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The second law of thermodynamics; memoirs by Carnot, Clausius, and Thomson. Translated and edited by W. F. Magie
Magie, William Francis, 1858-1943. New York : Harper, 1899. https://hdl.handle.net/2027/uc2.ark:/13960/t3zs2mg92
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" 2 7 1997

HARPER'S SCIENTIFIC MEMOIRS
EDITED BY
J. S. AMES, PH.D.
PROFESSOORF PHYSICSIN JOHNSHOPKINSUNIVERSITY
VI. THE SECOND LAW OF THERMODYNAMICS
SCIENCE & ENGINEERING LIBRARY
AUG 2 7 1997 PHYSICS COLLECTION
UCLA

THE SECOND LAW OF THERMODYNAMICS
MEMOIRS BY CARNOT, CLAUSIUS AND THOMSON
TRANSLATED AND EDITED
BY W. F. MAGIE, FH D.
PROFESSOORF PHYSICSIN PRINCKTOUNNIVERSITY

HARPER

NEW YORK AND LONDON
& BROTHERS PUBLISHERS
1899

HARPER'S SCIENTIFIC MEMOIRS. XDITKOBT
J. S. AMES, PH.D., i-BoruwokopriiTsioaIN JOHNSHOPKINVSNIVKEBITY.
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SICWYORKAMI LONDON: HARPER A BROTHERS,PUBLISHERS.
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PREFACE

AFTER the invention of the steam-engine in its present form by James Watt, the attention of engineers and of scientific men was directed to the problem of its further improvement. With this end in view, the young Sadi Carnot, in 1824, pub-

lished the Reflexions sur la Puissance Motrice du Feu, of which

this translation is given in this volume. In* this really great

memoir, Carnot examined the relations between heat and the

work done by heat used in an ideal engine, and by reducing

the problem to its simplest form and avoiding all special questions relating to details, he succeeded in establishing the condi-

tions upon which the economical working of all heat-engines depends. It is not necessary here to animadvert upon the use

made by Carnot of the substantial theory of heat, and the con-

sequent failure of the proof of his main proposition when the true nature of heat was appreciated. It is sufficient to say that though the proof was invalid, the proposition remained true, and carried with it the truth of such of Carnot's deduc-

tions as were based solely upon it.

Carnot's memoir remained for a long time unappreciated, and it was not until use was made of it by William Thomson
(now Lord Kelvin), in 1848, to establish an absolute scale of temperature, that the merits of the method proposed in it were recognized. In his first paper on this subject Thomson retained the substantial theory of heat, but the evidence in favor

of the mechanical theory became so strong that he soon after adopted the new view. Applying it to the questions treated

by Carnot, he found be proved by denying

that Carnot's the possibility

porfop"osthiteionpercpoeutuldal

no longer motion,"

and was led to lay down a second fundamental principle to serve

in the demonstration. This principle is now called the Second

Law of Thermodynamics.

A part of the memoir in which this

PREFACE

principle is stated :ml many of its consequences developed is given in this volume. It was publislu-d in March. is.M.
In the previous year Clansius published a discussion of the
same question as that treated by Thomson, in which he lays

down a principle for use in the demonstration of Cairnot's propositton. which, while not the same in form as Thomson's, is the

same in content, and ranks as another statement of the Second

Law of Thermodynamics.

\\\^ paper is also given in this vol-

ume. While not so powerful or so inclusive as Thomson's, it

deserves attention for the clearness and simplicity of its form.

Clausing followed up this paper by others, and subsequently

published a book in which the subject of Thermodynami--Lriven a systematic treatment, and in which he introduced and developed the important function called by him the entropy.
The science of Thermodynamics, founded by the labors of
these three illustrious men, has led to the most important de-

velopments in all departments of physical science. It lia-

pointed out relations among the properties of bodies which

could scarcely have been anticipated in any other way : it halaid the foundation for the Science of Chemical Physics ; and.

taken in connection with the kinetic theory of gases, as developed by Maxwell and Boltzmann, it has furnished a funeral view of the operations of the universe which is far in advance

of any that could have been reached by purely dynamical rea-

soning.

GENERAL CONTENTS

PAGE

Preface

v

Reflections on the Motive Power of Heat. By Sadi Carnot

3

Biographical Sketch of Carnot

60

Ou the Motive Power of Heat, and on the Laws which can be De-

duced from it for the Theory of Heat. By R. Clausius

65

Biographical Sketch of Clausius

107

Ill The Dynamical Theory of Heat. (Selected Portions.) By William Thomson (Lord Kelvin)

Biographical Sketch of Lord Kelvin

147

Bibliography

149

INDEX..

.. 151

REFLECTIONS ON THE MOTIVE POWER OF HEAT
BY
SADI CAENOT
Paris, 1824

CONTENTS
Ueat-enginet JbB of Temperature ReversibleProcesses "Carnot's Cycle" Efficiencya Function of Limiting Temperatures SpecificHeatsof Gate* Motive Ptoeerof Air, Steam,Alcohol Vapor GreatestEfficiency I ;,rinvt Typesof Machines Advantagesand Disadvantageosf Steam

PAGI 8 7 11 10 -2(\
21 40 49 62
55

REFLECTIONS ON THE MOTIVE POWER OF HEAT AND
ON ENGINES SUITABLE FOR DEVELOPING THIS POWER
BY
SADI CARNOT
IT is well known that heat may be nsed as a cause of motion, and that the motive power which may be obtained from it is very great. The steam-engine, now in such general use, is a manifest proof of this fact.
To the agency of heat may be ascribed those vast disturbances which we see occurring everywhere on the earth ; the movements of the atmosphere, the rising of mists, the fall of rain and other meteors,* the streams of water which channel the surface of the earth, of which man has succeeded in utilizing only a small part. To heat are due also volcanic eruptions and earthquakes. From this great source we draw the moving force necessary for our use. Nature, by supplying combustible material everywhere, has afforded us the means of generating heat and the motive power which is given by it, at all times and in all places, and the steam-engine has made it possible to develop and use this power.
The study of the steam-engine is of the highest interest, owing to its importance, its constantly increasing use, and the great changes it is destined to make in the civilized world. It has already developed mines, propelled ships, and dredged rivers and harbors. It forges iron, saws wood, grinds grain, spins and weaves stuffs, and transports the heaviest loads. In the future it will most probably be the universal motor, and
* [Any atmospJiericphenomenonwasformerly called a meteor.]

MKMuIKS ON

will furnish the power now obtained from animals, from water-

falls, and from air-currents.

Over the first of these motors it

has the advantage of economy, and over the other two the in-

calculable and that

itas dwvaonrktagneeetdhantevitercabne bienteursreudpteevde. rywhIefrein

and the

always. future

the steam-engine is so perfected as to render it less costly to

construct it and to supply it with fuel, it will unite all desir-

able qualities and will promote the development of the industrial arts to an extent which it is difficult to foresee. It is, in-

deed, not only a powerful and convenient motor, which can be set up or transported anywhere, and substituted for other motors already in use, but it lends to the rapid extension of those arts in which it is used, and it can even create arts hith-

erto unknown.

The most signal service which has been rendered to England by the steam-engine is that of having revived the working of

her coal-mines, which had languished and was threatened with extinction on account of the increasing ditlirulty of excavation

and extraction of the coal.* We may place in the second rank

the services rendered in the manufacture of iron, as much by furnishing an abundant supply of coal, which took the place of

it is it
is it l>y
a is

wood as the wood began to be exhan-te.l. a-

the powerful

machines of all kinds tho use of which

either facilitated or

made possible.

Iron and fire, as every one knows, are the mainstays of the

mechanical arts. Perhaps there not in all England single

industry whose existence not dependent <>nthese agents, and
which does not use them extensively. If England were to-day

to lose its steam-engines

would lose also its coal and iron,

and this loss would dry up all its sources of wealth and destroy

its prosperity

;

it

would annihilate this colossal power. The

destruction of its navy, which considers its strongest support,

would be, perhaps, less fatal. The safe and rapid navigation by means of steamships an

f

i> 4!
\ n

One may wifely say that the mining of coal has increasedtenfold since the Jnv.-nti. tho steamengine. Tin* mining of copper, <>tfin. mid <>f iron has increasedalmost as much. The effect produced half century ago in the mines of Knd.ind imw IH-III^r< -utedin the irnld and -liver mines of the New W<H|,|,the working of which was steadily declining. principally on account of the insulli. -ncy of the motors used for the excavation and extraction of the minerals.

THE SECOND LAW OF THERMODYNAMICS
entirely new art due to the steam-engine. This art has already made possible the establishment of prompt and regular communication on the arms of the sea, and on the great rivers of the old and new continents. By means of the steam-engine regions still savage have been traversed which but a short time ago could hardly have been penetrated. The products of civilization have been taken to all parts of the earth, which they would otherwise not have reached for many years. The navigation due to the steam-engine has in a measure drawn together the most distant nations. It tends to unite the peoples of the earth as if they all lived in the same country. In fact, to diminish the duration, the fatigue, the uncertainty and danger of voyages is to lessen their length.*
The discovery of the steam-engine, like most human inventions, owes its birth to crude attempts which have been attributed to various persons and of which the real author is not known. The principal discovery consists indeed less in these first trials than in the successive improvements which have brought it to its present perfection. There is almost as great a difference between the first structures where expansive force was developed and the actual steam-engine as there is between the first raft ever constructed and a man-of-war.
If the honor of a discovery belongs to the nation where it
acquired all its development and improvement, this honor cannot in this case be withheld from England : Savery, Newcomen, Smeaton, the celebrated Watt, Woolf, Trevithick, and other English engineers, are the real inventors of the steam-engine.
At their hands it received each successive improvement. It is
natural that an invention should be made, improved, and perfected where the need of it is most strongly felt.
In spite of labor of all sorts expended on the steam-engine, and in spite of the perfection to which it has been brought, its theory is very little advanced, and the attempts to better this state of affairs have thus far been directed almost at
random. The question has often been raised whether the motive power
* We speak of diminishing the danger of voyages ; in fact, though the use of the steam-enginein ships is attendedwith some dangers,theseare always exaggeratedand are compensatedfor by the ability of ships to keep a definite course,and to resist winds which would otherwise drive the vessel on the coast, or on shoals or reefs.
5

is f
; it

MEMOIRS OX

of heat is limited or not ;* whether there is a limit to the pos-

sible improvements of the steam-engine which, in the nature of the case, cannot be passed by any means ; or if, on the other

hand, these improvements are capable of indefinite extension. Inventors have tried for a long time, and are still trying, to

find whether there is not a more efficient agent than water Ky

which to develop the motive power of heat; whether, for e\-

ample, atmospheric air does not offer great advantages in this

respect. We propose to submit these questions to a critical

examination.

The phenomenon of the production of motion by heat hus not been considered in a sufficiently general way. It has leen

treated only in connection with machines whoso nature ami

mode of action do not admit of a full investigation of it. In

is,

such machines the phenomenon

a

in measure, imperfect and

incomplete

thus becomes difficult to recognize its principles

t

and study its laws. To examine the principle of lie production

of motion by heat in all its generality,

it

must be conceive.! in-

; it

dependently of any mechanism or of any particular agent

necessary to establish proofs applicable not only to steam-

engines but to all other heat-engines, irrespective of the work-

by it by
by

ing substance and the manner in which

acts.

The machines which are not worked

heat for instance,

those worked by men or animals, by water-falls, or air cur-

rents can be studied to their last details

the principles of

mechanics. All possible cases may be anticipated, all imagi-

nable actions are subject to general principles already well estahlished ;md applicable in all circumstances. The theory of

is a a it is

snch machines complete. Such theory evidently lacking

for heat-engines. We shall never possess until the laws of

physics are so extended and generalixed as to make known in

advance all the effects of heat acting in definite \\-.\\ on any

body whatsoever.

a 6
a

* The expression motive power here signifies the useful cfTcci that a

in

motor capable of producing. This effect may always he nnnsiin.1 in

terms of UP'. as

Ibe elevationof well known,

weight through by the product of

certain distance the weight ami the

; it is

In ijuii>i .1to-

is it

f I.

which

it

raised.

We distinguish hen- betweenthe stcnm-cngincand the heat-enginein

general.which can be worked by nny agent, and uot by water vapor only,

to realize the motive power of heat.

THE SECOND LAW OF THERMODYNAMICS
We shall take for granted in what follows a knowledge, at least a superficial one, of the various parts which compose an ordinary steam-engine. "We think it unnecessary to describe the fire-box, the boiler, the steam -chest, the piston, the condenser, etc.
The production of motion in the steam-engine is always accompanied by a circumstance which we should particularly notice. This circumstance is the re-establishment of equilibrium in the caloric* that is, its passage from one body where the temperature is more or less elevated to another where it is lower. What happens, in fact, in a steam-engine at work? The caloric developed in the fire-box as an effect of combustion passes through the wall of the boiler and produces steam, incorporating itself with the steam in some way. This steam, carrying the caloric with it, transports it first into the cylinder, where it fulfils some function, and thence into the condenser, where the steam is precipitated by coming in contact with cold water. As a last result the cold water in the condenser receives the caloric developed by combustion. It is warmed by means of the steam, as if it had been placed directly on the fire-box. The steam is here only a means of transporting caloric ; it thus fulfils the same office as in the heating of baths by steam, with the exception that in the case in hand its motion is rendered useful.
We can easily perceive, in the operation which we have just described, the re- establishment of equilibrium in the caloric and its passage from a hotter to a colder body. The first of these bodies is the heated air of the fire-box ; the second, the water of condensation. The re-establishment of equilibrium of the caloric is accomplished between them if not completely, at least in part ; for, on the one hand, the heated air after having done its work escapes through the smoke-stack at a much lower temperature than that which it had acquired by the combustion ; and, on the other hand, the water of the condenser, after having precipitated the steam, leaves the engine with a higher temperature than that which it had when it entered.
The production of motive power in the steam-engine is
* [Caloric u lieat consideredas an indestructiblesubstance. T/ie word is usedby Carnot interchangeablywith fen, fire, or heat.]
7

MKMOIRS ON
therefore not due to a real consumption of the caloric, but t //> transfer from a hotter /<>a n>1tli>frault/ that is to say, to tin- ivestablishment of its equilibrium, which is assumed to have been destroyed by a chemical action such as combustion, or by some other cause. We shall soon see that this principle is applicable to all engines operated by heat.
According to this principle, to obtain motive power it is not enough to produce heat ; it is also necessary to provide cold, without which the heat would be useless. For if there existed only bodies as warm as our furnaces, how would the condensation of steam be possible, and where could it be sent if it were once produced? It cannot be replied that it could be ejected into the atmosphere, as is done with certain engines,* since the atmosphere would not receive it. In the actual state of things the atmosphere acts as a vast condenser for the steam. because it is at a lower temperature ; otherwise it would soon be saturated, or, rather, would be saturated in advance.f
Everywhere where there is a difference of temperature, and where the re-establishment of equilibrium of the caloric can lie effected, the production of motive power is possible. Water vapor is one agent for obtaining this power, but it is not the only one ; all natural bodies can be applied to this purpose, for they are all susceptible to changes of volume, to successive contractions and dilatations effected by alternations of heat and cold ; they are all capable, by this change of volume, of overcoming resistances and thus of developing motive power. A
* Some high -pressureengines eject vapor into the atmosphere of condensing it. They are used mostly in places where it is difficult to procure a current of cold water sutn.-i.-ni t ctTcctcondensation.
t The existenceof water in a liquid state, which is here necessarilyassumed, since without it the steamengine could not be supplied, presupposesthe existenceof a pressurecapableof preventing it from evaporating, and consequentlyof a pressureequal to or greater than the tension of the
vapor at the temperatureof the water. If such a pressure were not ex-
erted by the atmospherea quantity of water vapor would instantly be produced sufficient to exert this pressure on itself, and this pressuremust always be overcomein ejecting the steam of the engine into the new atmosphere. This is evidently equivalent to overcoming the ton-inn whi.-ii is exertedby the vapor after it has beencondensedby the ordinary means.
If a very high temperaturewere to prevail at the surface of theearth, as
it almost certainly does in its interior, all the water of the oceanswould exist in the form of vapor in the atmosphere,and there would be no water in a liquid state.
8

THE SECOND LAW OF THERMODYNAMICS

solid body, such as a metallic bar, when alternately heated and

cooled, increases and diminishes in length and can move bod-

ies fixed at its extremities.

A liquid, alternately heated and

cooled, increases and diminishes in volume and can overcome

obstacles more or less great opposed to its expansion. An

aeriform fluid undergoes considerable changes of volume with changes of temperature ; if it is enclosed in an envelope capa-

ble of enlargement, such as a cylinder furnished with a piston, it

will produce movements of great extent. The vapors of all bod-

ies which are capable of evaporation, such as alcohol, mercury,

sulphur, etc., can perform the same function as water vapor. This, when alternately heated and cooled, will produce motive

power in the same way as permanent gases, without returning to

the liquid state. Most of these means have been proposed, several

have been even tried, though, thus far, without much success.

We have explained that the motive power in the steam-engine

is due to a re-establishment of equilibrium in the caloric ; this

statement holds not only for steam-engines but also for all heat-

engines that is to say, for all engines in which caloric is the

motor. Heat evidently can be a cause of motion only through

the changes of volume or of form to which it subjects the body ;

those changes cannot occur at a constant temperature, but are

due to alternations of heat and cold ; thus to heat any sub-

stance it is necessary to have a body warmer than it, and to

cool it, one cooler than it. We must take caloric from the

first of these bodies and transfer it to the second by means of

the intermediate body, which transfer re-establishes, or, at least,

tends to re-establish, equilibrium of the caloric.

At this point we naturally raise an interesting and important

question : Is the motive power of heat invariable in quantity,

or does it vary with the agent which one uses to obtain it

that is, with the intermediate body chosen as the subject of the

action of heat ?
It is clear that the question thus raised supposes given a cer-

tain quantity of caloric* and a certain difference of temperature.

* It is unnecessaryto explain here what is meant by a quantity of caloric or of heat (for we use the two expressions interchangeably),or to describe how thesequantities are measuredby the calorimeter; nor shall we explain the terms latent heat, degree of temperature,specific heat, etc. The readershould be familiar with theseexpressionsfrom his study of the elementarytreatisesof physics or chemistry.
9

M HMO IRS OX

{.* a

For example, we suppose that we have at onr disposal a body. . I .

maintained at the temperature 100 degrees, and another Im.ly. B, at degrees, and inquire what quantity of motive power will be produced by the transfer of a given quantity of caloric for example, of so much as is necessary to melt a kilogram of

ice from the first of these bodies to the second ; we inquire if this quantity of motive power is necessarily limited ; if it varies with the substance used to obtain it; if water vapor offers in

this respect more or less advantage than vapor of alcohol or of

mercury, than a permanent gas or than any other substance.

We shall try to answer these questions in the light of the con-

siderations already advanced.

We have previously called attention to the fact, which is self-
evident, or at least becomes so if we take into consideration the

changes of volume occasioned by heat, that wherrrrr difference of temperature tktproduction <i/nifin- jtmn'r Conversely, wherever this power can be employed,

it

is is

///>/> />ns*i/i/,: possible

a

to produce difference of temperature or to destroy the equilibrium of the caloric. Percussion and friction of bodies are

a

means of raising their temperature spontaneously* to higher degree than that of surrounding bodies, and consequently of

destroying existed.

that
It

is

equilibrium in the caloric which an experimental fact that the

had previously temperature of

gaseous fluids sion. This

is a

is

raised by compression and lowered by expansure method of changing the temperature of

bodies, and thus of destroying the equilibrium of the caloric in

the same substance, as often as we please. Steam, when used

:/>'a it is

a it is
by

in reverse way from that in which

used in the steam-

engine, can thus be considered as means of destroying the

equilibrium of the caloric. To be convinced of this,

only

necessary to notice attentively the way in which motive power developed by the action of heat on water vapor. Let us
consider two bodies, and B, each maintained ut constant

AA

temperature, that of being higher than that of

these two

bodies, which can either give up or receive heat without

change of temperature, perform the funetions of two indefinitely great reservoirs of calorie. We will call the first body

the source and the second the refrigerator.
If we desire to produce motive power

the transfer of

is a

a

a

[

* That

it,

withoutthecommunicationof heat.] 10

THE SECOND LAW OF THERMODYNAMICS
certain quantity of heat from the body A to the body B we may proceed in the following way :
1. We take from the body A a quantity of caloric to make steam that is, we cause A to serve as the fire-pot, or rather
as the metal of the boiler in an ordinary engine ; we assume the steam produced to be at the same temperature as the body A.
2. The steam is received into an envelope capable of enlargement, such as a cylinder furnished with a piston. We then increase the volume of this envelope, and consequently also the volume of the steam. The temperature of the steam falls when it is thus rarefied, as is the case with all elastic fluids ; let us assume that the rarefaction is carried to the point where the temperature becomes precisely that of the body B.
3. We condense the steam by bringing it in contact with B and exerting on it at the same time a constant pressure until it becomes entirely condensed. The body B here performs the function of the injected water in an ordinary engine, with the difference that it condenses the steam without mixing with it and without changing its own temperature.* The operations which we have just described could have been performed in a reverse sense and order. There is nothing to prevent the for-
* It will perhaps excite surprise that B, being at the sametemperature as the steam,can condenseit. Without doubt this is not rigorously possible, but the smallestdifference in temperature will determine condensation. This remark is sufficient to establish the propriety of our reasoning. In the sameway, in the differential calculus, to obtain an exact result it is sufficientto be able to conceive of the quantities neglectedas capable of being indefinitely diminished relativeto thequantities retained in the equation.
The body B condensesthe steamwithout changing its own temperature. We have assumedthat this body is maintained at a constanttemperature. The caloric is therefore taken from it as fast as it is given up to it by the steam. An example of such a body is furnished by the metallic walls of the condenserwhen the vapor is condensed in it by means of cold water applied to the outside, as is done in some engines. In the sameway the water of a reservoir can be maintainedat a constantlevel, if the liquid runs out at oneside as fast as it comesin at the other.
One could even conceive the bodies A and B such that they would remain of themselves at a constant temperaturethough losing or gaining quantities of heat. If, for example, the body A were a mass of vapor ready to condenseand the body B a massof ice ready to melt, thesebodies, as is well known, could give out or receivecaloric without changing their temperature.
11

MEMOIRS ON

mat ion of vapor by means of the caloric of the body B, and its

compression from the temperature of 11, in such a way that it acquires the temperature of the body A, and then its condensation in contact with A, under a pressure which is maintained

constant until it is completely liquefied.

In the first series of operations there is at the same time a

production of motive power and a transfer of caloric from the
body A to the body R ; in the reverse series there is at the same time an expenditure of motive power and a return of the caloric from B to A. Hut if in each case the sumo quantity of

vapor has been used, if there is no loss of motive power or of

caloric, the quantity of motive power produced in the first

case will equal the quantity expended in the second, and the quantity of caloric which in the first case passed from A to B

will equal the quantity which in the second case returns from B to A, so that an indefinite number of such alternating oper-

ations can be effected without the production of motive power

or the transfer of caloric from one body to the other. Now if

there were any method of using heat preferable to that whieh
we have employed, that is to say, if it were possible that the

caloric should produce, by any process whatever, a larger quan-

tity of motive power than that produced in our first series of

operations, it would be possible, by diverting a portion of this

power, to effect a return of caloric, by the method just indicated, from the body B to the body A that is, from the refrig-

erator to the source and thus to re-establish things in their

original state, and to put them in position to recommence an

operation exactly similar to the first one, and so on : there

would thus result not only the perpetual motion, but an indef-

inite creation of motive power without consumption of caloric

or of any other agent whatsoever. Such a creation is entirely

contrary to the ideas now accepted, to the laws of mechanics

and of sound physics ; it is inadmissible.*

We may hence con-

* The objection will perhapshere be made that perpetual motion has

only beendemonstratedlo be impossiblein the caseof mechanicalactions,

and that it may not be so when we employ the agency of lira:

iricity ; but can we conceiveof the phenomena of heat ami <.f !< triciiy

a* due lo they not

any other be subject

causethan somemotion of bodies,nml, as to the general laws of mechanics? 15-i !

such. -. <l.

slnmM \\,- not

know a potttriori that all the attempts made to produce perpetual motion

by any meanswhateverhavebeenfruitless ; that no truly perpetualinutiou

THE SECOND LAW OF THERMODYNAMICS

elude that the maximum motive power resulting from the use of steam is also the maximum motive power which can be obtained

by any other means. We shall soon give a second and more

rigorous demonstration of this law. What has been given

should only be regarded as
It may properly be asked,

a sketch (see in connection

page 15). with the

proposition

just stated, what is the meaning of the word maximum 9 How

can we know that this maximum is reached and that the

steam is used in the most advantageous way possible to produce

motive power ? Since any re-establishment of equilibrium in the caloric can

be used to produce motive power, any re-establishment of equilibrium which is effected without producing motive power should be considered as a veritable loss: now, with little re-

flection, we can see that any change of temperature which is not

due to a change of volume of the body can be only a useless re-
establishment of equilibrium in the caloric.* The necessary condition of the maximum is, then, that in bodies used to obtain

the motive power of heat, no change of temperature occurs which is

has ever been produced, meaning by that, a motion which continuesindefinitely without change in the body used as an agent?
The electromotiveapparatus (Volta's pile) has sometimesbeenconsidered capable of producing perpetual motion; the attempt has beenmade to realize it by the construction of the dry pile, which is claimed to be unalterable; but, in spite of all that has been done, the apparatus always deteriorates perceptibly when its action is sustained for some time with any
energy. The generaland philosophical acceptationof the words perpetualmotion
should comprehendnot only a motion capable of indefinite continuance after it has been started, but also the action of an apparatus, of a set of bodies, capableof creating motive power in an unlimited quantity, and of setting in motion successivelyall the bodiesof nature, if they are originally at rest,and of destroying in them the principle of inertia,and finally capable of furnishing in itself all the forces necessaryto move the entire universe, to prolong and to constantly accelerateits motion. Such would be
a real creation of motive power. If this were possible, it would be useless
to search for motive power in combustibles, in currents of water and
air. We should have at our disposal an inexhaustible source from which we could draw at will.
* We do not heretake into considerationany chemicalaction betweenthe bodiesusedto obtain the motive power of heat. The chemicalaction which occurs in the sourceis in a sensepreliminary, an fiction not designedto immediatelycreatemotive power, but to destroyequilibrium in the caloric, to produce a differencein temperaturewhich shall finally result in motion.
13

MEMOIRS ON

not due to a change of volume. Conversely, every time that this oon.lition is fulfilled, the maximum is attained.

This principle should not be lost sight of in the construction
of heat-engines. It is the foundation upon which they rest. If
it cannot be rigorously observed, it should at least be departed from as little as possible.

Any change of temperature which is not due to a change of

volume or to chemical action (which we provisionally assume not to occur in this case) is necessarily due to the direct transfer of caloric from a hotter to a colder body. This transfer takes

place principally at the points of contact of bodies at different

temperatures; thus such contacts should be avoided as much

as possible. They doubtless cannot be avoided entirely, but at

least care should be taken that the bodies brought in contact

should differ but little in temperature.

When we assumed in the previous demonstration that the caloric of the body A was used to produce steam, we supposed the

steam to be produced at the same temperature as that of the
body A; thus the only contact was between two bodies of equal

temperature ; the change of temperature which the steam after-

wards experienced was due to expansion and consequently to a

change of volume; finally condensation was effected without
contact of bodies of different temperatures. It was effected by

the exercise of a constant pressure on the steam brought in contact with the body B, at the same temperature as that of the body It. The condition of the maximum was thus fulfilled. In

reality things would not occur exactly as we have supposed. I n

order to effect a transfer of the caloric from one body to the other, the first must have the higher temperature ; but this dif-

ference may be supposed to be as small an we please ; wo may, in theory, consider it zero without invalidating the arirumt -nt.
A more valid objection may be made to our demonstration,

namely:

When we produce steam by taking caloric from the body .1.

Mini when this steam is afterward condensed by contact with

\\\ the body h, the water used to form it, which was a>snmed to In-.

at the beginning, at the temperature of the

. I. i-. at the

end it is

of the colder.

operation, at
If we wish

the to

temperature recommence

of the hody IS that is. an operation similar to

the first, to develop a new quantity of motive power with the

same instrument and the same steam, we must first re-establish

14

THE SECOND LAW OF THERMODYNAMICS
the original state of things and bring the water to the temperature which it had at first. This can no doubt be done by placing it immediately in contact with the body A ; but in that case there is contact between bodies of different temperatures and loss of motive power.* It would become impossible to perform the reverse operation that is, to cause the caloric used in raising the temperature of the liquid to return to the body A.
This difficulty can be removed by supposing the difference of temperature between the body A and the body B infinitely small ; the quantity of heat needed to bring the liquid back to its original temperature is also infinitely small and negligible relatively to that finite quantity which is needed to produce the steam.
The proposition being thus demonstrated for the case in which the difference of temperature of the two bodies is infinitely small may easily be extended to cover the general case. In fact, if we desire to produce motive power by the transfer of caloric from the body A to the body Z, the temperature of the latter body being very different from that of the former, we may imagine a series of bodies B, C, D . . .at temperatures intermediate between those of the bodies A and Z, and chosen in such a manner that the differences between A and B, B and C . . . shall be always infinitely small. The caloric which proceeds from A arrives at Z only after having passed through the bodies B, C, D . . . and after having developed in each of these transfers the maximum of motive power. The reverse operations are here all possible, and the reasoning on page 11 becomes rigor-
ously applicable. According to the views now established we may with pro-
*This kind of loss is always met with in steam-engines. In fact,,the water which supplies the boiler is always colder than that which it already contains, and hence a uselessre-establishmentof equilibrium in the caloric takes place betweenthem. It is easyto seea posteriori that this reestjiblishment of equilibrium entails a loss of motive power if we reflect that it would be possible to heat the water supply before injecting it by using it as water of condensation in a small accessoryengine, in which stenmtaken from the large boiler could be used and in which condensation would occur at a temperatureintermediate betweenthat of the boiler and that of the principal condenser. The force produced by the small engine would entail no expenditureof heat, since all that it would use would recuter the boiler with the water of condensation.
15

MEMOIRS ON
priety compare the motive power of heat with that of a waterfall; both have a maximum which cannot be surpassed, whatever may be, on the one hand, the machine used to receive the action of the water and whatever, on the other hand, the substance used to receive the action of the heat. The motive power of fulling water depends on the quantity of water and on the height of its fall; the motive power of heat depends also on the quantity of caloric employed and on that which might be named, which we, in fact, will call, its descent* that is to say. on t he difference of temperature of the bodies between which t he exchange of caloric is effected. In the fall of water the motive power is strictly proportional to the difference of level between the higher and lower reservoirs. In the fall of caloric the motive power doubtless increases with the difference of temperature between the hotter and colder bodies, but we do not know whether it is proportional to this difference. We do not know, for example, whether the fall of the caloric from 100 to 50 degrees furnishes more or less motive power than the fall of the same caloric from 50 degrees to zero. This is a question which we propose to examine later.
We shall give here a second demonstration of the fundamental proposition stated on page 13 and present this proposition in a more general form than we have before.
When a gaseous fluid is rapidly compressed its temperature rises, and when it is rapidly expanded its temperature falls. This is one of the best established facts of experience ; we shall take it as the basis of our demonstration, f When tin temperature of a gas is raised and we wish to bring it back to its
The matter here treatedbeing entirely new. we are obliged to employ expressionshitherto unused and which are not perhapsas clear ascould be desired.
f The facts of experience which best prove the change of temperature of a gas by compressionor expansionare the following :
1. The fall of temperatureindicated by a thermometerplaced under tinreceiverof an air pump in which a \.n mini is produced. This is very p<t rrptililr with a Bregucl thermometer; it may amount to upwards of 40 or 50 degrees. The cloud,which is formed in this operationseemsto in tin to ii,c cii<lriisalion of water vapor caii-cil hy the coolintr of the air.
2. The ignition of tinder in the so-called fire-syringe (pneumatic tinderbox), which is, as is well known, a small pump in which air may rapidly compressed.
The fall of temperatureindicated by thermometer placed in re16

8. a
!>< a

THE SECOND LAW OF THERMODYNAMICS
original temperature without again changing its volume, it is necessary to remove caloric from it. This caloric may also be removed as the compression is effected, so that the temperature of the gas remains constant. In the same way, if the gas is rarefied, we can prevent its temperature from falling, by furnishing it with a certain quantity of caloric. "We shall call the caloric used in such cases, when it occasions no change of temperature, caloric due to a change of volume. This name does not indicate that the caloric belongs to the volume; it does not belong to it any more than it does to the pressure, and it might equally well be called caloric due to a change ofpressure. We are ignorant of what laws it obeys in respect to changes of volume : it is possible that its quantity changes with the nature of the

ceptacle in which air has beencompressed,and from which it is allowed to

escapeby opening a stopcock.

4. The results of experimentson the velocity of sound. M. de Laplace

has shown that to harmonize these results with theory and calculation we

must assumethat air is heatedby a sudden compression.

The only fact which can be opposed to these is an experiment of MM.

Gay-Lussac and Welter, describedin the Annales de Chimie et dePhysique.
If a small opening is made in a large reservoir of compressedair, and the

bulb of a thermometeris placed in the current of air escapingthrough this

opening, no perceptiblefall of temperatureis indicated by the thermometer.

We may explain this fact in two ways :

1. The friction of the air against the walls of the opening through which

it escapesmay perhaps develop enough heat to be noticed ; 2. The air

which impinges immediately upon the bulb of the thermometer may re-

cover by its shock against the bulb, or rather by the detour which it is

forced to make by the encounter,a density equal to that which it had in

the receiver, somewhat in the same way as a current of water rises above

its level when it meetsa fixed obstacle.

The change of temperaturein gasesoccasionedby a changeof volume

may be consideredone of the most important facts in physics, becauseof

the innumerableconsequenceswhich it entails, and at the sametime as one

of the most difficult to elucidate and to measure by conclusive experi-

ments. It presentssingular anomalies in several cases.

Must we not attribute the coldness of the air in high regionsof the at-

mosphereto the lowering of its temperature by expansion? The reasons

hitherto given to explain this coldnessare entirely insufficient ; it has been

said that the air in high regions, receiving but a small amount of heat

reflectedby the earth, and itself radiating into celestial space,would lose

caloric and thus becomecolder ; but this explanation is overthrown when

we consider that at equal elevations the cold is as great or even greateron

elevatedplains than on the tops of mountainsor in parts of theatmosphere

distant from the earth.

B

17

MEMOIRS ON

gas, or with its density or with its temperature.

Experiment

has tanght us nothing on this subject; it has taught us only

that this caloric is developed in greater or less quantity by the

compression of elastic fluids.

This preliminary idea having been stated, let us imagine an

elastic fluid atmospheric air, for example enclosed in a cylin-

drical vessel abed (Fig. 1) furnished with a movable diaphragm or piston cd; let us assume also the two bodies A, B both at constant tem-

peratures, that of A being higher than that of B, and let us consider the series

of operations which follow :

//// 1. Contact of the body A with the

air contained in the vessel

<>wr ith

the wall of this vessel, which wall is

supposed to be a good conductor of

caloric. By means of this contact the

air attains the same temperature as the

body .4; cd is the position of the pis-

ton.

2. The piston rises gradually until

it takes the position

Contact al-

ef.

it is 1
i is

flKrr^

ways maintained between the air ami

^ the body A, and the temperature thus

remains constant during the rarefac-

tion. The body

A

furnishes the ca-

loric necessary to maintain constant

Fig.

temperature.

3. The body

A

is

removed and the

air no longer in contact with any body capable of supply-

ing

with caloric; the piston, however, continues to move

it it
ef it

and passes from the position

to the position gh. The air

rarefied without receiving caloric and its temperature falls.

Let us suppose that falls until

becomes equal to that of

the body />'; at this instant the piston ceases to move and

occupies the position ////. 4. The air brought in contact with the body

:/.'

it

is

com-

pressed by the piston as returns from the position //// to the

a

position nl. The air, however, remains at constant temperature on account of its contact with the body B, to which gives

it

up its caloric. 18

a is
is

THE SECOND LAW OF THERMODYNAMICS
5. The body B is removed and the compression of the air continued. The temperature of the air, which is now isolated, rises. The compression is continued until the air acquires the temperature of the body A. The piston during this time passes from the position cd to the position ik.
6. The air is again brought in contact with the body A\ the piston returns from the position ik to the position ef, and the
temperature remains constant. 7. The operation described in No. 3 is repeated, and then the
operations 4, 5, 6, 3, 4, 5, 6, 3, 4, 5, and so on, successively. In these various operations a pressure is exerted upon the
piston by the air contained in the cylinder ; the elastic force of this air varies with the changes of volume as well as with
the changes of temperature ; but we should notice that at equal volumes that is, for similar positions of the piston the temperature is higher during the expansions than during the compressions. During the former, therefore, the elastic force of the air is greater, and consequently the quantity of motive power produced by the expansions is greater than that which is consumed in effecting the compressions. Thus there remains an excess of motive power, which we can dispose of for any purpose whatsoever. The air has therefore served as a heat-engine ; and it has been used in the most advantageous way possible, for there has been no useless re-establishment of equilibrium in the caloric.
All the operations described above can be carried out in a direct and in a reverse order. Let us suppose that after the sixth step, when the piston is at ef, it is brought back to the position ik, and that, at the same time, the air is kept in contact with the body A ; the caloric furnished by this body dur-
ing the sixth operation returns to its source that is, to the body
A and the condition of things is the same as at the end of the fifth operation. If now we remove the body A and move the piston from ef to cd, the temperature of the air will fall as many degrees as it rose during the fifth operation and will equal that of the body B. A series of reverse operations to those above described could evidently be carried out ; it is only necessary to bring the system into the same initial state and in each
operation to carry out an expansion instead of a compression,
and conversely. The result of the first operation was the production of a cer19

MKMOIRS ON

tain quantity of motive power and the transfer of the caloric from the body -1 to the body B ; the result of the reverse opera-

tion would be the consumption of the motive power product- 1
and the return of the caloric from the body B to the body .1 :

so that the two series of operations in a sense annul or neutralize

each other.

The impossibility of making the caloric produce a larger

quantity of motive power than that which we obtained in our first series of operations is now easy to prove. It may be de-

monstrated by an argument similar to that used on paire 11.

The argument will have even a greater degree of rigor : the air

which serves to develop the motive power is brought back,

at the end of each cycle of operations, to its original condi-

tion, which was, as we noticed, not quite the case with the

steam.*

We have chosen atmospheric air as the agency employed t..

develop the motive power of heat; but it is evident that the

same reasoning would hold for any other gaseous substance, and

even for all other bodies susceptible of changes of temperature

by successive contractions and expansions

that is, for all

bodies in Nature, at least, all those which are capable of develop-

ing the motive power of heat. Thus we are led to establish this

general proposition :

The motive power \ of /ie<tt i* imli'i>i->nlrnt of the ay>

We implicitly assume-i,n our demonstration. tli.-U if a body expericncesany changes,ami returns exactly to its origin tl stair. vt. t a certain numtteroftransformations thnl is to say, to its original Matedeterminedby its density,its temperature,and its modeof aegrcpitioii : \v. a-Mime. I say, that the body contains the samequantity of hc:it as it containednl first, or. in other words, thatthcqtinniitiesof heatabsorbedand te|.-as<d in iitransformationsexactly compensateone another. Thi fact has never IN-CM called in question ; it was fttfirst admittedwiihoul consideriti-m and after wanis verified in many cases by exporimcnts with the ralorim--IIT. To deny it would he to overthrow UK*entire theory of heat, of wliich it is die i foundation. It may IK; rcmarkc.l, in passinp. that tin- fiin<l:inicntalprin i-ipli-soii which the theory of heat rests should l jfiven the nvM cueful i \ itnination. Severnlexperimental fatis si-<-mto Ix; almost inexplicable in Hi- actual Htaicof that theory. [The doubts here expressed as to the validity of the assumptionsmade with respect to the nature of li lvlyecmdfloih[|n/Mi/ym*mmdtionoausy(t-uiabfsirelpnhewoctteh'tsltelhmtmeoinotrndrueoeimtnintfoeoadtaaeun<rrneja,tioeciitfrtumtha/ie"l.iartwet.jCoeruckSrt.eni"oen\ntLouiffteetht<o"/s'em'aostsiur,'mptKioMn.]*,an.I
m

THE SECOND LAW OF THERMODYNAMICS

ployed to develop it ; its quantity is determined solely by the tem-

peratures of the bodies between which, in the final result,* the

transfer
It is

of the caloric understood in

occurs. this statement

that the method

used for

developing motive power, whatever it may be, attains the highest

perfection of which it is capable. This condition will be fulfilled, as we remarked above, if there is no change of tempera-

ture in the bodies which is not due to a change of volume or,
which amounts to the same thing differently expressed, if the temperatures of the bodies which come in contact with each

other are never perceptibly different.

Various methods of developing motive power may be adopted,

either by the use of different substances or of the same sub-

stance in different states; for example, by the use of a gas at

two different densities.

This remark leads us naturally to the interesting study of aeri-

form fluids, a study which will conduct us to new results con-

cerning the motive power of heat, and will give us the means

of verifying in some particular cases the fundamental proposi-

tion stated above, f It can easily be seen that our demonstration will be simplified
if we suppose the temperatures of the bodies A and B to be very

slightly different. Then the movements of the piston will

be very small during operations 3 and 5, and these operations

may be suppressed without perceptible influence on the de-

velopment of motive power. That is, a very small change of

volume ought to be sufficient to produce a very small change

of temperature, and this change of volume is negligible com-

pared with that of operations 4 and 6, which are unrestricted

in extent.
If we suppress operations 3 and 5 in the series above de-

scribed, it is reduced to the following:

1. Contact of the gas contained in abed (Fig. 2) with the body A, and passage of the piston from cd to ef;

2. with

Removal the body

of the body A, B, and return

contact of the gas enclosed in
of the piston from ef to cd ;

abcf

3. Removal of the body B, contact of the gas with the body

* [That is, upon tliecompletionof a cycleof operations.]
\ We shall supposein what follows that the reader is familiar with the
latest progressof modernphysics in the departmentsof heat and gases. 21

1KMOIRS ON A, and passage of the piston from cd to ef that is to say,
tition of the first operation, and so on. The motive power resulting from the operations 1, 2, 3, taken
together, will evidently be the difference between that which is produced by the expansion of the gas while its temperature equals that of the body A and that which is consumed to compress the gas while its temperature equals that of the body B.
Let us suppose that the operations 1 and 'I are performed with two gases which are chemically different, but which are subjected to the same pressure for example, that of the atiuos-
Pig.
phere ; these gases behave in the same circumstances in exactly the same way that is to say, their expansive forces, originally equal, remain so irrespective of changes of volume ami temperature, provided that these changes are the same in both. This is an evident consequence of the laws of Mariotte ami <>fM M. <iav-Lussiic and Dalton, which laws are eommoii t<>all elastic fluids, and in virtue of which the same relation- exist in all these fluids between the volume, expansive force, ami temperature. Since two different gases, taken at the same temperature and under the same pressure, should liehave alike under the same circumstances, they should produce eijual quantities of motive power when subjected to the operations above described. Now this implies, according to the fundamental proposition which we have established, that two equal quantities of caloric are employed in these operations that is, that the quantity of caloric transferred from the body A to the body

THE SECOND LAW OF THERMODYNAMICS

B is the same whichever of the two gases is used in the opera-

tions. The quantity of caloric transferred from the body A to the body B is evidently that which is absorbed by the gas in the increase of its volume, or that which it afterwards emits

during compression. We are thus led to lay down the follow-

ing proposition :

When a gas passes without change of temperature from one definite volume and pressure to another, the quantity of caloric ab-

sorbed or emitted is alivays the same, irrespective of the nature of the gas chosen as the subject of the experiment.
For example, consider 1 litre of air at the temperature of 100 degrees and under the pressure of 1 atmosphere*, if the

volume of this air is doubled, a certain quantity of heat must

if,

be supplied to it in order to maintain it at the temperature of 100 degrees. This quantity will be exactly the same instead of performing the operation with air, we use carbonic acid gas,

if

nitrogen, hydrogen, vapor of water, or of alcohol that is, we double the volume of litre of any one of these gases at the

if

by 1

temperature of 100 degrees and under atmospheric pressure. The same thing would be true, in the reverse sense, the

volume of the gas, instead of being doubled, were reduced one-

half compression.

The quantity of heat absorbed or set free by elastic fluids

during their changes of volume has never been measured by

direct experiment. Such an experiment would doubtless present great difficulties, but we have one result which for our pur-

is

poses nearly equivalent to

;it

this result has been furnished

by the theory of sound, and may be received with confidence be-

cause of the rigor of the demonstration by which established. It may be described as follows

has been

1:

Atmospheric air will rise in temperature degree centigrade

is is ;

when its volume reduced by y|^ by sudden compression.* The experiments on the velocity of sound were made in air

a

under pressure of 760 millimetres of mercury and at the tem-

G

perature of degrees and

it

is

only in these circumstances that

Poisson's statement

applicable. We shall, however, for the

it it

a

* M. Poisson, to whom we owe this statement,has shown that agrees very well with the resultsof an experiment by MM. Clement and Desormes on the behavior of air entering into vacuum or rather into slightly rarefied air. It agreesalso, very nearly, with result obtained by MM. GayLussac and Welter. (Seenote, p. 32.)

a

MEMOIRS ON

sake of convenience, consider it to hold at a tcmperatnre of

degrees, which is only slightly different.

Air compressed by -pj-y and so raised in temperature 1 degree

differs from air heated directly by the same amount only in its
density. If we call the original volume V, the compression 1>\ Y^-y reduces it to V 'j\j V. Direct heating under constant

^ pressure, according to the law of M. Gay-Lussuc, should in-

crease the volume of the air by

of that which it would have

at degrees; thus the volume of the air is in one process re-

T", is 1

duced to V -rfj and in the other increased to V+ 5^-7T. The difference between the quantities of heat present in tin-

air in the two cases evidently the quantity used to raise its

;

temperature directly by degree thus the quantity of heat ab-

],.,

sorbed by the air in passing from the volume

I'

,

V

to the

r+^Tis volume

equal to that which

is

necessary to raise

1

its temperature degree.

Let us now suppose that, instead of heating the air while sub-

it a

jected to constant pressure and able to expand freely, we en-

close in an envelope not capable of expansion, and then raise

its temperature

1

degree. The air thus heated degree dif-

1

fers from air compressed by yfy, by having its volume larger

by y^. Thus, then, the quantity of heat which the air gives up

by reduction of its volume by jfa equal to that which

re-

a V T ,r F, 1
is ?.T$I
is

quired to raise its temperature

As the differences,

fyJ

degree at constant volume.

and F+

are .-mall in

comparison with the volumes themselves, we may consider 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.

M sensibly proportional to the changes of volume.

\\

e thus

:

obtain the following relation

The quantity of heat required to raise the temperature of air

is

1

under constant pressure degree to the quantity required to

raise

it

1

rir jfr degree at constant volume in the ratio of the numbers

+

to rfy,

by

or. multiplying both terms 11G.2G7, in the ratio of the nnin-

;7+116 to 267.

is

This the ratio between the capacity for heat of air under constant pressure and its capacity at constant volume. If tin-

is < .
by

first of these two capacities expressed

unity the other will

be expressed by the number - - "

approximately, 0.700.

Ji

THE SECOND LAW OF THERMODYNAMICS

Their difference 10.700 or 0.300 will evidently express the quantity of heat which will occasion the increase of volume of the air when its temperature is raised 1 degree under constant
pressure. From the law of MM. Gay-Lussac and Daltou this increase
of volume will be the same for all other gases ; from the theorem demonstrated on page 23 the heat absorbed by equal increments of volume is the same for all elastic fluids; we are thus led to establish the following proposition :
The difference between the specific heat under constant pressure
and the specific heat at constant volume is the same for all gases. It must be noticed here that all the gases are assumed to be
taken at the same pressure for example, the pressure of the atmosphere and also that the specific heats are measured in
terms of the volumes.
Nothing is now easier than to construct a table of the specific heats of gases at constant volume with the aid of our knowl-
edge of their specific heats under constant pressure. We
present this table, the first column of which contains the results of direct experiments by MM. Delaroche and Berard on the specific heat of gases under atmospheric pressure. The second column contains the numbers in the first diminished by
0.300.
TABLE OF THE SPECIFIC HEAT OF GASES

GASES

SPECIFIC HEAT UN- SPECIFIC HEAT

DER CONSTANT AT CONSTANT

PRESSURE

VOLUME

Atmospheric air Hydrogen
Carbonic acid Oxygen . . Nitrogen . . . . Nitrons oxide
Olefiant gas Carbonic oxide . .

1 000

903

.

1 258

0.976

1 000

1 350

1.553

1.034

700 603
958
676
700 1 050 1 253 0.734

The numbers in the two columns are referred to the same unit, to the specific heat of atmospheric air under constant
pressure. The difference between the corresponding numbers in the
two columns being constant, the ratio between them should be 25

MEMOIRS ON

variable ; thus the ratio between the specific heats of gases under constant pressure and at constant volume varies for the
f different gases. \\ have seen that the temperature of the air when it under-
goes a sudden compression of t f ff of its volume rises 1 degree. That of other gases should also rise when they are similarly roiii|>ivss<_-<l. The temperature should rise, not equally for all, but in the inverse ratio of their specific heats at constant volume. In fact, the reduction of volume being, by hypothesis, always the .-aim-, the quantity of heat due to this reduction should also be always the same, and consequently should cans.' a rise of temperature depending only on the specific heat of the gas after its compression, and evidently in an inverse ratio to that specific heat. It is therefore easy to construct the table of elevations of temperature of the different gases for a com-
pression

TABLE OF THE ELEVATION OF Till: TKM I'KKATl UK OF &A8E8

1H K TO COMPRESSION

GASES

ELEVATION TI-KK Kon

AOKinTinKM. in'ioIa: \

OF
ONACNOHCtaaxyliiemttryrdrrfbbgrioooaoooesugnnngpntseiiehccnne.or.gixa.oca.ic.x..dsi.i..de.d....ea.......i...r..................................................................................................................................................................................................

VOLfMK OK o
1.000 1.160 0.730 1.035 1.000
0.< 0.558 0.955

,\

f

A second compression of Tfy of the new volume would, as we
shall soon see, again raise the temperature of these gases In an amount nearly equal to the first ; but this would not be the case for a third, a fourth, or a hundredth compression of tinsame sort. The capacity of gases for heat changes with their volume; it is quite possible that it changes also with their temperature.*
[It ,u found ty ReynauH (M.-in. !.- I'Acadfimle, asnrf.,p. 58) tl. tprtifit heat of the "permanent" gate* it independentof prtmure and tern-
i re.} ''!

THE SECOND LAW OF THERMODYNAMICS

We shall now deduce from the general proposition presented

on page 20 a second theorem which will be the complement of

that which has just been demonstrated.

Let us suppose that the gas contained in the cylinder abed

(Fig. 2) is transferred to the receptacle a'b'c'd' (Fig. 3), which is of equal height, but which has a different and larger base ; the

gas will increase in volume and diminish in density and elas-

tic force in the inverse ratio of the two volumes abed, a'b'c'd'.

The total pressure exerted on each piston, cd, c'd', will be the

same, for the surfaces of these pistons are in the direct ratio of

the volume.

Let us suppose that the operations described on page 21 as

performed on the gas contained in abed are performed on the

gas in a'b'c'd' that is, let us suppose that the piston c'd' is given

displacements equal in amplitude to those given the piston cd,

and that it occupies successively the positions c'd' correspond-

ing to cd, and e'f corresponding to ef.
subject the gas, by means of the two

At the same time let us bodies A, B, to the same

variations of temperature as those to which it was subjected

when enclosed in abed ; the total force exerted on the piston

will be the same in both cases at corresponding instants. This

results immediately from Mariotte's law* ; in fact, the densities

of the two gases are always in the same ratio for similar posi-

tions of the pistons, and, their temperatures being always equal,

the total pressures exerted on the pistons are always in the same
ratio. If this ratio is at any time that of equality, the pressures

will bo always equal.

Further, as the movements of the two pistons have equal am-

plitudes, the motive power they both produce will evidently be

the same, from which we may conclude, from the proposition

* Mariotte'slaw, upon which our demonstrationis based,is oneof thebestestablishcdphysical laws. It has served as a foundation for several theories verified by experiment, and which verify in their turn the laws on which they rest. We may also cite, as an important verification of Mariotte's law and also of the law of MM. Gay-Lussac and Dalton for a large range of temperature,the experiments of MM. Dnlong and Petit. (See Annales de Chimie et de Physique, Feb., 1818,vol. vii., p. 122.) We mav also cite the still more recentexperimentsof Davy and Faraday.
The theoremshere deduced would perhapsnot be exact if applied outside of certain limits either of density or of temperature. They should only be taken as true within the limits within which the laws of Mariotte, Gay-Lussac, and Dalton are themselvesestablished.
27

MEMOIRS ON

on page 20, that the quantities of heat used by each are equal

that is to say, that the same quantity of heat passes from .1 t<> // in each case.
The heat taken from the body A and given to the body

i>

/.'

nothing other than the heat absorbed by the expansion of the gas and afterwards set free by onipn-iuii. We are thus led

<

to establish the following theorem

:

nf

I' of
I".

II 1

lit'ii (lie I'olnitli' an flitxtir jlinil rlminiis. without r}/n)irjt<

tim/ierature, from

to "//// tin- /<<//////<,,/',/ i/nantitii

of

tin'

same gas, equal in weight and at tin' >v////- fn,ij r,tf >//. clntn<i>x

f

from IT toV', the quantities of heat ali.^,rl,,;l or .^7 !<>from nn-li

/>

will be equal when the ratio U' to

of

I

/mtl to tint!

of

U

toV.

This theorem may be stated in another form, as follows:

of

When a gas changes in volume without change fi-m/H-ratiin-

the quantities

of

it

heat which absorbs or gives KI> an- in arith-

metical progression when the increments or reductions

of

volume

are in geometrical /;/-o///v /'///.

a

When we compress one litre of air maintained at tempera-

{, it

ture of 10 degrees and reduce its volume to

J
a

litre, gives

a

out certain quantity of heat. This quantity will bo al\\a\-

if

the same we further reduce the volume from to from

2J

it J

.1

it

4

to and so on.
If, instead of compressing the air, we allow to expand to litres, litres,8 litres, etc., successively, we must supply \\iih
equal quantities of heat in order to keep its tomporuturc c,nst ant. This easily explains why the temperature of air risrs \\-hrn

is it

is

suddenly compressed. We know that this temperature

sufficient to ignite tinder and even to cause the air to become

luminous.

If we assume for the time lieing the specific heat

of air as constant,

in

spite of changes of volume and tempera-

ture, the temperature will increase in arithmetical pm^ressinn

^{ . it i>

as the volume diminished in geometrical progression. Start-

ing with this as given, and admitting that an elevation of inn

perature of degree corresponds to compression of r

easy to conclude that when air reduced to

fa

of its original

volume its temperature should rise about 300 degrees, which enough to ignite tinder.*

is

is a
1

it 1 is

it

l

When the volume mlur<i l,y ,}, that In. when becomes\\\ of
that which wa* at first the temperaturerisen <i<

THE SECOND LAW OF THERMODYNAMICS

The elevation of temperature would evidently be still greater if the capacity of the air for heat were to become less as its

volume diminishes ; now this is probable, and seems to be con-

firmed by the results of the experiments of MM. Delaroche and

Berard on the specific heat of air taken at different densities.

(See the Memoir published in the Annalesde Chimie et de Physique, vol. Ixxxv., pp. 72, 224.)
The two theorems given on pages 23 and 28 are sufficient for

the comparison of the quantities of heat absorbed or released in

the changes of volume of elastic fluids, whatever may be the den-

sity and chemical nature of these fluids, provided always that

they are taken and maintained at a certain invariable tempera-

ture ; but these theorems do not give us any means of compar-

ing quantities of heat absorbed or released by elastic fluids whose

volumes are changed at different temperatures.

Thus we do

not know the relation between the heat released by 1 litre of air

reduced in volume one-half when its temperature is kept at zero

and the heat released by the same litre of air reduced in volume

one -half when its temperature is kept at 100 degrees. The

knowledge of this relation is connected with the knowledge of

the specific heat of the gases at different degrees of tempera-

ture, and on other data which Physics, in its present state, can-

not furnish.

The second of our theorems affords a means of knowing by what law the specific heat of gases varies with their density.

Let us suppose that the operations described on page 21, instead of being performed with two bodies, A, B, whose temper-

is
i1
it ; )*. d

A now reduction of T|ff brings the volume to (Ul) 4. and tlie tempera-

ture should rise another degree.

After x such reductions the volume is (Hf aQ the temperatureshould

be higher by
If we set

x degrees.
(Hi^T ?.

and

tuke tue

logarithms

of

both

sides, we

find

x = 300 about.

If we set (Hf)* = we find that x = 80, which shows that the tem-

perature of air compressedto one half of its original volume rises 80 de-

All this dependenton the hypothesis that the specific heat of air does not change when the volume diminishes but if, for the reasonsgiven on pages31 and 32,we consider the specificheatof air compressedto one-half its volume as reduced in the ratio of 700 to 616,we must multiply 80 degrees by J^, which brings to 90 degrees.
29

MEMOIRS ON
atures differ by an infinitely small quantity, are performed with
byl. two bodies whose temperatures differ by a finite quant ity, say
In a complete cycle of operations the body A furnishes to the
elastic fluid a certain quantity of heat which may be dividt <! into two portions : 1, the quantity required to keep the temperature of the fluid constant during expansion ; 2, that required to change the temperature of the fluid from that of the
J. body B to that of the body A, after the fluid has been restored
to its original volume and is put in contact with the body 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 of which, b', is due to the cooling of the gas by the body B, the other, ', is that released by the gas during the reduction of its volume. The sum of these two
quantities is a'-f b' ', this should be equal to a -f- b, for after a
complete cycle of operations the gas returns exactly to its
original state.* It must have given up all the caloric with which it had at first been supplied. We then have
a+b=a'+b',

or,

a a' = b b'.

Now, from the theorem given on page 28, the quantities a and

a' are independent of the density of the gas, always provided

that the quantity of the gas by weight remains the -aim- ami

that the variations of volume are proportional to the original

volume. The difference a a' should satisfy the same con-

ditions, and consequently also the difference b' b, which is

equal to it. But b' is the caloric necessary to raise the temper-

ature of the gas contained in nhnl one degree (Fig. 2); b' is the

//. caloric released by the gaa when it is enclosed in

and its

temperature falls one degree. These quantities can serve as a

measure of the specific heats. We an- thus led to establish the

following proposition :

[The M* here madeof thecaloric theorytitiata the demonttrationand Uttdtto erroneouteondunont.}
M

THE SECOND LAW OF THERMODYNAMICS
The change made in the specific heat of a gas in consequence of a change of volume depends only upon the relation between the original volume and that which results from the change that is to say, the difference between the specific heats does not depend on the absolute magnitudes of the volumes but on their ratio.
This proposition may be stated in another way, namely : When the volume of a gas increases in geometrical progression its specific heat increases in arithmetical progression. Thus, if a is the specific heat of air taken at a given density, and a + h its specific heat when its density is one-half this, its specific heat will be a+'2h when its density is one-quarter this, a+'3h when its density is one-eighth this, and so on. The specific heats are here referred to weight. They are supposed to be taken at constant volume ; but, as we shall see, they would follow the same law if they were taken under constant pressure. In fact, to what cause is due the difference between the specific heats taken at constant volume and under constant pressure ? It is due to the caloric required in the latter case to produce the increase of volume. Now, by Mariotte's law, the increase of volume of a gas, for a given change of temperature, should be a definite fraction of the original volume, which fraction is independent of the pressure. From the theorem given on page 28,if the ratio between the original volume and the changed volume is given, the heat required to produce the increase of volume is determined thereby. It depends only on this ratio and on the quantity of the gas by weight. "We must then conclude that : The difference between the specific heat under constant pressure and that at constant volume is always the same, whatever the density of the gas, provided that the quantity of the gas by weight remains the same. These specific heats both increase as the density of the gas diminishes, but their difference does not change.* Since the difference between the two capaci-
* MM. Gay-Lussac and Welter have found by direct experiments,cited in the MecaniqueCelesteand in the Annales de Chimie et de Physique,July, 1822,page267,that the ratio betweenthe specificheatunder constantpressure and that at constant volume varies very little with the density of the gas. From what we have just seen,it is the differenceand not the ratio that should remain constant. However, as the specific heatsof gases,for a given weight, vary very little with their density, it is clear that the ratio also will experienceonly very small changes.
31

M KM Ml US ON

ties for heat is constant, when one increases in arithmetical progression the other will increase in a similar progression ; thus our law applies to specific heats taken under constant pressure.
We have tacitly supposed that the specific heat inn. with the volume. This increase is shown in the experiments of MM. Delaroche and Hi' rani; these physicists have found that the specific heat of air under the pressure of 1 meter of
mercury is 0.967 (see the memoir already referred to), taking as the unit the specific heat of the same weight of air under tinpressure of 0.7GO meter. From the law followed hy the specific
heats with respect to pressure, observations made of tln-m in two particular cases permit us to calculate them in all pos-
sible cases ; thus, by using the result of the experiment of M M. Delaroche and Berard, which has just been cited, we have con-
structed the following table of the specific heats of air under
various pressures :

-PM'IFIC HEAT,

PRESSURE THAT OK AIR UNDER

IN

ATMOSPHERIC

ATMOSPHERES

PRESSURE

BEING 1.

iXTiiW

1.840 1.756 L.678
,688

VAr

..~><>1 .I'M

,886

.'.'">'.' '[.;.

.084 .000

SPECIFIC HEAT.

PKKSsritK THAT OF AIK 1MU.Il

IN

ATMOSIMIKICK

\TMOH-IIKKI-.S

PRESSURE

aura i

1

l.(MM)

2

0.916

4

8

0.748

16

.'!

0.580

64

0.496

US

0.411

51SJ M'.'l

ii o!l60

1 1I i
is a
,: h is
'.' a
|j>

From the experiments of MM. specific heat of atmospheric air

Gay-Lussac and WclU-r. under constant pressure

ttloietrinattioaoi frtnhe

stant volume 1.8748, uumtx-r\\lii< nearly constant for all pr.

and for all temperatures. In the previous iliHriivsion we have l>c<>lidn,

by other considerations,to the number ~~na-j =1.44, \\ hieli dilTersfnun

this by fo, and we have usedthis number to hentsof gasesat constant volume. Neither

construct this taM.-

tableof
nr (lie

thesperiiic tai.l.- uivrn

on page 88 should lie consideredas accurate. Tln-y an- intcmlr.1mainly

to set forth the laws followed by the specific heatsof aeriform fluids.

THE SECOND LAW OF THERMODYNAMICS

The numbers in the first column are in geometrical progression, while those in the second are in arithmetical progression.
We have carried the table out to extreme compressions and
rarefactions. It is to be supposed that air, before attaining a
density 1024 times its ordinary density that is, before becoming more dense than water would be liquefied. The specific
heats vanish, and even become negative if we prolong the table beyond the last number given. It seems probable that the numbers in the second column decrease too rapidly. The experiments on which we based our calculation were made within too narrow limits to enable us to expect great exactness in the numbers obtained, especially in the extreme values.
Since, on the one hand, we know the law by which heat is
evolved by the compression of a gas, and, on the other, the law by which the specific heat varies with the volume, it will be
easy for us to calculate the increase of temperature of a gas compressed without loss of caloric.* In fact, the compression can be considered as consisting of two successive operations :
1, compression at a constant temperature, and, 2, restoration
of the caloric emitted. In the second operation the temperature will rise in the inverse ratio to the specific heat which
the gas acquires by the reduction of its volume. "We can de-
termine the specific heat by means of the law above demonstrated. From the theorem on page 28 the heat set free by
compression should be represented by an expression of the form
s = A + B log v,
s being the heat, v the volume of the gas after compression,
A and B arbitrary constants dependent on the original volume
of the gas, on its pressure, and on the units which are chosen. The specific heat, which varies with the volume in accordance
with the law just demonstrated, should be represented by an expression of the form,

A' and B' being arbitrary constants different from A and B. The increase of temperature which the gas receives by com-

pression is proportional to the ratio -, or to the ratio -p

".

hea* t[,Tahnids

demonstrationis erroneousin that it assumestJie,materiality of also tfie changeof specificheatwith volume. The conclusionsare

invalid.]

MEMOIRS ON

It may be represented it by /, we shall have

by this ratio itself; tlins,

~

A A

+ li
~ /'

log io

r

if we represent

If the original volume of the gas is 1 and the original u-nipt'ni-

ture zero, we shall have at the same time / = 0, log y = 0, ami

hence A

; t will then express not only the increase of tem-

perature, but the temperature itself above the thermometric zero.

We must not think that we can apply the formula just given

to very large changes in the volume of the gas. We have taken

the rise of temperature to be in the inverse ratio to the specific

heat, which implies that the specific heat is constant at all tem-

peratures. Large changes of volume in the gas occasion large

changes of temperature, and there is no evidence that the specific

heat is constant at different temperatures, especially when these

temperatures are widely separated from each other. This con-

stancy of specific heat is only an hypothesis assumed in the case

of gases from analogy, and verified fairly well for solids and li(j aids

within a certain range of the thermometric scale, but which the

experiments of MM. Pulongaml Petit have shown to IK- inexact

when extended to temperatures much above 100 degrees.*

We sec no reason to assumea priori the constancy of the specific

heat of bodies at various temperatures that is to say, to assume that

equal quantities of heat will produce equal increments in tin- temperature

of a body, even when neither the state in>rtlic density of the body U

clinnired; as. for example,in the caseof an elastic fluid enclosedin n riirul

envelope. Direct experimentson solid and liquid bodieshave proved that

betweenzero and 100 degreesequal im-r.im-nt* of heat produce nearly

equal increments MM Dulong and

of temperature; Petit (see AnnaU*

but de

the more Cliimi, ,t

recent experiments of <! l'/i?/*iqnf, February.

MUM-li. and April, 1818)Imve shown that this relation does not hold for

temperaturesmuch over 100 degrees,whether they are measured by the

mercury thermometeror by the air thermometer.

Not only do the specific heats not u-nwin the same at different t. in

peratures.but the ratios betweenthem do not remain the same; so Unit

DO thermometric scale can establish the constancy of all specific heatsat

the same time. It would be interesting to examine whether the same

irregularities would obtain in gaseoussubstances,but the necessaryexperi-

ments presentalmostinsurmountableditllculiL-r

It teems probable that the irregularities of the specific beats of solid

bodiesmay beattributedto latent heat,employedin producing a commence-

ment of fusion, a softening which in many cases becomesperceptible in

these bodies long before complete fusion occurs. We can support this

opinion by the following observation:From theexperimentsof MM. Dulong

and Petit, the increaseof specific hent with the temperatureis more r.ij.i.i

84

THE SECOND LAW OF THERMODYNAMICS

According to a law due to MM. Clement and Desormes,*

established by direct experiment, water vapor, under whatever pressure it is formed, always contains the same quantity of heat

in equal weights ; this amounts to saying that the vapor when compressed or expanded without loss of heat is always in such a condition as to saturate the space which it occupies, if it is originally in this condition. Water vapor in this condition can

thus be considered a permanent gas, and should follow all the

laws of gases.

Consequently, the formula

~

A + B\ogv A'+U'\ogv

should be applicable and should agree with the table of tensions
constructed by M. Dalton from his direct experiments. We can, in fact, satisfy ourselves that our formula, with a

suitable determination of the arbitrary constants, represents
very approximately the results of experiment. The unimportant discrepancies which we find in it are no more than may

reasonably be attributed to errors of observation.!

with solids than with liquids, though the latter have a larger dilatability.
If the causewhich we have proposedto account for this irregularity is the

real one, it would disappearentirely with gases.

* t

[/it To

hdaestebremeninsheothwenbayrbRiatrnakriynecoannsdtaCnlatsusAi,mBt,hAat,thBi's,

law doesnot from data

hold.] taken

from M. Dalton's table, we must begin by calculating the volume of the

vapor by meansof its pressureand temperature,the quantity of the vapor

by weight being always constant. This is madeeasy by the laws of Ma-

riotte and Gay-Lussac. The volume will be given by the equation

in which v is the volume, t the temperature,p the pressure,and ca constant quantity which dependson the weight of the vapor and the units chosen.
The following is the table of the volumes occupied by a gram of vapor formed at various temperaturesand consequentlyunder various pressures:

ORCEOTIGt RADDEEGREESORKTXHI'EKTKOESNPSMSIKNpIElO)OMRNCPILTULHRIMEYVEATPROERSOROTHPVEVAISOPLOLIUERTXRMOPEEPRSAESGSREADM

Deg. 20 40 60 80 100

Mm. 5.060 17.32 53.00 144.6 352.1 760.0

Lit. 185.0
58.2 20.4
7.96 8.47 1.70

The first two columns in this table are taken from the Traite dePhysique 85

MKMOIRS ON
We shall now retnrn to our principal subject, the motive power of heat, from which we have already digressed too far.
We have shown that the quantity of motive power developed by the transfer of caloric from one body to another depends essentially on the temperatures of the two bodies, but we have not discussed the relation between these temperatures and the
quantities of motive power produced. It would seem at tir.-t natural enough to suppose that for equal differences of tem-
perature the quantities of motive power produced are equal that is, for example, that a given quantity of caloric passing from a body, A, kept at 100 degrees, to a body, B, kept at 50 degrees would develop a quantity of motive power equal to that which would be developed by the transfer of the same caloric from a body, #, kept at 50 degrees to a body, C, kept at zero. Such a law would indeed be a very remarkable one, hut we. do not see sufficient reason to admit it a priori. We shall examine this question by a rigorous method.
Let us suppose that the operations described on page 21 are performed successively on two quantities of atmospheric air equal in weight and volume but taken at different temperat :in >. and let us suppose also that the differences of temperature l>e-

is

i., a

of M. Biot (vol. pp. 272 mid 531). The third calculated by menusof the above formula, mid from tin- expuriincntul fact that tin-vapor of water

under atmosphericpressureoccupies volume 1700times us gn -at a>ili.it

it

which occupies when in the liquid state.

By using three numbers from the first column and the corresponding

numbers from the third, we can easily determine the constants of our

equation

_

lotr

logc

it t A + n

We shall not enter into the dotnils of the calculation necessaryto determine tin-si-quantities; will be enough for us to say that the following

- B = 1000.

& = 8.80.

satisfy sufficiently well the prescriltedconditions, so that the equation

It is a is

= 19.04
expressesvery approximately the relation existing between the volume of the vnpor and its temperature.
to be noticed that the quantity IT positive and very small. \\ hirli tends to confirm the proposition tint the specific heat of an elastic fluid in-
with the volume, but at wry slow rate. 80

THE SECOND LAW OF THERMODYNAMICS

tween the bodies A and B are the same in both cases ; thus, for example, the temperatures of these bodies will be in one

case 100 and 100 h (h being infinitely small), and in the other, 1 and 1 h. The quantity of motive power produced is in each case the difference between that which the gas fur-

nishes by its expansion and that which must be used to restore it to its original volume. Now this difference is here the same
in both cases, as >vemay satisfy ourselves by a simple argument, which we do not think it necessary to give in full ; so that the motive power produced is the same. Let us now compare the quantities of heat used in the two cases. In the first case the quantity used is that which the body A imparts to the air in order to keep it at a temperature of 100 degrees during its expansion ; in the second, it is that which the same body imparts to it to maintain its temperature at 1 degree during an exactly
similar change of volume. If these two quantities were equal
it is evident that the law which we have assumed would follow. But there is nothing to prove that it is so ; we proceed to prove

that these quantities of heat are unequal. The air which we first supposed to occupy the space abed

(Fig. 2) and to be at a temperature of 1 degree, may be made to occupy the space abef, and to acquire the temperature of 100 degrees by two different methods :
TL. It may first be heated without change of volume, and then

expanded while its temperature is kept constant.
2. It may first be expanded while its temperature is kept constant, and then heated when it has acquired its new vol-

ume.

Let a and b be the quantities first of the two operations, and

of heat b' and

used ' the

successively in the quantities used in

the second ; as the final result of these two operations is the same, the quantities of heat used in each should be equal ; we

then obtain

from which we have

a'a=b b'.

We represent by a' the quantity of heat necessary to raise the temperature of the gas from 1 to 100 degrees when it occupies

the volume abef, and by a the quantity of heat necessary to

raise the temperature of the gas from 1 to 100 degrees when it

occupies the volume abed.

37

MEMOIRS ON

The density of the air is less in the first case than in the second, and from the experiments of MM. Delaroche and Beranl. already cited on page 32, its capacity for heut should be a little

greater. As the quantity a' is greater than the quantity a, b should be

is b',
:

greater than consequently, stating the proposition generally, we may say that

is of

The quantity of hrat <lnc t<>the clntmjr

volume

of a

gas be-

/">

comes greater as the teiiifn nttiirc rai*nl.

Thus, for example, more caloric

required to maintain at

a

100 degrees the temperature of certain quantity of air whose

a1

volume

doubled than to maintain at degree t)ie tempera-

ture of the same quantity of air during similar expansion.

These unequal quantities of heat will, however, as we have

seen, produce equal quantities of motive power for equal de-

scents of caloric occurring at different heights on the thermo-

metric scale; from which we may draw the following conclu-

:

sion

of

The descent caloric prmlnn's more motive power at lower de-

ffftU of temjH'raturr limn tit higher.* Thus given quantity of heat will develop more motive

a

a

power in passing from body whose temperature

kept at

degree to another whose temperature

kept at zero than

the temperatures of these two bodies had been lul and loo

respectively. It must be said that the difference should lie very

small

;

it

if

would be zero the capacity of air for heat remained

constant in spite of changes of density. According to the ex-

periments of MM. Delaroche and Beranl. this capacity \arie>

very little, so little, indeed, that the differences notiee<l miu'ht

strictly be attributed to errors of observation or to s..me eir-

cumstances which were not taken into account.
It would be out of the question for us, with the experimental
data at our command, to determine rigorously the law which

is is
if 1

[ it '
(' if
it
/
If
l.y / ]

* The prefffling drmon*tration erroMout in contequenfeof the.attump-

Hun of the materiality of heat. The ennclwtion inform becauteof theerroneoututeof a variabletpeciftcheatof air. tutoredeonttant.at Carnotpoint*out. theefficiencyihould br. the tame at all temperaturet. The ratio of thr n,>

correct,but only
<m///tf,t/it/.be con-
../.,.../ t->th,

heatvted hould beequal to thedifferenceof tempernt'tr,i,,<itti)>liebdy a eon-

tfiint. I/if C>riun fnitftiini." At M notekiunr, (hit

Con-

t.ii.t. but thereciprocalof theabtolutetemperatureof thetoureeof htat.

THE SECOND LAW OF THERMODYNAMICS
the motive power of heat varies at different degrees of the
thermometric scale. It is connected with the law of the varia-
tions of the specific heat of gases at different temperatures, which has not been determined with sufficient exactness.* We

* If we admit that the specificheat of a gas is constant when its volume

docs not change,but only its temperaturevaries, analysis would lead us to

a relation between the motive power and the therraometric degree. We

shall now examinethe way in which this may be done; it will also give us

an opportunity of showing how someof the propositions formerly estab-

lished should be statedin algebraic form.

Let / be the quantity of motive power produced by the expansionof a

given quantity of air changing from the volume 1 litre to the volume v

litres at constanttemperature. If v increasesby the infinitely small quan-

tity dv, r will increase by the quantity dr, which, from the nature of mo-

tive power, will expansive force

be equal to the increase of which the elastic fluid then

volume
has. If

do multiplied by p representsthe

the ex-

pansive force, we shall have the equation

(1)

dr=pdv.

Let us supposethe constant temperatureat which the expansionoccurs to

be equal to t degrees centigrade. Representing by q the elastic force of

the air at the same temperature,t, occupying the volume of 1 litre, we

lp shall have from Mariotte's law

=

,

from

which

p

=

-v

Now if Pis the elastic force of the sameair always occupying the volume

1, but at the temperaturezero, we shall have from M. Gay-Lussac's law

If, for the sake of brevity, we representby N the quantity ~T>, the equa-
tion will become

by using which we have, from equation (1),
v Considering t constant,and taking the integrals of the two terms, we ob-

If we supposethat r=Q when c=l, we shall have (7=0, from which

(2)

r = ^V(<+267)logc.

This is the motive power produced by the expansion of the air at the tem-

perature t, whose volume has changed from 1 to . If insteadof working

MHMOIRS OX

shall now endeavor to determine definitively the motive power of heat, and in order to verify our fundamental proposition

at the temperaturet we work in exactly the sameway at the temperature

t+dt, the power develorp+edw/ =illybe(t +(It+2G71log r.

Subtracting (8)

equation

(2)

we

obtain
Sr = N

log

oft.

Let be the quantity of heat used to keep the temperatureof the gns
constantduring its expansion. From the discussion on page21 ir will be

the power developedby thedescentof thequantity of heat<from the degree

t +dt to the degreet. Let representthe motive power developed by the

descent of a unit of heat from t degreesto zero; since from the gi-mml

principle establishedou page 21 this quantity n should depend only on t,

it may be representedby the function J-'t, fn.ni which u = Ft.

When t increasesand becomesi-\-<it.n becomesv +du, from which

Subtracting the precedingequation we have

du=F(t+dl)-Fl = Ftdt.

This is evidently the quantity of motive power produced by the descentof

a unit quantity of heal from the degreet + dt to the degree/.
If the quantity of heat, instead of being a unit, had been<. the motive

power produced would have been

(4)

edu = eF'tdt.

But eduis the sameas(r. both being the power developed by thedescentof

the quantity of heat <from the degree t +dt to the degree t ; consequently,

edu= Si;

and, from equations (3) and (4), eF'tdt

= JVlog//

;

or, dividing by Ftdt, and representing by T the fraction ^- . which is a function of t only, we have

The equation

e = T log c

is the analytical expressionof the law statedon page 28 ; it is the samefor

all gases,since the laws we have used are common i- all If we representby * the quantity of heal required to chant''1the volume

of the ait with which we arc working from 1 to r. and the temper-iture from

zero to t, the difference between and r will IK;thequantity of heat required

to changethe temperatureof the air, while its volume remains 1. from /<m

to t. This quantity dependson / only. It will be somefunction of /, and

we shall have, if we call it U,

If wo differentiatethis equation with rc*pe<-to t only anil representl.\ /' and I' the differential coefficientsof T and U, it will become

(5)

^

40

THE SECOND LAW OF THERMODYNAMICS
that is, to show that the quantity of motive power produced is really independent of the agent used we shall choose sev-

-37is nothing other than the specific heat of the gas at constant volume,
and our equation (5) is the analytical expressionof the law statedon page
31.If we suppose the specific heat to be constant at all temperatures an

hypothesiswhich was discussedon page 34 the quantity -r will be inde-

pTenadnedntUo' fmtu, satnadl,stoobseaintidsfeypeenqdueantitoonf

(5) t;

for we

two shall

particular values of v, then haveT' = C, a con-

stuntquantity. Multiplying T and C by dtaud integrating both sides we

find

T=Ct + Cl ;

but as T=jj^ we have N T
Multiplying both sidesby dt and integrating we obtain

or, changing the arbitrary constants, and remembering that Ft is zero when t = 0, we have

(6)

Pt = Alog(l +^.

The nature of the function Ft is thus determined,and may serve us as a meansof calculating the motive power developed by any descentof heat. But this last conclusion is basedon the hypothesisof the constancyof the specific heatof a gas whose volume doesnot change an hypothesiswhich experimenthas not yet sufficiently verified. Until thereare further proofs of its validity equation (6) can only be admitted for a small part of the thermometric scale.
The first term in equation (5) represents,as we have said, the specific heat of the air occupying the volume . Experiment has taught us that this specific heat varies only slightly in spite of considerable changes of volume, so 'that the coefficient T' of log v must be a very small quantity.
If we assumethat it is zero and multiply the equation T' = Q by dt and
then integrate,we have T= C, a constantquantity.
But

from which

f-M^.

from which we may conclude by a secondintegration that Ft = At + B. 41

MEMOIRS ON

if
7*0 a
T

is a T f

eral such agents

atmospheric air, water vapor, and alcohol

vapor. Let us take first atmospheric air. The operation is effected*

according to the method indicated on page 21. We make the
following hypotheses : The air is taken under atmospheric pressure ; the tempera-
ture of the body A is j^Vr f a degree above zero and that of the body B is zero. We see that the difference is, as it should be, very small. The increase of. the volume of the air in our operution will be ^fa+yfa of the original volume ; this is a very

small increase considered absolutely, but large relatively to the difference of temperature between A and B.
The motive power developed by the two operations described on page 21 taken together will be very nearly proportional to the

increase of volume and to the difference bet ween the two pressures exerted by the air when its temperature is 0.001 and zero.

According to the law of M. Gay-Lussac, this difference is

i g OF of the elastic force of the gas, or very nearly

the atmospheric pressure.

The pressure of the atmosphere

equal to that of

jrVoT column

,, ,i
is

^;

of water 10 <jfr meters high

of this pressure equal to

that of water column 8> oirg x 10.40 meters in height.

by

As for the increase of volume,

is,

hypothesis,

jii

is, it

of the original volume that

of the volume occupied

11 + by 1

kilogram of air at zero, which

is

equal to 0.77 cubic meters,

we take into account the specific gravity of air; thus the

product,

( Tl

+

lW O.T7II

*

1

l<UO

is 1

expresses the motive power developed. This power

here

estimated in cubic meters of water raised to the height of

meter.
If we carry out the multiplications indicated, we find for the

product 0.000000372. Let us now try to determine the quantity of heat used to ob-
tain this resultthat is, the quantity transferred from the body

A

to the body B. The body furnishes

is :

is

0,

;

t

when = D zero thus
Ft tlint to nay. the motive power produced exactly proportional to the descentof the caloric. This the analytical expression of the statement made on page 88.
42

is

A is

THE SECOND LAW OF THERMODYNAMICS

1. The heat required to raise the temperature of 1 kilogram of air from zero to 0.001.
2. The quantity required to maintain the temperature of the air at 0.001 when it undergoes an expansion of

The first of these quantities of heat may be neglected, as it is very small in comparison with the second, which is, from the discussion on page 24, equal to that required to raise the tem-

perature of 1 kilogram of air under atmospheric pressure 1

degree. The specific heat of air by weight is 0.267 that of water, from
the experiments of MM. Delaroche and Berard on the specific heat of gases. If, then, we take for the unit of heat the quantity

required to raise 1 kilogram of water 1 degree, the quantity required to raise 1 kilogram of air 1 degree will be 0.267. Thus the quantity of heat furnished by the body A is

0.267 unit. This quantity of heat is capable of producing 0.000000372 unit

of motive power by its descent from 0.001 to zero. For a descent one thousand times as great, or of one degree,
the motive power will be very nearly one thousand times as

great as this, or 0.000372.
Now if, instead of using 0.267 unit of heat, we use 1000 units,
ff the motive power produced will be given by the proportion ^Z^^-IOJLQ, from which -r = = 1.395 units.

Thus if 1000 units of heat pass from a body

whose temperature is kept at 1 degree to another at zero, they will produce by their action on air

1.395 units of motive power. We shall compare this result with that which

is obtained from the action of heat on water

vapor. Let us suppose that 1 kilogram of water is
contained in the cylinder abed (Fig. 4) between the base ab and the piston cd, and let us assume also the existence of two bodies, A, B, each

maintained at a constant temperature, that of
A being higher than that of B by a very small

quantity. We shall now imagine the following

Fig

operations :

43

MEMOIRS ON

1. Contact of the water with the body A, change of the

position of the piston from cd to ef, formation of vapor at the
temperature of the body A to fill the vacuum made 1>\ tilt-

increase of the volume. We shall assume the volume atn-t

to be large enough to contain all the water in a state of

vapor ; x'. Removal of the body A, contact of the vapor with the
body B, precipitation of a part of this vapor, decrease of its elastic force, return of the piston from efto ab, and liquefaction

of the rest of the vapor with the contact of the

bboydtyheB;effect

of

the

pressure

combined

3. Removal of the body B, new contact of the water with

the body A, return of the water to the temperature of this

body, a repetition of the first operation, and so on. The quantity of motive power developed in a complete cy-

cle of operations is measured by the product of the volume

of the vapor multiplied by the difference between its tensions at the temperatures of the body A and of the body B respec-

tively.

The heat used that is, that transferred from the body A to the body B is evidently the quantity which is required to

transform the water into vapor, always neglecting the small

li quantity necessary to restore the water from the tempt -rature
of the body to that of the body A. Let us suppose that the temperature of the body A is 100
degrees and that of the body B 99 degrees. From 1C. Dahoifs table the difference of these tensions will be 26 millimetres of
mercury or 0.36 meter of water. The volume occupied by tinvapor is 1700 that of the water, so that, if we use l kilo-ram, it will be 170Q litres or 1.700 cubic meters. Thus the motive

power developed is 1.700x0.36 = 0.611 unit

of the sort which we used before. The quantity of heat used is the quantity required to trans-
form the water into vapor, the water beini: already at a ti-m|H-rature of 100 degrees. This quantity has !><u. d<t<rmiind I>y experiment ; it has been found equal to A.*><decrees, or,
-leaking with greater precision, to 550 of our units of

heat Thus it. c,i i unit of motive power result from the use of 550

units of heat. 44

THE SECOND LAW OF THERMODYNAMICS

The quantity of motive power produced by 1000 units of heat

will be given by the proportion

550 0.611

=

10x00,'

f.rom

wh, ic. h,

x

=

-611
550

=

1.112.

Thus 1000 units of heat transferred from a body maintained

at 100 degrees to one maintained at 99 degrees will produce

1.112 units of motive power when acting on the water vapor.

The number 1.112 differs by nearly ^ from 1.395, which was the number previously found for the motive power developed by

1000 units of heat acting on air ; but we must remember that
in that case the temperature of the bodies A and B were 1

degree and zero, while in this case they are 100 and 99 degrees

respectively. The difference, is indeed the same, but the tem-

peratures on the thermometric scale are not the same. In order

to obtain an exact pomparison it would be necessary to calculate

the motive power developed by the vapor formed at 1 degree

and condensed at zero, and also to determine the quantity of

heat contained in the vapor formed at 1 degree. The law of

MM. Clement and Desormes, to which we referred on page 35,

gives us this information.

The heat used in turning water

into vapor (chaleur constituante) is always the same at whatever

temperature the vaporization occurs. Therefore, since 550

degrees of heat are required to vaporize the water at the tem-

perature of 100 degrees, we must have 550 4-100, or 650 degrees,

to vaporize the same weight of water at zero.

By using the data thus obtained, and reasoning in other

respects quite in the same way as we did when the water was

at 100 degrees, we readily see that 1.290 is the motive power

developed by 1000 units of heat acting on water vapor between

the temperatures of 1 degree and zero. This number approaches 1.395 more nearly than the other.
It only differs by y1^, which is not outside the limits of prob-

able error, considering the large number of data of different

sorts which we have found it necessary to use in making this

comparison. Thus our fundamental law is verified in a particular case.*

* In a memoir of M. Petit (Annales de Chimie et de Physique,July, 1818, page294) there is a calculation of the motive power of heat applied to air and to water vapor. The results of this calculation aremuch to the advantage of atmospheric air ; but this is owing to a very inadequate way of considering the action of heat.
45

MEMOIRS ON

is 1
it-

We shall now examine the case of heat acting on alcohol vapor. The method used in this case is exactly the same as in the

case of water vapor, but the data are different. Pure alcohol

boils under ordinary pressure at 78.7 centigrade. According to MM. Delaroche ami P.t'-rard, 1 kilogram of this substance absorbs 207 units of heat when transformed into vapor at this

same temperature, 78.7. The tension of alcohol vapor at 1 degree below its boiling-
point is diminished by .,'.,. und is J. less than atmospho

pressure (this at least the result of the experiments of M. Hi-tancour, an account of which was given in the second part of M. IVony's An-h i/tff /< lliidrnnUque, pages 180, !!">).*

We find, by use of these data, that the motive ]u>\vrr developed, in acting on kilogram of alcohol at the temperatures 77.7 and 78.7, would be 0.251 unit.

This results from the use of 207 units of heat. For 1000

units

we must set the proportion

J'1

-207
''.

= 1000 *C

,

. from

which

Z= 1.230.

This number

is a

little greater than 1.112, resulting from the use

it by

if

of water vapor at 100 and 99 degrees but \vi- assume the water

;

vapor to be employed at 78 and 77 degrees, we find,

the law

of MM. Cli'ment and Desormes, 1.212 for the motive power

by

produced

1000 units of heat. As we see, this number ap-

proaches 1.230 very nearly

;

it

only differs from by -j^.

;

* a

M. Dalton thought that he hiul discovered thnt the vapors of different,

liquids exhibited equal tensionstit temperatureson tlie tin-rm.mn-trie scale

equally distant from their boiling-points this law Is, however, not rigor-

is

ously, but only approximately, correct. The same true of tlie law of

the ratio of the latent heat of vapors to their densities (see ex

from memoir of M. C. Despreiz. Annale* de Chimit et de Ptiyrique, vol.

xvi.. p. 105.and vol. xxiv.. p. 828). Questions of this nectedwith thoserelating to the motive power of heat

kind are closelyconDavy and Km .<>.,

\

recently tried to recognize the changes of tension of liquefied gases for

small changes of temperature,after having made excellent experiments

a

on the liquefaction of gnsesby theeffect of considerablepressure,

t

had In view the use of new liquids in the production of motive p>\\,

p.

(see Annale* de Chimie et fa Physiqnt, January. 1*24. 80). From the theory given above we can predict that the useof theseliquids presentsno

advantagefor the economical use of heat. The advantage could only be realized at the low temperatureat which would be possible to work, and

it

it

by the use of sources from which, for this reason, would become pos-

sible to extract caloric.

46

THE SECOND LAW OF THERMODYNAMICS
We should have liked to have made other comparisons of this kind for example, to have calculated the motive power developed by the action of heat on solids and liquids, by the freezing of water, etc. ; but in the present state of Physics we are not able to obtain the necessary data.* The fundamental law which we wish to confirm seems, however, to need additional verifications to be put beyond donbt; it is based upon the theory of heat as it is at present established, and, it must be confessed, this does not appear to us to be a very firm foundation. New experiments alone can decide this question; in the mean time we shall occupy ourselves with the application of the theoretical ideas above stated, and shall consider them as correct in the examination of the various means proposed at the present time to realize the motive power of heat.
It has been proposed to develop motive power by the action of heat on solid bodies. The mode of procedure which most naturally presents itself to our minds is to firmly fix a solid body a metallic bar, for example by one of its extremities,
and to attach the other extremity to a movable part of the machine ; then by successive heating and cooling to cause the length of the bar to vary, and thus produce some movement.
Let us endeavor to decide if this mode of developing motive
power can be advantageous. We have shown that the way to get the best results in the production of motion by the use of heat is to so arrange the operations that all the changes of temperature which occur in the bodies are due to changes of volume. The more nearly this condition is fulfilled the better the heat will be utilized. Now, by proceeding in the manner just described, we are far from fulfilling this condition ; no change of temperature is here due to a change of volume ; but the changes are all due to the contact of bodies differently heated, to the contact of the metallic bar either with the body which furnishes the heat or with the body which absorbs it.
The only means of fulfilling the prescribed condition would be to act on the solid body exactly as we did on the air in the operations described on page 18, but for this we must be able to produce considerable changes of temperature solely by the change of volume of the solid body, if, at least, we desire to
* The data lacking are the expansive force acquired by solids and liquids for u given increase of temperature,and the quantity of heat absorbed or emittedduring changesin the volume of thesebodies.
47

MKNh'IRS ON

use considerable descents of caloric. Now this seems to be impracticable, for several considerations lead ns to think that

the changes in the temperature of solids or liquids by compres-

sion and expansion are quite small.

1. We often observe in engines (in heat-engines particularly) solid parts which are subjected to very considerable forces, some-

times in one sense and sometimes in another, and although those

forces are sometimes as great as the nature of the substances employed will permit, the changes in temperature are scarcely

perceptible. 2. In the process of striking medals, of rolling plates, or of

drawing wires, metals undergo the greatest compressions to

which we can subject them by the use of the hardest and most

resisting materials. Notwithstanding this the rise in temperature is not great, for if it were, the steel tools which we u.<-in

these operations would soon lose their temper. 3. We know that it is necessary to exert a very great force
on solids and liquids to produce in them a reduction of volume

comparable to that which they undergo by cooling (for ex-

ample, by a cooling from 100 degrees to zero). Now, cooling

requires a greater suppression of by a simple reduction of volume.

caloric than would
If this reduction

be n><|iiiiv<l were pro-

duced by mechanical means the heat emitted could not change

the temperature of the body as many degrees as the cooling.
It would, however, require the use of a force which would cer-

tainly be very considerable. Since solid bodies are susceptible

to but small changes of temperature by changes of volume, and since, moreover, the condition for the best use of heat, in the de-

velopment of motive power is that any change of temperature

should be due to a change of volume, solid bodies do not seem

to be well adapted to realize this power.
This is equally true in the case of liquids ; the same reasons could be given for rejecting them.*
We shall not speak hero of the practical difficulties, which are innumerable. The movements produced by the expansion and
compression of solids or liquids can only be very small. To

extend these movements we should be forced to use complicated

* The recentexperiment*of M. Oersted on the compressibility of W.H.T hnvc shown that for a pressureof 5 atmospheresthe temperatureof the liquid undergoes no perceptible change. (See Annalet d Chimie et de Phyrique. February, 1828,p. 192.)
48

THE SECOND LAW OF THERMODYNAMICS

mechanisms and also materials of the greatest strength to trans-

mit enormous pressures ; and, finally, the successive operations conld only proceed very slowly compared with those of the

ordinary heat-engine, so that even large and expensive ma-

chines would produce only insignificant results.
Elastic fluids, gases, or vapors are the instruments peculiarly fitted for the development of the motive power of heat ; they unite all the conditions necessary for this service ; they may be

easily compressed, and possess the property of almost indefinite

expansion ; changes of volume occasion in them great changes of temperature, and finally they are very mobile, can be easily
and quickly heated and cooled, and readily transported from

one place to another, so that they are able to produce rapidly

the effects expected of them.

We can easily conceive of many machines fitted for the de-

velopment of the motive power of heat by the use of elastic

fluids, but however they are constructed in other respects, the

following conditions must not be lost sight of : 1. The temperature of the fluid should first be raised to the
highest degree possible, in order to obtain a great descent of

caloric and consequently a great production of motive power. 2. For the same reason the temperature of the refrigerator

should be as low as possible. 3. The operations must be so conducted that the transfer of
the elastic fluid from the highest to the lowest temperature

should be due to an increase of volume that is, that the cool-

ing of the gas should occur spontaneously by the effect of ex-

pansion. The limits to which the temperature of the fluid can be

raised in the first operation are determined only by the tem-

perature of combustion ; they are very much higher than ordinary temperatures. The limits of cooling are reached in the

temperature of the coldest bodies which we can conveniently use in large quantities ; the body most used for this purpose is

the water available at the place where the operation is car-

ried on. As to the third condition, it introduces difficulties in the

realization of the motive power of heat, when the object is to

profit by great differences of temperature, that is to utilize great descents of caloric. For in that case the gas must change

from a very high temperature to a very low one, by expansion,

D

49

MEMOIRS ON

which requires a great change of volume and density. To

effect this the gas must at first be subjected to a very high pressure, or it must acquire by expansion an enormous volume,

either of which conditions is difficult to realize. Tin- first

necessitates the use of very strong vessels to contain the gas

when it is at a high pressure and temperature; the second re-

quires the use of vessels of a very large size. In fact, these are the principal obstacles in the way of profit-

ably using in steam-engines a large portion of the motive power

of heat. We are of necessity limited to the use of a small de-

scent of caloric, although the combustion of coal furnishes us with the means of obtaining a very great one. In the use of

steam-engines the elastic fluid is rarely developed at a pressure

higher than atmospheres, which pressure corresponds to near-
ly lf0 degrees centigrade, and condensation is rarely effected at

a temperature much below 40 degrees; the descent of caloric

from 100 to 40 degrees is 120 degrees, while we can obtain by

combustion a descent of from 1000 to 2000 degrees.

To conceive of this better, we shall recall what we have previously called the descent of caloric : It is the transfer of heat from a body, A, at a high temperature to a body, 11,whose tem-

perature is lower. We say that the descent of caloric is lun

a

a />' is

J degrees or 1000 degrees when the difference of temperature

between the bodies

ami

100 or 1000 decrees. In

6 is

steam-engine working under pressure of atmospheres the

temperature of the boiler 100 degrees. This the tempera-

is is

a

ture of the body

A ;

it

maintained by contact with the fur-

a

nace at constant temperature of lH" decree*, ami affords

continual supply of the heat necessary to the formation of

steam.

a

a is by

The condenser the body /?;

is

it

is

maintained

means of

current of cold water at an almost constant temperature of 4<>

degrees, and continually absorbs the caloric which rarrie.l t<>

by the steam from the body A. The difference of temperature between these two bodies 16040, or 120 degrees

is

it

;

is

is

for this reason that we say that the descent of caloric in this

case 120 doL'

i<

1

il

capable of producing by combustion

higher -m-

t

peratlire than lIMM) decrees, ami the temperature, of the e.uld

is

water which we ordinarily use about 1<d>ecrees, so that we can

a

easily obtain descent of caloric of 1000 degrees, of which only

50

it

THE SECOND LAW OF THERMODYNAMICS
120 degrees are utilized by steam-engines, and even these 120 degrees are not all used to advantage ; there are always considerable losses due to useless re-establishments of equilibrium in the caloric.
It is now easy to perceive the advantage of those engines
which are called high-pressure engines over those in which the pressure is lower : this advantage depends essentially upon the power of utilizing a larger descent of caloric. The steam being produced under greater pressure is also at a higher temperature, and as the temperature of condensation is always nearly the same the descent of caloric is evidently greater.
But to obtain the most favorable results from high-pressure engines the descent of caloric must be used to the greatest advantage. It is not enough that the steam should be produced at a high temperature, but it is also necessary that it should attain a sufficiently low temperature by its expansion alone. It should thus be the characteristic of a good steam-engine not only that it uses the steam under high pressure, but that it uses it under successive.pressures which are very variable, very different from each other, and progressively decreasing.*
*This principle, which is the real basis of the theoryof the steam-engine, has beendevelopedwith great clearness by M. Clement in a memoir presentedto the Academy of Sciences a few years ago. This memoir has neverbeen printed, and I owe my acquaintancewith it to the kindness of the author. In it not only is this principle established,but applied to various systemsof engines actually in use ; the motive power of each is calculated by the help of the law cited on p. 35 and comparedwith the results of experiment. This principle is so little known or appreciatedthat Mr. Perkins, the well-known London mechanician, recently constructed an enginein which the steam, formed under a pressureof 35 atmospheres,a pressure never before utilized, experienced almost HO expansion, as we may easily be convinced by the slightest knowledge of the engine. It is composedof a single cylinder, which is very small, and at each stroke is entirely filled with steamformed under a pressureof 35 atmospheres. The steamdoes no work by expansion, for thereis no room for the expansion to take place : it is condensedas soonas it passesout of the small cylinder. It acts only under a pressureof 35 atmospheres,and not, as the bestusage would require, under progressivelydecreasing pressures. This engine of Mr. Perkins does not realize the hopes which it at first excited. It was claimed that the economyof coal in this machinewas T9ff greater than in the best machines of Watt, and that it also possessedother advantages over them. (See Annales de Ghimie et de Physique, April, 1823,p. 429.) These assertionshave not been verified. Mr. Perkins's engine may nevertheless be considered a valuable invention in that it has proved it to be
51

MEMOIRS ON

In order to show, to a certain extent, a /W^r/Wi the advantage of high-pressure engines, let us assume that the strain formed under atmospheric pressure is contained in H cylimln-

feasiblc to use steam under much higher pressures than ever before, and becausewhen properly modified it may lead us to really useful tesults.
Wait, to whom we <>\va.l-most nil the great improvementsin the strum engine, and who has brought thesemachinesto a stateof perfection which can hardly IK? surpassed, was the first to use steam under pr.>i:i-e**ivrly decreasingpressures. In many cases he checked the introduction of the steaminto the cylinder at one-half, one-third, or one-quarterof tin- stroke of the piston, which was thus completed under a picssiirc which constant ly diminished. The first enginesworking on this principle dale from 177s Walt had conceived the idea in 1769,and took out a patent for it in 17^,'
A table annexedto Watt's patent is here presented. In it lie supposes the vapor to enter the cylinder during the first quarter of the stroke of the piston, and he then calculatesthe meanpressureby dividing thestroke in;.. twenty parts:
l-ARTOWr TIIKPACTYHLFIRNOD*TEURBHBADOFTHK DKRKAT8OIKTPOARPLKMMKfR8K8O1rR-THlUUKMtV.A1POKT,H

0.05\ 0.101 Steam entering

,\11.0n0o0,,}/

i) i:> o-jo

freely from the boiler.

M1..0-0000V, Total pressure.

Quarter

.0.25 0.80

;i.ooo ) 0.810

085

0.714

0.40

0625

Half.

0.45 .0ii.5V0, 060 0.65 0.70 aO.S7O8 0.85

' The steamcut off.
and moving the piston by expansion aloue.

o 560 0o.5:0, 0..Half 0417 0.885 0875 0.888..One

the original third.

pressure.

0o.I8M11

0.90

Bottom of 095

cylinder .1.00

0.250..One-quarter.

Total

11 588

Mean pressure,11'888= 0.579.

On this showing he remarks that the meanpressureis more thnn half of

the original pressure,so that a quantity of steamequal t ne quarter \\ dl

produce an effect greater than one-half [freely intrmlncctlfmm tin hnlrr

until theendof theftrolce}. Watt hereassumesthat the i

\p:m-in

<>tfhe steamis

in

accordancewith

Mariotte's law. This assumption should not be considered correct, be-

THE SECOND LAW OF THERMODYNAMICS

cal vessel, abed (Fig. 5). under the piston cd, which at first

touches the base ab; the steam, after moving the piston from

ab to cd, will subsequently act in a manner with

which we need not occupy ourselves. Let us

suppose that after the piston has reached cd it is

forced loss of

down any of

to its

ecfalworiitch.outIteswciallpebeofcosmteparems,seodr

into the space abef, and its density, elastic force, and temperature will all increase together.
If the steam, instead of being formed under

atmospheric pressure, were produced exactly in

the state in which it is when compressed into a

abef, and if, after having moved the piston from

#%.5

ab to ef by its introduction into the cylinder, it

should move it from ef to cd solely by expansion,- the motive

power produced would be greater than in the first case. In

fact, an equal movement of the piston would take place under

the influence of a higher pressure, although this would be va-

riable and even progressively decreasing. The steam would require for its formation a precisely equal

quantity of caloric, but this caloric would be used at a higher

temperature.
It is from considerations of this kind that engines with two cylinders (compound engines) were introduced, which were invented by Mr. Hornblower and improved by Mr. Woolf. With

cause,on the one hand thetemperatureof the elasticfluid is lowered by expansion, and on the other there is nothing to show that a part of this fluid does not condenseby expansion. Watt should also have taken into account the force necessaryto expel the steam remaining after condensation, whose quantity is greater in proportion as the expansion has been carried further. Dr. Robinson added to Watt's work a simple formula to calculate the effect of the expansion of steam,but this formula is affected by the sameerrors to which we have just called attention. It has, chaolwcuelvaetiro, nbeseunffuicsieefnutllytoexcaocntsttorubcetoorfsuinsefiunrnpirsahcintigcetheWmTewhitahveatmhoeuagnhstoitf worth while to recall these facts becausethey are little known, especially in France. Engines have been constructed there after the modelsof inventors but without much appreciation of the principles on which these modelswere made. The neglect of theseprinciples has often led to grave faults. Engines which were originally well conceived have deteriorated in the hands of unskilful constructors, who, wishing to introduce unimportant improvements,have neglected fundamental considerationswhich they did not know enough to appreciate.
53

MEMOIRS ON

respect to the economy of fuel, they are considered the best en-

gines. They are composed of a small cylinder, which at each

stroke of the piston is more or less and often entirely tilled with steam, and of a second cylinder, of a capacity usually

fonr times as great, which receives only the steam which has already been used in the first one. Thus the volume of the steam at the end of this operation is at least four times its original volume. It is carried from the second cylinder direct-

ly into the condenser; but it is evident that it could he carried into a third cylinder four times as large as the second, where its

volume would become sixteen times its original volume. The chief obstacle to the use of a third cylinder of this kind is tin-

large space which it requires, and the size of the openings

which are necessary to allow the steam to escape.*

We shaU say nothing more on this subject, our object ii<>t being to discuss the details of construction of heat-engines.

These should be treated in a separate work. exists at present, at least in France, f

No such work

* It is easyto perceivethe advantagesof having two cylinders, for when there is only one the pressureon the pistonwill vary very much he!ween the beginning and end of the stroke. Also, all the portions of the machinede signed to transmit the action must be strong enough to re-ist tin- first impulse, and filled together pcifectly so as to avoid sudden motions by \\liii-h they might bedamagedand which would soon wear them out. Tins \\onld be specially true of the walking beam, the supports, tin- conm<-ting-rod, the crank, and of the first cog-wheels. In these parts tin- irreirnlarity of the impulse would be roost felt and would do tin- most damage The steam-chestwould also have to lie strong enough to resist tin- i pressureemployed,and large enough to contain the v:i|>orwhen its volume is increased. If two cylinders are used the capacity of the first need not be great, so that it is easyto give it the strength required, while the second must be large but need not be very strong.
Engines with two cylinders have been planned on proper principles hut have often fallen far short of yielding a good iwulis n.smiL'lit have been expectedof them. Thi- is the case principally IK-CMu*e the dimer>the different parts are difficult to arrange and are often not in proper proportion to each other. There are no good models of the-e engines,while there are excellent onesof those constructed after Watt's plan To this is due the it regularity which we observe in the effects produced by the former. whil thoseproduced by the latter are almost uniform.
4In the work entitled De la fifc*MM Vintral. by M. Heron de V ill, .fosse, vol. iii , p. .V) itq., we find a good description of the steamermine- M..W used in mining. The subject Ir.s b. en treatedwilh sufficient ful England in the Kneyflnptnlia Brittinnica. Some of the data which we have employed have beentaken from the latter work.

54

THE SECOND LAW OF THERMODYNAMICS

"While the expansion of the steam is limited by the dimensions of the vessels in which it dilates, the degree of condensation at which it is possible to begin to use it is only limited by the resistance of the vessels in which it is generated namely, of the boilers. In this respect we are far from having reached the possible limits. The character of the boilers in

general use is altogether bad; although the tension of the steam

is rarely carried in them beyond 4 to 6 atmospheres, they often
burst and have caused serious accidents. It is no doubt quite

possible to avoid such accidents and at the same time to make

the tension of the steam greater than that commonly employed. Besides the high - pressure engines with two cylinders of
which we have been speaking, there are also high-pressure engines with one cylinder. Most of these have been constructed by two skilful English engineers, Messrs. Trevithick and Viv-
ian. They use the steam under a very high pressure, sometimes 8 or 10 atmospheres, but they have no condenser. The steam, after its entrance into the cylinder, undergoes a certain

expansion, but its pressure is always greater than that of the atmosphere. When it has done its work, it is ejected into the
atmosphere. It is evident that this mode of procedure is en-

tirely equivalent, with respect to the motive power produced,

to condensing the steam at 100 degrees, and that we lose a part of the useful effect, but engines thus worked can dispense with

the condenser and air-pump. They are less expensive than the

others, and are not so complicated ; they take less room, and can be used where it is not possible to obtain :i current of cold water sufficient to effect condensation. In such places they possess

an incalculable advantage, since no others can be used. They are used principally in England to draw wagons for the carriage of coal on railroads, either in the interior of mines or on the

surface.

Some remarks may still be made on the use of permanent

gases and vapors other than water vapor in the development of

the motive power of heat.

Various attempts have been made to produce motive power by the action of heat on atmospheric air. This gas, in com-

parison with water vapor, presents some advantages and some

disadvantages, which we shall now examine.
1. It has this notable advantage over water since for the same volume it has a much smaller
55

vapor, that capacity for

MEMOIRS <>N

heat it cools more for an equal expansion, as is proved by what

we have previously said. We have seen the importance of effect-

ing the greatest possible changes of temperature by changes of

volume alone.

2. Water vapor can be formed only by the aid of a boiler, while atmospheric air can be heated directly by combustion within itself. Thus a considerable loss is avoided, not only in
the quantity of heat, but also in its thermometric degree. This advantage belongs exclusively to atmospheric air ; the other gases do not possess it; they would be even more difficult to

heat than water vapor.
3. In order to produce a great expansion of the air, and to cause thereby a great change of temperature, it would be necessary to subject it in the first place to rather a high pressure, to compress it by an air-pump or by some other means before heating it. This operation would require a special apparatus which does not form a part of the steam-engine. In it the water is
in a liquid state when it enters the boiler, and requires only a

small force-pump to introduce it. 4. The cooling of the vapor by the contact of the refrigerat-

ing body is more rapid and easy than the cooling of air could
be. It is true that we have the resource of eject inn it into the. atmosphere. This procedure would have the further advan-

tage that we could then dispense with a refrigerator, not always at our disposal, but in that case the air

which is
must nt

expand so far that its pressure is lower than that of the a;

phere. 5. One of the most serious drawbacks to the employment of
steam is that it cannot be used at high temperatures except with vessels of extraordinary strength. This is not true <>afir. for which there is no necessary relation between its tempera-
ture and elastic force. The air, then, seems bettor fitted than

steam to realize the motive power of the descent of caloric at

high temperatures; perhaps at low temperatures water vapor

would be preferable. We can even conceive of the possibility

of making the same heat act successively in air and in water

vapor. perature

All
of

that would be necessary the air sufficiently high,

would after it

be to keep the tem-
had Wn nsi-d. and

instead of ejecting it immediately into the atmosphere, to sur-

it, if it

round a steam-boiler with as

came directly from the

fire-box. N

THE SECOND LAW OF THERMODYNAMICS

The use of atmospheric air for the development of the mo-

tive power of heat presents very great practical difficulties which,

however, may not be insurmountable.

These difficulties once

overcome, it will doubtless be far superior to water vapor.*

As for other permanent gases, they should be finally rejected ;

they have all the inconveniences of atmospheric air without

any of its advantages.

The same may be said of other vapors in comparison with

water vapor.

* Among the attempts madeto develop the motive power of heat by the use of atmosphericair, we should notice particularly those of MM. Niepce, which were madein France severalyears ago by meansof an apparatus, called by the inventors pyreolophore. This instrument consists essentially of a cylinder, furnished with a piston, and tilled with atmospheric air at ordinary density. Into this is projected some combustible substance in a highly attenuatedform, which remains in suspensionfor a momentin the air and is then ignited. The combustion produces nearly the sameeffect as if the elastic fluid were a mixture of air and combustible gas of air and carburetted hydrogen, for example a sort of explosion occurs and a suddenexpansion of the elastic fluid, which is made use of by causing it to act altogether against the piston. This moves through a certain distance, and the motive power is thus realized. There is nothing to prevent a renewal of the air and a repetition of the first operation. This very ingenious engine, which is especially interesting on account of the novelty of its principle, fails in an esseniialparticular. The substanceusedfor the combustible (lycopodium powder, that which is used to produceflameson the stage)is so expensive, that all other advantagesare outweighed, and unfortunately it is difficult to make use of a moderatelycheap combustible, for it requires a substancethat is very finely pulverized, in which the ignition is prompt, is propngntedrapidly, and which leaveslittle or no residue.
Instead of following MM. Niepce's operationsit would seemto us better to compress the air by air-pumps and to conduct it through a perfectly sealed fire-box into which the combustible is introduced in small quantities by some mechanismwhich is easy to conceive of; to allow it to develop its action in a cylinder with a pistonor in any other envelopecapable of enlargement; to eject it finally into the atmosphere,or even to pass it under a steam-boilerin order to utilize its remaining heat.
The chief difficulties which we should have to meet in this mode of operation would be the enclosure of the fire-box in a sufficiently solid envelope, thesuitablecontrol of the combustion,the maintenanceof a moderate temperaturein the severalparts of the engine, and the prevention of O; rapid deteriorationof the cylinder and piston. We do not consider these difficulties insurmountable.
It is said that successfulattemptshavebeenmadein England to develop motive power by the action of heat on atmosphericair. We do not know what theseare, if, indeed, they have really beenmade.
57

MEMOIRS ON

1.1

It would no doubt be preferable if there were an abundant

supply of a liquid which evaporated at a higher temperature

than water, the specific heat of whose vapor was less for equal

volume, and which did not injure the metals used in the con-

struction of an engine ; but no such body exists in natim-. The use of alcohol vapor has been suggested, and engines

have even been constructed in order to make it possible, in \\hii-li the mixture of the vapor with the water of condensation

is avoided by applying the cold body externally instead of in-

troducing it into the engine.
It was thought that alcohol vapor possessed a notable advan-

tage on account of its having a greater tension than that of

water vapor at the same temperature. We see in this only an-

other difficulty to be overcome. The principal defect of water

vapor is its excessive tension at a high temperature, and this

defect is still more marked in alcohol vapor. As for the ad-

vantage which it was believed to have with respect to a larger

production of motive power, we know from the principles stated

above that they are imaginary.

Thus it is with the use of water vapor and atmospheric air

that the future attempts to improve the steam-engine shun

be made. AH efforts should be directed to utilize by means of

these agents the largest possible descents of caloric.

by

We shall conclude

showing how far we are from the reali-

1

\\ by

zation,

means already known, of all the motive power of the

combustibles.
A kilogram of coal burned in the calorimeter furnishes

quantity of heat capable of raising the tempi-rat uru of about

7000 kilograms of water degree that is, from the definition

it

given (page 43) furnishes 7000 units of heat. The largest

by

descent of caloric which can be realized measured

is

the dif-

ference of the temperature produced by combustion and that
of the refrigerating body. It difficult to see any limit to the

is is

temperature of combustion other than that at which the com-

bination of the combustible with oxygen

is

effected. Let us

t< is

assume, however, that this limit 1000 degrees, which

cer-

tainly within the bounds of truth. We shall assume the m

peratnre of the refrigerator to be degrees.

In >4r).

have calculated approximately (page

the quantity of

by

motive power developed

1000 units of at in passing from

the temperature 100 to the temperature 99, and have found

a

THE SECOND LAW OF THERMODYNAMICS

it to be 1.112 units, each equal to 1 meter of water raised 1

meter.
If the motive power were proportional to the descent of caloric, if it were the same for each thermometric degree, nothing would

be easier than to estimate it from 1000 to degrees. Its value

would be

1.112 x 1000 = 1112.

But as this law is only approximate, and perhaps at high temperatures departs a good deal from the truth, we can only make a very rough estimate. Let us suppose the number 1112 to be reduced one-half that is, to 560.
Since one kilogram of coal produces 7000 units of heat, and since the number 560 is referred to 1000 units, we must multiply it by 7, which gives us
7 x 560 = 3920,

which is the motive power of one kilogram of coal. In order to compare this theoretical result with the results
of experiment, we shall inquire how much motive power is actually developed by one kilogram of coal in the best heat-engines
known. The engines which have thus far offered the most advanta-
geous results are the large engines with two cylinders used in the pumping out of the tin and copper mines of Cornwall. The best results which they have furnished are as follows : Sixtyfive million pounds of water have been raised one English foot
by the burning of one bushel of coal (the weight of a bushel is 88 Ibs.). This result is equivalent to raising 195 cubic meters of water one meter by the use of one kilogram of coal, which
consequently produces 195 units of motive power.*

* The result given here was furnished by an engine whose large cylin-

der out

was 35 inches in diameter,with a 7-foot stroke ; it is used to one of the mines of Cornwall, called " Wheal Abraham." This

pump result

should in a way be consideredasan exception,for it only was accomplished for a short time during one month. A product of 30 million Ibs. raised

one English foot by a bushel ered to be an excellentresult

of for

coal weighing 88 a steam-engine.

Ibs. It is

is generally consid-( sometimesreached

by the engines made on Watt's system,but has rnrely been exceeded.

This result expressedin French units is equal to 104000kilograms raised

one meter by the burning of one kilogram of coal.

By what we ordinarily understandas one horse-power in thecalculation

MEMOIRS ON

195 units are only one-twentieth of 3920, the theoretical max-

imum ; consequently only ^ of the motive power of the combus-

tible has been utilized.

We have, moreover, chosen our example from among the

best steam-engines known. Most of the others have been vrry

inferior. For example, G'haillot's engine raises '-.'<c>ubic meters of water 33 meters in consuming 30 kilograms of coal, which is

equivalent to 22 units of motive power to 1 kilogram, a result

nine times less than that cited above, and one hundred and

eighty times less than the theoretical maximum.

We should not expect ever to employ in practice all the motive power of the combustibles used. The efforts which one

would make to attain this result would be even more harmful
than useful if they led to the neglect of other important considerations. The economy of fuel is only one of the conditions

which should be fulfilled by steam-engines ; in many cases it is

only a secondary consideration.

It must often yield the prece-

dence to safety, to the solidity and durability of the engine, to the space which it must occupy, to the cost of its construction, etc. To be able to appreciate justly in each case the consider-

ations of convenience and economy which present themselves,

to be able to recognize the most important of those which are

only subordinate, to adjust them all suitably, and finally to

reach the best result by the easiest method such should be the

power of the man who is called on to direct and co-ordinate the

labors of his fellow-men, and to make them concur in attaining

a useful purpose.

BIOGRAPHICAL SKETCH
NICOLAS-LEON A RD-S ADI CARXOT was born in Paris on June 1, 17% ; the son of the illustrious engineer, soldier, and statesman who played so prominent a part in the history of France during the Revolution. He was educated at the Keolc I'i.l\ -
of the efficiency of steam-engines,a 10 horse-power engine should raise 10 x 75, or 750 kilograms 1 meter iu a second, or 750 x 8600= 3700000 kilograms 1 meterin an hour.
If we suppose eacli kilogram of coal to raise 104000kilograms, it is necessaryto divide 2700000by 104000to find the coal burned in one hur
by the 10horse-powerengine,which gives us YoY = 28 kilograms. Hut it is very rare that a 10horse-powerengineconsumeslessthan26kilograms of coal an hour.
60

THE SECOND LAW OF THERMODYNAMICS
technique, and served for several years as an officer of engineers and on the general staff. His inclinations towards the study of science were so strong that they controlled the whole course of his life. While still engaged in his profession he devoted such time as he could spare to scientific investigations, and he at last resigned from the army in order to obtain more leisure for studious pursuits. He died of the cholera on August 24, 1832. The memoir on the "Motive Power of Heat" is the only one which he published. It shows that he possessed a mind able to penetrate to the heart of a question, and to invent general methods of reasoning. The extracts from his note-book, published by his brother, indicate that he was also fertile in devising experiments. It is interesting to note that the doubt of the validity of the substantial theory of heat, expressed by him in his memoir, developed later into complete disbelief, and that he not only adopted the mechanical theory of heat, but planned experiments to test it similar to those of Joule, and calculated that the mechanical equivalent of heat is equal to 370 kilogrammeters.

ON THE MOTIVE POWER OF HEAT, AND ON THE LAWS WHICH CAN BE DEDUCED FROM IT FOR THE THEORY OF HEAT
BY
R. CLAUSIUS
(Poggendorff's Annalen, vol. Ixxix., pp. 376and 500. 1850)

CONTEXTS
Work of Carnot and Clapeyron Dynamical Theoryof Heat Equivalenceof Heat and Work Camot't Cycle Application to Changeof State SecondLate of Thermodynamics Carnot't /*/<// Application of Clapeyron'*Equation MechanicalEquivalent,of Heat

MM 65 66 67
78 88 90 .):{ 105

ON THE MOTIVE POWER OF HEAT, AND ON THE LAWS WHICH CAN BE DEDUCED FROM IT FOR THE THEORY OF HEAT
BY
R. CLAUSIUS

SINCE heat was first used as a motive power in the steam-
engine, thereby suggesting from practice that a certain quantity of work may be treated as equivalent to the heat needed to produce it, it was natural to assume also in theory a definite rela-
tion between a quantity of heat and the work which in any possible way can be produced by it, and to use this relation in drawing conclusions about the nature and the laws of heat it-
self. In fact, several fruitful investigations of this sort have already been made ; yet I think that the subject is not yet ex-
hausted, but on the other hand deserves the earnest attention
of phvsicists, partly because serious objections can be raised to
the conclusions that have already been reached, partly because other conclusions, which may readily be drawn and which will
essentially contribute to the establishment and completion of the theory of heat, still remain entirely unnoticed or have not yet been stated with sufficient definiteness.
The most important of the researches here referred to was that of S. Carnot,* and the ideas of this author, were afterwards given analytical form in a very skilful way by Clapeyron.f Carnot showed that whenever work is done by heat and

* Reflexionssur la puissancemotricedu developpecr ettepuissance,par 8. Carnot.

feu, etsur Us machinespropresa
Paris, 1824. I have not been

able to obtain a copy of this book, and am acquaintedwith it only through

the work of Clapeyron and Thomson, from the latter of whom are quoted

the extractsafterwards given.

f Journ. deI'ficolePolytechniquev, ol. xix. (1834),andPogg. Ann., vol. lix.

E

65

MKMoIKS ON

no permanent change occurs in the condition of the work in <r

body, a certain quantity of heat passes from a hotter to a c>|.|rr body. In the steam-engine, for example, by means of the

steam which is developed in the boiU'r and precipitated in the

condenser, heat is transferred from the grate to the eonden> T.

This transfer he considered as the heat change, correspond in;: to the work done, lie says expressly that no heat is lost in tin-

padrodcses:s,"

but that This fact

the is

quantity of not doubted

heat remains unchanged, ; it was assumed at first

and with-

out investigation, and then established in many cases by calorimetric measurements. To deny it would overthrow tin- whole theory of heat, of which it is the foundation." I am nut awan-. however, that it has been sufficiently proved by experiment

that no loss of heat occurs when work is done ; it may, perhaps. on the contrary, be asserted with more correctness that even if such a loss has not been proved directly, it has yet been sh>wn

by other facts to be not only admissible, but even highly prob-
able. If it be assumed that heat, like a substance, cannot

diminish in quantity, it must also be assumed that it cannot increase. It is, however, almost impossible to explain the heat produced by friction except as an increase in the quantity of heat. The careful investigations of Joule, in whieh heat is

produced in several different ways by the application of me-

chanical work, have almost certainly proved not only the pos-

sibility of increasing the quantity of heat in any ciivumst

but also the law that the quantity of heat developed i- proportional to the work expended in the operation. To this it mu.-t be added that other facts have lately become known \\hich support the view, that heat is not a substance, but con>i.-i> in a
motion of the least parts of bodies. If this view is correct, it

is admissible to apply to heat the general mechanical principle that a motion may be transformed into work, and in siu-h a manner that the loss of //.->rira is proportional to the work ac-

complished.

These facts, with which Carnot also was well acquainted, and

the importance of which he has expressly re -oirnixi-d. almost compel us to accept the equivalence between heat and work, on the modified hypothesis that the accomplishment of work requires not merely a change in the >iistribution of heat, hut also an actual consumption of heat, ami that, conversely, heat can

be developed again by the exp' enditure of work. M

THE SECOXD LAW OF THERMODYNAMICS

In a memoir recently published by Holtzmann,* it seems at

first as if the author latter point of view.

intended He says

to (p.

co7n) s:id"erThtheeacmtiaotnteroffrtohme

this heat

supplied to the gas is either an elevation of temperature, in

conjunction with an increase in its elasticity, or mechanical

work, or a combination of both, and the mechanical work is

the equivalent of the elevation of temperature. The heat can

only be measured by its effects ; of the two effects mentioned the mechanical work is the best adapted for this purpose, and
it will accordingly be so used in what follows. I call the unit

of heat the heat which by its entrance into a gas can do the mechanical work a that is, to use definite units, which can lift a

kilograms through 1 meter." Later (p. 12) he also calculates the numerical value of the constant a in the same way as Mayer

had already done,f and obtains a number which corresponds

with the heat equivalent obtained by Joule in other entirely

different ways. In the further extension of his theory, how-

ever, in particular in the development of the equations from

which his conclusions are drawn, he proceeds exactly as Clapey-

ron did, so that in this part of his work he tacitly assumes that

the quantity of heat is constant.

The difference between the two methods of treatment has

been much more clearly grasped by W. Thomson, who has ex-

tended Carnot's discussion by the use of the recent observations
of Regnault on the tension and latent heat of water vapor. J He

speaks of the obstacles which lie in the way of the unrestricted

assumption of Carnot's theory, calling special attention to the researches of Joule, and also raises a fundamental objection

which may be made against it. Though it may be true in any case of the production of work, when the working body has re-

turned to the same condition as at first, that heat passes from a warmer to a colder body, yet on the other hand it is not gener-

ally true that whenever heat is transferred work is done. Heat

can be transferred by simple conduction, and in all such cases,
if the mere transfer of heat were the true equivalent of work,

there would be a loss of working power in Nature, which is

hardly conceivable. Nevertheless, he concludes that in the

* Ueberdie Warmeund Elasticit&t der Oaseund Dampfe, von C. Holtzmann, Mannheim, 1845: also Pogg. Ann., vol. 72a.
f Ann. der Chem.und Pfiarm. of WOhler and Liebig, vol. xlii., p. 239. J Transactionsof UieRoyal Societyof Edinburgh, vol. xvi.
67

MEMOIRS ON

present state of the science the principle adopted by Carnot is

still to he taken as the of the motive power of

most heat,

probable saying :

"baIfsiswefoarbaanndionnvetshtiigsatpiorinn-

ciple, we meet with innumerable other difficulties insuperable

without further experimental investigation, and an entire it--

I construction of the theory of heat from its foundation."* believe that we should not be daunted by these difficulties,

but rather should familiarize ourselves as much as possible

with the consequences of the idea that heat is a motion, since

it is only in this way that we can obtain the means \\here\\iih
to confirm or to disprove it. Then, too, I do not think the

difficulties are so serious as Thomson does, since even though

we must make some changes in the usual form of presentation,
yet I can find no contradiction with any proved facts. It is
not at all necessary to discard Caruot's theory entirely, a step which we certainly would find it hard to take, since it

has to some extent been conspicuously verified by experience.
A careful examination shows that the new method does not

stand in contradiction to the essential principle of Carnot, but

only to the subsidiary statement that no hmt /> l^t, sim-e in the production of work it may very well be the case that at the

same time a certain quantity of heut is consumeti and another

quantity transferred from a hotter to a colder body, and hoth

quantities of heat stand in a definite relation to the work that is done. This will appear more plainly in the sequel, and it

will there be shown that the consequences drawn fmm the two

assumptions are not only consistent with one another, but are

even mutually confirmatory.

1. CONSEQUENCES OF THE PRINCIPLE OP THE EQl'IY \U.\- K
OF HEAT A Xli \\oKK
We shall not consider here the kind of motion which ran lie conceived of as taking place within bodies, further than to assume in general that the particle-; of l>odie>an- in motion, ami that their heat is the measure of their ris viva, or rath, r still more generally, we shall only lay down a principle condition.-.! by that assumption as a fundamental prineiple, in the words: In all cases in which work is produced by the agency of li.at. a quantity of heat is consumed which is proportional to the
* Math, and Phyt. I\ijr, vol. p. 119,note.
68

I.,

THE SECOND LAW OF THERMODYNAMICS

work done ; and, conversely, by the expenditure of an equal

quantity of work an equal quantity of heat is produced.

Before we proceed to the mathematical treatment of this

principle, some immediate consequences may be premised

which affect our whole method of treatment, and which may

be understood without the more definite demonstration which

will be given them later by our calculations. It is common to speak of the total heat of bodies, especially

of gases and vapors, by which term is understood the sum of

the free and latent heat, and to assume that this is a quantity

dependent only on the actual condition of the body considered,
so that, if all its other physical properties, its temperature, its

density, etc., are known, the total heat contained in it is com-

pletely determined. This assumption, however, is no longer admissible if our principle is adopted. Suppose that we are

given a body in a definite state for example, a quantity of gas

with the temperature t and the volume t' and that we subject

it to various changes of temperature and volume, which are

such, however, as to bring it at last to its original state again.

According to the common assumption, its total heat will again be the same as at first, from which it follows that if during

one part of its changes heat is communicated to it from with-

out, the same quantity of heat must be given up by it in the

other part of its changes. Now with every change of volume

a certain amount of work must be done by the gas or upon

it it

since by its expansion

overcomes an external pressure, and

since its compression can be brought about only by an exertion
of external pressure. If, therefore, among the changes to which

has been subjected there are changes of volume, work must

be done upon

and by it. It not necessary, however, that

is it
it is

at the end of the operation, when

again brought to its

original state, the work done by shall on the whole equal that done upon it, so that the two quantities of work shall

counterbalance each other. There may be an excess of one or

the other of these quantities of work, since the compression

a a

may take.place at higher or lower temperature than the ex-

pansion, as will be more definitely shown later on. To this

it it

excess of work done by the gas or upon there must corre-

spond, by our principle, proportional excess of heat consumed

or produced, and the gas cannot give up to the surrounding

medium the same amount of heat as receives.

it,

it

MKMnlRS OX

The same contradiction to the ordinary assumption about the total hi-at may be presented in another way. If tin- gas at

/ and i' is brought to the higher temperature /, and tin- larger

volume t>j, the quantity of heat which must be imparted to it

is, on that assumption, independent of the way in which the

change is brought about ; from our principle, however, it is dif-

ferent, according as the gas is first heated wink- its vohun

is constant, and then allowed to expand at the <<instant tem-

peratnre /,, or is first expanded at the constant temperature/,,.

and then heated, or as the expansion and heating aiv inter-

changed in any other way or even occur together, since in all

these cases the work done by the gas is different.
In the same way, if a quantity of water at the temperature

/ is changed into vapor at the temperature /, and of the

volume r,. it will make a difference in the amount of lu-at

needed if the water as such is first heated to /, and then

evaporated, or if it is evaporated at / and the vapor then

if it
/-, /,.

brought to the required volume and temperature.

and or

finally

the evaporation occurs at any intermediate tempera-

ture.

From these considerations and from the immediate applica-

tion of the principle, may easily be seen what conception

must he formed of luti-nt heat. I'sing again the example al-

ready employed, we distinguish in the quantity of heat which

must be imparted to the water during its changes the t'rtr and the Inti-nt heat. Of these, however, we may consider only the

former as The latter

really present not merely,

in the as its

vapor name

that has l.ecii formed.
implies. r//, >/,/,,/ from

our perception, hut

n<nr}n/< jm:cnf

nmsitinnl during

the changes in doing work.

In the heat consumed we must still introduce distinction

that to say, the work done of two kinds, l-'irst. there

certain amount of work done in overcoming the mutual attrac-

tions of the particles of the water, and in M-parating them to

such a distance from one another. that they are in the Mate of

vapor. Secondly, the vapor during its evolution iyu>t push

back an external pressure in order to make room for itself.

The former work we shall call the i/i/rnml. the latter the t-

W ln-H'il work, and shall partition the latent heat accordingly.

can make no difference with respect to the //,/,

work

It
/ /,,
/

whether the evaporation goes on at or at

or at any intcr-

70

is is
it is is ; it is :i i> a

THE SECOND LAW OF THERMODYNAMICS
mediate temperature, since we must consider the attractive force of the particles, which is to be overcome, as invariable.*
The external work, on the other hand, is regulated by the pressure as dependent on the temperature. Of course the same is true in general as in this special example, and therefore if it was said above that the quantity of heat which must be imparted to a body, to bring it from one condition to another, depended not merely on its initial and final conditions, but also on the way in which the change takes place, this statement refers only to that part of the latent heat which corresponds to the external work. The other part of the latent heat, as also the free heat, are independent of the way in which the changes take place.
If now the vapor at t l and v, is again transformed into water,
work will thereby be expended, since the particles again yield to their attractions and approach each other, and the external pressure again advances. Corresponding to this, heat must be produced, and the so-called liberated heat which appears during the operation does not merely come out of concealment but is actually made new. The heat produced in this reversed operation need not be equal to that used in the direct one, but that part which corresponds to the external work may be greater or less according to circumstances.
We shall now turn to the mathematical discussion of the subject, in which we shall restrict ourselves to the consideration of the permanent gases and of vapors at their maximum density, since these cases, in consequence of the extensive knowledge we have of them, are most easily submitted to calculation, and besides that are the most interesting.
Let there be given a certain quantity, say a unit of weight, of a permanent gas. To determine its present condition, three magnitudes must be known : the pressure upon it, its volume,
' * It cannot he raised,as an objection to this statement,that water at t, has less cohesion than at.tt, and that therefore less work would be needed to overcomeit. For a certain amount of work is used in diminishing the cohesion, which is done while the water as such is heated,and this must be reckoned in with that done during the evaporation. It follows at once that only a part of the heat, which the water takes up from without whilfe it is being heated,is to be consideredas free heat, while the remainder is used in diminishing the cohesion. This view is also consistentwith the circumstance that water has so much greatera specific heat than ice, and probably also than its vapor.
71

MEMOIRS ON

and its temperature. These magnitudes are in a mutual relationship, which is expressed by the combined laws of Mariotte and Gay-Lussac*, and may be represented by the equation :

(I.)

pv=R(a+t),

where p, v, and t represent the pressure, volume, and tem-

perature of the gas in its present condition, a is a constant.

the same for all gases, and

also constant, which in its

* 'o a is

if /,' is

complete form

is

~^i jo *><a>nt

are tne corresponding

values of the three magnitudes already mentioned for any other

condition of the gas. This last constant in so far different

for the different gases that

it

is

inversely proportional to their

specific gravities.
It true that Regnault has lately shown, by very careful

a

is

is

investigation, that this law not strictly accurate, yet the de-

partures from

are in the case of the permanent gases \erv

it

it is

small, and only become of consequence in the case of those gases which can be condensed into liquids. From this seems to follow that the law holds with greater accuracy the more

removed the gas from its condensation point with respect to

pressure and temperature. We may therefore, while the accuracy of the law for the permanent gases in their ordinary

is a
it is

condition

so great that can be treated as complete in most

investigations, think of limiting condition for each gas. in

which the accuracy of the law actually complete. We shall.

in what follows, when we treat the permanent gases as sin-h.

assume this ideal condition.

According to the concordant investigations of Regnault and
- Magnus, the value of for atmospheric air equal to 0.003G65,

the temperature

reckoned in centigrade degrees from the

freezing-point.

Since, however, as has been mentioned, the

gases do not follow the M. and G-. law exactly, the same value

- of will not always be obtained,

the measurements are

made in different circumstances.

The number here given

holds for the case when air taken at under the pressure of

one atmosphere, and heated to 100 at constant volume, ami the

* This law will hereaflcr. for brevity, be called tin- M. ami <} law, anil Muriuiu-'a law will be called the M. law.
78

if
is is
if
is

THE SECOND LAW OF THERMODYNAMICS

increase of its expansive force observed. If, on the other hand,
the pressure is kept constant, and the increase of its volume observed, the somewhat greater number 0.003670 is obtained.
Further, the numbers increase if the experiment is tried
under a pressure higher than the atmospheric pressure, while
they diminish somewhat for lower pressures. It is not there-
fore possible to decide with certainty on the number which should be adopted for the gas in the ideal condition in which naturally all differences must disappear ; yet the number 0.003665 will surely not be far from the truth, especially since this number very nearly obtains in the case of hydrogen, which probably approaches the most nearly of all the gases the ideal condition, and for which the changes are in the opposite sense
to those of the other gases. If we therefore adopt this value

of

we obtain

In consequence of equation (I.) we can treat any one of the three magnitudes p, v, and t for example, p as a function of
the two others, v and t. These latter then are the independent
variables by which the condition of the gas is fixed. We shall
now seek to determine how the magnitudes which relate to the
quantities of heat depend on these two variables.
If any body changes its volume, mechanical work will in general be either produced or expended. It is, however, in most cases
impossible to determine this exactly, since besides the external work there is generally an unknown amount of internal work done. To avoid this difficulty, Carnot employed the ingenious method
already referred to of allowing the body to undergo its various changes in succession, which are so arranged that it returns at
last exactly to its original condition. In this case, if internal work is done in some of the changes, it is exactly compensated for in the others, and we may be sure that the external work,
, which remains over after the changes are completed, is all the work that has been done. Clapeyron has represented this process graphically in a very clear way, arid we shall follow his presentation now for the permanent gases, with a slight alteration
rendered necessary by our principle.
In the figure, let the abscissa oe represent the volume and the ordinate ea the pressure on a unit weight of gas, in a condition in which its temperature = t. We assume that the gas
73

MEMOIRS ON

is contained in an extensible envelope, which, however, cannot
exchange heat with it. If, now. it is allowed to expand in this envelope, its temperature would fall if no heat were imparted to it. To avoid this, let it be put in contact, during its expansion, with a body, A, which is kept at the constant tempera-
ture t, and which imparts

just so much heat to the

gas that its temperature also

remains equal tot. During

this expansion at constant

temperature, its pressure

diminishes according to the

M. law, and may be repre-

sented by the ordinate of a

curve, ab, which is a por-

Fig. I

tion of an equilateral hyperbola. When the volume

of the gas has increased in this way from oe to of, tin- body .1

is removed, and the expansion is allowed to continue without the introduction of more heat. The temperature will

then fall, and the pressure diminish more rapidly than before.

The law which is followed in this part of the process may Inrepresented by the curve be. When the volume of the gas has

increased in this way from of to Of/, and its temperature has

fallen from t to T, we it again to its original

begin to volume

compress
oe. If it

it, in order to restore were left to itself its

temperature would airain rise. This, however, we do not permit, but bring it in contact with a body, li, at the constant tem-

perature r, to which it at once gives up the heat that is pro-

duced, so that it keeps the temperature T ; and while it is in contact with this body we compress it so far (by the amount ////)

that the remaining compression he is exactly sufficient to raise its temperature from T to /. if during this last coinpn -<-ion it gives up no heat. During the former compression the piv.-smv
increases according to the M. law. and is represented l,\ tic portion cd of an equilateral hyperbola. During the latter, on

the other hand, the increase is more rapid and is represented

by the curve da. This curve must end exactly at n. for since

at the end of the operation the volume and temperature have again their original values, the same must be true of the

pressure also, which is a function of them both. The gas is

74

THE SECOND LAW OF THERMODYNAMICS

therefore in the same condition again as it was at the begin-

ning. Now, to determine the work produced by these changes, for

the reasons already given, we need to direct our attention only

to the external work. During the expansion the gas does work, which is determined by the integral of the product of the dif-

ferential of the volume into the corresponding pressure, and

is therefore represented geometrically by the quadrilaterals

eabf and fbcg. During the compression, on the other hand, work is expended, which is represented similarly by the quadrilaterals (jcdk and hdae. The excess of the former quantity of

work over the latter is to be looked on as the whole work produced during the changes, and this is represented by the quad-

rilateral abed.
If the process above described is carried out in the reverse

order, the same magnitude, abed, is obtained as the excess of

the work expended over the work done.
In order to make an analytical application of the method just described, we will assume that all the changes which the

gas undergoes are infinitely small. We may then treat the

curves obtained as straight lines, as they are represented in the

accompanying figure. \Ve may also, in determining the area

of the quadrilateral abed, consider it a par-

allelogram, since the

error arising there-

from can only be a quantity of the third

order, while the area

itself is a quantity of

the second order. On

this assumption, as

may easily be seen,

the area may be represented by the product

Fig.2

ef.bk, if k is the point in which the ordinate bf cuts the lower

side of the quadrilateral.

The magnitude bk is the increase

of the pressure, while the gas at the constant volume of has

its temperature raised from r to t that is, by the differential

t T= dt. This magnitude may be at once expressed by the

aid of equation (I.) in terms of v and t, and is 75

MEMO IKS ON
* WT
If, further, we denote the increase of volume ef by dv, we obtain the area of the quadrilateral, and so, also,

(1)

The work done =

We must now determine the heat consumed in these changes.
The quantity of heat which must be communicated to a gas, while it is brought from any former condition in a definite way to that condition in which its volume = v and its temperature = t, may be called Q, and the changes of volume in the above process, which must here be considered separately, may l>erepresented as follows: efby dv, fig by d'v, eh by St>a, nd fgby 3'r.
During an expansion from the volume oe v to the volume of v 4- dv at the constant temperature /, the gas must receive the quantity of heat

di-

and correspondingly, during an expansion from oh v + 2t to

[8^-K-l* og = v + 2r + d'v at the temperature t
heat,

dl, the quantity of

In the case before us this latter quantity must be taken as
negative in the calculation, because the real process was a compression instead of the expansion assumed. Pnrini: tin- expansion from of to og and the compression from oh to oe, the gas h:is neither gained nor lost heat, and hence the quantity of
,*,,,/ heat which the gas has received in excess of that which it has
given up that is, the heat m,

The magnitudes Iv and d'v must be eliminate'! fn"n this expression. For this purpose we have first, immediately from the inspection of the figure, the following equation :
l-'r<>inthe condition that during the compression from // to ml therefore also conversely during an expansion from <><
to oh occurring under the same conditions, and similarly dur76

THE SECOND LAW OF THERMODYNAMICS
ing the expansion from of to og, both of which occasion a fall of temperature by the amount dt, the gas neither receives nor gives up heat, we obtain the equations

Eliminating from these three equations and equation (2) the three magnitudes d'v, cv, and I'v, and also neglecting in the development those terms which, in respect of the differentials, are of a higher order than the second, we obtain

(3)

The

heat consumed

=

-LL dt

(\(-dv

1 }

d4v- (\-(artr }I J dvdt.

If we now return to our principle, that to produce a certain

amount of work the expenditure of a proportional quantity of

heat is necessary, we can establish the formula
_ The heat consumed .

The work done
where A is a constant, whicli denotes the heat equivalent for the unit of work. The expressions (1) and (3) substituted in
this equation give

R.dvdt
v or

(IL)

dt \dv) dv \dt

We may consider this equation as the analytical expression of our fundamental principle applied to the case of permanent
gases. It shows that Q cannot be a function of v and t, if these variables are independent of each other. For if it were, then by the well-known law of the differential calculus, that if a function of two variables is differentiated with respect to botji of them, the order of differentiation is indifferent, the right-
hand side of the equation should be equal to zero. The equation may also be brought into the form of a complete
differential equation, 77

MK MM IKS ON
r in which is an arbitrary function of v and /. This differen-
tial equation is naturally not integrable, but becomes so only if a second relation is given between the varial>K-s. by which / may be treated as a function of t'. The reason for this is found in the last term, and this corresponds exactly to the >.rlirnal work done during the change, since the dilTerential of this work is ]><h\from which we obtain
if we eliminate/) by means of (I.). We have thus obtained from equation (Il.rt) what was in-
troduced before as an immediate consequence of onr principle. that the total amount of heat received by the gas durin.ir a change of volume and temperature can be separated into two parts, one of which, T, which comprises the //>, heat that has entered and the heat consumed in doing internal work, if any such work has been done, has the properties which are commonly assigned to the total heat, of being a function of / and /. and of being therefore fully determined by the initial and final conditions of the gas, between which the transformation has taken place; while the other part, which comprises the heat consumed in doing external work, in dependent not only on the terminal conditions, but on the whole course of the changes between these conditions.
Before we undertake to prepare this equation for further conclusions, we shall develop the analytical e\|iiv-.-ion of our fundamental principle for the case of vapors at their maximum density.
In this case we have no right to apply the M. and (i. law, and so must restrict ourselves to the principle alone. In order to obtain an equation from it, we again use the method given by Carnot and graphically presented by Clapcyion. \\ith a slight modification. Consider a liquid contained in a \c-el impenetrable by heat, of which, however, only a part is filled by the liqni I. while the rest is left free for the vapor, \\hirh is at the maximum density corresponding to its temperature./. Tho total volume of both liquid and vapor is represented in the accompanying figure by the abscissa oe, and the pressure of the
78

THE SECOND LAW OF THERMODYNAMICS

vapor by the ordinate ea. Let the vessel now yield to the pressure and enlarge in volume while the liquid and vapor are in contact with a body, A,

at the constant temperature /. As the volume increases,

more liquid evaporates, but

the heat which thus becomes

latent is supplied from the body A, so that the temper-

ature, and so also the press-

ure, of the vapor remain
unchanged. If in this way

is t
is is
ff3

the total volume is increased

from oe to of, an amount of

Fi

external work done which

represented by the rectangle eabf. Now remove the body A and let the vessel increase in volume still further, while heat

can neither enter nor leave it. In this process the vapor already

present will expand, and also new vapor will be produced, and

in consequence the temperature will fall and the pressure dimin-

ish. from

Let this process go on until the temperature
to r, and the volume has become oy. If the

has fall

changed of press-

ure during this expansion

represented by the curve be, the

external work done in the process =fbcg. Now diminish the volume of the vessel, in order to bring the

liquid with its vapor back to its original total volume, oe; and

let this compression take place, in part, in contact with the body

B at the temperature r, into which body all the heat set free

by the condensation of the vapor will pass, so that the temper-

ature remains constant and = r, in part without this body, so

that the temperature rises. Let the operation be so managed

is is

that the first part of the compression

carried out only so far

(to oh) that the volume he still remaining

exactly such that

compression through

it

will raise the temperature from to

r

again. During the former diminution of volume the pressure

remains invariable, = gc, and the external work employed

equal to the rectangle gcdh. During the latter diminution of

is is ,

volume the pressure increases and represented by the curve

da, which must end exactly at the point

since the original

pressure, ea, must correspond to the original temperature,

The work employed in this last operation

= hdae. At the

79

is t

t.

MEMOIRS ON

end of the operation the liquid and vapor are again in the same condition as at the beginning, so that the excess of tin- fj-tt-nml work done over that employed is also the total work done. It

is represented by the quadrilateral abed, and its area must also

be set equal to the heat consumed during the same time.

For our purposes we again assume that the changes just de-

scribed are infinitely small, and on this assumption represent

the whole process by the accompanying figure, in which the

curves ad and be which occur in

Fig. 3 have become straight lines.

So far as the area of the quadrilat-

eral abed is concerned, it may again

be considered a parallelogram, and

may be represented by the product
ef.bk. If, now, the pressure of tin;

/a Pig 4

'

vapor at the temperature t and at its maximum tension is represented

by/?, and if the temperature ditTcr-

ence t T is represented by */-.

hftve

=$*.

The line ef represents the increase of volume, which occurs in consequence of the passage of a certain quantity of liquid,

which may be called dm, over into vapor. Representing now

the volume of a unit weight of the vapor at its maximum <!en-

sity at the temperature t by *, and the volume of the same

quantity of liquid at the temperature / by 9, we have evidently
ef-(s- )dm,

- and consequently the area of the quadrilateral, or

(5)

The work done = (* ef^dmdt.

In order to represent the quantities of heat concerned, we will introduce the following symbols. The quantity of heat which becomes latent when a unit weight of the liquid evapo-
rates at the temperature / and under the corresponding pressure, is called r, and the specific heat of the liquid is called

Both of these quantities, as well as also *, a, and ^, are to be
considered functions of t. Finally, let us denote by //// tinquantity of heat which must be imparted to a unit weight of
80

it t.
a -fit is

THE SECOND LAW OF THERMODYNAMICS

the vapor if its temperature is raised from t to t dt, while

so compressed that

it

is

again at the maximum density for this

temperature without the precipitation of any part of it. The

quantity

7* is

likewise a function of

It will, for the pres-

ent, be left undetermined whether has positive or negative

value.
If we now denote by

/i

the mass of liquid originally present

in the vessel, and by tit the mass of vapor, and further by dm

the mass which evaporates during the expansion from oe to of,

and by d'm the mass which condenses during the compression

from og to oh, the heat which becomes latent in the first opera-

tion and taken from the body

rdm,

and that which
to the body B

set free in the second operation and given

is is is j
A is is is

it is
p /*
of p of

In the other expansion and in the other compression heat neither gained nor lost, so that, at the end of the process,

(6)

The heat consumed=rdm(r

jrdt}d'm.

In this expression the differential d'm must be replaced by dm

and dt. For this purpose we make use of the conditions under

which the second expansion and the second compression oc-

curred. The mass of vapor, which condenses during the com-

pression from oh to oe, and which would be evolved by the cor-

responding expansion from oe to oh, may be represented by Sm,

and that which

evolved by the expansion from

to og by

I'm. We then have at once, since at the end of the process the

same mass of liquid and the same mass of vapor m must be

present as at the beginning, the equation

dm + Km d'm -f Im.

Further, we obtain for the expansion from oe to oh, since in
the temperature of the mass of liquid and the mass of vapor m must be lowered by dt without the emission of heat, the

equation

and similarly for the expansion from

to og, by substituting

fidm and m+ dm for and m, and Z'm for m,

81

M KMOIRS ON

If from these three equations and (6) we eliminate
tudes d'm, 3m, and I'm, and reject terms of higher the second, we have

the magniorder than

(7) The formulas (?) and (5) must now he connected in the same way as that used in the case of the permanent gases, that

it is,

and we obtain as the analytical expression of the fundamental

principle in the case of vapors at their maximum density the

equation

e+ is

(III.)

-h = A(*-.)'%.

If, instead of using our principle, we adopt the assumption that the quantity of heat conx/anf. \ve must replace (III.), as

appears from (7), by

(8)

is r^
if
is a
a

This equation has been used, not exactly in the same form,

at least in its general sense, to obtain value for tin- magni-

tude A. So long as Watt's law

considered true for water.

that the sum of the free and latent heats of quantity of vapor

at its maximum density equal for all temperatures, and that

therefore

rf/+'=;
must be concluded that for this liquid 7<=0. This conclu-

sion has, in fact, often been stated us con-ret, in that

has

been said that

quantity of vapor at its maximum dmsit v.

and then compressed or expanded in ves>i-l impermeable hy heat, remains at its maximum .im.-itv. \\\\\ sinee Ki-nault*

has corrected Watt's law by substituting for the approximate

relation

,/..

if a t. 7* is a it It

the equation (8) gives for the value 0.305.

would there-

fore follow that the quantity of vapor formerly considered in

mm. de rAcad.. xxl., the 9tb and 10.1,M, moires.

it it

THE SECOND LAW OF THERMODYNAMICS

the vessel impermeable by heat would be partly condensed by

compression, and on expansion would not remain at the maximum density, since its temperature would not fall in a way to
correspond to the diminution of pressure. It is entirely different if we replace equation (8) by (III.).
The expression on the right-hand side is, from its nature, always positive, and it therefore follows that h must be less than 0.305. It will subsequently appear that the value of this expression is

so great that h is negative. We must therefore conclude that the quantity of vapor before mentioned is partly condensed,

not by compression, but by expansion, and that by compression

its temperature rises at a greater rate than the density increases, so that it does not remain at its maximum density.
It must be admitted that this result is exactly opposed to the common view already referred to ; yet I do not believe that
it is contradicted by any experimental fact. Indeed, it is more

consistent than the former view with the behavior of steam

as observed by Pambour.

Pambour* found that the steam

which issues from a locomotive after it has done its work

always has the temperature at which the tension, observed

at the same time, is a maximum.

From this it follows

either that A=0, as it was once thought to be, because this

assumption agreed with Watt's law, accepted as probably true, or that h is negative. For if h were positive, the temperature

of the vapor, when released, would be too high in comparison

with its tension, and that could not have escaped Pambour's
notice. If, on the other hand, h is negative, according to our

former statement, there can never arise from this cause too

low a temperature, but a part of the steam must become liquid, so as to maintain the rest at the proper temperature. This

part need not be great, since a small quantity of vapor sets free

on condensation a relatively large quantity of heat, and the water formed will probably be carried on mechanically by the rest of the steam, and will in such researches pass unnoticed, the more likely as it might be thought, if it were to be observed, that it was water from the boiler carried out mechanically.
The results thus far obtained have been deduced from the fundamental principle without any further hypothesis. The equation (II. a) obtained for permanent gases may, however, be

* Traite desLocomotivess,ecoudedition,and TheoriedesMachinesd Vupeur, second edition.

MEMOIRS ON

made much more fruitful by the help of an obvious subsidiary

hypothesis.

The gases show in their various relation

pecially in the relation expressed by the M. and (J. law be-

tween volume, pressure, and temperature, so great a regularity of behavior that we are naturally led to take the view that the mutual attraction of the particles, which acts within solid ami

liquid bodies, no longer acts in gases, so that while in the case

of other bodies the heat which produces expansion must over-

come not only the external pressure but the internal attraction as well, in the case of gases it has to do only with the external
pressure. If this is the case, then during the expansion of a

gas only so much heat becomes latent as is used in doing external work. There is, further, no reason to think that a gas. if it expands at constant temperature, contains more ir<-<h- eat
than before. If this be admitted, we have the law : a per-

it /.<
is

manent gas, when expanded at constant frt/t/irrti/nrr, tnkcs /</> only xo much heat ns mnsumed in tl<n'n<r/.rfrrntil imrk ilnrimi the expansion. This law probably true for any <:aswith the

same degree of exactness as that attained by the M. and (J. law

applied to it.

From this

- follows at once that

since, as already noticed, R

dv represents

the external

work

a) v
('

done during the expansion dv. It follows that the fimetion which occurs in (II. does not contain /. and the equation
therefore takes the form

(ll.b)

dQ=cdt+AR

<i'\

by is
(J. f, <

a is a it
/

c

where can be function of only. It even probable that

this magnitude r, which represents the speciii.- heat of the gag

at constant volume,

constant.

Now in order to apply this equation to special cases, we must

introduce the relation between the variable-

and /-. which

is

obtained from the conditions of each separate ,-i-e. mi> the

equation, and so make

integrable. We shall here .n.-ider

a

only few simple examples of this sort, which are eiiher in-

teresting in themselves or become so comparison with other

theorems already announced.

84

THE SECOND LAW OF THERMODYNAMICS
We may first obtain the specific heats of the gas at constant
volume and at constant pressure if in (II. b) we set #=const., and j3 = const. In the former case, dv=Q, and (II. b) becomes

In the latter case, we obtain from the condition j9=const., by

the help of equation (I.),

, Rdt

*>=

,

or and this, substituted

dv

tit

v

in (II. b), gives

if we denote by c' the specific heat at constant pressure.

It appears, therefore, that the difference of the two specific heats of any gas is a constant magnitude, AR. This magni-

tude also involves a simple relation among the different gases.

- The complete expression for R is

) , where p , v , and t

are any three corresponding values of p, v, and t for a unit
of weight of the gas considered, and it therefore follows, as has already been mentioned in connection with the adoption of equation (I.), that R is inversely proportional to the specific

gravity of the gas, and hence also that the same statement must
hold for the difference c' c=AR, since A is the same for all

gases.
If we reckon the specific heat of the gas, not with respect to
the unit of weight, but, as is more convenient, with respect to the unit of volume, we need only divide c and c' by v , if the
volumes are taken at the temperature t and pressure j . Des-
ignating these quotients by / and y, we obtain

(11)

y' I'o

In this last quantity nothing appears which is dependent on

the particular nature of the gas, and the difference of the specific heats referred to the unit of volume is therefore the same for all gases.
This law was deduced by Clapeyron from Carnot's theory, 85

MEMOIRS ON

though the constancy of the difference c' c, which we have deduced before, is not found in his work, where the expression given for it still has the form of a function of the temperature.
If we divide equation (11) on both sides by y, we have

(12)

*-l=-.-J^-,

in which k, for the sake of brevity, is used for the quotient ,

or, what amounts to the same thing, for the quotient. This

quantity has acquired special importance in science from the

theoretical discussion by Laplace of the propagation of sound

in air. The excess of fin's quotient over unity is therefore. /<// tIn-

different yases, same at constant

inversely volume,

/n-ojtortiontil to tin- sped
ifike$tttn r>f<mil lo tin

lie Ite/i/s of tin' unit ,,f ruhnne.

This law has, in fact, been found by Dulong from experiment*

to be so nearly accurate that he has assumed it, in view of its

theoretical probability, to be strictly accurate, and has therefore employed it, conversely, to calculate the specific heats of
the different gases from the values of k determined by observation. It must, however, be remarked that the law is only theoretically justified when the M. and G. law holds, whii-h is not the case with sufficient exactness for all the gases employed

by Dulong.
If it is now assumed that the specific heat of gases at con-
stant volume f is constant, which has been stated al>ove to bo very probable, the same follows for the specific heat at constant pressure, and consequently the quotient of the tir

C heats c

k is a constant.

This law, which Poisson has already

assumed as correct on the strength of the experiment! of (layLussac and Welter, and has made the basis of his investigations

on the tension and heat of gases, f is therefore in ^<><a\ greement with our present theory, while, it would not bo possible

on Carnot's theory as hitherto developed.
If in equation (II. A) we set @=const., we obtain the follow-
ing equation between v and / :

Ann. df Chim. et de Phyt., xli.. and Po^g. Ann., xvi. f Traiti de Mecanique,secondedition, vol. ii., p. 646.
M

THE SECOND LAW OF THERMODYNAMICS

(13)
which gives, if c is considered constaiit,

v '.(a+t)=const.,

or, since from

equation

(10),^ACH

=

c' C

1 = # 1,

v*~ l (rt-M)=const.

/, of

HMen,ceanwde/,,have,

if

v

,t

, and
a +j

j

t-\v (14)

a+

are three corresponding o

values

If we substitute in this relation the pressure p first for v and

then for t by means of equation (I.), we obtain

These are the relations which hold between volume, temper-
ature, and pressure, if a quantity of gas is compressed or expanded within an envelope impermeable by heat. These equations agree precisely with those which have been developed by Poisson for the same case,* which depends upon the fact that

he also treated k as a constant.
Finally, if we set t = const, in equation (Il.b), the first term

on the right drops out, and there remains

(17)

dQ=AR f^dv,

from which we have
Q = AR (a + t) log v + const., or, if we denote by v , p , t , and Q the values of v, p, t, and
Q, which hold at the beginning of the change of volume,

(18)

Q-Q =AR(a + t )\og?-.

If From this follows the law also developed by Carnot :

a gas

changes its volume without changing Us temperature, the quanti-

ties of heat evolved or absorbed are in arithmetical progression, while the volumes are in geometrical progression.

* Traite deMecanique,vol. ii., p. 647. 87