Chapter II: Notes of Sadi Carnot on Mathematics, Physics, and Other Subjects

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.

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).

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.

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.

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.