Chapter III: Appendix (2)

If, some day, the steam-engine shall be so perfected that it can be set up and supplied with fuel at small cost, it will combine all desirable qualities, and will afford to the industrial arts a range the extent of which can scarcely be predicted. It is not merely that a powerful and convenient motor that can be procured and carried anywhere is substituted for the motors already in use, but that it causes rapid extension in the arts in which it is applied, and can even create entirely new arts.

The most signal service that the steam-engine has rendered to England is undoubtedly the revival of the working of the coal-mines, which had declined, and threatened to cease entirely, in consequence of the continually increasing difficulty of drainage, and of raising the coal.[5] We should rank second the benefit to iron manufacture, both by the abundant supply of coal substituted for wood just when the latter had begun to grow scarce, and by the powerful machines of all kinds, the use of which the introduction of the steam-engine has permitted or facilitated.

Iron and heat are, as we know, the supporters, the bases, of the mechanic arts. It is doubtful if there be in England a single industrial establishment of which the existence does not depend on the use of these agents, and which does not freely employ them. To take away to-day from England her steam-engines would be to take away at the same time her coal and iron. It would be to dry up all her sources of wealth, to ruin all on which her prosperity depends, in short, to annihilate that colossal power. The destruction of her navy, which she considers her strongest defence, would perhaps be less fatal.

The safe and rapid navigation by steamships may be regarded as an entirely new art due to the steam-engine. Already this art has permitted the establishment of prompt and regular communications across the arms of the sea, and on the great rivers of the old and new continents. It has made it possible to traverse savage regions where before we could scarcely penetrate. It has enabled us to carry the fruits of civilization over portions of the globe where they would else have been wanting for years. Steam navigation brings nearer together the most distant nations. It tends to unite the nations of the earth as inhabitants of one country. In fact, to lessen the time, the fatigues, the uncertainties, and the dangers of travel—is not this the same as greatly to shorten distances?[6]

The discovery of the steam-engine owed its birth, like most human inventions, to rude attempts which have been attributed to different persons, while the real author is not certainly known. It is, however, less in the first attempts that the principal discovery consists, than in the successive improvements which have brought steam-engines to the condition in which we find them to-day. There is almost as great a distance between the first apparatus in which the expansive force of steam was displayed and the existing machine, as between the first raft that man ever made and the modern vessel.

If the honor of a discovery belongs to the nation in which it has acquired its growth and all its developments, this honor cannot be here refused to England. Savery, Newcomen, Smeaton, the famous Watt, Woolf, Trevithick, and some other English engineers, are the veritable creators of the steam-engine. It has acquired at their hands all its successive degrees of improvement. Finally, it is natural that an invention should have its birth and especially be developed, be perfected, in that place where its want is most strongly felt.

Notwithstanding the work of all kinds done by steam-engines, notwithstanding the satisfactory condition to which they have been brought to-day, their theory is very little understood, and the attempts to improve them are still directed almost by chance.

The question has often been raised whether the motive power of heat[7] is unbounded, whether the possible improvements in steam-engines have an assignable limit,—a limit which the nature of things will not allow to be passed by any means whatever; or whether, on the contrary, these improvements may be carried on indefinitely. We have long sought, and are seeking to-day, to ascertain whether there are in existence agents preferable to the vapor of water for developing the motive power of heat; whether atmospheric air, for example, would not present in this respect great advantages. We propose now to submit these questions to a deliberate examination.

The phenomenon of the production of motion by heat has not been considered from a sufficiently general point of view. We have considered it only in machines the nature and mode of action of which have not allowed us to take in the whole extent of application of which it is susceptible. In such machines the phenomenon is, in a way, incomplete. It becomes difficult to recognize its principles and study its laws.

In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent. It is necessary to establish principles applicable not only to steam-engines[8] but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated.

Machines which do not receive their motion from heat, those which have for a motor the force of men or of animals, a waterfall, an air-current, etc., can be studied even to their smallest details by the mechanical theory. All cases are foreseen, all imaginable movements are referred to these general principles, firmly established, and applicable under all circumstances. This is the character of a complete theory. A similar theory is evidently needed for heat-engines. We shall have it only when the laws of Physics shall be extended enough, generalized enough, to make known beforehand all the effects of heat acting in a determined manner on any body.

We will suppose in what follows at least a superficial knowledge of the different parts which compose an ordinary steam-engine; and we consider it unnecessary to explain what are the furnace, boiler, steam-cylinder, piston, condenser, etc.

The production of motion in steam-engines is always accompanied by a circumstance on which we should fix our attention. This circumstance is the re-establishing of equilibrium in the caloric; that is, its passage from a body in which the temperature is more or less elevated, to another in which it is lower. What happens in fact in a steam-engine actually in motion? The caloric developed in the furnace by the effect of the combustion traverses the walls of the boiler, produces steam, and in some way incorporates itself with it. The latter carrying it away, takes it first into the cylinder, where it performs some function, and from thence into the condenser, where it is liquefied by contact with the cold water which it encounters there. Then, as a final result, the cold water of the condenser takes possession of the caloric developed by the combustion. It is heated by the intervention of the steam as if it had been placed directly over the furnace. The steam is here only a means of transporting the caloric. It fills the same office as in the heating of baths by steam, except that in this case its motion is rendered useful.

We easily recognize in the operations that we have just described the re-establishment of equilibrium in the caloric, its passage from a more or less heated body to a cooler one. The first of these bodies, in this case, is the heated air of the furnace; the second is the condensing water. The re-establishment of equilibrium of the caloric takes place between them, if not completely, at least partially, for on the one hand the heated air, after having performed its function, having passed round the boiler, goes out through the chimney with a temperature much below that which it had acquired as the effect of combustion; and on the other hand, the water of the condenser, after having liquefied the steam, leaves the machine with a temperature higher than that with which it entered.

The production of motive power is then due in steam-engines not to an actual consumption of caloric, but _to its transportation from a warm body to a cold body_, that is, to its re-establishment of equilibrium—an equilibrium considered as destroyed by any cause whatever, by chemical action such as combustion, or by any other. We shall see shortly that this principle is applicable to any machine set in motion by heat.

According to this principle, the production of heat alone is not sufficient to give birth to the impelling power: it is necessary that there should also be cold; without it, the heat would be useless. And in fact, if we should find about us only bodies as hot as our furnaces, how can we condense steam? What should we do with it if once produced? We should not presume that we might discharge it into the atmosphere, as is done in some engines;[9] the atmosphere would not receive it. It does receive it under the actual condition of things, only because it fulfils the office of a vast condenser, because it is at a lower temperature; otherwise it would soon become fully charged, or rather would be already saturated.[10]

Wherever there exists a difference of temperature, wherever it has been possible for the equilibrium of the caloric to be re-established, it is possible to have also the production of impelling power. Steam is a means of realizing this power, but it is not the only one. All substances in nature can be employed for this purpose, all are susceptible of changes of volume, of successive contractions and dilatations, through the alternation of heat and cold. All are capable of overcoming in their changes of volume certain resistances, and of thus developing the impelling power. A solid body—a metallic bar for example—alternately heated and cooled increases and diminishes in length, and can move bodies fastened to its ends. A liquid alternately heated and cooled increases and diminishes in volume, and can overcome obstacles of greater or less size, opposed to its dilatation. An aeriform fluid is susceptible of considerable change of volume by variations of temperature. If it is enclosed in an expansible space, such as a cylinder provided with a piston, it will produce movements of great extent. Vapors of all substances capable of passing into a gaseous condition, as of alcohol, of mercury, of sulphur, etc., may fulfil the same office as vapor of water. The latter, alternately heated and cooled, would produce motive power in the shape of permanent gases, that is, without ever returning to a liquid state. Most of these substances have been proposed, many even have been tried, although up to this time perhaps without remarkable success.

We have shown that in steam-engines the motive power is due to a re-establishment of equilibrium in the caloric; this takes place not only for steam-engines, but also for every heat-engine—that is, for every machine of which caloric is the motor. Heat can evidently be a cause of motion only by virtue of the changes of volume or of form which it produces in bodies.

These changes are not caused by uniform temperature, but rather by alternations of heat and cold. Now to heat any substance whatever requires a body warmer than the one to be heated; to cool it requires a cooler body. We supply caloric to the first of these bodies that we may transmit it to the second by means of the intermediary substance. This is to re-establish, or at least to endeavor to re-establish, the equilibrium of the caloric.

It is natural to ask here this curious and important question: Is the motive power of heat invariable in quantity, or does it vary with the agent employed to realize it as the intermediary substance, selected as the subject of action of the heat?

It is clear that this question can be asked only in regard to a given quantity of caloric,[11] the difference of the temperatures also being given. We take, for example, one body _A_ kept at a temperature of 100° and another body _B_ kept at a temperature of 0°, and ask what quantity of motive power can be produced by the passage of a given portion of caloric (for example, as much as is necessary to melt a kilogram of ice) from the first of these bodies to the second. We inquire whether this quantity of motive power is necessarily limited, whether it varies with the substance employed to realize it, whether the vapor of water offers in this respect more or less advantage than the vapor of alcohol, of mercury, a permanent gas, or any other substance. We will try to answer these questions, availing ourselves of ideas already established.

We have already remarked upon this self-evident fact, or fact which at least appears evident as soon as we reflect on the changes of volume occasioned by heat: _wherever there exists a difference of temperature, motive power can be produced_. Reciprocally, wherever we can consume this power, it is possible to produce a difference of temperature, it is possible to occasion destruction of equilibrium in the caloric. Are not percussion and the friction of bodies actually means of raising their temperature, of making it reach spontaneously a higher degree than that of the surrounding bodies, and consequently of producing a destruction of equilibrium in the caloric, where equilibrium previously existed? It is a fact proved by experience, that the temperature of gaseous fluids is raised by compression and lowered by rarefaction. This is a sure method of changing the temperature of bodies, and destroying the equilibrium of the caloric as many times as may be desired with the same substance. The vapor of water employed in an inverse manner to that in which it is used in steam-engines can also be regarded as a means of destroying the equilibrium of the caloric. To be convinced of this we need but to observe closely the manner in which motive power is developed by the action of heat on vapor of water. Imagine two bodies _A_ and _B_, kept each at a constant temperature, that of _A_ being higher than that of _B_. These two bodies, to which we can give or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator.

If we wish to produce motive power by carrying a certain quantity of heat from the body _A_ to the body _B_ we shall proceed as follows:

(1) To borrow caloric from the body _A_ to make steam with it—that is, to make this body fulfil the function of a furnace, or rather of the metal composing the boiler in ordinary engines—we here assume that the steam is produced at the same temperature as the body _A_.

(2) The steam having been received in a space capable of expansion, such as a cylinder furnished with a piston, to increase the volume of this space, and consequently also that of the steam. Thus rarefied, the temperature will fall spontaneously, as occurs with all elastic fluids; admit that the rarefaction may be continued to the point where the temperature becomes precisely that of the body _B_.

(3) To condense the steam by putting it in contact with the body _B_, and at the same time exerting on it a constant pressure until it is entirely liquefied. The body _B_ fills here the place of the injection-water in ordinary engines, with this difference, that it condenses the vapor without mingling with it, and without changing its own temperature.[12]

The operations which we have just described might have been performed in an inverse direction and order. There is nothing to prevent forming vapor with the caloric of the body _B_, and at the temperature of that body, compressing it in such a way as to make it acquire the temperature of the body _A_, finally condensing it by contact with this latter body, and continuing the compression to complete liquefaction.

By our first operations there would have been at the same time production of motive power and transfer of caloric from the body _A_ to the body _B_. By the inverse operations there is at the same time expenditure of motive power and return of caloric from the body _B_ to the body _A_. But if we have acted in each case on the same quantity of vapor, if there is produced no loss either of motive power or caloric, the quantity of motive power produced in the first place will be equal to that which would have been expended in the second, and the quantity of caloric passed in the first case from the body _A_ to the body _B_ would be equal to the quantity which passes back again in the second from the body _B_ to the body _A_; so that an indefinite number of alternative operations of this sort could be carried on without in the end having either produced motive power or transferred caloric from one body to the other.

Now if there existed any means of using heat preferable to those which we have employed, that is, if it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body _B_ return to the body _A_ from the refrigerator to the furnace, to restore the initial conditions, and thus to be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without consumption either of caloric or of any other agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible.[13] We should then conclude that _the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever_. We will soon give a second more rigorous demonstration of this theory. This should be considered only as an approximation. (See page 59.)

We have a right to ask, in regard to the proposition just enunciated, the following questions: What is the sense of the word _maximum_ here? By what sign can it be known that this maximum is attained? By what sign can it be known whether the steam is employed to greatest possible advantage in the production of motive power?

Since every re-establishment of equilibrium in the caloric may be the cause of the production of motive power, every re-establishment of equilibrium which shall be accomplished without production of this power should be considered as an actual loss. Now, very little reflection would show that all change of temperature which is not due to a change of volume of the bodies can be only a useless re-establishment of equilibrium in the caloric.[14] The necessary condition of the maximum is, then, _that in the bodies employed to realize the motive power of heat there should not occur any change of temperature which may not be due to a change of volume_. Reciprocally, every time that this condition is fulfilled the maximum will be attained. This principle should never be lost sight of in the construction of heat-engines; it is its fundamental basis. If it cannot be strictly observed, it should at least be departed from as little as possible.

Every change of temperature which is not due to a change of volume or to chemical action (an action that we provisionally suppose not to occur here) is necessarily due to the direct passage of the caloric from a more or less heated body to a colder body. This passage occurs mainly by the contact of bodies of different temperatures; hence such contact should be avoided as much as possible. It cannot probably be avoided entirely, but it should at least be so managed that the bodies brought in contact with each other differ as little as possible in temperature. When we just now supposed, in our demonstration, the caloric of the body _A_ employed to form steam, this steam was considered as generated at the temperature of the body _A_; thus the contact took place only between bodies of equal temperatures; the change of temperature occurring afterwards in the steam was due to dilatation, consequently to a change of volume. Finally, condensation took place also without contact of bodies of different temperatures. It occurred while exerting a constant pressure on the steam brought in contact with the body _B_ of the same temperature as itself. The conditions for a maximum are thus found to be fulfilled. In reality the operation cannot proceed exactly as we have assumed. To determine the passage of caloric from one body to another, it is necessary that there should be an excess of temperature in the first, but this excess may be supposed as slight as we please. We can regard it as insensible in theory, without thereby destroying the exactness of the arguments.

A more substantial objection may be made to our demonstration, thus: When we borrow caloric from the body _A_ to produce steam, and when this steam is afterwards condensed by its contact with the body _B_, the water used to form it, and which we considered at first as being of the temperature of the body _A_, is found at the close of the operation at the temperature of the body _B_. It has become cool. If we wish to begin again an operation similar to the first, if we wish to develop a new quantity of motive power with the same instrument, with the same steam, it is necessary first to re-establish the original condition—to restore the water to the original temperature. This can undoubtedly be done by at once putting it again in contact with the body _A_; but there is then contact between bodies of different temperatures, and loss of motive power.[15] It would be impossible to execute the inverse operation, that is, to return to the body _A_ the caloric employed to raise the temperature of the liquid.

This difficulty may be removed by supposing the difference of temperature between the body _A_ and the body _B_ indefinitely small. The quantity of heat necessary to raise the liquid to its former temperature will be also indefinitely small and unimportant relatively to that which is necessary to produce steam—a quantity always limited.

The proposition found elsewhere demonstrated for the case in which the difference between the temperatures of the two bodies is indefinitely small, may be easily extended to the general case. In fact, if it operated to produce motive power by the passage of caloric from the body _A_ to the body _Z_, the temperature of this latter body being very different from that of the former, we should imagine a series of bodies _B_, _C_, _D_ ... of temperatures intermediate between those of the bodies _A_, _Z_, and selected so that the differences from _A_ to _B_, from _B_ to _C_, etc., may all be indefinitely small. The caloric coming from _A_ would not arrive at _Z_ till after it had passed through the bodies _B_, _C_, _D_, etc., and after having developed in each of these stages maximum motive power. The inverse operations would here be entirely possible, and the reasoning of page 52 would be strictly applicable.

According to established principles at the present time, we can compare with sufficient accuracy the motive power of heat to that of a waterfall. Each has a maximum that we cannot exceed, whatever may be, on the one hand, the machine which is acted upon by the water, and whatever, on the other hand, the substance acted upon by the heat. The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the _height of its fall_,[16] that is to say, the difference of temperature of the bodies between which the exchange of caloric is made. In the waterfall the motive power is exactly proportional to the difference of level between the higher and lower reservoirs. In the fall of caloric the motive power undoubtedly increases with the difference of temperature between the warm and the cold bodies; but we do not know whether it is proportional to this difference. We do not know, for example, whether the fall of caloric from 100 to 50 degrees furnishes more or less motive power than the fall of this same caloric from 50 to zero. It is a question which we propose to examine hereafter.

We shall give here a second demonstration of the fundamental proposition enunciated on page 56, and present this proposition under a more general form than the one already given.

When a gaseous fluid is rapidly compressed its temperature rises. It falls, on the contrary, when it is rapidly dilated. This is one of the facts best demonstrated by experiment. We will take it for the basis of our demonstration.[17]

If, when the temperature of a gas has been raised by compression, we wish to reduce it to its former temperature without subjecting its volume to new changes, some of its caloric must be removed. This caloric might have been removed in proportion as pressure was applied, so that the temperature of the gas would remain constant. Similarly, if the gas is rarefied we can avoid lowering the temperature by supplying it with a certain quantity of caloric. Let us call the caloric employed at such times, when no change of temperature occurs, _caloric due to change of volume_. This denomination does not indicate that the caloric appertains to the volume: it does not appertain to it any more than to pressure, and might as well be called _caloric due to the change of pressure_. We do not know what laws it follows relative to the variations of volume: it is possible that its quantity changes either with the nature of the gas, its density, or its temperature. Experiment has taught us nothing on this subject. It has only shown us that this caloric is developed in greater or less quantity by the compression of the elastic fluids.

FIG. 1.
]

This preliminary idea being established, let us imagine an elastic fluid, atmospheric air for example, shut up in a cylindrical vessel, _abcd_ (Fig. 1), provided with a movable diaphragm or piston, _cd_. Let there be also two bodies, _A_ and _B_, kept each at a constant temperature, that of _A_ being higher than that of _B_. Let us picture to ourselves now the series of operations which are to be described:

(1) Contact of the body _A_ with the air enclosed in the space _abcd_ or with the wall of this space—a wall that we will suppose to transmit the caloric readily. The air becomes by such contact of the same temperature as the body _A_; _cd_ is the actual position of the piston.

(2) The piston gradually rises and takes the position _ef_. The body _A_ is all the time in contact with the air, which is thus kept at a constant temperature during the rarefaction. The body _A_ furnishes the caloric necessary to keep the temperature constant.

(3) The body _A_ is removed, and the air is then no longer in contact with any body capable of furnishing it with caloric. The piston meanwhile continues to move, and passes from the position _ef_ to the position _gh_. The air is rarefied without receiving caloric, and its temperature falls. Let us imagine that it falls thus till it becomes equal to that of the body _B_; at this instant the piston stops, remaining at the position _gh_.

(4) The air is placed in contact with the body _B_; it is compressed by the return of the piston as it is moved from the position _gh_ to the position _cd_. This air remains, however, at a constant temperature because of its contact with the body _B_, to which it yields its caloric.

(5) The body _B_ is removed, and the compression of the air is continued, which being then isolated, its temperature rises. The compression is continued till the air acquires the temperature of the body _A_. The piston passes during this time from the position _cd_ to the position _ik_.

(6) The air is again placed in contact with the body _A_. The piston returns from the position _ik_ to the position _ef_; the temperature remains unchanged.

(7) The step described under number 3 is renewed, then successively the steps 4, 5, 6, 3, 4, 5, 6, 3, 4, 5; and so on.

In these various operations the piston is subject to an effort of greater or less magnitude, exerted by the air enclosed in the cylinder; the elastic force of this air varies as much by reason of the changes in volume as of changes of temperature. But it should be remarked that with equal volumes, that is, for the similar positions of the piston, the temperature is higher during the movements of dilatation than during the movements of compression. During the former the elastic force of the air is found to be greater, and consequently the quantity of motive power produced by the movements of dilatation is more considerable than that consumed to produce the movements of compression. Thus we should obtain an excess of motive power—an excess which we could employ for any purpose whatever. The air, then, has served as a heat-engine; we have, in fact, employed it in the most advantageous manner possible, for no useless re-establishment of equilibrium has been effected in the caloric.

All the above-described operations may be executed in an inverse sense and order. Let us imagine that, after the sixth period, that is to say the piston having arrived at the position _ef_, we cause it to return to the position _ik_, and that at the same time we keep the air in contact with the body _A_. The caloric furnished by this body during the sixth period would return to its source, that is, to the body _A_, and the conditions would then become precisely the same as they were at the end of the fifth period. If now we take away the body _A_, and if we cause the piston to move from _ef_ to _cd_, the temperature of the air will diminish as many degrees as it increased during the fifth period, and will become that of the body _B_. We may evidently continue a series of operations the inverse of those already described. It is only necessary under the same circumstances to execute for each period a movement of dilatation instead of a movement of compression, and reciprocally.

The result of these first operations has been the production of a certain quantity of motive power and the removal of caloric from the body _A_ to the body _B_. The result of the inverse operations is the consumption of the motive power produced and the return of the caloric from the body _B_ to the body _A_; so that these two series of operations annul each other, after a fashion, one neutralizing the other.

The impossibility of making the caloric produce a greater quantity of motive power than that which we obtained from it by our first series of operations, is now easily proved. It is demonstrated by reasoning very similar to that employed at page 56; the reasoning will here be even more exact. The air which we have used to develop the motive power is restored at the end of each cycle of operations exactly to the state in which it was at first found, while, as we have already remarked, this would not be precisely the case with the vapor of water.[18]

We have chosen atmospheric air as the instrument which should develop the motive power of heat, but it is evident that the reasoning would have been the same for all other gaseous substances, and even for all other bodies susceptible of change of temperature through successive contractions and dilatations, which comprehends all natural substances, or at least all those which are adapted to realize the motive power of heat. Thus we are led to establish this general proposition:

_The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of the caloric._

We must understand here that each of the methods of developing motive power attains the perfection of which it is susceptible. This condition is found to be fulfilled if, as we remarked above, there is produced in the body no other change of temperature than that due to change of volume, or, what is the same thing in other words, if there is no contact between bodies of sensibly different temperatures.

Different methods of realizing motive power may be taken, as in the employment of different substances, or in the use of the same substance in two different states—for example, of a gas at two different densities.

This leads us naturally to those interesting researches on the aeriform fluids—researches which lead us also to new results in regard to the motive power of heat, and give us the means of verifying, in some particular cases, the fundamental proposition above stated.[19]

We readily see that our demonstration would have been simplified by supposing the temperatures of the bodies _A_ and _B_ to differ very little. Then the movements of the piston being slight during the periods 3 and 5, these periods might have been suppressed without influencing sensibly the production of motive power. A very little change of volume should suffice in fact to produce a very slight change of temperature, and this slight change of volume may be neglected in presence of that of the periods 4 and 6, of which the extent is unlimited.

If we suppress periods 3 and 5, in the series of operations above described, it is reduced to the following:

(1) Contact of the gas confined in _abcd_ (Fig. 2) with the body _A_, passage of the piston from _cd_ to _ef_.

FIG. 2. FIG. 3.
]

(2) Removal of the body _A_, contact of the gas confined in _abef_ with the body _B_, return of the piston from _ef_ to _cd_.

(3) Removal of the body _B_, contact of the gas with the body _A_, passage of the piston from _cd_ to _ef_, that is, repetition of the first period, and so on.

The motive power resulting from the _ensemble_ of operations 1 and 2 will evidently be the difference between that which is produced by the expansion of the gas while it is at the temperature of the body _A_, and that which is consumed to compress this gas while it is at the temperature of the body _B_.

Let us suppose that operations 1 and 2 be performed on two gases of different chemical natures but under the same pressure—under atmospheric pressure, for example. These two gases will behave exactly alike under the same circumstances, that is, their expansive forces, originally equal, will remain always equal, whatever may be the variations of volume and of temperature, provided these variations are the same in both. This results obviously from the laws of Mariotte and MM. Gay-Lussac and Dalton—laws common to all elastic fluids, and in virtue of which the same relations exist for all these fluids between the volume, the expansive force, and the temperature.

Since two different gases at the same temperature and under the same pressure should behave alike under the same circumstances, if we subjected them both to the operations above described, they should give rise to equal quantities of motive power.

Now this implies, according to the fundamental proposition that we have established, the employment of two equal quantities of caloric; that is, it implies that the quantity of caloric transferred from the body _A_ to the body _B_ is the same, whichever gas is used.

The quantity of caloric transferred from the body _A_ to the body _B_ is evidently that which is absorbed by the gas in its expansion of volume, or that which this gas relinquishes during compression. We are led, then, to establish the following proposition:

_When a gas passes without change of temperature from one definite volume and pressure to another volume and another pressure equally definite, the quantity of caloric absorbed or relinquished is always the same, whatever may be the nature of the gas chosen as the subject of the experiment._

Take, for example, 1 litre of air at the temperature of 100° and under the pressure of one atmosphere. If we double the volume of this air and wish to maintain it at the temperature of 100°, a certain quantity of heat must be supplied to it. Now this quantity will be precisely the same if, instead of operating on the air, we operate upon carbonic-acid gas, upon nitrogen, upon hydrogen, upon vapor of water or of alcohol, that is, if we double the volume of 1 litre of these gases taken at the temperature of 100° and under atmospheric pressure.

It will be the same thing in the inverse sense if, instead of doubling the volume of gas, we reduce it one half by compression. The quantity of heat that the elastic fluids set free or absorb in their changes of volume has never been measured by any direct experiment, and doubtless such an experiment would be very difficult, but there exists a datum which is very nearly its equivalent. This has been furnished by the theory of sound. It deserves much confidence because of the exactness of the conditions which have led to its establishment. It consists in this:

Atmospheric air should rise one degree Centigrade when by sudden compression it experiences a reduction of volume of ¹⁄₁₁₆.[20]

Experiments on the velocity of sound having been made in air under the pressure of 760 millimetres of mercury and at the temperature of 6°, it is only to these two circumstances that our datum has reference. We will, however, for greater facility, refer it to the temperature 0°, which is nearly the same.

Air compressed ¹⁄₁₁₆, and thus heated one degree, differs from air heated directly one degree only in its density. The primitive volume being supposed to be _V_, the compression of ¹⁄₁₁₆ reduces it to _V_ − ¹⁄₁₁₆ _V_.

Direct heating under constant pressure should, according to the rule of M. Gay-Lussac, increase the volume of air ¹⁄₂₆₇ above what it would be at 0°: so the air is, on the one hand, reduced to the volume _V_ − ¹⁄₁₁₆ _V_; on the other, it is increased to _V_ + ¹⁄₂₆₇ _V_.

The difference between the quantities of heat which the air possesses in both cases is evidently the quantity employed to raise it directly one degree; so then the quantity of heat that the air would absorb in passing from the volume _V_ − ¹⁄₁₁₆ _V_ to the volume _V_ + ¹⁄₂₆₇ _V_ is equal to that which is required to raise it one degree.

Let us suppose now that, instead of heating one degree the air subjected to a constant pressure and able to dilate freely, we inclose it within an invariable space, and that in this condition we cause it to rise one degree in temperature. The air thus heated one degree will differ from the air compressed ¹⁄₁₁₆ only by its ¹⁄₁₁₆ greater volume. So then the quantity of heat that the air would set free by a reduction of volume of ¹⁄₁₁₆ is equal to that which would be required to raise it one degree Centigrade under constant volume. As the differences between the volumes _V_ − ¹⁄₁₁₆ _V_, _V_, and _V_ + ¹⁄₂₆₇ _V_ are small relatively to the volumes themselves, we may regard the quantities of heat absorbed by the air in passing from the first of these volumes to the second, and from the first to the third, as sensibly proportional to the changes of volume. We are then led to the establishment of the following relation:

The quantity of heat necessary to raise one degree air under constant pressure is to the quantity of heat necessary to raise one degree the same air under constant volume, in the ratio of the numbers

¹⁄₁₁₆ + ¹⁄₂₆₇ to ¹⁄₁₁₆;

or, multiplying both by 116 × 267, in the ratio of the numbers 267 + 116 to 267.

This, then, is the ratio which exists between the capacity of air for heat under constant pressure and its capacity under constant volume. If the first of these two capacities is expressed by unity, the other will be expressed by the number (267)/(267 + 116), or very nearly 0.700; their difference, 1 − 0.700 or 0.300, will evidently express the quantity of heat which will produce the increase of volume in the air when it is heated one degree under constant pressure.

According to the law of MM. Gay-Lussac and Dalton, this increase of volume would be the same for all other gases; according to the theory demonstrated on page 87, the heat absorbed by these equal increases of volume is the same for all the elastic fluids, which leads to the establishment of the following proposition:

_The difference between specific heat under constant pressure and specific heat under constant volume is the same for all gases._

It should be remarked here that all the gases are considered as taken under the same pressure, atmospheric pressure for example, and that the specific heats are also measured with reference to the volumes.

It is a very easy matter now for us to prepare a table of the specific heat of gases under constant volume, from the knowledge of their specific heats under constant pressure. Here is the table:

TABLE OF THE SPECIFIC HEAT OF GASES. ───────────────────────┬───────────────────────┬─────────────────────── NAMES OF GASES. │ Specific Heat under │Specific Heat at Const. │ Const. Press. │ Vol. ───────────────────────┼───────────────────────┼─────────────────────── Atmospheric Air, │ 1.000 │ 0.700 Hydrogen Gas, │ 0.903 │ 0.603 Carbonic Acid, │ 1.258 │ 0.958 Oxygen, │ 0.976 │ 0.676 Nitrogen, │ 1.000 │ 0.700 Protoxide of Nitrogen, │ 1.350 │ 1.050 Olefiant Gas, │ 1.553 │ 1.253 Oxide of Carbon, │ 1.034 │ 0.734 ───────────────────────┴───────────────────────┴───────────────────────

The first column is the result of the direct experiments of MM. Delaroche and Bérard on the specific heat of the gas under atmospheric pressure, and the second column is composed of the numbers of the first diminished by 0.300.

The numbers of the first column and those of the second are here referred to the same unit, to the specific heat of atmospheric air under constant pressure.

The difference between each number of the first column and the corresponding number of the second being constant, the relation between these numbers should be variable. Thus the relation between the specific heat of gases under constant pressure and the specific heat at constant volume, varies in different gases.

We have seen that air when it is subjected to a sudden compression of ¹⁄₁₁₆ of its volume rises one degree in temperature. The other gases through a similar compression should also rise in temperature. They should rise, but not equally, in inverse ratio with their specific heat at constant volume. In fact, the reduction of volume being by hypothesis always the same, the quantity of heat due to this reduction should likewise be always the same, and consequently should produce an elevation of temperature dependent only on the specific heat acquired by the gas after its compression, and evidently in inverse ratio with this specific heat. Thus we can easily form the table of the elevations of temperature of the different gases for a compression of ¹⁄₁₁₆.

TABLE OF THE ELEVATION OF TEMPERATURE<BR>OF _Gases through the Effect of Compression_. ──────────────────────┬──────────────────────────────────────────────── NAMES OF GASES. │ Elevation of Temperature for a Reduction of │ Volume of ¹⁄₁₁₆. ──────────────────────┼──────────────────────────────────────────────── │ ° Atmospheric Air, │ 1.000 Hydrogen Gas, │ 1.160 Carbonic Acid, │ 0.730 Oxygen, │ 1.035 Nitrogen, │ 1.000 Protoxide of Nitrogen,│ 0.667 Olefiant Gas, │ 0.558 Carbonic Oxide, │ 0.955 ──────────────────────┴────────────────────────────────────────────────

A second compression of ¹⁄₁₁₆ (of the altered volume), as we shall presently see, would also raise the temperature of these gases nearly as much as the first; but it would not be the same with a third, a fourth, a hundredth such compression. The capacity of gases for heat changes with their volume. It is not unlikely that it changes also with the temperature.

We shall now deduce from the general proposition stated on page 68 a second theory, which will serve as a corollary to that just demonstrated.

Let us suppose that the gas enclosed in the cylindrical space _abcd_ (Fig. 2) be transported into the space _a′b′c′d′_ (Fig. 3) of equal height, but of different base and wider. This gas would increase in volume, would diminish in density and in elastic force, in the inverse ratio of the two volumes _abcd_, _a′b′c′d′_. As to the total pressure exerted in each piston _cd_, _c′d′_, it would be the same from all quarters, for the surface of these pistons is in direct ratio to the volumes.

Let us suppose that we perform on the gas inclosed in _a′b′c′d′_ the operations described on page 70, and which were taken as having been performed upon the gas inclosed in _abcd_; that is, let us suppose that we have given to the piston _c′d′_ motions equal to those of the piston _cd_, that we have made it occupy successively the positions _c′d′_ corresponding to _cd_, and _e′f′_ corresponding to _ef_, and that at the same time we have subjected the gas by means of the two bodies _A_ and _B_ to the same variations of temperature as when it was inclosed in _abcd_. The total effort exercised on the piston would be found to be, in the two cases, always the same at the corresponding instants. This results solely from the law of Mariotte.[21] In fact, the densities of the two gases maintaining always the same ratio for similar positions of the pistons, and the temperatures being always equal in both, the total pressures exercised on the pistons will always maintain the same ratio to each other. If this ratio is, at any instant whatever, unity, the pressures will always be equal.

As, furthermore, the movements of the two pistons have equal extent, the motive power produced by each will evidently be the same; whence we should conclude, according to the proposition on page 68, that the quantities of heat consumed by each are the same, that is, that there passes from the body _A_ to the body _B_ the same quantity of heat in both cases.

The heat abstracted from the body _A_ and communicated to the body _B_, is simply the heat absorbed during the rarefaction of the gas, and afterwards liberated by its compression. We are therefore led to establish the following theorem:

_When an elastic fluid passes without change of temperature from the volume U to the volume V, and when a similar ponderable quantity of the same gas passes at the same temperature from the volume U′ to the volume V′, if the ratio of U′ to V′ is found to be the same as the ratio of U to V, the quantities of heat absorbed or disengaged in the two cases will be equal._

This theorem might also be expressed as follows:

_When a gas varies in volume without change of temperature, the quantities of heat absorbed or liberated by this gas are in arithmetical progression, if the increments or the decrements of volume are found to be in geometrical progression._

When a litre of air maintained at a temperature of ten degrees is compressed, and when it is reduced to one half a litre, a certain quantity of heat is set free. This quantity will be found always the same if the volume is further reduced from a half litre to a quarter litre, from a quarter litre to an eighth, and so on.

If, instead of compressing the air, we carry it successively to two litres, four litres, eight litres, etc., it will be necessary to supply to it always equal quantities of heat in order to maintain a constant temperature.

This readily accounts for the high temperature attained by air when rapidly compressed. We know that this temperature inflames tinder and even makes air luminous. If, for a moment, we suppose the specific heat of air to be constant, in spite of the changes of volume and temperature, the temperature will increase in arithmetical progression for reduction of volume in geometrical progression.

Starting from this datum, and admitting that one degree of elevation in the temperature corresponds to a compression of ¹⁄₁₁₆, we shall readily come to the conclusion that air reduced to ¹⁄₁₄ of its primitive volume should rise in temperature about 300 degrees, which is sufficient to inflame tinder.[22]

The elevation of temperature ought, evidently, to be still more considerable if the capacity of the air for heat becomes less as its volume diminishes. Now this is probable, and it also seems to follow from the experiments of MM. Delaroche and Bérard on the specific heat of air taken at different densities. (See the Mémoire in the _Annales de Chimie_, t. lxxxv. pp. 72, 224.)

The two theorems explained on pp. 72 and 81 suffice for the comparison of the quantities of heat absorbed or set free in the changes of volume of elastic fluids, whatever may be the density and the chemical nature of these fluids, provided always that they be taken and maintained at a certain invariable temperature. But these theories furnish no means of comparing the quantities of heat liberated or absorbed by elastic fluids which change in volume at different temperatures. Thus we are ignorant what relation exists between the heat relinquished by a litre of air reduced one half, the temperature being kept at zero, and the heat relinquished by the same litre of air reduced one half, the temperature being kept at 100°. The knowledge of this relation is closely connected with that of the specific heat of gases at various temperatures, and to some other data that Physics as yet does not supply.

The second of our theorems offers us a means of determining according to what law the specific heat of gases varies with their density.

Let us suppose that the operations described on p. 70, instead of being performed with two bodies, _A_, _B_, of temperatures differing indefinitely small, were carried on with two bodies whose temperatures differ by a finite quantity—one degree, for example. In a complete circle of operations the body _A_ furnishes to the elastic fluid a certain quantity of heat, which may be divided into two portions: (1) That which is necessary to maintain the temperature of the fluid constant during dilatation; (2) that which is necessary to restore the temperature of the fluid from that of the body _B_ to that of the body _A_, when, after having brought back this fluid to its primitive volume, we place it again in contact with the body _A_. Let us call the first of these quantities _a_ and the second _b_. The total caloric furnished by the body A will be expressed by _a_ + _b_.

The caloric transmitted by the fluid to the body _B_ may also be divided into two parts: one, _b′_, due to the cooling of the gas by the body _B_; the other, _a′_, which the gas abandons as a result of its reduction of volume. The sum of these two quantities is _a′_ + _b′_; it should be equal to _a_ + _b_, for, after a complete cycle of operations, the gas is brought back exactly to its primitive state. It has been obliged to give up all the caloric which has first been furnished to it. We have then

_a_ + _b_ = _a′_ + _b′_;

or rather,

_a_ − _a′_ = _b′_ − _b_.

Now, according to the theorem given on page 81, the quantities _a_ and _a′_ are independent of the density of the gas, provided always that the ponderable quantity remains the same and that the variations of volume be proportional to the original volume. The difference _a_ − _a′_ should fulfil the same conditions, and consequently also the difference _b′_ − _b_, which is equal to it. But _b′_ is the caloric necessary to raise the gas enclosed in _abcd_ (Fig. 2) one degree; _b′_ is the caloric surrendered by the gas when, enclosed in _abef_, it is cooled one degree. These quantities may serve as a measure for specific heats. We are then led to the establishment of the following proposition:

_The change in the specific heat of a gas caused by change of volume depends entirely on the ratio between the original volume and the altered volume._ That is, the difference of the specific heats does not depend on the absolute magnitude of the volumes, but only on their ratio.

This proposition might also be differently expressed, thus:

_When a gas increases in volume in geometrical progression, its specific heat increases in arithmetical progression._