Scientific Memoirs/1/Memoir on the Motive Power of Heat

by Benoît Paul Émile Clapeyron

SCIENTIFIC MEMOIRS.

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VOL. I.— PART III.

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Article XV.

Memoir on the Motive Power of Heat; by E. Clapeyron, Mining Engineer.

From the Journal de l'Ecole Royale Polytechnique; Paris; vol. xiv. p. 153 et seq.

§ 1.

Few questions are more worthy of fixing the attention of geometers and natural philosophers than those which relate to the constitution of gases and vapours: the functions they exercise in nature, and the advantages which industry derives from them, account sufficiently for the numerous and important labours of which they have been the object: this vast question, however, is far from being exhausted. The law of Mariotte and that of Gay-Lussac, which establish the relations existing between the volume, the pressure, and the temperature of a given quantity of any gas, have both long since obtained the assent of scientific men. The experiments recently made by MM. Arago and Dulong leave no doubt of the accuracy of the first of those laws within very extended limits of pressure; but these important results give no information respecting the quantity of heat which the gases contain, and which is disengaged by pressure or diminution of temperature,—they do not give the law of the specific calorics when the pressure and the volume are constant. This part of the theory of heat, however, has been the object of profound researches, among which we may cite those of MM. La Roche and Bérard on the specific caloric of gases. Lastly, M. Dulong, in a memoir which he published under the title of Recherches sur la Chaleur Spécifique des Fluides Elastiques, has established by experiments free from all objection, that equal volumes of all elastic fluids at the same tempeature and under the same pressure, being suddenly compressed or dilated by the same fraction of their volume, disengage or absorb the same absolute quantity of heat.

Laplace, and subsequently M. Poisson, have made public some very remarkable theoretical researches on this subject; but they rest upon hypothetical data which appear liable to objection. It is admitted in them that the ratio of the specific caloric when the volume remains constant to the specific caloric under a constant pressure, is invariable, and that the quantities of heat absorbed by gases are proportional to their temperatures.

I will finally quote among the works which have appeared on the theory of heat, one by M. S. Carnot, published in 1824, under the title of Reflexions sur la Puissance Motrice du Feu. The idea taken for the basis of his researches appears to me fertile and incontestible: his demonstrations are founded on the absurdity which arises from admitting the possibility of producing absolutely either the motive force or the heat.

The various theorems to which this new method of reasoning conducts us may be enunciated as follows:

1. When a gas without change of temperature passes from a determined volume and pressure to another volume and pressure equally determined, the quantity of caloric absorbed or lost is always the same, whatever be the nature of the gas subjected to experiment.

2. The difference between the specific heat under a constant pressure and the specific heat at a constant volume is the same for all gases.

3. When a gas varies in volume without change of temperature, the quantities of heat absorbed or disengaged by that gas are in arithmetical progression, if the increments or diminutions of volume are in geometrical progression.

This new mode of demonstration appears to me worthy of fixing the attention of geometers; it is, in my opinion, free from every objection, and it has acquired additional importance since its verification by the labours of M. Dulong, in which the truth of the first theorem which I have recited is demonstrated by experiment.

I think it will be of some interest to revive this theory: M. S. Carnot, dispensing with mathematical analysis, arrives, by a series of delicate reasonings difficult to apprehend, at results easily deducible from a more general law, which I shall endeavour to establish. But before entering upon the subject, it will be useful to return to the fundamental axiom upon which the researches of M. Carnot are founded, and which will be my starting point also.

§ II.

It has long been remarked that heat may be employed to develop motive force, and reciprocally that by motive force we may develop heat. In the first case we should observe that there is always a passage of a determinate quantity of caloric from a body at a given temperature to another body at a lower temperature; thus in the steam-engine, the production of the mechanical force is attended by the passage of a part of the heat which is developed by combustion in the furnace, the temperature of which is very high, into the water in the condenser, the temperature of which is much lower.

Reciprocally, it is always possible to render the passage of caloric from a hot to a cold body useful for the production of a mechanical force: to obtain this it is sufficient to construct a machine resembling an ordinary steam-engine, in which the heated body serves to produce steam and the cold one to condense it. It results from this that there is a loss of vis viva, of mechanical force, or of quantity of action, whenever immediate contact takes place between two bodies of different temperatures, and heat passes from the one into the other without traversing an intermediate body; therefore in every machine intended to make efficient the motive force developed by heat, there is a loss of power whenever a direct communication of heat takes place between bodies of different temperatures, and consequently the maximum of the effect produced cannot be obtained but by means of a machine in which only bodies of equal temperature are brought into contact. Now the knowledge we possess of the theory of gases and vapours shows the possibility of attaining this object.

Let us, then, suppose two bodies retained, one at a temperature $T$, and another at an inferior temperature $t$; such, for example, as the sides of a steam-boiler, in which the heat developed by combustion constantly supplies the place of that which the steam produced carries away; and the condenser of the common atmospheric engine, in which a current of cold water removes, every moment, both the heat which the steam loses in condensing, and that which belongs to its proper temperature. For the sake of simplicity we will call the first body $A$ and the second $B$.

Let us now take any gas whatever, at the temperature $T$, and bring it into contact with the source of heat $A$, representing its volume $V_{0}$ by the absciss $A\;B$, and its pressure by the ordinate $C\;B$ (fig 1).

If the gas is inclosed in an extensible vessel, and which is allowed to extend in a void space in which it cannot lose heat either by radiation or by contact, the source of heat $A$ will supply it, from moment to moment, with the quantity of caloric which its increase of volume renders latent, and it will preserve the same temperature $T$. Its pressure, on the contrary, will diminish according to the law of Mariotte. The law of this variation may be represented by a curve $C\;E$, of which the volumes will be the abscisses, and the corresponding pressures the ordinates.

Supposing the dilatation of the gas to continue until the volume $A\;B$ has become $A\;D$, and that the pressure corresponding to this new volume is $D\;E$, the gas during its dilatation will have developed a quantity of mechanical action, which will have for its value the integral of the product of the pressure by the differential of the volume, and which will be represented geometrically by the area comprised between the axis of the abscisses, the two coordinates $C\;B$, $D\;E$, and the portion of a hyperbola $C\;E$.

Fig. 1

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Supposing, again, that the body $A$ is removed and that the dilatation of the gas continues in an inclosure impermeable to heat; then a part of its sensible caloric becoming latent, its temperature will diminish and its pressure will continue to decrease in a more rapid manner and according to an unknown law, which law might be represented geometrically by a curve $E\;F$, the abscissæ of which would be the volumes of the gas, and the ordinates the corresponding pressures: we will suppose that the dilatation of the gas has continued until the successive reductions which its sensible caloric experiences have reduced the temperature $T$ of the body $A$ to the temperature $t$ of the body $B$; its volume will then be $A\;G$, and the corresponding pressure $F\;G$. It will also be evident from the same reasoning, that the gas during this second part of its dilatation will develop a quantity of mechanical action represented by the area of the mixtilinear trapezium $D\;E\;F\;G$.

Now that the gas is brought to the temperature $t$ of the body $B$, let us bring them together: if we compress the gas in an inclosure impermeable to heat, but in contact with the body $B$, the temperature of the gas will tend to rise by the evolution of latent heat rendered sensible by compression, but will be absorbed in proportion by the body $B$, so that the temperature of the gas will remain equal to $t$. The pressure will increase according to the law of Mariotte: it will be represented geometrically by the ordinates of a hyperbola $K\;F$, and the corresponding abscisses will represent the corresponding volumes. Suppose the compression to be increased until the heat disengaged and absorbed by the body $B$ is precisely equal to the heat communicated to the gas by the source $A$ during its dilatation in contact with it in the first part of the process. Let then the volume of gas be $A\;H$, and the corresponding pressure $H\;K$: the gas in this state contains the same absolute quantity of heat that it did at the moment of commencing the process, when it occupied the volume $A\;B$ under the pressure $C\;B$. If therefore we remove the body $B$ and continue to compress the gas in an inclosure impermeable to heat, until the volume $A\;H$ is reduced to the volume $A\;B$, its temperature will successively increase by the evolution of the latent caloric, which the compression converts into sensible caloric. The pressure will increase in a corresponding ratio; and when the volume shall be reduced to $A\;B$, the temperature will become $T$, and the pressure $B\;C$. In fact, the successive states which the same weight of gas experiences are characterized by the volume, the pressure, the temperature, and the absolute quantity of caloric which it contains: two of these four quantities being known, the other two become known as consequences of the former; thus in the case in question the absolute quantity of heat and the volume having become what they were at the beginning of the process, we may be certain that the temperature and pressure will also be the same as before. Consequently, the unknown law according to which the pressure will vary when the volume of gas is reduced in its inclosure impermeable to heat, will be represented by a curve $K\;C$, which will pass through the point $C$, and in which the abscisses always represent the volumes, and the ordinates the pressures.

However, the reduction of the gaseous volume from $A\;G$ to $A\;B$ will have consumed a quantity of mechanical action which, for the reasons we have stated above, will be represented by the two mixtilinear trapeziums $F\;G\;H\;K$ and $K\;H\;B\;C$. If we subtract from these two trapeziums the two first, $C\;B\;D\;E$ and $E\;D\;G\;F$, which represent the quantity of action during the dilatation of the gas, the difference, which will be equal to the sort of curvilinear parallelogram $C\;E\;F\;K$, will represent the quantity of action developed in the circle of operations which we have just described, and after the completion of which the gas will be precisely in the same state in which it was originally. Still, however, the entire quantity of heat furnished by the body $A$ to the gas during its dilatation by contact with it, passes into the body $B$ during the condensation of the gas, which takes place by contact with it.

Here, then, we have mechanical force developed by the passage of caloric from a hot to a cold body, and this transfer is effected without the contact of bodies of different temperatures.

The inverse operation is equally possible: thus, we take the same volume of gas $A\;B$ at the temperature $T$ and under the pressure $B\;C$, inclose it in an envelope impermeable to heat, and dilate it until its temperature, gradually diminishing, becomes equal to $t$; we continue the dilatation in the same envelope,—but after having introduced the body $B$, which is at the same temperature,—and carry on the operation until the body $B$ has restored to the gas the heat which it had received in the preceding operation. We next remove the body $B$, and condense the gas in an inclosure impermeable to heat until its temperature again becomes equal to $T$. We then introduce the body $A$, which possesses the same temperature, and continue the reduction of volume until all the heat taken from the body $B$ is transferred to the body $A$. The gas will then be found to have the same temperature and to contain the same absolute quantity of heat as at the beginning of the operation, whence we may conclude that it occupies the same volume and is subjected to the same pressure.

Here the gas passes successively, but in an inverse order, through all the states of temperature and pressure through which it had passed in the first series of operations; consequently the dilatations become compressions, and reciprocally, but they follow the same law. Further, the quantities of action developed in the first case are absorbed in the second, and reciprocally; but they retain the same numerical values, for the elements of the integrals which compose them are the same.

We thus see that by causing heat to pass, in the manner first indicated, from a body retained at a determinate temperature, into a body retained at an inferior temperature, we develop a certain quantity of mechanical action, which is equal to the quantity which must be consumed in order to cause the same quantity of heat to pass from a cold to a hot body, by the inverse process we have subsequently described.

We may arrive at a similar result by converting any liquid into vapour. We take the liquid and bring it into contact with the body $A$ in an extensible envelope impermeable to heat, and suppose the temperature of the liquid to be equal to the temperature $T$ of the body $A$. Upon the axis of the abscisses $A\;X$ (fig. 2.) we describe a quantity $A\;B$ equal to the volume of the liquid, and upon a line parallel to the axis of the ordinates $A\;Y$, a quantity $B\;C$ equal to the pressure of the vapour of the liquid, which corresponds to the temperature $T$.

If we increase the volume of the liquid, a portion of it will pass into the state of vapour; and as the source of heat $A$ furnishes the latent caloric necessary to its formation, the temperature will remain constant and equal to $T$. Then if quantities representing the successive volumes occupied by the mixture of liquid and vapour are described upon the axis of the abscisses, and the corresponding values of the pressures are taken for ordinates, as the pressure remains constant, the curve of the pressures will here be reduced to a right line $C\;E$ parallel to the axis of the abscisses.

When a certain quantity of vapour has been formed, and the mixture of liquid and vapour occupies a volume $A\;D$, the body $A$ may be withdrawn and the dilatation continued. A fresh quantity of liquid will then pass into the gaseous state, and a part of the sensible caloric becoming latent, the temperature of the mixture will diminish as well as

Fig. 2

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the pressure. Suppose the dilatation to be continued until the temperature diminishing gradually becomes equal to the temperature $t$ of the body $B$; let $A\;F$ be the volume and $F\;G$ the pressure corresponding to it. The law of the variation of the pressure will be given by a curve $E\;G$, which will pass through the points $E$ and $G$.

During this first part of the operation which we are describing, a quantity of action will be developed represented by the surface of the rectangle $B\;C\;E\;D$, and that of the mixtilinear trapezium $E\;G\;F\;D$.

We will now bring forward the body $B$, put it in contact with the mixture of liquid and vapour, and successively reduce its volume; a part of the vapour will pass into the liquid state, and as the latent heat disengaged in its condensation will be absorbed by the body $B$, the temperature will remain constant and equal to $t$. We shall thus continue to reduce the volume until all the heat furnished by the body $A$ in the first part of the operation has been conveyed to the body $B$.

Let $A\;H$ then be the volume occupied by the mixture of vapour and liquid; the corresponding pressure will be $K\;H$ equal to $G\;F$: the temperature remaining equal to $t$, during the reduction of the volume from $A\;F$ to $A\;H$, the law of the pressure between these two limits will be represented by the line $K\;G$ parallel to the axis of the abscisses.

Arrived at this point, the mixture of vapour and liquid upon which we are operating, which occupies the volume $A\;H$ under a pressure $KH$, at a temperature $t$, possesses the same absolute quantity of heat that the liquid possessed at the commencement of the operation; if, therefore, we remove the body $B$, and continue the condensation, in a vessel impermeable to heat, until the volume again becomes equal to $A\;B$, we shall have the same quantity of matter occupying the same volume, and possessing the same quantity of heat as at the commencement of the operation: its temperature and its pressure ought, therefore, also to be the same as at that epoch; the temperature will thus again become equal to $T$, and the pressure equal to $C\;B$. The law of the pressures during this last part of the operation, will therefore be given by a curve passing through the points $K$ and $C$; and the quantity of action absorbed during the reduction of the volume from $A\;F$ to $A\;B$, will be represented by the rectangle $F\;H\;K\;G$ and the mixtilinear trapezium $B\;C\;K\;H$, If, then, we deduct from the quantity of action developed during the dilatation, that which is absorbed during the compression, we shall have for the difference the surface of the mixtilinear parallelogram $C\;E\;G\;K$, which will represent the quantity of action developed during the entire series of the operations that we have described, and at the conclusion of which the liquid employed will be found in its primitive state.

But it is necessary to remark that all the caloric communicated by the body A has passed to the body B, and that this transmission has taken place without there having been any other contact than that between bodies of the same temperature.

It might be proved in the same manner as for the gases, that by repeating the same operation in an inverse order, the heat of the body $B$ may be made to pass to the body $A$, but that this result will only be obtained by the absorption of a quantity of mechanical action, equal to that which has been developed in the passage of the same quantity of caloric from the body $A$ to the body $B$.

From what precedes, it results that a quantity of mechanical action and a quantity of heat passing from a hot to a cold body, are quantities of the same nature, and that it is possible to substitute the one for the other reciprocally; in the same manner as in mechanics a body falling from a certain height, and a mass endowed with a certain velocity, are quantities of the same order, and can be transformed one into the other by physical agents.

Hence also it follows that the quantity of action $F$ developed by the passage of a certain quantity of heat $C$, from a body $A$ maintained at a temperature $T$, to a body $B$ maintained at a temperature $t$, by one of the processes that we have just indicated, is the same, whatever be the gas or the liquid employed, and is the greatest that it is possible to realize.

Suppose that by causing the quantity of heat $C$ of the body $A$ to pass to the body $B$, by some other process, it was possible to realize a larger quantity of mechanical action $F'$; we should employ one part of it $F$, to restore to the body $A$ from the body $B$ the quantity of heat $C$, by one of the two means that we have just described. The vis viva $F$ employed for this purpose would be equal, as we have seen, to that which would be developed in the passage of the same quantity of heat $C$, from the body $A$ to the body $B$; it is therefore, according to the hypothesis, smaller than $F'$; a quantity of action $F'-F$, would therefore be produced, which would be created absolutely and without consumption of heat; an absurd result, which would imply the possibility of creating either force or heat in a gratuitous and indefinite manner. It appears to me that the impossibility of such a result might be accepted as a fundamental axiom of mechanics: the demonstration by pulleys, that Lagrange has given, of the principle of virtual velocities, against which no one has attempted to raise an objection, rests upon an analogous principle. In the same manner it may be proved that no gas or vapour exists which, employed in the processes described to transmit the heat of a hot body to a cold one, is capable of developing a larger quantity of action than any other gas or vapour.

We shall therefore admit the following principles as the basis of our researches.

Caloric passing from one body to another maintained at a lower temperature may cause the production of a certain quantity of mechanical action; there is a loss of vis viva whenever bodies of different temperature come into contact. The maximum effect will be produced when the passage of the caloric from the hot to the cold body takes place by one of the methods which we have just described. We may add, that the effect will be found to be independent of the chemical nature, of the quantity, and of the pressure of the gas or liquid employed; so that the maximum quantity of action, which the passage of a determinate quantity of heat from a hot to a cold body can develop, is independent of the nature of the agents which serve to realize it.

§III.

We shall now translate analytically the various operations that have been described in the preceding paragraph; we shall deduce from them the expression of the maximum quantity of action produced by the passage of a given quantity of heat from a body maintained at a determinate temperature, to another body maintained at a lower temperature, and we shall arrive at new relations between the volume, the pressure, the temperature, and the absolute quantity of heat or latent caloric of solid, liquid, or gaseous bodies.

Let us return to the two bodies $A$ and $B$, and suppose that the temperature $t$ of the body $B$ is lower by the infinitely small quantity $dt$, than the temperature $t$ of the body $A$. We shall suppose in the first instance that a gas serves for the transmission to the body $B$, of the caloric of the body $A$. Let $v_{0}$ be the volume of the gas under the pressure $p_{0}$ at a temperature of $t_{0}$; let $p$ and $v$ be the volume and the pressure of the same weight of gas at the temperature $t$ of the body $A$. The law enunciated by Mariotte, combined with that of Gay-Lussac, establishes between these different quantities the relation

$pv={\frac {p_{0}v_{0}}{267+t_{0}}}(267+t),$

or, for simplicity,

${\begin{aligned}&{\frac {p_{0}v_{0}}{267+t_{0}}}=R:\\&pv=R(267+t).\end{aligned}}$

The body $A$ is brought into contact with the gas. Let $me=v$, $ae=p$ (fig. 3.). If the gas be allowed to expand by the infinitely small quantity $dv=eg$, the temperature will remain constant, in consequence of the presence of the source of heat $A$; the pressure will diminish, and become equal to the ordinate $bg$. We now remove the

Fig. 3.

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body $A$, and allow the gas to expand, in an inclosure impermeable to heat, by the infinitely small quantity $gh$, until the heat becomes latent, reduces the temperature of the gas by the infinitely small quantity $dt$, and thus brings it to the temperature $t-dt$ of the body $B$. In consequence of this reduction of temperature, the pressure will diminish more rapidly than in the first part of the operation, and will become $ch$. We now take the body $B$, and reduce the volume $mh$ by the infinitely small quantity $fh$, calculated in such a manner that during this compression the gas may transmit to the body $B$ all the heat it has derived from the body $A$ during the first part of the operation. Let $fd$ be the corresponding pressure; that done, we remove the body $B$, and continue to compress the gas until it is again reduced to the volume $me$. The pressure will then again be equal to $ae$, as we have shown in the preceding paragraph; and in the same manner also it will be proved, that the quadrilateral figure $a\;b\;c\;d$ will be the measure of the quantity of action produced by the transmission to the body $B$, of the heat derived from the body $A$, during the expansion of the gas.

Now it is easy to show that this quadrilateral figure is a parallelogram; this results from the infinitely small values assigned to the variations of the volume and pressure: let us conceive that perpendiculars are erected upon each point of the plane upon which the quadrilateral figure $a\;b\;c\;d$ is traced, and that on each of them, commencing at their foot, are described two quantities $T$ and $Q$, the first equal to the temperature, the second to the absolute quantity of heat possessed by the gas, when the volume and the pressure have the value assigned to them by the absciss $v$ and the ordinate $p$ which correspond to each point.

The lines $ab$ and $cd$ belong to the projections of two curves of equality of temperature, passing through two points infinitely near, taken upon the surface of temperatures; $ab$ and $cd$ are therefore parallel: $ad$ and $bc$ will be also projections from two curves, for which $Q=\mathrm {const.}$, and which would also pass through two points infinitely near, taken upon the surface $Q=f(pv)$; these two elements are therefore also parallel. The quadrilateral figure $a\;b\;c\;d$ is therefore a parallelogram, and it is easy to see that its area may be obtained by multiplying the variation of the volume during the contact of the gas with the body $A$ or the body $B$, that is to say, $eg$, or its equal $fh$, by $bn$, the difference of the pressures supported during these two operations, and corresponding to the same value of the volume $v$. Now, $eg$, or $fh$, being the differentials of the volume, are equal to $dv$; $bn$ will be obtained by differentiating the equation $pv=R(267+v)$, supposing $v$ constant; we shall then have $bn=dp=R{\frac {d\;t}{v}}$. The expression of the quantity of action developed will therefore be $R{\frac {dt\;dv}{v}}$.

It remains to determine the quantity of heat necessary to produce this effect: it is equal to that which the gas has derived from the body $A$, whilst its volume has increased by $dv$, at the same time preserving the same temperature $t$. Now $Q$ being the absolute quantity of heat possessed by the gas, ought to be a certain function of $p$ and of $v$, considered as independent variables; the quantity of heat absorbed by the gas will therefore be

$dQ={\frac {dQ}{dv}}dv+{\frac {dQ}{dp}}dp;$

but the temperature remaining constant during the variation of the volume, we have

$vdp+pdv=0,\;\mathrm {whence} \;dp=-{\frac {p}{v}}dv,$

and consequently

$dQ=\left({\frac {dQ}{dv}}-{\frac {p}{v}}{\frac {dQ}{dp}}\right)dv.$

If we divide the effect produced by this value of $dQ$, we shall have

${\frac {R\;dt}{v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}}}$

for the expression of the maximum effect which can be developed by the passage of a quantity of heat equal to unity, from a body maintained at the temperature $t$ to a body maintained at the temperature $t-dt$.

We have shown that this quantity of action developed is independent of the agent which has served to transmit the heat; it is therefore the same for all the gases, and is equally independent of the ponderable quantity of the body employed: but there is nothing that proves it to be independent of the temperature; $v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}$ ought therefore to be equal to an unknown function of $t$, which is the same for all the gases.

Now by the equation $pv=R(267+t)$, $t$ is itself the function of the product $pv$; the partial differential equation is therefore

$v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}=F(p.v),$

having for its integral

$Q=f(p.v)-F(p.v)\log[(hyp)p].$

No change is effected in the generality of this formula by substituting for these two arbitrary functions of the product $pv$, the functions $B$ and $C$ of the temperature, multiplied by the coefficient $R$; we shall thus have

$Q=R(B-C\log p).$

That this value of $Q$ satisfies all the conditions to which it is subject may be easily verified; in fact we have

${\begin{aligned}&{\frac {dQ}{dv}}=R\left({\frac {dB}{dt}}{\frac {p}{R}}-\log p{\frac {dC}{dt}}{\frac {p}{R}}\right)\\&{\frac {dQ}{dp}}=R\left({\frac {dB}{dt}}{\frac {v}{R}}-\log p{\frac {dC}{dt}}{\frac {v}{r}}-C{\frac {1}{p}}\right);\end{aligned}}$

whence

$v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}=CR,$

and consequently

${\frac {Rdt}{v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}}}={\frac {dt}{C}}.$

The function $C$ by which the logarithm of the pressure in the value of $Q$ is multiplied is, as we see, of great importance; it is independent of the nature of the gases, and is a function of the temperature alone; it is essentially positive, and serves as a measure of the maximum quantity of action developed by the heat.

We have seen that of the four quantities $Q$, $t$, $p$, and $v$, two being known, the other two follow from them; they ought therefore to be united together by two equations; one of them,

$pv=R(267+t),$

results from the combined laws of Mariotte and Gay-Lussac. The equation

$Q=R(B-C\log p),$

deduced from our theory, is the second. However, the numerical determination of the alterations produced in the gases, when the volume and the pressure are varied in an arbitrary manner, requires a knowledge of the functions $B$ and $C$.

We shall see upon another occasion that a value approaching to the function $C$ may be obtained through a considerable extent of the thermometrical scale; besides, being determined for one gas it will be determined for all. As to the function $B$, it may vary in different gases; however, it is probable that it is the same for all the simple gases: that they all have the same capacity for heat, is at least the apparent result of the indications of experiment.

Let us return to the equation

$Q=R(B-C\log p).$

We will compress a gas occupying the volume $v$, under the pressure $p$, until the volume becomes $v'$, and allow it to cool till the temperature sinks to the same point. Let $p'$ be the new value of the pressure; let $Q'$ be the new value of $Q$; we shall have

$Q-Q'=RC\log {\frac {p'}{p}}=RC\log {\frac {v}{v'}}.$

The function $C$ being the same for all the gases, it is evident that equal volumes of all the elastic fluids, taken at the same temperature and under the same pressure, being compressed or expanded by the same fraction of their volume, disengage or absorb the same absolute quantity of heat. This law M. Dulong has deduced from direct experiment.

This equation shows also that when a gas varies in volume without change of temperature, the quantities of heat absorbed or disengaged by this gas are in arithmetical progression, if the increments or reductions of volume are in geometrical progression. M. Carnot enunciates this result in the work already cited.

The equation

$Q-Q'=RC\log({\frac {v}{v'}})$

expresses a more general law; it includes all the circumstances by which the phænomenon can be affected, such as the pressure, the volume, and the temperature.

In fact, since

$R={\frac {p_{0}v_{0}}{267+t_{0}}}={\frac {pv}{267+t}},$

we have

$Q-Q'={\frac {267+t}{pv}}C\log {\frac {v}{v'}}.$

This equation exhibits the influence of the pressure; it shows that equal volumes of all the gases, taken at the same temperature, being compressed or expanded by the same fraction of their volume, disengage or absorb quantities of heat proportionate to the pressure.

This result explains why the sudden entrance of the air into the vacuum of the air-pump does not disengage a sensible quantity of heat. The vacuum of the air-pump is nothing but a volume of gas $v$, of which the pressure $p$ is very small; if atmospheric air be admitted, its pressure $p$ will suddenly become equal to the pressure of $p'$ of the atmosphere, its volume $v$ will be reduced to $v'$, and the expression of the heat disengaged will be

$C{\frac {pv}{267+t}}\log {\frac {v}{v'}}=C{\frac {pv}{267+t}}\log {\frac {p'}{p}}.$

The heat disengaged by the reentrance of atmospheric air into the vacuum will therefore be what this expression becomes when $p$ is there made very small; it is then found that $\log {\frac {p'}{p}}$ becomes very great, but the product of $p$ by $\log {\frac {p'}{p}}$ is not the less small on that account; in fact we have

$p\log {\frac {p'}{p}}=p\log p'-p\log p=p(\log p'-\log p),$

a quantity which converges towards zero when $p$ diminishes.

The quantity of heat disengaged will therefore be small in proportion to the feebleness of pressure in the recipient, and it will be reduced to zero when the vacuum is perfect.

We shall add that the equation

$Q=R(B-C\log p)$

gives the law of the specific calorics at a constant pressure and volume.

The expression of the first is

$R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right);\qquad \qquad$

of the second,

$R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p-C{\frac {1}{p}}{\frac {d\,p}{d\,t}}\right),$

equal to

$R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p-{\frac {C}{267+t}}\right).$

The first may be obtained by differentiating $Q$ with relation to $t$, supposing $p$ constant; the second, by supposing $v$ constant. If we take equal volumes of different gases at the same temperature and under the same pressure, the quantity $R$ will be the same for all; and accordingly we see that the excess of specific caloric at a constant pressure, over the specific caloric of a constant volume, is the same for all, and equal to ${\frac {R}{267+t}}C$.

§ IV.

The same method of reasoning applied to vapours enables us to establish a new relation between their latent caloric, their volume, and their pressure.

We have shown in the second paragraph how a liquid passing into the state of vapour may serve to transmit the caloric from a body maintained at a temperature $T$, to a body maintained at a lower temperature $t$, and how this transmission develops the motive force.

Let us suppose that the temperature of the body $B$ is lower by the infinitely small quantity $d\,t$ than the temperature of the body $A$. We have seen that if $c\,b$ (fig. 4.) represents the pressure of the vapour

Fig. 4.

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of the liquid corresponding to the temperature $t$ of the body $A$, and $f\,g$ that which corresponds to the temperature $t-d\,t$ of the body $B$; $b\,h$ the increase of volume due to the vapour formed in contact with the body $A$, $h\,k$ that which is due to the vapour formed after the body $A$ has been removed, the formation of which has reduced the temperature by the quantity $d\,t$; we have seen, I say, that the quantity of action developed by the transmission of the latent caloric furnished by the body $A$, [and transmitted] from that body to the body $B$, is measured by the quadrilateral figure $c\,d\,e\,f$. Now this surface is equal, if we neglect the infinitely small quantities of the second order, to the product of the volume $c\,d$ by the differential of the pressure $d\,h-e\,k$. Naming $p$ the pressure of the vapour of the liquid corresponding to the temperature $t$, $p$ will be a function of $t$, and we shall have $d\,h-e\,k={\frac {d\,p}{d\,t}}d\,t$. $c\,d$ will be equal to the increase of volume produced in water when it passes from the liquid into the gaseous state, under the pressure $p$, at a corresponding temperature. If we call $\rho$ the density of the liquid, $\delta$ that of the vapour, and $v$ the volume of the vapour formed, $\delta v$ will be its weight, and ${\frac {\delta v}{\rho }}$ will be the volume of the liquid evaporated. The increase of volume owing to the formation of a volume $v$ of vapour will therefore be

$v\left(1-{\frac {\delta }{\rho }}\right).$

The effect produced will therefore be

$\left(1-{\frac {\delta }{\rho }}\right)v{\frac {d\,p}{d\,t}}d\,t.$

The heat, by means of which this quantity of action has been produced, is the latent caloric of the volume $v$ of vapour formed; let $k$ be a function of $t$ representing the latent caloric contained in the unity of volume of the vapour furnished by the liquid subjected to experiment, at a temperature $t$, and under a corresponding pressure, the latent caloric of the volume $v$ will be $kv$, and the ratio of the effect produced to the heat expended will be expressed by

${\frac {\left(1-{\frac {\delta }{\rho }}\right){\frac {d\,p}{d\,t}}d\,t}{k}}.$

We have demonstrated that it is the greatest which can possibly be obtained; that it is independent of the nature of the liquid employed, and the same as that obtained by the employment of the permanent gases: now we have seen that this is expressed by ${\frac {d\,t}{C}}$, $C$ being a function of $t$ independent of the nature of the gases; we shall therefore also have

${\frac {\left(1-{\frac {\delta }{\rho }}\right){\frac {d\,p}{d\,t}}}{k}}={\frac {1}{C}},\quad {\mbox{whence}}\;k=\left(1-{\frac {\delta }{\rho }}\right){\frac {d\,p}{d\,t}}C.$

With regard to the generality of vapours, the ratio ${\frac {\delta }{\rho }}$ of the density of the vapour to that of the liquid from which it is formed may be neglected before it arrives at unity, so long as the temperature is not very high; we shall have therefore, sensibly,

$k=C{\frac {d\,p}{d\,t}}.$

This equation expresses that the latent caloric contained in equal volumes of the vapour of different liquids at the same temperature, and under the corresponding pressure, is proportional to the coefficient ${\frac {d\,p}{d\,t}}$ of the pressure with regard to the temperature.

Whence it results, that the latent caloric contained in the vapours of liquids which commence boiling only at high temperatures, as mercury for example, is very feeble, since for these vapours the quantity ${\frac {d\,p}{d\,t}}$ is very small.

We shall not insist upon the consequences which result from the equation

$k=\left(1-{\frac {\delta }{\rho }}\right){\frac {d\,p}{d\,t}}C.$

We shall simply remark that if, as everything leads us to believe, $C$ and ${\frac {d\,p}{d\,t}}$ do not become infinite for any value of the temperature, $k$ will become null when we have $\delta =\rho$, that is, that when the pressure is strong enough, and the temperature sufficiently elevated to render the density of the vapour equal to that of the liquid, the latent caloric is reduced to zero.

§ V.

Variation is produced in the volume of all the substances of nature by changes in the temperature and pressure to which they are subjected; liquids and solids are amenable to this law, and serve equally to develop the motive power of heat; it has been proposed to substitute them for the vapour of water, in order to render this motive force available; they have even occasionally been advantageously employed, particularly when it was necessary to develop a very considerable momentaneous effort, exerted within narrow limits.

In bodies of these kinds, as in the gases, it may be remarked, that of the four quantities, the volume $v$, the pressure $p$, the temperature $T$, and the absolute quantity of heat $Q$, two being determined, the others are deducible from them; if then we take two of them, $p$ and $v$ for example, as independent variables, the two others $T$ and $Q$ may be considered as functions of the former two.

In what manner the quantities $T$, $p$, and $v$, vary with respect to each other may be ascertained by direct experiments upon the elasticity and dilatability of bodies; it is thus that Mariotte's law relative to the elasticity of the gases, and Gay Lussac's relative to their dilatability, lead to the equation

$pv=R(267+t);$

all that remains is to determine $Q$ in functions of $p$ and $v$.

A relation exists between the functions $T$ and $Q$, which may be deduced from principles analogous to those which we have just established. Let us increase the temperature of the body by the infinitely small quantity $d\,T$, and at the same time prevent the increase of the volume; the pressure will then be augmented; if we represent the volume $v$ by the absciss $a\,b$ (fig. 5), and the primitive pressure by the ordinate $b\,d$, this

Fig. 5.

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augmentation of pressure may be represented by the quantity $d\,f$, which will be of the same order as the increase of temperature $d\,T$ to which it is owing, that is infinitely small.

Now we will take a source of heat $A$, maintained at the temperature $T+d\,T$, and allow the volume $v$ to increase by the quantity $b\,c$; the presence of the source $A$, maintained at the temperature $T+d\,T$, prevents the reduction of the temperature. During this contact, the quantity $Q$ of heat that the body possesses will increase by the quantity $d\,Q$, which will be derived from the source $A$. We will afterwards remove the source $A$, and the given body will become cool by the quantity $d\,T$, at the same time retaining the volume $a\,c$. The pressure will then diminish by the infinitely small quantity $g\,e$.

The temperature of the body being thus reduced to $T$, which is that of the source of heat $B$, we will take $B$, and reduce the volume of the body by a quantity $b\,c$, in such a manner that all the heat developed by the diminution of volume may be absorbed by the body $B$, and the temperature remain equal to its primitive value $T$. The volume $V$ also again becoming the same as it was at the commencement of the operation, it is certain that the pressure will return to its primitive value $b\,d$, as will also the absolute quantity of heat $Q$.

If we now connect the four points $f$, $g$, $e$, $d$ by right lines we shall form a quadrilateral figure, the area of which will measure the quantity of action developed during the operation described. Now it is easy to see that $f\,g$ and $d\,c$ are two elements infinitely near, described upon two curves infinitely near, the equations of which will be $T+d\,T=\mathrm {const.}$, and $T=\mathrm {const.}$ They ought therefore to be considered as parallel; the two ordinates which terminate the quadrilateral figure in the other direction being also parallel, the figure is parallelogrammical, and measures $b\,c\times d\,f$.

Now $f\,d$ is nothing but the increase experienced by the pressure $p$, the volume $v$ remaining constant, and $T$ becoming $T+d\,T$. We have therefore

$d\,f={\frac {d\,p}{d\,T}}d\,T,$

whence

$f\,d={\frac {1}{\frac {d\,T}{d\,p}}}d\,T.$

And $b\,c$ being the increase of volume $d\,v$

$f\,d\times b\,c={\frac {d\,v.\,d\,T}{\frac {d\,T}{d\,p}}}.$

It only remains to determine the heat consumed in the production of this quantity of mechanical action.

We have first raised the temperature of the body subjected to experiment by the quantity $d\,T$ without changing its primitive volume $v$; afterwards, when it had become $v+d\,v$, we have lowered its temperature by the same quantity $d\,T$ without varying its primitive volume $v+d\,v$. Now it may easily be seen that this double operation can be effected without loss of heat; let us suppose that $n$ being a number indefinitely great, the interval of temperature $d\,T$ be divided into a number $n$ of new intervals ${\frac {d\,T}{n}}$, and that we have $n+1$ sources of heat maintained at the temperatures $T$, $T+{\frac {d\,T}{n}}$, $T+{\frac {2d\,T}{n}}$,........ $T+{\frac {(n-1)d\,T}{n}}$ and $T+d\,T$.

To raise the temperature of the body upon which we are operating from $T$ to $T+d\,T$, we bring it successively into contact with the second, the third, and the $(n+1)$th of these sources, until it has acquired the temperature of each of them. When, on the contrary, the volume $v$ of the body being increased by $d\,v$, we wish to give it the temperature $T$, we bring it successively into contact with the $n$th, the $(n-1)$th, and the first of these sources, until it has acquired the temperature of each of them. We then return to these sources the heat that has been borrowed from them in the first part of the operation; for it is not necessary to attend to the differences of an order of inferior magnitude, arising from changes that may have been produced in the specific caloric of the body, in consequence of the variations of $v$ and $Q$.

Nothing therefore will have been lost or gained by any of these sources, excepting always the source of which the temperature is $T+d\,T$, which will have lost the heat necessary to elevate the temperature of the body upon which we are operating from $T+{\frac {(n-1)\;d\,T}{n}}$ to $T+d\,T$, and the source maintained at the temperature $T$, which will have acquired the heat necessary to reduce the temperature of the same body from $T+{\frac {d\,T}{n}}$ to $T$. If we suppose $n$ to be infinitely great, these quantities of heat may be neglected.

We see, therefore, that when the body in question, (its temperature being thus reduced to $T$,) is brought into contact with the source of heat $B$, the heat communicated to it from the source $A$ will be all it has gained from the commencement of the operation. In consequence of the reduction of its volume in contact with the body $B$, it will be found at its original volume and temperature; the quantities $Q$ and $P$ will therefore have re-assumed their primitive value; it is therefore certain that all the heat borrowed from the source $A$, and nothing but that heat, will have been given to the body $B$.

Whence it results that the effect produced,

${\frac {d\,v\;d\,T}{\frac {d\,T}{d\,p}}}$

is owing to the transmission of the heat absorbed by the body subjected to the experiment during its contact with the source $A$, and which has afterwards flowed into the source $B$.

The temperature having remained constant during the contact with the source $A$, it follows that the variations $d\,p$ and $d\,v$ of the pressure and the volume are connected by the relation

${\frac {d\,T}{d\,p}}d\,p+{\frac {d\,T}{d\,v}}d\,v=0$

These variations $d\,p$ and $d\,v$ occasion a variation in the absolute quantity of heat $Q$, the expression of which is

$d\,Q={\frac {d\,Q}{d\,p}}d\,p+{\frac {d\,Q}{d\,v}}d\,v=d\,v\left[{\frac {d\,Q}{d\,v}}-{\frac {d\,Q}{d\,p}}{\frac {\left({\frac {d\,T}{d\,v}}\right)}{\left({\frac {d\,T}{d\,p}}\right)}}\right];$

such is the quantity of heat consumed in the production of the effect that we have just calculated. The effect produced by a quantity of heat equal to unity will therefore be

${\frac {d\,T}{{\frac {d\,Q}{d\,v}}.\quad {\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}.\quad {\frac {d\,T}{d\,v}}.}}$

It will be shown, as in the case of the gases, that this effect produced, is the largest which it is possible to realize; and as all the substances of nature may be employed, in the manner that has just been indicated, to produce this maximum effect, it is necessarily the same for all.

When this theory has been applied specially to the gases, we have called ${\frac {1}{C}}$ the coefficient of $d\,T$ in the expression of this maximum quantity of action; the equation therefore of all the substances of nature, solid, liquid, or gaseous, will be

${\frac {d\,Q}{d\,v}}\centerdot {\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}\centerdot {\frac {d\,T}{d\,v}}=C$

in which $C$ is a function of the temperature which is the same for all.

For the gases we have

$T=-267+{\frac {1}{R}}pv,$

whence we deduce

${\frac {d\,T}{d\,p}}={\frac {v}{R}}\quad {\mbox{and}}\;\;{\frac {d\,T}{d\,v}}={\frac {p}{R}}.$

The preceding equation applied to the gases takes therefore the form

$v{\frac {d\,Q}{d\,v}}-p{\frac {d\,Q}{d\,p}}=RC=F(p,v):$

it is the equation at which we have already arrived, and of which the integral is

$Q=R(B-C\log p);$

that of the general equation

${\frac {d\,Q}{d\,v}}{\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}{\frac {d\,T}{d\,v}}=C$

is of the form

$Q=F(T)-C\phi (p,v);$

$F(T)$ is an arbitrary function of the temperature, and $\phi (p,v)$ a particular function satisfying the equation

${\frac {d\,T}{d\,v}}{\frac {d\,\phi }{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi }{d\,v}}=1$

(See the note appended to this Memoir.)

We shall now deduce various consequences from the general equation at which we have arrived.

We have previously seen that when we compress a body by the quantity $d\,v$, the temperature remaining constant, the heat disengaged by the condensation is equal to

$d\,Q=d\,v\left[{\frac {d\,Q}{d\,v}}-{\frac {d\,Q}{d\,p}}{\frac {\left({\frac {d\,T}{d\,v}}\right)}{\left({\frac {d\,T}{d\,p}}\right)}}\right];$

and as

${\frac {d\,Q}{d\,v}}{\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}{\frac {d\,T}{d\,v}}=C,$

the preceding expression takes the form

$d\,Q=d\,v{\frac {C}{\left({\frac {d\,T}{d\,p}}\right)}}=-d\,p{\frac {C}{\left({\frac {d\,T}{d\,v}}\right)}}$

This last equation may be put under the form

$d\,Q=-d\,p\;C{\frac {d\,v}{d\,T}};$

${\frac {d\,v}{d\,T}}$ is the differential coefficient of the volume with regard to the temperature, the pressure remaining constant.

We thus arrive at this general law, which is applicable to all the substances of nature, solid, liquid, or gaseous: If the pressure supported by different bodies, taken at the same temperature, be augmented by a small quantity, quantities of heat will be disengaged from it, which will be proportional to their dilatability by heat.

This result is the most general consequence deducible from this axiom: that it is absurd to suppose that motive force or heat can be created gratuitously and absolutely.

§ VI.

The function of the temperature $C$ is, as we see, of great importance in consequence of the part it sustains in the theory of heat: it enters into the expression of the latent caloric which is contained in all substances, and which is disengaged from them by pressure. Unfortunately no experiments have been made from which we can determine the values of this function, corresponding to all the values of the temperature. To obtain $t=0$ we must proceed in the following manner.

M. Dulong has shown that the air, and all the other gases taken at the temperature of 0°, and under the pressure $0^{m}\centerdot 76$ of mercury, when compressed by ${\frac {1}{267}}$ of their volume, disengage a quantity of heat, capable of elevating the same volume of atmospheric air by $0.421$.

Suppose that we operate upon a kilogramme of air occupying a volume $v=0.770$ of a cubic metre, under the pressure of the atmosphere $p$, equivalent to $10230$ kilogrammes upon a square metre; we have

$pv=R(267+t),$

and

$Q=R(B-C\log p).$

If a variation be suddenly effected in $v$ by an infinitely small quantity $d\,v$, without there being any variation in the absolute quantity of heat $Q$, we shall have

$p\;d\,v+v\;d\,p=R\;d\,t,$

and

$0=R\left({\frac {d\,B}{d\,t}}-\log p{\frac {d\,C}{d\,t}}\right)-RC{\frac {d\,p}{p}},$

or preferably

${\frac {d\,t}{C}}R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right)=R{\frac {d\,p}{p}}={\frac {R}{p}}\left({\frac {R\;d\,t-p\;d\,v}{v}}\right)={\frac {R\;d\,t-p\;d\,v}{267+t}}.$

Now $R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right)$ being the partial differential of $Q$ in respect of $t$, $p$ remaining constant, is nothing else than the specific caloric of the air at a constant pressure; it is the number of unities of heat necessary to elevate a kilogramme of air under atmospheric pressure by one degree; we have therefore

$R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right)=0.267.$

Then substituting $-{\frac {v}{267}}$ for $d\,v$, and $0.421$ for $d\,t$, we arrive lastly at

${\frac {1}{C}}=1.41.$

{{{2}}}

This is the maximum effect producible by a quantity of heat, equal to that which would elevate by 1° a kilogramme of water taken at zero, passing from a body maintained at 1° to a body maintained at 0°. It is expressed in kilogrammes raised one metre high.

Having the value of $C$, which corresponds to $t=0$, it is interesting to know, setting out from this point, whether $C$ increases or decreases, and in what proportion. An experiment of MM. De Laroche and Bérard upon the variations experienced by the specific caloric of the air, when the pressure is varied, enables us to calculate the value of the differential coefficient ${\frac {d\,C}{d\,t}}$.

In fact, according to our formulas, the specific caloric of the air under two pressures $p$ and $p'$ differs by $R{\frac {d\,C}{d\,t}}-\log {\frac {p}{p'}}$; rendering this quantity equal to the difference of the specific calorics, as it has been deduced from the results of MM. De Laroche and Bérard; taking the mean of two experiments, we find

${\frac {d\,C}{d\,t}}=0.002565.$

In these experiments the air entered into the calorimeter at the temperature of $96^{\circ }.90$, and quitted it at that of $22^{\circ }.83$; $0.002565$ is therefore the mean value of the differential coefficient ${\frac {d\,C}{d\,t}}$ between these two temperatures.

From this result we learn, that between these two limits the function $C$ increases, though very slowly; consequently the quantity ${\frac {1}{C}}$ diminishes; whence it follows that the effect produced by the heat diminishes at high temperatures, though very slowly.

The theory of vapours will furnish us with new values of the function $C$ at other temperatures. Let us return to the formula

${\frac {1}{C}}={\frac {\left(1-{\frac {\delta }{\rho }}\right){\frac {d\,p}{d\,t}}}{k}},$

which we have demonstrated in paragraph IV. If we neglect the density of the vapour before that of the fluid, this formula will be reduced to

${\frac {1}{C}}={\frac {\frac {d\,p}{d\,t}}{k}}.$

We may remark in passing, that at the temperature of ebullition ${\frac {d\,p}{d\,t}}$ is nearly the same for all vapours; $C$ itself varies little with the temperature, so that $k$ is nearly constant. This explains the observations of certain philosophers, who have remarked that at the boiling point, equal volumes of all vapours contain the same quantity of latent caloric; but we see at the same time that we are only approximating to this law, since it supposes that $C$ and ${\frac {d\,p}{d\,t}}$ are the same for all vapours at the boiling point.

From the experiments made by several philosophers we are enabled to calculate the values corresponding to the boiling point of $k$ and ${\frac {p}{d\,t}}$, for different liquids; we can therefore deduce from them the corresponding values of ${\frac {1}{C}}$.

NAMES of Liquids.	Value in atmospheres of ${\frac {d\,p}{d\,t}}$ at the temperature of ebullition.	Density of the vapour at the temp. of ebullition, the density of the air being 1.	Quantity of latent heat contained in a kilogramme of vapour.	Temperature of ebullition.	Corresponding values of ${\frac {1}{C}}$
Sulphuric Ether	${\frac {1}{28.12}}$	$2.280$	$90.8$	$35.5$	$1.365$
Alcohol	${\frac {1}{25.19}}$	$1.258$	$207.7$	$78.8$	$1.208$
Water	${\frac {1}{29.1}}$	$0.451$	$543.0$	$100.$	$1.115$
Essence of Turpentine	${\frac {1}{30}}$	$3.207$	$76.8$	$156.8$	$1.076$

These results confirm, in a striking manner, the theory that we are explaining; they show that $C$ is slowly augmented with the temperature, as has been already stated: we have seen that for $t=o$, ${\frac {1}{C}}=1.41$, whence $C=0.7092$; this result is deduced from experiments upon the velocity of sound.

We here find, starting from experiments upon the vapour of water, for $t=100^{\circ }$, ${\frac {1}{C}}=1.115$, whence $C=0.8969$; $C$ is therefore increased from 0 at 100° to $0.187$, which gives as the mean of the differential coefficient between these two limits

${\frac {d\,C}{d\,t}}=0.00187.$

The mean of the two experiments performed by MM. De Laroche and Bérard gives us, between the limits $22^{\circ }\centerdot 83$ and $96\centerdot 90$, for the mean value of ${\frac {d\,C}{d\,t}}$ the quantity $0.002565$.

These two results differ little from each other, and their divergence will be sufficiently explained, by reflecting on the number and the variety of the experiments whence the data on which they are founded are derived.

There is another means of calculating the values of ${\frac {1}{C}}$, between extended limits of the temperature in an approximative manner; for this it is necessary to admit, that the quantity of caloric contained in a given weight of the vapour of water is the same whatever be the temperature and the corresponding pressure; and still further, that the laws relative to the compression and the dilatation of the permanent gases are equally applicable to vapours: adopting these laws, towards which we have only approximated, the formula

${\frac {1}{C}}={\frac {\frac {d\,p}{d\,t}}{K}}$

will express $K$ in function of $t$; ${\frac {d\,p}{d\,t}}$ may be deduced from 0 to 100° from experiments long since made by several philosophers, and from 100° to 224° from recent experiments of MM. Arago and Dulong.

Thus we find for			Values of ${\frac {1}{C}}.$
$t=0\centerdot$	${\frac {1}{C}}=1\centerdot 586$		$1\centerdot 410$
$t=35\centerdot 5$	${\frac {1}{C}}=1\centerdot 292$	We have already found for the values of ${\frac {1}{C}}$ corresponding to the same values of $t$.	$1\centerdot 365$
$t=78\centerdot 8$	${\frac {1}{C}}=1\centerdot 142$		$1\centerdot 208$
$t=100\centerdot$	${\frac {1}{C}}=1\centerdot 102$		$1\centerdot 106$
$t=156\centerdot 8$	${\frac {1}{C}}=1\centerdot 072$		$1\centerdot 078$

These last, deduced from experiments upon sound, the vapours of sulphuric ether, alcohol, water, and essence of turpentine, accord with the first in a satisfactory manner.

These remarkable coincidences, obtained by numerical operations performed upon a great variety of data, furnished by phænomena of many different kinds, appear to us to add greatly to the evidence of our theory.

§ VII.

The function $C$ is, as we have seen, of great importance: it is the connecting link of all the phænomena produced by heat upon solid, liquid, or gaseous bodies. It is greatly to be desired that experiments of the most precise exactitude, such as the researches upon the propagation of sound in gases taken at different temperatures, were instituted, in order to establish this function with all the requisite certainty. It would conduce to the determination of several other important elements of the theory of heat, with regard to which we know nothing, or have arrived by our experiments at very insufficient approximations only. In this number may be included the heat disengaged by the compression of solid or liquid bodies; the theory that we have enunciated enables us to determine it numerically for all the values of the temperature for which the function $C$ is known in a manner sufficiently exact, that is to say, from $t=0$ to $t=224^{\circ }$.

We have seen that the heat disengaged by the augmentation of pressure $d\,p$ is equal to the dilatation by the heat of the body subjected to experiment, multiplied by $C$. With regard to the air taken at zero, the quantity of heat disengaged may be directly deduced from the experiments upon sound in the following manner.

M. Dulong has shown that a compression of ${\frac {1}{267}}$ raises the temperature of a volume of air taken at zero by $0^{\circ }\centerdot 421$. Now the $0.267$ unity of heat necessary to elevate a kilogramme of air taken at zero under a constant pressure by 1°, are equal to the heat necessary to maintain the temperature of the gas dilated by ${\frac {1}{267}}$ of its volume at zero, above the heat necessary to elevate the dilated volume, maintained constant, by 1°; the last is equal to ${\frac {1}{0.421}}$ of the first; their sum is therefore equal to the first multiplied by $1+{\frac {0}{0.421}}$; the former therefore, that is the heat necessary to maintain the temperature of 1 kil. of air, dilated by ${\frac {1}{267}}$ of its volume, at zero, is equal to $(0.267):\left(1+{\frac {1}{0.421}}\right)$, or to $0.07911$.

We arrive at the same results by the application of the formula

$Q=R(B-C\log p),$

whence

$d\,Q=RC{\frac {d\,v}{v}},$

putting $C={\frac {1}{410}}$, and observing that a diminution of volume of ${\frac {1}{267}}$ corresponds to an increase of pressure equal to ${\frac {1}{267}}$ of an atmosphere.

Knowing the quantity of heat disengaged from gases by compression, to ascertain that which a similar pressure would disengage from any substance whatever, from iron for example, we write the proportion: $0.07911$ of heat disengaged by a volume of air equal to $0.77$ of a cubic metre, subjected to an increase of pressure equal to ${\frac {1}{267}}$ of an atmosphere, is to that disengaged from the same volume of iron under the same circumstances as $0\centerdot 00375$, the cubic dilatability of the air, is to $0\centerdot 00003663$, the cubic dilatability of iron. For the second term of the proportion we find the number $0\centerdot 0007718$. Now a volume of $0^{m}\centerdot 77$ of iron weighs $5996$ kilogrammes; the heat disengaged by one kilogramme will therefore be ${\frac {0\centerdot 007718}{5996}}$; for the pressure of an atmosphere it will be $267$ times more considerable, or equal to $0\centerdot 00003436$; the division of this number by the specific caloric of the iron referred to that of water gives the quantity of the elevation of the temperature of the iron by the pressure of an atmosphere; it is, we see, too feeble to be appreciated by our thermometrical instruments.

§ VIII.

We shall not further insist upon the consequences to the theory of heat of the results enunciated in this Memoir; but it may, perhaps, be useful to add a few words upon the employment of heat as a motive force. M. S. Carnot, in the work already cited, appears to have established the true basis of this important part of practical mechanics.

High or low pressure engines without detent (détente) bring into use the vis viva developed by the caloric contained in the vapour, in its passage from the temperature of the boiler to that of the condenser; the high pressure engines without condenser act as if provided with a condenser at a temperature of 100°. In the latter, therefore, all that is brought into use is the passage of the latent caloric contained in the vapour, from the temperature of the boiler to the temperature 100°. As to the sensible caloric of the vapour, it is entirely lost in all the engines without detent.

The sensible caloric is in part utilized in the engines with detent, in which the temperature of the vapour is allowed to sink: the cylindrical envelope, the use of which in Woulf's engines with two cylinders is to maintain the vapour at a constant temperature, though very useful to diminish the limits of the variation of the motive force acting upon the pistons, has an injurious influence as to the quantity of effect produced compared to the consumption of fuel.

To render useful all the motive force at our disposal, the detent should be continued until the temperature of the vapour be reduced to that of the condenser; but practical considerations, suggested by the manner in which the motive force of fire is employed in the arts, prevent the attainment of this limit.

We have elsewhere shown that the employment of gases, or of any other liquid than water, between the same limits of temperature, could add nothing to the results already obtained; but from the preceding considerations it follows, that the temperature of the fire being from 1000° to 2000° higher than that of the boilers, there is an enormous loss of vis viva in the passage of the heat from the furnace into the boiler. It is therefore only from the employment of caloric at high temperatures, and from the discovery of agents proper to realize its motive force, that important improvements may be expected in the art of utilizing the mechanical power of heat.

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NOTE.

The integral of the general equation

${\frac {d\,Q}{d\,v}}\centerdot {\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}{\frac {d\,T}{d\,v}}=C,$

is, as we have seen,

$Q=F(T)-C\phi (p,z)$

(1)

$F(T)$ is an arbitrary function of the temperature $T$, varying from one body to another; $C$ is a function of the temperature which is the same for all the substances of nature, and $\phi (p,v)$ is a particular function of $p$, and of $v$ satisfying the equation

${\frac {d\,T}{d\,v}}\centerdot {\frac {d\,\phi }{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi }{d\,v}}=1$

(2)

This function $\phi$ may be determined in the following manner. Let

$\phi =\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}+\phi ',$

be substituted in the equation (2), it will be

${\frac {d\,T}{d\,v}}{\frac {d\,\phi '}{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi '}{d\,v}}={\frac {d\,T}{d\,p}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}};$

this equation is satisfied by putting

$\phi '=\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}+\phi '';$

$\phi ''$ satisfying the equation

${\frac {d\,T}{d\,v}}{\frac {d\,\phi ''}{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi ''}{d\,v}}={\frac {d\,T}{d\,p}}{\frac {d}{d\,v}}\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}},$

we shall have equally

$\phi ''=\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}+\phi '''.$

We thus see that $\phi (pv)$ is given by a series of terms, each of which is obtained by means of the preceding one, by differentiating it in respect to $v$, multiplying by the ratio ${\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}$, and integrating the result in respect of $p$. The first term of this series being $\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}$, it is evident that the value of $\phi$ may be easily obtained; substituting this value in the equation (1), we have for the expression of the general integral of the partial differential equation

${\frac {d\,Q}{d\,v}}{\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}{\frac {d\,T}{d\,v}}=C$

the formula

$Q=F(T)-C\left|{\begin{aligned}&\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}\\&+\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}\\&+\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right.$

This equation gives the law of the specific calorics, and of the heat disengaged by the variations of the volume and of the pressure of all the substances of nature, when the relation which exists between the temperature, the volume, and the pressure is known.