Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XX

by Josiah Willard Gibbs

XX.

RUDOLF JULIUS EMANUEL CLAUSIUS.

[Proceedings of the American Academy, new series, vol. xvi, pp. 458–465, 1889.]

Rudolf Julius Emanuel Clausius was born at Cöslin in Pomerania, January 2, 1822. His studies, after 1840, were pursued at Berlin, where he became Privat-docent in the University, and Instructor in Physics in the School of Artillery. He was Professor of Physics at Zürich in the Polytechnicum (1855–67) and in the University (1857–67), at Würzburg (1867–69), and finally at Bonn (1869–88), where he died on the 24th of August, 1888.

His literary activity commenced in 1847, with the publication of a memoir in Crelle's Journal, "Ueber die Lichtzerstreuung in der Atmosphäre, und über die Intensität des durch die Atmosphäre reflectirten Sonnenlichts."^[1] This was immediately followed by other writings relating to the same subject, two of which were subsequently translated from Poggendorff's Annalen^[2] for Taylor's Scientific Memoirs. A treatise entitled "Die Lichterscheinungen der Atmosphäre" formed part of Grunert's "Beiträge zur meteorologischen Optik."

An entirely different subject, the elasticity of solids, was discussed in his paper (1849), "Ueber die Veränderungen, welche in den bisher gebräuchlichen Formeln für das Gleichgewicht und die Bewegung fester Körper durch neuere Beobachtungen nothwendig geworden sind."^[3]

But it was with questions of quite another order of magnitude that his name was destined to be associated. The fimdamental questions concerning the relation of heat to mechanical effect, which had been raised by Rumford, Carnot, and others, to meet with little response, were now everywhere pressing to the front.

"For more than twelve years," said Regnault in 1853, "I have been engaged in collecting the materials for the solution of this question:—Given a certain quantity of heat, what is, theoretically, the amount of mechanical effect which can be obtained by applying the heat to evaporation, or the expansion of elastic fluids, in the various circumstances which can be realised in practice?"^[4] The twenty-first volume of the Memoirs of the Academy of Paris, describing the first part of the magnificent series of researches which the liberality of the French government enabled him to carry out for the solution of this question, was published in 1847. In the same year appeared Helmholtz's celebrated memoir, "Ueber die Erhaltung der Kraft." For some years Joule had been making those experiments which were to associate his name with 6ne of the fundamental laws of thermodynamics and one of the principal constants of nature. In 1849 he made that determination of the mechanical equivalent of heat by the stirring of water which for nearly thirty years remained the unquestioned standard. In 1848 and 1849 Sir William Thomson was engaged in developing the consequences of Carnot's theory of the motive power of heat, while Professor James Thomson in demonstrating the effect of pressure on the freezing point of water by a Carnot's cycle, showed the flexibility and the fruitfulness of a mode of demonstration which was to become canonical in thermodynamics. Meantime Rankine was attacking the problem in his own way, with one of those marvellous creations of the imagination of which it is so difficult to estimate the precise value.

Such was the state of the question when Clausius published his first memoir on thermodynamics: "Ueber die bewegende Kraft der Wärme, und die Gesetze, welche sich daraus für die Wärmelehre selbet ableiten lassen."^[5]

This memoir marks an epoch in the history of physics. If we say, in the words used by Maxwell some years ago, that thermodynamics is "a science with secure foundations, clear definitions, and distinct boundaries,"^[6] and ask when those foundations were laid, those definitions fixed, and those boundaries traced, there can be but one answer. Certainly not before the publication of that memoir. The materials indeed existed for such a science, as Clausius showed by constructing it from such materials, substantially, as had for years been the common property of physicists. But truth and error were in a confusing state of mixture. Neither in France, nor in Germany, nor in Great Britain, can we find the answer to the question quoted from Regnault. The case was worse than this, for wrong answers were confidently urged by the highest authorities. That question was completely answered, on its theoretical side, in the memoir of Clausius, and the science of thermodynamics came into existence. And as Maxwell said in 1878, so it might have been said at any time since the publication of that memoir, that the foundations of the science were secure, its definitions clear, and its boundaries distinct.

The constructive power thus exhibited, this ability to bring order out of confusion, this breadth of view which could apprehend one truth without losing sight of another, this nice discrimination to separate truth from error,—these are qualities which place the possessor in the first rank of scientific men.

In the development of the various consequences of the fundamental propositions of thermodynamics, as applied to all kinds of physical phenomena, Clausius was rivalled, perhaps surpassed, in activity and versatility by Sir William Thomson. His attention, indeed, seems to have been less directed toward the development of the subject in extension, than toward the nature of the molecular phenomena of which the laws of thermodynamics are the sensible expression. He seems to have very early felt the conviction, that behind the second law of thermodynamics, which relates to the heat absorbed or given out by a body, and therefore capable of direct measurement, there was another law of similar form but relating to the quantities of heat (Le., molecular vis viva) absorbed in the performance of work, external or internal.

This may be made more definite, if we express the second law in a mathematical form, as may be done by saying that in any reversible cyclic process which a body may undergo

$\int {\frac {dQ}{t}}=0,$

where $dQ$ is an elementary portion of the heat imparted to the body, and $t$ the absolute temperature of the body, or the portion of it which receives the heat. Or, without limitation to cyclic processes, we may say that for any reversible infinitesimal change,

$dQ=t\,dS,$

where $S$ denotes a certain function of the state of the body, called by Clausius the entropy. The element of heat may evidently be divided into two parts, of which one represents the increase of molecular vis viva in the body, and the other the work done against forces, either external or internal. If we call these parts $dH$ and $dQ_{w},$ we have

$dQ=dH+dQ_{w}.$

Now the proposition of which Clausius felt so strong a conviction was that for reversible cyclic processes

$\int {\frac {dQ_{w}}{dt}}=0,$

and that for any reversible infinitesimal change

$dQ_{w}=t\,dZ,$

where $Z$ is another function of the state of the body, which he called the disgregation, and regarded as determined by the positions of the elementary parts of the body without reference to their velocities. In this respect it differed from the entropy. An immediate consequence of these relations is that for any reversible cyclic process

$\int {\frac {dH}{t}}=0,$

and therefore that $H,$ the molecular vis viva of the body, must be a function of the temperature alone. This important result was expressed by Clausius in the following words: "Die Menge der in einem Körper wirklich vorhandenen Wärme ist nur von seiner Temperatur und nicht von der Anordnung seiner Bestandtheile abhängig."

To return to the equation

$dQ_{w}=t\,dZ.$

This expresses that heat tends to increase the disgregation, and that the intensity of this tendency is proportional to the absolute temperature. In the words of Clausius: "Die mechanische Arbeit, welche die Wärme bei irgend einer Anordnungsänderung eines Korpers thun kann, ist proportional der absoluten Temperatur, bei welcher die Aenderung geschieht."

Such in brief and in part were the views advanced by Clausius in 1862, in his memoir, "Ueber die Anwendung des Satzes von der Aequivalenz der Verwandlungen auf die innere Arbeit."^[7] Although they were advanced rather as a hypothesis than as anything for which he could give a formal proof, he seems to have little doubt of their correctness, and his confidence seems to have increased with the course of time.

The substantial correctness of these views cannot now be called in question. The researches especially of Maxwell and Boltzmann have shown that the molecular vis viva is proportional to the absolute temperature, and Boltzmann has even been able to determine the precise nature of the functions which Clausius called entropy and disgregation.^[8] But the anticipation, to a certain extent, at so early a period in the history of the subject, of the ultimate form which the theory was to take, shows a remarkable insight, which is by no means to be lightly esteemed on account of the acknowledged want of a rigorous demonstration. The propositions, indeed, as relating to quantities which escape direct measurement, belong to molecular science, and seem to require for their complete and satisfactory demonstration a considerable development of that science. This development naturally commenced with the simplest case involving the characteristic problems of the subject,—the case, namely, of gases. The origin of the kinetic theory of gases is lost in remote antiquity, and its completion the most sanguine cannot hope to see. But a single generation has seen it advance from the stage of vague surmises to an extensive and well established body of doctrine. This is mainly the work of three men, Clausius, Maxwell, and Boltzmann, of whom Clausius was the earliest in the field, and has been called by Maxwell the principal founder of the science.^[9] We may regard his paper (1867), "Ueber die Art der Bewegung, welche wir Wärme nennen,"^[10] as marking his definite entrance into this field, although many points were incidentally discussed in earlier papers.

This was soon followed by his papers, "Ueber die mittlere Länge der Wege, welche bei der Molecularbewegung gasförmiger Körper von den einzelnen Molecülen zurückgelegt werden,"^[11] and "Ueber die Wärmeleitung gasförmiger Körper."^[12]

A very valuable contribution to molecular science is the conception of the virial, defined in his paper (1870), "Ueber einen auf die Wärme anwendbaren Satz,"^[13] where he shows that in any case of stationary motion the mean vis viva of the system is equal to its virial.

In the mean time, Maxwell and Boltzmann had entered the field. Maxwell's first paper, "On the Motions and Collisions of perfectly elastic Spheres,"^[14] was characterized by a new manner of proposing the problems of molecular science. Clausius was concerned with the mean values of various quantities which vary enormously in the smallest time or space which we can appreciate. Maxwell occupied himself with the relative frequency of the various values which these quantities have. In this he was followed by Boltzmann. In reading Clausius, we seem to be reading mechanics; in reading Maxwell, and in much of Boltzmann's most valuable work, we seem rather to be reading in the theory of probabilities. There is no doubt that the larger manner in which Maxwell and Boltzmann proposed the problems of molecular science enabled them in some cases to get a more satisfactory and complete answer, even for those questions which do not at first sight seem to require so broad a treatment.

Boltzmann's first work, however (1866), "Ueber die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie,"^[15] was in a line in which no one had preceded him, although he was followed by some of the most distinguished names among his contemporaries. Somewhat later (1870) Clausius, whose attention had not been called to Boltzmann's work, wrote his paper, "Ueber die Zurückführung des zweiten Hauptsatzes der mechanischen Wärmetheorie auf allgemeine mechanische Principien."^[16]

The point of departure of these investigations, and others to which they gave rise, is the consideration of the mean values of the force-function and of the vis viva of a system in which the motions are periodic, and of the variations of these mean values when the external influences are changed. The theorems developed belong to the same general category as the principle of least action, and the principle or principles known as Hamilton's, which have to do, explicitly or implicitly, with the variations of these mean values.

Among other papers of Clausius on this subject, we may mention the two following: "eber einen neuen mechanischen Satz in Bezug auf stationäre Bewegung"^[17] (1873), and "Ueber den Satz vom mittleren Ergal und seine Anwendimg auf die Molecularbewegungen der Gase"^[18] (1874).

The first problem of molecular science is to derive from the observed properties of bodies as accurate a notion as possible of their molecular constitution. The knowledge we may gain of their molecular constitution may then be utilized in the search for formulas to represent their observable properties. A most notable achievement in this direction is that of van der Waals, in his celebrated memoir, "On the Continuity of the Gaseous and Liquid States." To this part of the subject belong the following papers of Clausius: "Ueber das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur,"^[19] and "Ueber die theoretische Bestimmung des Dampfdruckes und der Volumina des Dampfes und der Flüssigkeit" (two papers).^[20]

Another matter in which Clausius showed his originality and power was the vexed subject of electrodynamics, as treated in his memoir, "Ueber die Ableitung eines neuen electrodynamischen Grundgesetzes."^[21] Various points in the theory of electricity in which the principles of thermodynamics or of molecular science were involved, had previously been treated in different papers, of which the earliest appeared in 1852,^[22] while the doctrine of the potential (electrical and gravitational) was treated in a separate book, which appeared in 1859, with the title, "Die Potentialfunction und das Potential, ein Beitrag zur mathematischen Physik." This subsequently went through several editions, in which it was revised and enlarged. All these subjects, with others, were brought together in a single volume, "Die mechanische Behandlung der Electricität," which appeared in 1879, forming the second volume of his "Mechanische Wärmetheorie."^[23] Later papers on electricity related to the principles of electrodynamics,^[24] electrical and magnetic units,^[25] and dynamo-electric machines.^[26]

The Royal Society's catalogue of scientific papers, and the excellent indices to the Annalen der Physik und Chemie, in which Clausius's work usually appeared, render it unnecessary to enumerate in detail his scientific papers. The list, indeed, would be a long one. The Royal Society's catalogue gives seventy-seven titles for the years 1847–1873. Subsequently twenty-five papers have appeared in the Annalen alone, and about half as many others elsewhere.

But such work as that of Clausius is not measured by counting titles or pages.a His true monument lies not on the shelves of libraries, but in the thoughts of men, and in the history of more than one science.

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1. Vol. xxxiv, p, 122, and vol. xxxvi, p. 186.
2. Vol. lxxvi, pp. 161 and 188.
3. Pogg, Ann., vol. lxxvi, p. 46 (1849).
4. Comptes Rendus, voL xxxvi, p. 676.
5. Read in the Berlin Academy, February 18, 1850, and published in the March and April numbers of Poggendorff's Annalen.
6. Nature, voL xvii, p. 257.
7. Pogg. Ann., vol. cxvi, p. 73. See also vol. cxxvii, p. 477 (1866).
8. Sitzungsberichte Wien. Akad., vol. lxiii, p. 728 (1871).
9. Nature, vol. xvii, p. 278.
10. Pogg. Ann., vol. c, p. 353 (1857).
11. Ibid., vol. cv, p. 239 (1858). See also Wied. Ann., vol. x, p. 92.
12. Ibid., vol. cxv, p. 1 (1862).
13. Ibid., vol. cxli, p. 124. See also Jubelband, p. 411.
14. Phil. Mag., vol. xix, p. 19 (1860).
15. Sitzungsbericthe Wien. Akad., vol. liii, p 165.
16. Pogg. Ann., vol. cxlii, p. 433.
17. Ibid., vol. cl, p. 106.
18. Ibid., Ergänzungsband vii, p. 215.
19. Wied. Ann., vol. ix, p. 337 (1880).
20. Ibid., vol. xiv, p. 279 and p. 692 (1881).
21. Crelle's Journal, vol. lxxxii, p. 85 (1877).
22. "Ueber das mechanische Aequivalent einer electrischen Entladung und die dabei stattfindende Erwärmung des Leitungsdrahtes." Pogg. Ann., voL lxxxvi, p. 337. "Ueber die bei einem stationiären electrischen Strome in dem Leiter gethane Arbeit und erzeugte Warme." Pogg. Ann., vol. lxxxvii, p. 415 (1852). "Ueber die Anwendung der mechanischen Wärmetheorie anf die thermoelectrischen Erscheinungen." Pogg. Ann., vol. xc, p. 513 (1853). "Ueber die Electricitätsleitung in Electrolyten." Pogg. Ann., vol. ci, p. 338 (1857).
23. The first volume of this work appeared in 1876, and contained the general theory with the more immediate consequences of the two fundamental laws. The third volume haa not yet appeared, but it is expected very soon, edited by Professor Planck and Dr. Pulfrich. In a certain sense this work may be regarded as a second edition of an earlier one (1864 and 1867), which consisted of a reprint of papers and had the title "Abhandlungen über die mechanische Wärmetheorie.".
24. Wied. Ann., vol. x, p. 608; vol. xi, p. 604.
25. Ibid., voL xvi, p. 529; voL xvii, p. 713.
26. Ibid., vol. xx, p. 353; vol. xxi, p. 385.