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:: Fundamental Paper ::


Mechanical Theory of Heat
Rudolf Clausius

:: Introduction ::
Table of Contents
Fourth Memoir
Fifth Memoir
Sixth Memoir
Ninth Memoir

This URL contains a founding paper for the Science of Human Thermodynamics

Mechanical Theory of Heat
Rudolf Clausius lived from 1822-1888. He was born to a large German family where he was the last born of six brothers among other siblings.  Here he was educated at a small private school in which his father was the principle.  Following private schooling, Clausius moved on to the Gymnasium in Stettin where he remained until he had completed his schooling in 1840.  From here, Clausius entered the University of Berlin in which he first entertained thoughts of a degree in history, but eventually settled on a degree in mathematics and physics which he completed in 1844.  He then spent a probationary year at the Frederic-Werder Gymnasium teaching advanced classes in mathematical physics.  His PhD dissertation, which proposed an explanation for the blue color of the sky, the red colors seen at sunrise and sunset, and the polarization of light, was completed by 1848 at Halle University.  
  Rudolf Clausius [1822-1888]

** Communicated to the Naturforschende Gesellschaft of Zurich, Jan. 27th, 1862; published in the Viertaljahrschrift of this Society, vol. vii. P. 48; in Poggendorff’s Annalen, May 1862, vol. cxvi. p. 73; in the Philosophical Magazine, S. 4. vol. xxiv. pp. 81, 201; and in the Journal des Mathematiques of Paris, S. 2. vol. vii. P. 209.

*** Read at the Philosophical Society of Zurich on the 24th of April, 1865, published in the Vierteljahrsschrift of this society, Bd. x. S. 1.; Pogg. Ann. July, 1865, Bd. cxxv. S. 353; Journ. de Liouville, 2e ser. t. x. p. 361.

Clausius’ first and most famous paper, of sixteen in total, published in 1850, was a treatise on the mechanical theory of heat, entitled "On the Motive Power of Heat and on the Laws which can be deduced from it for the Theory of Heat."  In this famous paper, Clausius set forth the argument that whenever work is done by heat, a certain quantifiable amount of permanent change occurs in the working body, the "working body" being typically a cylindrical body of steam or liquid.  This was in direct contrast to Sadi Carnot, the French physicist who in 1824, in his founding thermodynamic paper "Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power", reasoned that whenever work is done by heat, no permanent change occurs in the working body.  It was in this famous paper, and nine memoirs to follow, that Clausius began to develop the concept of entropy, which accounts for changes that occur in the condition of working body whenever work is done by heat.  

In 1850, and over the next fifteen years, Clausius wrote nine memoirs on various aspects of the motive power of heat, with focus on what he called a “modification to the first fundamental theorem”.  In 1865, these collected memoirs were published in a book entitled Mechanical Theory of Heat
Carnot, Sadi. (1824). “Reflections on the Motive Power of Fire.” (55 pages)
Clapeyron, Emile. (1834). “Memoir on the Motive Power of Heat.” (35 pages)
Clausius, Rudolf. (1850). “On the Motive Power of Heat” (43 pages)   
Being this situation as it is, the IoHT has taken selected excerpts from an original 1865 copy of Clausius' Mechanical Theory of Heat, containing the essential points in the development of the concept of entropy, and posted them online (below) for public consumption.  Before reading the following excerpts, however, it will please the mind of the reader to first read the three papers mentioned above, as found in the adjacent book.  These three papers are the founding stones of the science of thermodynamics.  In addition, Clausius' 1857 paper, "On the Nature of the Motion which we call Heat", which helped to establish the kinetic theory of gases, is found online (click here) and should be read as well.

Essentially, Carnot’s 1824 paper puts forward the hypothesis that heat and work are equivalent and gives us a verbal description of an engine cycle, in which reversibility is assumed.  Clapeyron’s 1834 paper graphically analyzes Carnot’s engine cycle.  Lastly, Clausius’ 1850 paper, which is his first memoir, argues that the first fundamental theorem in the mechanical theory of heat, in the form of approximately Q = W, as presented by Carnot in a verbal manner and by Clapeyron in graphical manner, needs amendment in that it does not account for changes in the constitution of the working body.  By this, Clausius argued that Carnot’s theorem did not account for the fact that real-life processes are "irreversible"; meaning that during any sort of transformation the molecules of the working body do work on each other, i.e. change their arraignments as the working body progresses from the "initial state" to the "final state" of each step of the Carnot cycle.  Subsequently, this work that the molecules of the body do on each other, which Clausius terms “internal work”, needed to be energetically accounted for in the energy balance equation.  From this argument is where the concept of entropy arose.

To note, as the first memoir of Clausius is available in book form, we will not reproduce it here.   Secondly, the second and third memoirs as found in the 1865 book Mechanical Theory of Heat are essentially discussions on how to calculate heat capacities and vapour pressures.  As such, they will not be reproduced here.  Thirdly, the fourth and the sixth memoirs are where the essence of entropy takes form, both mathematically and conceptually.  The main parts of these memoirs, as well as an excerpt of the fifth memoir, are shown below.  Both the seventh and the eight memoirs will not be reproduced here; to note, however, the eight memoir essentially contains verbal arguments for the proposal that the entropy of the universe tends towards a maximum.

R. Clausius,
Professor of physics in the University of Zurich
Zurich, August 1864

Edited by
T. Archer Hirst, F.R.S.,
Professor of Mathematics in University College, London.

With introduction by
Professor Tyndall.
London, May 1867

Mathematical Introduction (1858):
On the Treatment of Differential Equations which are not Directly Integrable, pp. 1-13.

First Memoir (1850):
On the Moving Force of Heat and the Laws of Heat Which May be Deduced Therefrom, pp. 14-69

Second Memoir (1851):
On the Deportment of Vapour During its Expansion Under Different Circumstances, pp. 90-100

Third Memoir (1851):
On the Theoretic Connexion of Two Empirical Laws Relating to the Tensions and the Latent Heat of Different Vapours, pp. 104-110

Fourth Memoir (1854):
On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat, pp. 111-135

Fifth Memoir (1856):
On the Application of the Mechanical theory of Heat to the Steam-Engine, pp. 136-207

Sixth Memoir (1862):
On the Application of the Theorem of the Equivalence of Transformations to Interior Work, pp. 215-250

Seventh Memoir (1863):
On an Axiom in the Mechanical Theory of Heat, pp. 267-289.

Eighth Memoir (1863):
On the Concentration of Rays of Heat and Light, and on the Limits of its Action, pp. 290-326.

Ninth Memoir (1865):
On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat, pp. 327-365.

In my memoir “On the Moving Force of Heat, &c.”, I have shown that the theorem of the equivalence of heat and work, and Carnot’s theorem, are not mutually exclusive, by that, by a small modification of the latter, which does not affect its principle, they can be brought into accordance.  With the exception of this indispensable change, I allowed the theorem of Carnot to retain its original form, my chief objection then being, by the application to the two theorems to special cases, to arrive at conclusions which, according as they involved known or unknown properties of bodies, might suitably serve as proofs of the truth of the theorems, or as examples of their fecundity.

This form, however, although it may suffice for the deduction of the equations which depend upon the theorem, is incomplete, because we cannot recognize therein, with sufficient clearness, the real nature of the theorem, and its connexion with the first fundamental theorem.  The modified form in the following pages will, I think, better fulfill this demand, and in its applications well be found very convenient.

Before proceeding to the examination of the second theorem, I may be allowed a few remarks on the first theorem, so far as this is necessary for the supervision of the whole.  It is true that I might assume this as known from my former memoirs or from those of other authors, but to refer back would be inconvenient; and besides this, the exposition I shall here give is preferable to my former one, because it is at once more general and more concise.

Theorem of the equivalence of Heat and Work

Whenever a moving force generated by heat acts against another force, and motion in the one direction of the other ensues, positive work is performed by the one force at the same time that negative work is done by the other.  As this work has only to be considered as a simple quantity in calculation, it is perfectly arbitrary, in determining its sign, which of the two forces is chosen as the indicator.  Accordingly in researches which have a special reference to the moving force of heat, it is customary to determine the sigh by counting as positive the work done by heat in overcoming any other force, and as negative the work done by such other force.  In this manner the theorem of the equivalence of heat and work, which forms only a particular case of the general relation between vis viva and mechanical work, can be briefly enunciated thus:
On a modified form of the second fundamental theorem in the mechanical theory of heat
[December 1854]
and the passage of the quantity of heat Q from the temperature t1 to the temperature t2, has the equivalence-value:
wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.

(skipping three pages)

The equation:
is the analytical expression, for all reversible cyclical processes, of the second fundamental theorem in the mechanical theory of heat.
(from page 141)

The two fundamental theorems, which hold good in every cyclical process, are represented by the following equations:

Q = A*W(I)

A = the thermal equivalence of the unit of work.
W = the external work performed during the cyclical process.
dQ = an element of the same, whereby any heat withdrawn from the body is to be considered as an imparted negative quantity of heat.  The integral in the second equation is extended over the whole quantity Q.
T = absolute temperature.
N = the equivalence-value of all the uncompensated transformations involved in a cyclical process.

On the Application of the Theorem of the Equivalence of Transformations to Interior Work**
In a memoir published in the year 1854, wherein I sought to simplify to some extent the form of the developments I had previously published, I deduced, form my fundamental proposition that heat cannot, by itself, pass from a colder into a warmer body, a theorem which is closely allied to, but does not entirely coincide with, the one first deduced by S. Carnot from considerations of a different kind, based upon the older views of the nature of heat.  It has reference to the circumstances under which work can be transformed into heat, and conversely, heat converted into work; and I have called it the Theorem of the equivalence of Transformations.  I did not, however, there communicate the entire theorem in the general form in which I had deduced it, but confined myself on that occasion  to the publication of a part which can be treated separately form the rest, and is capable of more strict proof.

In general, when a body changes its state, work is performed externally and internally at the same time, the exterior work having reference to the forces which extraneous bodies exert upon the body under consideration, and the interior work to the forces exerted by the constituent molecules of the body in question upon each other.  The interior work is for the most part so little known, and connected with another equally unknown quantity (in fact with the increase of heat actually present in the body) in such a way, that in treating of it we are obliged in some measure to trust to probabilities; whereas the exterior work is immediately accessible to the observation and measurement, and thus admits of more strict treatment.  Accordingly, since, in my former paper, I wished to avoid everything that was hypothetical, I entirely excluded the interior work, which I was able to do by confining myself to the consideration of cyclical process—that is to say, operations in which the modifications which the body undergoes are so arranged that the body finally returns to its original condition.  In such operations the interior work which is performed during the several modifications, partly in a positive sense and partly in a negative sense, neutralizes itself, so that nothing but exterior work remains, for which the  theorem in question can then be demonstrated with mathematical strictness, starting for the above-mentioned fundamental proposition.

I have delayed till now the publication of the remainder of my theorem, because it leads to consequence which is considerably at variance with the ideas hitherto generally entertained of the heat contained in bodies, an I therefore thought it desirable to make still further trial of it.  But as I have become more and more convinced in the course of years that we must not attach too great weight to such ideas, which in part are founded more upon usage than upon a scientific basis, I feel that I ought to hesitate no longer, buy to submit to the scientific public the theorem of the equivalence of transformations in its complete form, with the theorems which attach themselves to it.  I venture to hope that the importance which these theorems, supposing them to be true, possess in connexion with the theory of heat will be though of justify their publication in their present hypothetical form.

I will, however, at once distinctly observe that, whatever hesitation may be felt in admitting the truth of the following theorems, the conclusions arrived at in my former paper, in reference to cyclical processes, are not at all impaired.

1. I will begin by briefly stating the theorem of the equivalence of transformations, as I have already developed it, in order to be able to connect with it the following considerations:

When a body goes through a cyclical process, a certain amount of exterior work may be produced, in which case a certain quantity of heat must be simultaneously expended; or, conversely, work my be expended and a corresponding quantity of heat may by gained.  This may be expressed by saying: Heat can be transformed into work, or work into heat, by a cyclical process.

There may also be another effect of a cyclical process: heat may be transferred form one body to another, by the body which is undergoing modification absorbing heat form the one body and giving it out again to the other.  In this case the bodies between which the transfer of heat takes place are to be viewed merely as heat reservoirs, of which we are not concerned to know anything except the temperatures.  If the temperatures of the two bodies differ, heat passes, either from a warmer to a colder body, or from a colder to a warmer body, according to the direction in which the transference of heat takes place.  Such a transfer of heat may also be designated, for the sake of uniformity, a transformation, inasmuch as it may be said that heat of one temperature is transformed into heat of another temperature.

The two kinds of transformations that have been mentioned are related in such a way that one presupposes the other, and that they can mutually replace each other.  If we call transformations which can replace each other equivalent, and seek the mathematical expressions which determine the amount of the transformations in such a manner that the equivalent transformations become equal in magnitude, we arrive at the following expression: If the quantity of heat Q of the temperature t is produced from work, the equivalence-value of this transformation is:
And if the quantity of heat Q passes from a body whose temperature is t1 into another whose temperature is t2, the equivalence-value of this transformation is:
Where T is a function of the temperature which is independent of the kind of process by means of which the transformation is effected, and T1 and T2 denote the values of this function which correspond to the temperatures t1 and t2.  I have shown by separate considerations that T is in all probability nothing more than the absolute temperature.

These two expressions further enable us to recognize the positive or negative sense of the transformations.  In the first, Q is taken as positive when work is transformed into heat, and as negative when heat is transformed into work.  In the second, we may always take Q as positive, since the opposite senses of the transformations are indicated by the possibility of the difference [1/T2 – 1/T1] being either positive or negative.  It will thus be seen that the passage of heat from a higher to a lower temperature is to be looked upon as a positive transformation, and its passage form a lower to a higher temperature as a negative transformation.

If we represent the transformations which occur in a cyclical process by these expressions, the relation existing between them can be stated in a simple and definite manner.  If the cyclical process is reversible, the transformations which occur therein must be partly positive and partly negative, and the equivalence-values of the positive transformations must be together equal to those of the negative transformations, so that the algebraic sum of all the equivalence-values become = 0.  If the cyclical process is not reversible, the equivalence values of the positive and negative transformations are not necessarily equal, but they can only differ in such a way that the positive transformations predominate.  The theorem respecting the equivalence-values of the transformations may accordingly be stated thus: The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.

The mathematical expression for this theorem is as follows.  Let dQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation:
must be true for every reversible cyclical process, and the relation:
must hold good for every cyclical process which is in any way possible.

2. Although the necessity of this theorem admits of strict mathematical proof if we start from the fundamental proposition above quoted, it thereby nevertheless retains an abstract form, in which it is with difficulty embraced by the mind, and we feel compelled to seek for the precise physical cause, of which this theorem is a consequence.  Moreover, since there is no essential difference between interior and exterior work, we may assume almost with certainty that a theorem which is generally applicable to exterior work cannot be restricted to this alone, but that, where exterior work is combined with interior work, it must be capable of application to the latter alone.

Considerations of this nature led me, to assume a general law respecting the dependence of the active force of heat on temperature, among the immediate consequences of which is the theorem of the equivalence of transformations in its more complete form, and which at the same time leads to other important conclusions.  This law I will at once quote, and will endeavour to make its meaning clear by the addition of a few comments.  As for the reasons for supposing it to be true, such as do not at once appear form its internal probability will gradually become apparent in the course of this paper.  It is as follows:
Mechanical Theory of Heat

price $11.95 (click here)
where K is a constant depending on the unit, hitherto left undetermined, according to which Z is to be measured.

(skipping about 8 pages worth of derivation and discussion)

7. We will now investigate the manner in which, from equation (II), it is possible to arrive at the equation (I) previously given in Art. 1, which equation must hold, according to the fundamental theorem that I have already enunciated, for every reversible cyclical process.

When the successive changes of condition constitute a cyclical process, the disgregation of the body is the same at the end of the operation as it was at the beginning, and hence the following equation must be good:
Equation (II) is hereby transformed into:
In order that this equation may accord with equation (I), namely:
the following equation must hold for every reversible cyclical process:
It is this equation which leads to the consequence referred to in the Art. as at variance with commonly received views.  It can, in fact, be proved that, in order that this equation may be true, it is at once necessary and sufficient to assume the following theorem:
On Several Convenient Forms of the Fundamental Equations of
the Mechanical Theory of Heat***
(from page 354 of the ninth memoir)

The other magnitude to be here noticed is connected with the second fundamental theorem, and is contained in equation (IIa).  In fact if, as equation (IIa) asserts, the integral:
Vanishes whenever the body, starting from any initial condition, returns thereto after its passage through any other conditions, then the expression dQ/T under the sign integration must be the complete differential of a magnitude which depends only on the present existing condition of the body, and not upon the way by which t reached the latter.  Denoting his magnitude by S, we can write
or, if we conceive this equation to be integrated for any reversible process whereby this body can pass from the selected initial condition to its present one, and denote at the same time by So the value which the magnitude S has in that initial condition,
This equation is to be used in the same way for determining S as equation (58) was for defining U.  The physical meaning of S has already been discussed in the Sixth Memoir.

(after a series of derivations)

we obtain the equation:
We might call S the transformation content of the body, just as we termed the magnitude U its thermal and ergonal content.  But as I hold it to be better terms for important magnitudes from the ancient languages, so that they may be adopted unchanged in all modern languages, I propose to call the magnitude S the entropy of the body, from the Greek word τροπη, transformation.  I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied their physical meanings, that a certain similarity in designation appears to be desirable.

(from last paragraph of the last page (365))


1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum.

The forces which here enter into consideration may be divided into two classes: those which the atoms of a body exert upon each other, and which depend, of course, upon the nature of the body, and those which arise from the foreign influences to which the body may be exposed.  According to these two classes of forces which have to be overcome, of which the latter are subject to essentially different laws, I have divided the work done by heat into interior and exterior work.

With respect to the interior work, it is easy to see that when a body, departing from its initial condition, suffers a series of modifications and ultimately returns to its original state, the quantities of interior work thereby produced must exactly cancel one another.  For if any positive or negative quantity of interior work had remained, it would it must have produced an opposite exterior quantity of work or a change in the existing quantity of heat; and as the same process could be repeated any number of times, it would be possible, according to the sign, either to produce work or heat continually from nothing, or else to lose work or heat continually, without obtaining any equivalent; both of which cases are universally allowed to be impossible.  But if at every return of the body to its initial condition the quantity of interior work is zero, it follows, further, that the interior work corresponding to any given change in the condition of the body is completely determined by the initial and final conditions of the latter, and is independent of the path pursued in passing form one condition to the other.  Conceive a body to pass successively in different ways from the first to the second condition, but always to return in the same manner to its initial state.  It is evident that the quantities of interior work produce along the different paths must all cancel the common quantity produced during the return, and consequently must be equal to each other.

It is otherwise with the exterior work.  With the same initial and final conditions, this can vary just as much as the exterior influences to which the body may be exposed can differ.

Let us now consider at once the interior and exterior work produced during any given change of condition.  If opposite in sign they may partially cancel each other, and what remains must then be proportional to the simultaneous change which has occurred in the quantity of existing heat.  In calculation, however, it amounts to the same thing if we assume an alteration in the quantity of heat equivalent to each of the two kinds of work.  Let Q be the quantity of heat which must be imparted to a body during its passage, in a given manner, for one condition to another, and heat withdrawn from the body being counted as an imparted negative quantity of heat.  Then Q may be divided into three parts, of which the first is employed in increasing the heat actually existing in the body, the second in producing the interior, and the third in producing the exterior work.  What was before stated of the second part also applies to the first – it is independent of the path pursued in the passage of the body from one state to another: hence both parts together may be represented by one function U, which we know to be completely determined by the initial and final states of the body.  The third part, however, the equivalent of exterior work, can like this work itself, only be determined when the precise manner in which the changes of conditions took place is known. If W be the quantity of exterior work, and A the equivalent of heat for the unit of work, the value of the third part will be A*W, and the first fundamental theorem will be expressed by the equation:

Q = U + A*W(I)

When the several changes are of such a nature that through them the body returns to its original conditions, or when, as we shall in the future express it, these changes form a cyclical process, we have:

U = 0

and the foregoing equation becomes:

Q = A*W

(skipping about three paragraphs talking about pressure-volume external work)

The work done during an increment of volume dv will be the pdv.  Hence the work done during a simultaneous increase of t and v is: 

dW = pdv

and when we apply this to equation (I), we obtain:

dQ = dU + A*pdv(II)

(skipping two pages of derivation on how to eliminate U from equation (II) using a few derivatives)

On this account I will not here pursue the subject further, but pass on to the consideration of the second fundamental theorem in the mechanical theory of heat.

Theorem of the equivalence of transformations

Carnot’s theorem, when brought into agreement with the first fundamental theorem, expresses a relation between two kinds of transformations, the transformation of heat into work, and the passage of heat form a warmer to a colder body, which may be regarded as the transformation of heat at a higher, into heat at a lower temperature.  The theorem, as hitherto used, may be enunciated in some such manner as the following:
Mechanical work may be transformed into heat, and conversely heat into work, the magnitude of the one being always proportional to that of the other.
In deducing this theorem, however, a process is contemplated which is too simple a character; for only two bodies losing or receiving heat are employed, and it is tacitly assumed that one of the two bodies between which the transformation of heat takes place is the source of the heat which is converted into work.  Now by previously assuming, in this manner, a particular temperature of the heat converted into work, the influence which a change of this temperature has upon the relation between the two quantities of heat remains concealed, and therefore the theorem in the above form is incomplete.

It is true this influence may be determined without great difficulty by combining the theorem in the above limited form with the first fundamental theorem, and thus completing the former by the introduction of the results thus arrived at.  But by this indirect method the whole subject would lose much of its clearness and facility of supervision, and on this account it appears to me preferable to deduce the general form of the theorem immediately from the same principle which I have already employed in my former memoir, in order to demonstrate the modified theorem of Carnot.

This principle, upon which the whole of the following development rests, is as follows:
In all cases where a quantity of heat is converted into work, and where the body effecting this transformation ultimately returns to its original condition, another quantity of heat must necessarily be transferred from a warmer to a colder body; and the magnitude of the last quantity of heat, in relation to the first, depends only upon the temperature of the bodies between which heat passes, and not upon the nature of the body effecting the transformation.
Everything we know concerning the interchange of heat between two bodies of different temperature confirms this; for heat everywhere manifests a tendency to equalize differences of temperature, and therefore to pass in contrary direction, i.e. from a warmer to colder bodies.  Without further explanation, therefore, the truth of this principle will be granted.

The principle may be more briefly expressed thus: Heat cannot by itself pass from a colder to a warmer body; the words “by itself”, however, here requires explanation.  Their meaning will, it is true, be rendered sufficiently clear by the exposition contained in the present memoir, nevertheless it appears desirable to add a few word here in order to leave no doubt as to the signification and comprehensiveness of the principle.

In the first place, the principle implies that in the immediate interchange of heat between two bodies by conduction and radiation, the warmer body never receives more heat from the colder one than it imparts to it.  The principle holds, however, not only for process of this kind, but for all others by which a transmission of heat can be brought about between two bodies of different temperature, amongst which process must be particularly noticed those wherein the interchange of heat is produced by means of one or more bodies which, on changing their condition, either receive heat from a body, or impart heat to other bodies.

On considering the results of such processes more closely, we find that in one and the same process heat may be carried from a colder to warmer body and another quantity of heat transferred from a warmer to a colder body without any other permanent change occurring.  In this case we have not a simple transmission of heat from a colder to a warmer body, or an ascending transmission of heat, as it may be called, but two connected transmission of opposite characters, one ascending and the other descending, which compensate each other.  It may, moreover, happen that instead of a descending transmission of heat accompanying, in the one and the same process, the ascending transmission, another permanent change may occur which has the peculiarity  of not being reversible without either becoming replaced by a new  permanent change of a similar kind, or producing a descending transmission of heat.  In this case the ascending transmission of heat may be said to be accompanied, not immediately, but mediately, by a descending one, and the permanent change which replaces the latter may be regarded as a compensation for the ascending transmission.

Now it is to these compensations that our principle refers; and with the aid of this conception the principle may be also expressed thus: an uncompensated transmission of heat from a colder to a warmer body can never occur.  The term “uncompensated” here expresses the same idea that was intended to be conveyed by the words “by itself” in the previous enunciation of the principle, and by he expression “without some other change connected therewith, occurring at the same time” in the original text. 

(skipping about thirteen pages)

The second fundamental theorem in the mechanical theory of heat may thus be enunciated:*

If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q of the temperature t from work, has the equivalence-value:
In order to understand the significance of this law, we require to consider more closely the processes by which heat can perform mechanical work.  These processes always admit of being reduced to the alteration in some way or another of the arrangement of the constituent parts of the body.  For instance, when bodies are expanded by heat, their molecules being thus separated from each other: in this case the mutual attractions of the molecules on the one hand, and external opposing forces on the other, in so far as any such are in operation, have to be overcome.  Again, the state of aggregation of bodies is altered by heat, solid bodies rendered liquid, and both solid and liquid bodies being rendered aeriform: here likewise internal forces, and in general external forces also, have to be overcome.  Another case which I will also mention, because it differs so widely from the foregoing, and therefore shows how various are the modes of action which have here to be considered, is the transfer of electricity form one body to the other, constituting the thermo-electric current, which takes place by the action of heat on two heterogeneous bodies in contact.

In the cases first mentioned, the arrangements of the molecules is altered.  Since, even which a body remains in the same state of aggregation, its molecules do not retain fixed in varying position, by are constantly in a state of more of less extended motion, we may, when speaking of the arrangement of the molecules at any particular time, understand either the arrangement which would result from the molecules being fixed in the actual position they occupy at the instant in question, or we may suppose such an arrangement that each molecule occupies its mean position.  Now the effect of heat always tend to loosen the connexion between the molecules, and so to increase their mean distances from one another.  In order to be able to represent this mathematically, we will express the degree in which the molecules of a body are separated from each other, by introducing a new magnitude, which we will call the disgregation of the body, and by help of which we can define the effect of heat as simply tending to increase the disgregation.  The way in which a definite measure of this magnitude can be arrived at will appear from the sequel.

In the case last mentioned, an alteration in the arrangement of the electricity takes place, an alteration which can be represented and taken into calculation in a way corresponding to the alteration of the position of the molecules, and which, when it occurs, we will consider as always included in the general expression change of arrangement, or change of disgregation.

In is evident that each of the changes that have been named may also take place in the reverse sense, if the effect of the opposing forces is greater than that of the heat.  We will assume as likewise self-evident that, for the production of work, a corresponding quantity of heat must always be expended, and conversely, that, by the expenditure of work, an equivalent quantity of heat must be produced.

3. If we know consider more closely the various cases which occur in relation to the forces which are operative in each of them, the case of the expansion of a permanent gas presents itself as particularly simple.  We may conclude from certain properties of the gases that the mutual attraction of their molecules at their mean distance is very small, and therefore that only a very slight resistance is offered to the expansion of a gas, so that the resistance of the sides of the containing vessel must maintain equilibrium with almost the sole effect of the heat.  Accordingly the externally sensible pressure of a gas forms an approximate measure of the separative force of the heat contained in the gas; and hence, according to the foregoing law, this pressure must be nearly proportional to the absolute temperature.  The internal probability of the truth of this result is indeed so great, that many physicists since Gay-Lussac and Dalton have without hesitation presuppose this proportionality, and have employed it for calculating the absolute temperature.

In the above-mentioned case of thermo-electric action, the force which exerts an action contrary to that of the heat is likewise simple and easily determined.  For at the point of contact of two heterogeneous substances, such as quantity of electricity is driven from the one to the other by the action of the heat, that the opposing force resulting form the electric tension suffices to hold the force exerted by the heat in equilibrium.  Now in a former memoir “On the application of the Mechanical Theory of Heat to the Phenomena of Thermal Electricity” (Poggendorft’s Annalen, vol. xc. P. 513), I have shown that, in so far as changes in the arrangement of the molecules are not produced at the same time by the changes of temperature, the difference of tension produced by heat must be proportional to the absolute temperature, as is required by the foregoing law.

In the other cases that are quoted, as well as in most others, the relations are less simple, because in them an essential part is played by the forces exerted by the molecules upon one another, forces which, as yet, are quite unknown.  It results, however, from the mere consideration of the external resistances which heat is capable of overcoming, that in general its force increases with the temperature.  If we wish, for instance, to prevent the expansion of a body by means of external pressure, we are obliged to employ a greater pressure the more the body is heated; hence we may conclude, without having a knowledge of the interior forces, that the total amount of the resistances which can be overcome in expansion, increases with the temperature.  We cannot, however, directly ascertain whether it increases exactly in the proportion required by the foregoing law, without knowing the interior forces.  On the other hand, if this law be regarded as proved on other grounds, we may reverse the process, and employ it for the determination of the interior forces exerted by the molecules.

The forces exerted upon one another by the molecules are not of so simple a kind that each molecule can be replaced by a mere point; for many cases occur in which it can be easily seen that we have not merely to consider the distances of the molecules, but also their relative positions.  If we take, for example, the melting of ice, there is no doubt that interior forces, exerted by the molecules upon each other, are overcome, and accordingly increase of disgregation takes place; nevertheless the centres of gravity of the molecules are on the average not so far removed from each other in the liquid water as they were in the ice, for the water is the more dense of the two.  Again, the peculiar behaviour of water in contracting when heated above 0° C., and only beginning to expand when its temperature exceeds 4°, shows that likewise in liquid water, in the neighbourhood of its melting-point, increase of disgregation is not accompanied by increase of the mean distances of its molecules.

In the case of the interior forces, it would accordingly be difficult—even if we did not want to measure them, but only to represent them mathematically—to find a fitting expression for them which would admit of a simple determination of the magnitude.  This difficulty, however, disappears if we take into calculation, not the forces themselves, but the mechanical work which, in any change of arrangement, is required to overcome them.  The expressions for the quantities of work are simpler than those for the corresponding forces; for the quantities of work can be all expressed, without further secondary statements, by the numbers which, having reference to the same unit, can be added together, or subtracted from one another, however various the forces may be to which they refer.

It is therefore convenient to alter the form of the above law by introducing, instead of the forces themselves, the work done in overcoming them.  In this form it reads as follows:
In all cases in which the heat contained in a body does mechanical work by overcoming resistances, the magnitude of the resistances which it is capable of overcoming is proportional to the absolute temperature.
4. The law does not speak of the work which the heat does, but of the work which it can do; and similarly, in the first form of the law, it is not of the resistances which the heat overcomes, by of those which it can overcome that mention is made.  This distinction is necessary for the following reasons:

Since the exterior forces which act upon a body while it is undergoing a change of arrangement my vary very greatly, it may happen that the heat, while causing a change of arrangement, has not to overcome the whole resistance which it would be possible for it to overcome.  A well-known and often-quoted example of this is afforded by a gas which expands under such conditions that it has not to overcome an opposing pressure equal to its own expansive force, as, for instance, when the space filled by the gas is made to communicate with another which is empty, or contains a gas of a lower pressure.  In order in such cases to determine the force of the heat, we must evidently not consider the resistance which actually is overcome, but that which can be overcome.

Also, in changes of arrangement of the opposite kind, that is, where the action of heat is overcome by the opposing forces, a similar distinction may require to be bade, but in this case only as far as this—that the total amount of the forces by which the action of the heat is overcome my be greater than the active force of the heat, but not smaller.

Cases in which these differences occur may be thus characterized.  When a change of arrangement takes place so that the force and counterforce are equal, the change can likewise take place in the reverse direction under the influence of the same force.  But if it occurs so that the overcoming force is greater than that which is overcome, the change cannot take place in the opposite direction under the influence of the same forces.  We may say that the change has occurred in the first case in a reversible manner, and in the second case in an irreversible manner.

Strictly speaking, the overcoming force must always be more powerful than the force which it overcomes; but as the excess of force does not require to have any assignable value, we may think of it as becoming continually smaller and smaller, so that its value may approach to naught as nearly as we please.  Hence it may be seen that the case in which the changes take place reversibly is a limit which in reality is never quite reacted, by to which we can approach as nearly as we please.  We may therefore, in theoretical discussions, still speak of this case as one which really exists; indeed, as a limiting case it possesses special theoretical importance.

I will take this opportunity of mentioning another process in which this distinction is likewise to be observed.  In order for one body to impart heat to another by conduction or radiation (in the case of radiation, wherein mutual communication of heat takes place, it is to be understood that we speak here of a body which gives out more heat than it receives), the body which parts with heat must be warmer that the body which takes up heat; and hence the passage of heat between two bodies of different temperature can take place in one direction only, and not in the contrary direction.  The only case in which the passage of heat can occur equally in both directions is when it takes place between bodies of equal temperature.  Strictly speaking, however, the communication of heat from one body to another of the same temperature is not possible; but since the difference of temperature may be as small as we please, the case in which it is equal to nothing, and the passage of heat accordingly reversible, is a limiting case which may be regarded as theoretically possible.

5. We will now deduce the mathematical expression for the above law, treating in the first place the case in which the change of condition undergone by the body under consideration takes place reversibly.  The result at which we shall arrive for this case will easily admit of subsequent generalizations, so as to include also the cases in which a change occurs irreversibly.

Let the body be supposed to undergo in infinitely small change of condition, whereby the quantity of heat contained in it, and also the arrangement of its constituent particles, may be altered.  Let the quantity of heat contained in it be expressed by H, and the change of this quantity by dH.  Further, le the work, both interior and exterior together, performed by the heat in the change of arrangement be denoted by dL, a magnitude which may be either positive or negative according as the active force of the heat overcomes the forces acting in the contrary direction, or is overcome by them.  We obtain the heat expended to produce this quantity of work by multiplying the work by the thermal-equivalent of a unit of work which we may call A; hence it is AdL.

The sum dH + AdL is the quantity of heat which the body must receive from without, and must accordingly withdraw from another body during the change of condition.  We have, however, already represented by dQ the infinitely small quantity of heat imparted to another body by the one which is undergoing modification, hence we must represent in a corresponding manner, by –dQ, the heat which it withdraws from another body.  We thus obtain the equation:

-dQ = dH + AdL


dQ + dH + AdL = 0

Note: In my previous memoirs I have separated from one another the interior and the exterior work performed by the heat during the change of condition of the body.  If the former be denoted by dI, and the latter by dW, the above equation becomes:

dQ + dH + AdI + AdW = 0

Since, however, the increase in the quantity of heat actually contained in a body, and the heat consumed by interior work during a change of condition, are magnitudes of which we commonly do not know the individual values, but only the sum of those values, and which resemble each other in being fully determined as soon as we know the initial and final conditions of the body, without our requiring to know how it has passed from the one to the other, I have thought it advisable to introduce a function which shall represent the sum of these two magnitudes, and which I have denoted by U.  Accordingly:

dU = dH + AdI

and hence the foregoing equation becomes:

dQ + dU + AdW = 0

and if we suppose the last equation integrated for any finite alteration of condition, we have:

Q + U + AW = 0

These are the equations which I have used in my memoirs published in 1850 and in 1854, partly in the particular form in which they are here given, with no other difference than that I there took the positive and negative quantities of heat in the opposite sense to what I have done here, in order to attain greater correspondence with the equation (I) given in Art. 1.

(Skipping a paragraph of foot-notes on how W. Thomson and Kirchhoff have used and defined U)

In order now to be able to introduce the disgregation also into the formulae, we must first settle how we are to determine it as a mathematical quantity.

(Skipping about three paragraphs)

Accordingly, let Z be the disgregation of the body, and dZ an infinitely small change of it, and let dL be the corresponding infinitely small quantity of work, we can then put:

dL = KTdZ
The mechanical work which can be done by heat during any change of the arrangement of a body is proportional to the absolute temperature at which this change occurs.
(skipping 4 pages)

9. I believe, indeed, that we must extend the application of this law, supposing it to be correct, still further, and especially to chemical combinations and decompositions.

The separation of chemically combined substances is likewise an increase of the disgregation, and the chemical combination of previously isolated substances is a diminution of their disgregation; and consequently these processes may be brought under considerations of the same class as the formation or precipitation of vapour.  That in this case also the effect of heat is to increase the disgregation, results from many well-known phenomena, many compounds being decomposable by heat into their constituents—as, for example, mercuric oxide, and, at very high temperatures, even water.  To this it might perhaps be objected that, in other case, the effect of increased temperature is to favor the union of two substances—that, for instance, hydrogen and oxygen do not combine at low temperatures, but do so easily at higher temperatures.  I believe, however, that the heat exerts here only a secondary influence, contributing to bring the atoms into such relative positions that their inherent forces, by virtue of which they strive to unite, are able to come into operation.  Heat itself can never, in my opinion, tend to produce combination, but only, and in every case, decomposition.

The quantity of heat actually present in a body depends only on its temperature, and not on the arrangement of its constituent particles.
See Also

Scarce original printing copies, when available, sell for $500-1000 dollars or more. The textbook, however, can be read as a Google books copy and recently became available: in 2008 in paperback and hardcover as a reprint (by BiblioBazaar at Amazon, 22 USD), as pictured adjacent, or by Kessinger Publishing, 27 USD.

Prior to this, only the first memoir by Clausius, of 1850, was available as a book-in-print, published by Dover in 1960, entitled: Reflections on the motive power of fire by Sadi Carnot and other papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius Edited with an Introduction by E. Mendoza (shown below) which is a, 152-page, three-volume collection containing the following three founding thermodynamic papers: