 0.50 „ 0.500 Half original pressure. 0.55 „ 0.454 0.60 „ 0.417 0.65 „ 0.385 0.70 „ 0.375 0.75 „ 0.333 One third. 0.80 „ 0.312 0.85 „ 0.294 0.90 „ 0.277 0.95 „ 0.262 Bottom of cylinder 1.00 „ 0.025 Quarter. Total, 11.583
Mean pressure (11.583)/(20) = 0.579.
On which he remarked, that the mean pressure is more than half the original pressure; also that in employing a quantity of steam equal to a quarter, it would produce an effect more than half.
Watt here supposed that steam follows in its expansion the law of Mariotte, which should not be considered exact, because, in the first place, the elastic fluid in dilating falls in temperature, and in the second place there is nothing to prove that a part of this fluid is not condensed by its expansion. Watt should also have taken into consideration the force necessary to expel the steam which remains after condensation, and which is found in quantity as much greater as the expansion of the volume has been carried further. Dr. Robinson has supplemented the work of Watt by a simple formula to calculate the effect of the expansion of steam, but this formula is found to have the same faults that we have just noticed. It has nevertheless been useful to constructors by furnishing them approximate data practically quite satisfactory. We have considered it useful to recall these facts because they are little known, especially in France. These engines have been built after the models of the inventors, but the ideas by which the inventors were originally influenced have been but little understood. Ignorance of these ideas has often led to grave errors. Engines originally well conceived have deteriorated in the hands of unskilful builders, who, wishing to introduce in them improvements of little value, have neglected the capital considerations which they did not know enough to appreciate.
NOTE H.—The advantage in substituting two cylinders for one is evident. In a single cylinder the impulsion of the piston would be extremely variable from the beginning to the end of the stroke. It would be necessary for all the parts by which the motion is transmitted to be of sufficient strength to resist the first impulsion, and perfectly fitted to avoid the abrupt movements which would greatly injure and soon destroy them. It would be especially on the working beam, on the supports, on the crank, on the connecting-rod, and on the first gear-wheels that the unequal effort would be felt, and would produce the most injurious effects. It would be necessary that the steam-cylinder should be both sufficiently strong to sustain the highest pressure, and with a large enough capacity to contain the steam after its expansion of volume, while in using two successive cylinders it is only necessary to have the first sufficiently strong and of medium capacity,—which is not at all difficult,—and to have the second of ample dimensions, with moderate strength.
Double-cylinder engines, although founded on correct principles, often fail to secure the advantages expected from them. This is due principally to the fact that the dimensions of the different parts of these engines are difficult to adjust, and that they are rarely found to be in correct proportion. Good models for the construction of double-cylinder engines are wanting, while excellent designs exist for the construction of engines on the plan of Watt. From this arises the diversity that we see in the results of the former, and the great uniformity that we have observed in the results of the latter.
NOTE I.—Among the attempts made to develop the motive power of heat by means of atmospheric air, we should mention those of MM. Niepce, which were made in France several years ago, by means of an apparatus called by the inventors a pyréolophore. The apparatus was made thus: There was a cylinder furnished with a piston, into which the atmospheric air was introduced at ordinary density. A very combustible material, reduced to a condition of extreme tenuity, was thrown into it, remained a moment in suspension in the air, and then flame was applied. The inflammation produced very nearly the same effect as if the elastic fluid had been a mixture of air and combustible gas, of air and carburetted hydrogen gas, for example. There was a sort of explosion, and a sudden dilatation of the elastic fluid—a dilatation that was utilized by making it act upon the piston. The latter may have a motion of any amplitude whatever, and the motive power is thus realized. The air is next renewed, and the operation repeated.
This machine, very ingenious and interesting, especially on account of the novelty of its principle, fails in an essential point. The material used as a combustible (it was the dust of Lycopodium, used to produce flame in our theatres) was so expensive, that all the advantage was lost through that cause; and unfortunately it was difficult to employ a combustible of moderate price, since a very finely powdered substance was required which would burn quickly, spread rapidly, and leave little or no ash.
Instead of working as did MM. Niepce, it would seem to us preferable to compress the air by means of pumps, to make it traverse a perfectly closed furnace into which the combustible had been introduced in small portions by a mechanism easy of conception, to make it develop its action in a cylinder with a piston, or in any other variable space; finally, to throw it out again into the atmosphere, or even to make it pass under a steam-boiler in order to utilize the temperature remaining.
The principal difficulties that we should meet in this mode of operation would be to enclose the furnace in a sufficiently strong envelope, to keep the combustion meanwhile in the requisite state, to maintain the different parts of the apparatus at a moderate temperature, and to prevent rapid abrasion of the cylinder and of the piston. These difficulties do not appear to be insurmountable.
There have been made, it is said, recently in England, successful attempts to develop motive power through the action of heat on atmospheric air. We are entirely ignorant in what these attempts have consisted—if indeed they have really been made.
NOTE J.—The result given here was furnished by an engine whose large cylinder was 45 inches in diameter and 7 feet stroke. It is used in one of the mines of Cornwall called Wheal Abraham. This result should be considered as somewhat exceptional, for it was only temporary, continuing but a single month. Thirty millions of lbs. raised one English foot per bushel of coal of 88 lbs. is generally regarded as an excellent result for steam-engines. It is sometimes attained by engines of the Watt type, but very rarely surpassed. This latter product amounts, in French measures, to 104,000 kilograms raised one metre per kilogram of coal consumed.
According to what is generally understood by one horse-power, in estimating the duty of steam-engines, an engine of ten horse-power should raise per second 10 × 75 kilograms, or 750 kilograms, to a height of one metre, or more, per hour; 750 × 3600 = 2,700,000 kilograms to one metre. If we suppose that each kilogram of coal raised to this height 104,000 kilograms, it will be necessary, in order to ascertain how much coal is burnt in one hour by our ten-horse-power engine, to divide 2,700,000 by 104,000, which gives ²⁷⁰⁰⁄₁₀₄ = 26 kilograms. Now it is seldom that a ten-horse-power engine consumes less than 26 kilograms of coal per hour.
