Popular Science Monthly/Volume 74/May 1909/Josiah Willard Gibbs and his Relation to Modern Science I

by Fielding Hudson Garrison

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JOSIAH WILLARD GIBBS AND HIS RELATION TO MODERN SCIENCE^[1]

By FIELDING H. GARRISON, M.D.

ASSISTANT LIBRARIAN, ARMY MEDICAL LIBRARY, WASHINGTON, D. C.

THE scientific papers of the late Professor Willard Gibbs, of Yale University, which have been brought together in a memorial edition by his pupil and colleague, Professor Bumstead, furnish one of the most remarkable examples in existence of the value and fruitfulness of mathematical methods in scientific investigation. Originally printed in the scientific transactions of his native state, some of these papers have, by reason of the speedy exhaustion of their first imprints, been much sought after, but for many years practically inaccessible, except in French and German translations.

Gibbs was not, like Edison, Langley, Rowland, the inventor, experimenter or expert in delicate measurements, nor was he the great all round physicist, like Maxwell, Helmholtz or Lord Kelvin. He was essentially and almost exclusively the mathematician, whose special function was not the discovery of isolated facts or new methods of experimental procedure, but the introduction of new currents of ideas; and it was the severe and rigorous form in which his ideas were cast that for a long period of time retarded their general adoption by the scientific world. If we accept Cayley's view that theoretical dynamics is in reality a branch of pure mathematics,^[2] then the opus magnum of Gibbs, his survey of heterogeneous equilibrium, may be fairly accounted a legitimate triumph for pure mathematics.

The enormous growth of biological science in the nineteenth century has somewhat overshadowed the importance of the deductive and analytic methods which were the very life of the science of the past, and although mathematics, beginning with primitive man's attempt to count, lies at the basis of all his exact knowledge of the material world, its true function has not always been appreciated or even understood. The synthetic or Baconian method, of which we have such supreme examples in the work of Galileo and Darwin, must always appeal by its very simplicity to scientific men, since, instead of indulging in special assumptions and hypotheses, it has obtained from nature, by observation and experiment alone, facts which, as in the Darwinian theory, can be concentrated upon some special proposition to be induced with the surety of Moltke's tactical device, Getrennt marschieren vereint schlagen. As science aims only to classify, predict and control phenomena, it has no absolute philosophic certainty except as a logical interpretation of the empirical facts of man's experience, and of the relative limitations of mathematical deduction and physical induction it has been well said that "the former breaks down on the subtlety of nature, the latter on its imperceptibility."^[3] Galileo's telescope, Leeuwenhoek's microscope, Lavoisier's balance, Kirchhoff's spectroscope, are doubtless of more practical value, but certainly not of more scientific importance than formal and symbolic logic, the calculus, determinants, quaternions, vector analysis or the improved formulation of dynamics. Without induction, it is true, no new facts; but without deductive methods there could be no interpretation of these facts, nor would scientists have the means of predicting other facts which go beyond experience, or of controlling phenomena. Yet an authority so open-minded as Professor Huxley, who seems to have confused mathematical methods with the scholastic reasoning and bigotry which opposed the great cause he championed, seldom lost an opportunity to say hard things about the science "which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation."^[4] In Professor Sylvester's brilliant and memorable reply to some of Huxley's after-dinner denunciations,^[5] "the most eloquent of mathematicians" retorted upon his adversary that his chaffing might have been more guarded "had his speech been made before instead of after dinner,"^[6] and went on to show that the maligned science employs not only imagination and invention, but observation and experiment at need. Even mathematicians, Sylvester pointed out, occasionally make discoveries, as witness Eisenstein's discovery of invariants, which was happened upon by purely physical observation "just as accidentally and unexpectedly as M. du Chaillu might meet a gorilla in the country of the Fantees."^[7] The touchstone of the matter lies in the one really telling remark that Huxley made about it, viz., that mathematics will not yield correct results if applied to erroneous data.^[8] The advance of modern science is largely bound up with the perfection of instruments of precision, and we have learned from the teaching of Lord Kelvin and the writings of Poincaré to recognize that mathematical science is itself a powerful instrument of precision, which, if applied in the right way to data of the right sort, may yield important results. Biologists like Loeb, Francis Galton, Karl Pearson, Driesch, physiologists like Helmholtz and Chauveau, psychologists like Wundt and Fechner, the leading physical chemists, van't Hoff, Arrhenius, Roozeboom, Ostwald, van der Waals, van Laar, Nernst, Le Chatellier, Bancroft, have all employed mathematics as a necessary part of their equipment, and more especially has knowledge been advanced by physicists like Maxwell, Lord Kelvin, Helmholtz, Hertz, the Curies, Stokes, J. J. Thomson and Gibbs, who have got at the more imperceptible aspects of nature by deductive methods "interlaced with physical induction and experience." In the discovery of radium by the Curies all the processes up to the use of pitchblende were inductive; after that every step taken was pure deduction, based upon the a priori assumption of an unknown substance. Maxwell could predict the existence of electromagnetic waves from his equations^[9] at least twenty-five years before their actual demonstration by Hertz^[10] and Gibbs's algebraic statement of the theorems of chemical statics was far in advance of their laboratory verification.

Ostwald, in his interesting "Biologic des Naturforschers,"^[11] has divided men of science into two classes: The classicists (Klassiker), men like Newton, Lagrange, Gauss, Harvey, who, dealing with a limited number of ideas in their work, seek formal perfection and attain it, leaving no school of followers behind them, but only the effect of the work itself; and the romanticists (Romantiker) who, like Liebig, Faraday, Darwin, Maxwell, are bold explorers in unknown fields, men fertile in ideas, leaving many followers and many loose ends of unfinished work which others complete. In the logical perfection of his work and in his unusual talent for developing a theme in the most comprehensive and exhaustive manner, Gibbs was emphatically the Klassiker. But in the scientific achievement of his early manhood he showed something of the spirit of the Romantiker also. His mathematical theory of chemical equilibrium was, as we have seen, far in advance of any experimental procedure known or contemplated at the time of its publication, and, although some of his predecessors, like James Thomson, Massieu, Horstmann, had come within sight of the new land and even skirted its shores, Gibbs, with the adventurous spirit of the true pioneer, not only conquered and explored it, but systematically surveyed it, living to see part of his territory occupied by a thriving band of workers, the physical chemists. Cayley, in his report on theoretical dynamics in 1857,^[12] expressed his conviction that the science of statics "does not admit of much ulterior development." The work of Gibbs has added to it the immense field of chemical equilibrium and wherever "phases," "heterogeneous systems," "chemical and thermodynamic potentials," or "critical states" are mentioned he has left his impress upon modern scientific thought. It is not without reason then, that Ostwald has called this mathematician "the founder of chemical energetics," asserting that "he has given new form and substance to chemistry for another century at least."^[13]

Josiah Willard Gibbs was born in New Haven, Conn., on February 11, 1839. His father, who was descended from Sir Henry Gibbs, of Honington, Warwickshire, was professor of sacred literature in Yale College during the years 1821-61, and was esteemed for unusual scholarship in his day. The son, like many other mathematicians, showed early aptitude for linguistic as well as for mathematical studies, and, entering Yale in 1854, was graduated in 1858, after winning many prizes and distinctions in Latin and mathematics. He began to teach mathematics and physics at Yale in 1863, having received his doctor's degree in that year. During 1866-69 he traveled in Europe, studying his chosen subjects at Paris, Berlin and Heidelberg, and hearing the lectures of Magnus, Kirchhoff and Helmholtz. In July, 1871, two years after his return, he was appointed professor of mathematical physics in Yale College, about the same time that Clerk Maxwell assumed similar duties in the Cavendish Laboratory, at Cambridge. This position Professor Gibbs held until his death, April 28, 1903. Professor Gibbs devoted his whole life to his work, the interests of his university and his pupils, and apart from the earlier years of travel and some excursions into the field of controversy, his was the quiet and uneventful career of the typical man of science. During his period of productive activity, 1873-1902, he made important contributions to the electromagnetic theory of light, multiple algebra and vector analysis, astronomy, theoretical and statistical dynamics, but his enduring fame rests chiefly upon his work in thermodynamics, the science which, in the words of his English biographer, he reduced "to its canonical form." He was a member of most of the important scientific societies of the world, was Rumford medalist of the American Academy of Arts and Sciences in 1881 and in 1901, the Royal Society, of London, conferred its highest distinction, the Copley medal, upon Professor Gibbs, as being "the first to apply the second law of thermodynamics to the exhaustive discussion of the relation between chemical, electrical and thermal energy and capacity for external work."^[14]

"The history of thermodynamics," says Maxwell, "has an especial interest as the development of a science within a short time and by a small number of men, from the condition of a vague anticipation of nature to that of a science with secure foundations, clear definitions and exact boundaries."^[15] Its development falls conveniently into three stages: (1) The derivation of the two laws governing thermal transformations of energy by Carnot, Rankine, Mayer, Joule, Clausius and Kelvin. (3) The deductive application of the second law to all physico-chemical phenomena by Gibbs. (3) The application of probabilities and statistical methods to the kinetic theory of gases by Clausius, Kelvin, Maxwell and Boltzmann and the final derivation of the theorems and equations of thermodynamics by statistical induction from the average behavior of mechanical systems by Gibbs.

"It must not be thought that heat generates motion or motion heat (though in some respects this is true) but the very essence of heat or the substantial self of heat is motion and nothing else.^[16] In this sentence from the Novum Organum it is clear that Bacon, like Descartes, Count Rumford, Sir Humphry Davy and Young, had a more or less definite notion of the dynamic nature of heat and its convertibility into work. But the exact science which treats of heat as a mode of energy begins with the publication, in^ 1884, of the "Réflexions sur la puissance motrice du feu" of Sadi Carnot, whom Lord Kelvin calls the "profundest thinker in thermodynamic philosophy" in the first half of his century.^[17] In this little work we have the first treatment of the heat engine as a reversible "cycle of operations," a mechanism which can be worked backward with its every action reversed; and such a system is now known everywhere as a "Carnot cycle." Carnot compared the motor power of heat to a fall of water.^[18] As the power of the waterfall depends upon its height and the quantity of fluid employed, so the motor power of heat depends, not upon the nature of the working substance, but upon the quantity of heat employed and the difference in temperature between its source (the boiler) and the sink (or exhaust cylinder) to which it flows. A heat motor, then, requires a hot body and a cold body; the ideally perfect engine would be completely reversible and the efficiency of engines working between the same limits of temperature is the same. In other words, heat can not perform work except by spontaneous flow from a higher to a lower temperature. This is Carnot's principle, from which is derived the first statement of the second law of thermodynamics, that as water flows towards the sea-level, but never backwards to its source, so heat can not flow from a colder to a warmer body. But Carnot, like every one else in his day, still thought that heat (calorique), like the water in the waterfall, was an indestructible, material substance and that the quantity of heat given out by the exhaust chamber of the engine is exactly the same as that taken in at the boiler. Although his posthumous papers indicate that he corrected this view before his death, he assumed that if we could find some way to consume the heat of a given body without the necessity of conveying it to a colder body, we might create motor power without fuel or obtain work from nothing, which would be perpetual motion. As late as 1865 an authority like Rankine^[19] still believed that heat is of material essence, and when in 1842-7 the labors of Robert Mayer and of Joule established the mechanical equivalent of heat and Helmholtz^[20] in 1847 showed that the first law of thermodynamics, the principle of conservation of energy, is applicable to all physical phenomena, it was found difficult to reconcile this principle with Carnot's tacit assumption that heat is unchangeable and indestructible. Even a physicist like William Thomson^[21] (the late Lord Kelvin) confessed himself baffled by the problem in 1849 and turned aside to establish his "absolute scale of temperature," without which further progress in the science would have been impossible; but his brother James Thomson, one of the earlier pioneers of physical chemistry, was able, by an implicit denial of Carnot's assumption, to predict and prove that the freezing point of water would be lowered by pressure (1849).^[22] The difficulty was, at length, settled in 1850 by Clausius, whose memoir "On the motor power of heat," "marks," says Gibbs, "an epoch in the history of physics," for before its publication, "truth and error were in a confusing state of mixture,"^[23] and "wrong answers were confidently urged by the highest authorities."

To Clausius we owe the doctrine, foreshadowed by Bacon, that the heat of a body is the rapid movement, or vis viva, of its molecules; the kinetic theory of gases and the molecular theory of electrolysis, since extended by Arrhenius into the doctrine of electrolytic or ionic dissociation. Clausius showed that part of the heat in a Carnot cycle is converted into available mechanical energy and consumed as work, while the rest of the heat can not be so utilized, because it exists in a completely diffused state. The perpetual motion which might be obtained from utilizing the heat of surrounding objects is impossible because such heat being completely diffused is, in Lord Kelvin's phrase, unavailable thermal energy. So the fallacious principle of the conservation of heat became merged into the doctrine of conservation of energy; the reconciliation between the first and second laws, which, like the Kantian antinomies, had seemed mutually contradictory, was effected, and Clausius reasoned that heat can not flow from a colder to a warmer body without compensation, that is, without the intervention of external forces. Meanwhile Lord Kelvin, to whom we owe our ideas and definitions of intrinsic and available energy, was able, in 1852, to shadow forth that comprehensive form of the second law afterwards stated by him as a physical law of irreversibility, according to which there is a universal tendency in nature towards irrevocable dissipation of energy.^[24] From this time on progress in the science was rapid. The mathematical part of the theory was improved by the introduction of the scalar value which Rankine called the "thermodynamic function"^[25] and Clausius the "entropy"^[26] of a body, a variable quantity, momentary increase or decrease of which indicates (in a reversible physico-chemical transformation) whether heat is leaving or entering the body at that moment, irrespective of its temperature or previous condition. The temperature of a body, although measured by arbitrary standards, is in reality a non-measurable "intensity" or quality of the body, depending upon whether it is capable of giving up or receiving heat, i. e., upon its dynamic potentiality; and, in practise, addition of heat to a body may change its physical state but does not necessarily alter its temperature; nor does a change of temperature, as Trevor has recently insisted,^[27] necessarily imply absorption or development of heat; but the entropy of a body is a definite measurable "capacity," and has been compared by Trevor to the weight in a mechanical system. For instance, imagine a frictionless, reversible mechanical system, such as a weight suspended by a cord passing over a pulley, and let this weight by its fall to the ground do a certain amount of work, such as raising a body attached to the other end of the cord. The potential energy of the system is measured by the height of the weight above ground, and when the weight falls, the available energy of the system decreases at each point and moment of the descent, while the unavailable energy undergoes a corresponding increase point for point. On reversing the operation and raising the weight, the available energy of the system is seen to increase while the unavailable energy decreases (i. e., increases in a negative direction). So, in any reversible thermodynamic system, the entropy at any moment is an index, determinant, or coefficient of the relative amount of unavailable energy it possesses. When the temperature in an isolated reversible system is constant, as in jacketed steam, the system is "isothermal" and the entropy may vary at any instant; but if a reversible system be so isolated that no heat can enter or leave the body, the temperature might vary but the entropy would be constant, and such systems, of which we have an approximation in the insulated cylinder of an engine, were called "adiabatic" by Rankine and "isentropic" by Gibbs.

There are no mathematical or ideally reversible systems in existence, although we have natural approximations to them in the motions of the heavenly bodies and in certain chemical reactions, or human approximations in reversible heat engines or reversible electric apparatus; the spontaneous processes of nature are always irreversible, proceeding irrevocably in a definite direction with no negative or reversed dissipation of energy. In spontaneous, irreversible flow of heat from a warmer to a colder body, the entropy or unavailable thermal energy of the system increases inevitably to a maximum. In other words, the entropy of a system is a criterion of its loss of efficiency or available energy during irreversible change, and it follows, in the memorable and aphoristic statement of the first and second laws by Clausius, that, while the energy of the universe is constant, its entropy (or that part of its energy which is unavailable) tends to a maximum and can never decrease:

Die Energie der Welt ist constant,
Die Entropie der Welt strebt einem Maximum zu.

With this important generalization, which is the motto of Gibbs's principal memoir, the first stage of thermodynamics ends. By stating the second law as irreversible increase of entropy in natural processes and by adopting some definite standard of the latter, all exact or scalar relations in thermodynamics can be treated as shown by Rankine, Clausius, and Gibbs, in a precise and definite manner.^[28] But the Helmholtz-Kelvin statement of the first and second laws as conservation and dissipation of energy enables us to apply these principles in the broadest and most philosophical way. Lord Kelvin extended the application of the second law to cosmic physics and, with Boltzmann, to predictions as to the ultimate thermal death of the earth. Meanwhile Clausius, Maxwell and Boltzmann began to apply the second law to the kinetic theory of gases, a phase of the subject which belongs essentially to the last stage of its development. Maxwell in particular emphasized the important point that since the heat of a body is the kinetic energy of its molecular motions, the second law is in reality not a mathematical but a statistical truth. It can not, says Maxwell, be reduced to a form as axiomatic as that of the first law, but stands upon a lower plane of probability, because it depends upon the motions of millions of molecules of which we can not get hold of a single one.^[29] Could we reduce ourselves to molecular dimensions, and with the gift of molecular vision trace the movements of individual molecules, the distinction between work and heat would

Thomson's "available energy," with the statement that Clausius meant by it that part of the energy which can not be converted into work. As Gibbs pointed out, this is entirely incorrect. The entropy of a body is a definite physical property of the body itself, and can not be measured by the same unit as energy. If $dQ$ represent the amount of heat imparted to a body at any point and $T$ its absolute temperature at that point, Clausius has shown that $dQ/T$ represents the infinitesimal change of entropy at that point for any given moment. The total (fliange of entropy of any reversible chemical system in passing from an initial state $a$ to a final state $b$ would then be

$\Sigma \ \int _{a}^{b}\ {\frac {dQ}{T}},$

and for a reversible (Carnot) cycle the mathematical statement of the second law is the "Carnot-Clausius equation":

$\int \left({\frac {dH}{T}}\right)=0.$

This means that the positive and negative entropies of the system in passing from $a$ to $b$ and in reversing backwards from $b$ to $a$ must balance each other. Or as Gibbs has expressed it, "The second law requires (for a reversible cycle) that the algebraic sum of all the heat received from external bodies, divided, each portion thereof, by the absolute temperature at which it is received shall be zero." The criterion of irreversible processes is the "inequality of Clausius"

$\int \left({\frac {dQ}{T}}\right)<0$

which implies that the phenomenon will proceed irrevocably or irreversibly in a definite direction, entropy increasing or available energy dissipating to a maximum until a final state of rest or equilibrium (uniformly distributed temperature) is attained. Reversible thermodynamics deals, then, with equations; irreversible thermodynamics with inequalities, because in reversible processes the total entropy of a system remains unchanged while in irreversible processes it continually increases.

vanish, and nothing would remain but the motions of material systems and the laws of mechanics. Hence the second law must either be obtained from our actual experience "with real bodies of sensible magnitude," or else derived a posteriori, as shown by Boltzmann, Helmholtz and Gibbs, from averages of the hypothetical motions of mechanical systems. In aid of this conception of the problem, Maxwell introduced his whimsical notion of the "sorting demon," a being endowed with molecular vision, who would be able through intelligence alone to sort or direct the molecular movements at will and so reverse the action of the second law on occasion.^[30]

The second stage of thermodynamics begins in 1872-3 with August Friedrich Horstmann's application of the entropy principle to problems of chemical dissociation.^[31] In October, 1873, Horstmann announced the condition for chemical equilibrium to be that of maximum entropy,^[32] and in December of the same year Gibbs, in a modest footnote, stated that the condition for thermodynamic equilibrium in a chemical system at constant temperature and pressure is that the function now universally known as the thermodynamic potential should be a minimum.^[33] In 1875 Lord Rayleigh stated that dissipation of energy is a sufficient if not a necessary condition for chemical change,^[34] and in October, 1875, appeared the first installment of Gibbs's memoir of three hundred pages on chemical equilibrium, which, by its applications of the entropy principle to all physico-chemical or energetic phenomena, has become a true scientific classic doing for the second law what Helmholtz, in his treatise on the conservation of energy, had previously done for the first.

Gibbs began his work in thermodynamics in 1873, with two important papers on diagrams and surfaces.^[35] In the first of these he made a careful and thoroughgoing study of all the diagrams that might be of use or value in thermodynamics, the best known being that upon which volume and pressure are erected to scale as coordinates, derived from the familiar "Watts' indicator diagram found upon every steam engine. Of the new diagrams which Gibbs introduced, he attached most importance to the volume-entropy diagram, because it tells more about the physical properties of a working substance than about the heat employed or the work done. But the most important of Gibbs's innovations for practical engineering purposes is the temperature-entropy diagram, which represents the efficiency of a Carnot cycle as a simple rectangular figure and is, he points out, "nothing more nor less than a geometrical representation of the second law of thermodynamics." The area of the Watts diagram represents the work done by the engine; the area of the Gibbs diagram represents the heat it has received and, either upon separate blackboards or upon "quadrant diagrams," the two taken together have proved invaluable in teaching thermodynamics to engineers. As the indicator diagram tells the engineer what he wants to know about the work done upon the piston, the efficiency of the valves and passages and the total horse power of the engine, the entropy diagram gives him the heat taken in or given out and shows directly the losses of efficiency from such heat wastes as wire-drawing of steam, incomplete expansion, etc. Professor John Perry says that the thermodynamics of heat engines is revealed by the entropy diagram "as it can be revealed in no other way," and he describes how "a man almost illiterate, innocent of algebra, can use his $t,\Phi$ diagram of water steam or air or ammonium anhydride, obtaining in a few minutes answers to problems which the mathematical engineers of years ago spent days in solving."^[36] In England the temperature-entropy diagram has been found very useful in "engine testing laboratories," and its ultimate adoption is due to the persistent crusade of Mr. Macfarlane Gray, late chief engineer of the Royal Navy, who introduced it independently in 1880 as the "theta-phi" ($\Theta \,\Phi$) diagram. American engineers should not forget that this diagram was first described in scientific literature by Professor Willard Gibbs,^[37] who clearly pointed out its advantages, in visualizing the second law, for teaching purposes and its use and significance when attached to heat-engines. In his second memoir^[38] Gibbs extends his graphical methods to three-dimensional space, the first example of which was the volume-pressure-temperature diagram employed by James Thomson^[39] in 1871. The first solid diagram described by Gibbs had for its coordinates, volume, entropy and energy and is now generally known as the "thermodynamic surface." It is a solid model or relief-map, affording a bird's-eye view of the chemico-physical changes of a system at constant temperature and pressure as it passes through the coexistent states of solid, liquid, vapor or gas. Maxwell, who had himself written learnedly of diagrams, immediately recognized the importance of this paper, part of which he incorporated as a chapter in his "Theory of Heat,"^[40] and, shortly before his death, he sent Gibbs a copy of a model of the thermodynamic surface constructed to scale with his own hands.^[41] These solid diagrams have played a great part in the elaborate studies of the continuity of gaseous and liquid states by Van der Waals and his pupils, of which we have recently witnessed the final triumph in the liquefaction of helium.

During the years 1875-8, Gibbs published the work which is his chief title to fame, his memoir "On the Equilibrium of Heterogeneous Substances."^[42] This treatise deals, as the title implies, with the statics of chemical substances which, as gases, vapors, liquids or solids, are in actual physical contact with each other, whether influenced or modified by gravity, osmosis, catalysis, capillarity or electromotive force or existing under such varied aspects as gaseous mixtures, liquid films, "solid solutions," or crystals. For the first time chemical substances are treated as continuous or contiguous "phases" of "matter in mass" acted upon, like mechanical systems, by forces having "potentials"—a new way of looking at things which has since become the definite view-point of physical chemistry. As Larmor says, "Gibbs made a clean sweep of the subject, and workers in the modern experimental science of physical chemistry have returned to it again and again to find their empirical principles forecasted in the light of pure theory, and to derive fresh inspiration for new departures."^[43] Some of its theorems, as the Helmholtz doctrine of free energy,^[44] Konowalow's theorem of indifferent points,^[45] Curie's theory of "crystal habit,^[46] were rediscovered by later investigators in ignorance of the earlier work. Indeed the primary intention of Gibbs's memoir, to treat chemical changes as a branch of mechanics, was not, at first, clearly understood, the long deferred review in the "Fortschritte der Physik"^[47] merely listing its contents. Maxwell, however, with the same cordial recognition which he had shown to Rowland, grasped its significance at once, incorporated some of its results in his memoir on "Diffusion,"^[48] and in an appreciative discourse before the Cambridge Philosophical Society in 1876 declared that the methods of Gibbs "seemed to throw a new light upon thermodynamics." Copies of the work were consequently much prized and sought after in England about this time; but the most substantial recognition of Gibbs's work was to come from Holland, where a long line of physical chemists, van der Waals, Roozeboom, van't Hoff, Lorentz, Schreinemakers, Stortenbeker, van Laar, Hoitsema, Kamerlingh Onnes, have developed his ideas with very substantial additions to their own fame. Parallel with the work of these men and the development of the important laws of Goldberg and Waage, van't Hoff and Arrhenius, the science of physical chemistry, which DuBois Reymond called "the chemistry of the future," came into being under the leadership of Ostwald in Germany and (since 1896) of Professor Bancroft in America. With the gradual recognition of the significance of "reversible reactions"^[49] and of Sainte-Claire-Deville's doctrine of chemical dissociation, the algebraic formulæ of Gibbs became slowly converted into working theories of physical chemistry. In 1892 Ostwald translated Gibbs's papers as "Thermodynamische Studien" and part of them were rendered into French in 1899 by Le Chatellier. The purely mathematical part of Gibbs's theory has been developed in extension by the labors of Duhem, Paul Sorel, Trevor, Bancroft, van der Waals, Larmor and Bryan. Roozeboom, van der Waals and Bancroft have made the widest applications of his ideas to chemistr}, while their best interpretation from the dynamic or energetic point of view is that of Ostwald^[50] and of Larmor,^[51] Although a genial and engaging writer in his discourse on "Multiple Algebra" and his biographical sketches, the strictly scientific papers of Gibbs are not, like those of Maxwell, Boltzmann and Hertz, attractive reading. Indeed, it has been said of his memoir on equilibrium that Ostwald is one of the few people in the world who ever read every word of it, for the student is repelled, not so much by its bristling quickset of some seven hundred formula as by the severe and austere reasoning and a literary style that is swift in movement and (doubtless from the very nature of the subject matter) tense and dry in quality. Although endowed with the scientific imagination of a man of genius, Gibbs's strong point in demonstration was unusual quickness of intelligence and great capacity for the rigors of formal logic, and the rapid movement of his mind as he clears away the underbrush and covers the vast area of his new territory is wearying and even confusing to the reader. The intention of the present resume is frankly journalistic, aiming only to emphasize such points in the Gibbsian theory as have been thrown into strongest relief by their relation to recent science. These are:

The General Equation of Thermodynamics.—In applying the laws of dynamics to thermal phenomena, Clausius had shown that if we differentiate with respect to the volume of a body, we obtain its pressure with reversed sign; if we differentiate with respect to its entropy we obtain its temperature on the thermodynamic scale; the energy of the body can then be expressed as a function of its volume and entropy, the differential coefficients with respect to the latter being the pressure (with negative sign) and the temperature. Gibbs has extended these principles to the formulation of a fundamental equation of thermodynamics, in which the new departure is taken of introducing the masses of the chemical components of a system as variables, the differential coefficients in this case being certain new conceptions which he terms the "potentials" of the substances considered. From this equation most of the principles and formulae of thermodynamics can be deduced. It lies at the basis of the new aggregate of sciences called "energetics"^[52] as well as of mathematical chemistry, in which all spontaneous changes of substance or state are regarded as more or less direct consequences of the second law. The equations of Clausius and Gibbs, although exceedingly general and difficult of application to chemistry, are exact, representing the physical facts.^[53]

The Chemical Potentials.—In the fundamental equation of Gibbs we distinguish two classes of variables, of which the volume, entropy and masses of the component substances are looked upon as magnitudes or capacities, while the temperature, the pressure and the potentials are to be thought of as qualitative, being non-measurable, nonadditive physical intensities of the system considered. Thus the

If $\epsilon ,t,\eta ,p$ and $v$ represent the energy, temperature, entropy, pressure and volume of a homogeneous substance respectively, the equation of Clausius may be written $d\epsilon =td\eta -pdf$. It is applicable to all one-component systems, such as steam in a boiler. The equation of Gibbs, which is applicable to any chemical system whatever, is written

$d\epsilon =td\eta -pdv+\mu _{1}dm_{1}+\mu _{2}dm_{2}\ldots +\mu _{n}dm_{n},$

where $\mu _{1},\mu _{2}\ldots$ denote the chemical potentials, and $m_{1},m_{2}$ the masses of the chemical components of the system. pressure of a substance connotes the intensity with which it tends to expand, its temperature the intensity with which it tends to part with heat, while the potential of a given chemical component represents (in Maxwell's acute interpretation) the intensity with which it tends to expel itself from the mass or compound containing it.^[54] Mathematically the Gibbsian potential, which Maxwell thought " likely to become very important in the history of chemistry," has been identified by Larmor with the marginal available energy per unit mass of substance at constant temperature,^[55] depending upon the percentage composition of the substance rather than its actual quantity. The chemical potentials may be regarded, not unlike the potentialities of an individual, as definite intensities which set things going, and as such their close relationship to the surface energies and surface tensions of biological science is obvious. As to the ultimate nature of the forces bound up with these potentials, whether due in the last analysis to electronic stresses or rotational stresses in the ether simply, we know little or nothing. Thermodynamic (or "energetic") doctrine rests upon the simple idea that mechanical, thermal, chemical and electric forces are different modes of energy, continually changing and passing into one another in an apparently elusive way, and is more concerned with their dynamic effects than with their actual nature.

(To he continued)

1. J. Willard Gibbs, "Scientific Papers," 2 vols., New York and London, 1908.
2. "Report British Association for the Advancement of Science," 1884, 20.
3. "Lectures on the Method of Science," Oxford, 1905, 12.
4. Huxley, "Lay Sermons," New York, 1871, 168.
5. Ibid., 66.
6. Nature, London, 1869-70, I., 237.
7. Ibid., 238.
8. "Mathematics may be compared to a mill of exquisite workmanship which grinds you stuff to any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data." Huxley, "Aphorisms and Reflections," London, 1907, 93.
9. Phil. Tr., 1865., CLV., 497-501.
10. "Tagebl. d. Versamml. d. deutsch. Naturf. u. Aerzte 1889," Heidelberg, 1890, 144-9.
11. Deutsche Rev., 1907, XXXII., Pt. I., 16.
12. "Report British Association for the Advancement of Science," 1857.
13. Am 28. April 1903 verschied im 64. Lebensjahre der Schöpfer der chemischen Energetik, J. Willard Gibbs. Der allgemeinen Chemie hat er für ein Jahrhundert Form und Gehalt gegeben." Ztschr. f. phys. Chem., 1903, XLIII., 760.
14. Nature,London, 1901-2 LXV.,107-8
15. Ibid., 1877-8, XVII., 257.
16. Bacon, "Novum Organum," English translation of 1850, p. 165.
17. Kelvin, "Popular Lectures," London, 1894, Vol. II., 460.
18. "On peut comparer la puissance motrice de la chaleur à celle d'une chute d'eau:. . . la puissance motrice d'une chute d'eau dépend de la hauteur et de la quantité du liquide; la puissance motrice de la chaleur dépend aussi de la quantité de calorique employé, et de ce que nous appellerons le hauteur de sa chute, c'est à dire de la différence de température des corps entre lesquels se fait l'échange du calorique." Carnot, "Reflexions," 1824, 15.
19. Rankine, Phil. Mag., 1865, 244.
20. HeImholtz, "Ueber die Erhaltung der Kraft," Berlin, 1847.
21. Sir W. Thomson, Tr. Roy. Soc. Edinb., 1849, XVI., 543.
22. J. Thomson, Ibid., 575-80.
23. Gibbs, Proc. Am. Acad. Arts and Sci., 1888-9, n. s., XVI., 459.
24. W. Thomson, Phil. Mag., 1852, IV., 304. "Available energy is energy which we can direct into any required channel. Dissipated energy is energy which we can not lay hold of and direct at pleasure, such as the energy of the confused agitation of molecules which we call heat." Maxwell, sub voce "Diffusion."
25. Rankine, Phil. Tr., 1854, CXLIV., 126.
26. Clausius, Poggend. Ann., 1855, CXXV., 390.
27. "When a mass of air is adiabatically compressed or when it expands into a vacuum, the temperature of the mass changes, but no heat is added to it. When heat is added to a block of metal, the temperature of the block rises. When heat is added to a mass of liquid water and overlying water vapor supporting a constant pressure, the temperature of the mass is not altered. Heat may be added to a mixture of potassium sulphocyanate and water in the process of forming a mixture, and the temperature fall. . . . The quantity of 'heat' added to a body in a change of its thermodynamic state is the work absorbed or absorbable by the body through direct intervention of a change of the temperature of another body. This is all that a 'quantity of heat' means. To assume it to mean a quantity of an imponderable fluid, or a quantity of the kinetic energy of hypothetical and inaccessible particles is to replace direct statement of physical facts, made with the aid of clearly defined terms, by a hypothetical interpretation of the facts." J. E. Trevor, Jour. Phys. Chem., 1908, XII., 316.
28. Maxwell and Tait originally used the term "entropy" as a synonym of
29. Maxwell, Nature, London, 1877-8, XVII., 279.
30. For a description of the Maxwell demon and the powers ascribed to him see Lord Kelvin's paper in Nature, 1879, 126.
31. Horstmann, Ann. d. Chem. u. Pharm., 1872, 8. Suppl.-Bd., 112-33.
32. Horstmann, Ibid., 1873, CLXX., 192-210.
33. Gibbs, Tr. Connect. Acad., Dec, 1873, II., foot-note to p. 393.
34. Lord Rayleigh, Proc. Roy. Inst., 1875, VII., 388.
35. Tr. Connect. Acad., 1873, II., 309-42, 382-404. Translated into French as "Diagrammes et surfaces thermodynamiques," Paris, C. Naud, 1903. Translated into German by Ostwald in 1892.
36. Nature, London, 1902-3, LXVII., 604.
37. Gibbs, Tr. Connect. Acad., April, 1873, II., 317-25. The equivalent of an entropy diagram was laid down and described by the Belgian physicist M. Belpaire in 1872 (Bull. Acad. roy. d. sc, Brux., 1872, 2. s., XXXIV., 520-6), but his treatment of the matter is so sketchy and slight in comparison with the exhaustive and illuminative handling of Gibbs that it seems negligible. The mere plotting of the diagram itself is nothing, for it was for years implicit in Rankine's algebraic use of the "thermodynamic function" ($\Phi$) as a coordinate (1854), and to this day the British unit of entropy is called a "Rank."
38. Tr. Connect. Acad., 1873, II., 382-404.
39. J. Thomson, Proc. Roy. Soc. Lond., 1871, XX., 1.
40. Maxwell, "Theory of Heat," London, 1902, 204-8.
41. "Copies of this model were distributed by Maxwell evidently with a certain amount of playful mystery, for each recipient thought that he was the happy possessor of one of (at most) three. The writer knows of six at least, and possibly there are more." C. G. K. in Nature, 1907, LXXXV., 361.
42. Tr, Connect. Acad., 1875-8, III., 108-248; 343-594. Abstract by Gibbs in Am. J. Sc, 1878, 3. s., XVI., 441-458.
43. "Larmor, "Encycl. Britan.," 10th ed., 1902, IV., 172.
44. Helmholtz, Sitzungsb. d. k. preuss. Akad. d. Wissetisch., XXII.
45. Konowalow, Wied. Ann., 1881, XIV., 48.
46. Curie, Bull. Soc. Min., 1885, VIII., 145.
47. Fortschr. d. Physik., 1878, XXXIV., 198.
48. "Encycl. Britan.," 9th ed., VII., 214-21.
49. The difference between reversible and irreversible chemical processes could hardly be better indicated than in the following comparison of van't Hoff: "Kill a chicken and prepare chicken soup; it would then be very difficult to get your chicken again. This is because preparing chicken soup is not reversible. On the contrary, let water evaporate or freeze, it will be easy to reproduce the water" (J. Phys. Chem., 1905, IX., 87). The distinction between reversible and irreversible reactions is thus a physico-chemical or thermodynamic conception, depending, like the operations of mechanical systems, upon the initial conditions, under which the phenomenon takes place.
50. See Ostwald, "Lehrb. d. allg. Chemie," Leipzig, 1896, II., 2. Th., 114-5.
51. See "Encycl. Britan.," 10th ed., XXVIII., sub voce Energetics.
52. Thoughout this paper, "energetics," thermodynamics and physical chemistry are regarded as practically identical in scope, in the original sense in which Gibbs referred to all material systems as "actually thermodynamic," or Ostwald to "das glänzendste Gebiet der heutigen Physik und Chemie, die reine Thermodynamik, oder da dieser Name viel zu eng ist, die reine Energetik."
53. 53
54. See the report of Maxwell's lecture in Am. J. Sc., 1877, 3. s., XIII., 380, which is fuller than the one given in his collected writings.
55. "Encycl. Britan.," 10th ed., XXVIII., 168.