# Gibbs, Josiah Willard

# Gibbs, Josiah Willard

(*b*. New Haven, Connecticut, 11 February 1839; *d*. New Haven, 28 April 1903)

*theoretical physics*.

Gibbs was the only son among the five children of Josiah Willard Gibbs and Mary Anna Van Cleve Gibbs. His father was a noted philologist, a graduate of Yale and professor of sacred literature there from 1826 until his death in 1861. The younger Gibbs grew up in New Haven and graduated from Yale College in 1858, having won a number of prizes in both Latin and mathematics. He continued at Yale as a student of engineering in the new graduate school, and in 1863 he received one of the first Ph.D. degrees granted in the United States. After serving as a tutor in Yale College for three years, giving elementary instruction in Latin and natural philosophy, Gibbs left New Haven for further study in Europe. By this time both his parents and two of his sisters were dead, and Gibbs traveled with his two surviving older sisters, Anna and Julia. He spent a year each at the universities of Paris, Berlin, and Heidelberg, attending lectures in mathematics and physics and reading widely in both fields. These European studies, rather than his earlier engineering education, provided the foundation for his subsequent career.

Gibbes returned to New Haven in June 1869. He never again left America and rarely left even New Haven except for his annual summer holidays in northern New England and a very occasional journey to lecture or attend a meeting. Gibbs never married and lived all his life in the house in which he had grown up, less than a block away from the college buildings, sharing it with Anna, Julia, and Julia’s family. In July 1871, two years before he published his first scientific paper, Gibbs was appointed professor of mathematical physics at Yale. He held this position without salary for the first nine years, living on his inherited income. It was during this time that he wrote the memoirs on thermodynamics that constitute his greatest contribution to science. Gibbs had no problem about declining a paid appointment at Bowdoin College in 1873, but he was seriously tempted to leave Yale in 1880, when he was invited to join the faculty of the new Johns Hopkins University at Baltimore. Only then did Yale provide a salary for Gibbs, as tangible evidence of the esteem in which he was held by his colleagues and of his importance to the university, but this salary was still only two thirds of what Johns Hopkins had offered him. Gibbs stayed on at Yale nevertheless and continued teaching there until his death, after a brief illness, in the spring of 1903.

Gibbs’s first published paper did not appear until he was thirty-four years old, and it displays his unique mastery of thermodynamics. If there are even earlier signs of Gibbs’s intellectual power, they must be sought in his previous engineering work. His doctoral thesis, “On the Form of the Teeth of Wheels in Spur Gearing,” certainly shows Gibbs’s unusually strong geometrical ability. He generally preferred “the niceties of geometrical reasoning” to analytical methods in his later work, and this is true in his thesis. The style of this early work also shows the same “austerity,” the same “extreme economy (one might almost say parsimony) in the use of words,”^{1} that made his later memoirs so difficult to read. Some of his engineering work, such as the design of an improved railway car brake, for which he received a patent in 1866, is hard to relate to the concerns of the future master of theoretical physics-but this is not true of all of it. After his return from Europe, Gibbs designed a new form of governor for steam engines by suitable mounting of a second pair of massive balls on the simple Watt governor. This new arrangement was planned to increase the responsiveness of the system to a change in the engine’s running speed. Although Gibbs went no further with this invention than having a model of it built in the department workshop, it is interesting nevertheless: the problems of dynamic equilibrium and stability of this particular mechanical device foreshadow the related questions of equilibrium and stability that he would soon raise and answer for general thermodynamic systems.

When Gibbs first turned his attention to thermodynamics in the early 1870’s, that science had already achieved a certain level of maturity. Rudolf Clausius had taken the essential step in 1850, when he argued that two laws and not just one are needed as the basis of a theory of heat. Only a year before that, William Thomson had been writing about the “very perplexing question,” and the associated “innumerable” and “insuperable” difficulties, of choosing *the* correct axiom for the theory.^{2} Should one hold fast to Carnot’s postulate (that heat must pass from a hot body to a colder one when work is done in a cyclic process), even though Carnot’s results seemed to depend on his use of the caloric theory of heat? Or should one accept the interconvertibility of heat and work, since James Joule’s new experimental evidence clearly favored the mechanical theory of heat? Clausius showed that, despite the apparent need to choose one law or the other, both were necessary and both could be maintained without contradiction. One had only to drop Carnot’s inessential requirement that heat itself be conserved. This change did have one major implication. The proof of Carnot’s theorem (that the maximum motive power of heat depends solely on the temperatures between which the heat is transferred) now had to appeal to a new axiom: that heat “everywhere exhibits the tendency to annul temperature differences, and therefore to pass from a *warmer* body to a *colder* one.”^{3} Clausius’ memoir demonstrated how one could develop a thermodynamics starting with both the equivalence of heat and work and his new axiom. These two laws of thermodynamics were restated in slightly different from a year later by Thomson, who proceeded to apply them to a variety of physical problems, including thermoelectricity.

Clausius tried as hard as he could to find the essence of the second law of thermodynamics, since he felt unable at first to “recognize…, with sufficient clearness, the real nature of the theorem.”^{4} This search led finally in 1865 to his most concise and ultimately most fruitful formulation of the two laws, the formulation Gibbs later used as the motto of his greatest work: “The energy of the universe is constant. The entropy of the universe tends to a maximum.”^{5} The two basic quantities, internal energy and entropy, were, in effect, defined by the two laws of thermodynamics. The internal energy *U* is that function of the state of the system whose differential is given by the equation expressing the first law,

where đ*Q* and đ*W* are, respectively, the heat added to the system and the external work done on the system in an infinitesimal process. For a simple fluid the work đW is given by the equation

where *P* is the pressure on the system and *V* is its volume. Neither the heat đ*Q* nor the work đ*W* is the differential of a function of state, and the inexactness or nonintegrability of these differentials is indicated by the symbol đ, whose use for this purpose goes back to Carl Neumann’s lectures in the early 1870’s.^{6} The entropy S is that state function whose differential is given by the equation

valid for reversible processes, where *T* is the absolute temperature. For irreversible processes equation (3) is replaced by the inequality

The power and importance of the entropy concept were certainly not evident to Clausius’ contemporaries upon the publication of his 1865 paper. Clausius himself considered it to be a summarizing concept and thought that the true physical significance of the second law was better expressed in terms of the disgregation, a concept he sought to interpret mechanically. Entropy plays no particular role in the thermodynamics texts by Georg Krebs (1874) and Carl Neumann (1875). The word was picked up by Peter Guthrie Tait in his *Sketch of Thermodynamics* (1868), but he completely changed its meaning, using it to denote available energy rather than the quantity Clausius had intended. Tait’s misinterpretation was taken up and repeated by James Clerk Maxwell in his *Theory of Heat* (1871). The confusion and uncertainty about the thermodynamic significance of entropy were only aggravated by the bitter priority disputes between Clausius and Tait (on behalf of Thomson) that raged in the early 1870’s. The basic structure of modern thermodynamics was implicitly present in the work of both Clausius and Thomson, but it was certainly not clearly visible to most writers on the subject.

It was in this context that Gibbs’s first scientific paper, “Graphical Methods in the Thermodynamics of Fluids,” appeared in 1873. His mastery and his quiet assurance in this paper are as remarkable as his scientific insight. Gibbs assumed from the outset that entropy is one of the essential concepts to be used in treating a thermodynamic system, along with energy, temperature, pressure, and volume. He immediately combined the first three equations given above to obtain the form

a relation that contains only the state variables of the system, the process-dependent heat and work having been eliminated. As Gibbs pointed out, an equation expressing the internal energy of the system in terms of its entropy and volume could appropriately be called its fundamental equation; for equation (5) would then allow one to determine the two equations of state, expressing temperature and pressure as functions of the pair, volume and entropy. These remarks were the starting point for Gibbs’s later work, but in this first paper he limited himself to a discussion of what could be done with geometrical representations of thermodynamic relationships in two dimensions.

James Watt’s indicator diagram, in which pressure and volume are plotted on the two coordinate axes, had been in use for thermodynamic purposes since émile Clapeyron’s memoir of 1834. But Gibbs showed how other choices for the coordinate variables would produce representations even more useful for thermodynamic purposes; the temperature-entropy diagram, for example, had many advantages in the study of cyclic processes. The form of the basic equation (5), expressing both laws of thermodynamics, suggested that the volume-entropy diagram might be best suited for general thermodynamic considerations, and Gibbs discussed it in more detail. He also showed how some of the interrelations among the curves describing, respectively, states of equal pressure, equal temperature, equal energy, and equal entropy were independent of how the thermodynamic diagram was constructed and followed directly from the stability of equilibrium states.

In his second paper, which appeared later in 1873, Gibbs extended his geometrical discussion to three dimensions by analyzing the properties of the surface representing the fundamental thermodynamic equation of a pure substance. The thermodynamic relationships could be brought out most clearly by constructing the surface using entropy, energy, and volume as the three orthogonal coordinates. Gibbs pointed out that, as a consequence of equation (5), the temperature and pressure of the body in any state was determined by the plane tangent to the surface at the corresponding point, since one has the equations

and

This way of representing the thermodynamic properties of a body in thermodynamic equilibrium could be used just as well when different parts of the body were in different states (for example, a mixture of liquid and gas or of two different crystalline forms of the same pure substance). Gibbs showed how one could use the thermodynamic surface to discuss the coexistence of the various phases (liquid, solid, and gas) of a pure substance and the stability of these states under given conditions of temperature and pressure. One feature of particular interest was the critical point—the state at which liquid and gas become identical-a phenomenon discovered experimentally in carbon dioxide by Thomas Andrews only a few years earlier.

These early papers, as well as Gibbs’s major memoir on thermodynamics that soon followed them, appeared in the *Transactions of the Connecticut Academy of Arts and Sciences*, a new and relatively obscure journal whose nonlocal circulation consisted largely of exchanges with other learned societies, including some 140 outside the United States. Gibbs did not count on finding his potential readers among those who checked the contents of the *Transactions*; he sent copies of his papers to an impressive list of scientists in many countries, a list that probably included all those he thought might really read and understand his work.^{7} One of them was James Clerk Maxwell, who proved to be Gibbs’s most enthusiastic and most influential reader. He immediately accepted Gibbs’s clarification of what Clausius had intended by the term “entropy,” correcting the error in his own *Theory of Heat* accordingly and informing Tait about his mistake.

Maxwell found Gibbs’s use of geometric rather than algebraic arguments particularly attractive, since he too preferred geometric insight to calculations, even when others found algebraic procedures decidedly simpler. He was sufficiently impressed by Gibbs’s paper on the thermodynamic surface to include a discussion of this subject in the fourth edition of *Theory of Heat* (1875) and actually to construct a model of the thermodynamic surface for water, which he sent to Gibbs. He talked about Gibbs’s work to his colleagues at Cambridge and recommended it to his friends. “Read Prof. J. Willard Gibbs on the surface whose coordinates are Volume Entropy and Energy,” he wrote to Tait, and then added for the benefit of his rather chauvinistic friend, “He has more sense than any German.”^{8} Maxwell even started to generalize Gibbs’s thermodynamics of a pure substance to include the case of heterogeneous mixtures. This proved to be quite unnecessary and was dropped when he received a set of galley proofs of Gibbs’s new memoir containing this generalization and a great deal more.

“On the Equilibrium of Heterogeneous Substances” contains Gibbs’s major contributions to thermodynamics. In this single memoir of some 300 pages he vastly extended the domain covered by thermodynamics, including chemical, elastic, surface, electromagnetic, and electrochemical phenomena in a single system. The basic idea had been foreshadowed in his two earlier papers, in which Gibbs had directed his attention to the properties characterizing the equilibrium states of simple systems rather than to the heat and work exchanged in particular kinds of processes. In the abstract of his memoir that Gibbs published in the American Journal of Science in 1878, he began by stating the simple but profound idea underlying his work:

It is an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system that when the entropy of the system has reached a maximum, the system will be in a state of equilibrium. Although this principle has by no means escaped the attention of physicists, its importance does not appear to have been duly appreciated. Little has been done to develop the principle as a foundation for the general theory of thermodynamic equilibrium.

^{9}

Gibbs formulated the criterion for thermodynamic equilibrium in two alternative and equivalent ways. “For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy [entropy], the variation of its entropy [energy] shall either vanish or be negative [positive].”^{10} The bracketed form immediately indicates that thermodynamic equilibrium is a natural generalization of mechanical equilibrium, both being characterized by minimum energy under appropriate conditions. The consequences of this criterion could then be worked out as soon as the energy of the system was expressed in terms of the proper variables. Gibbs’s first and probably most significant application of this approach was to the problem of chemical equilibrium. The result of his work was described by Wilhelm Ostwald as determining the form and content of chemistry for a century to come, and by Henri Le Chatelier as comparable in its importance for chemistry with that of Antoine Lavoisier.

The simplest case is that of a homogeneous phase—a liquid or gas, for example-containing *n* independent chemical species S_{1}, ..., S_{n} whose masses m_{1}, ..., m_{n} can be varied. Gibbs modified the basic equation (5) to include the change of internal energy due to a change in the mass of any of the chemical components by writing it in the form

Here *dm _{i}* is the change in mass (conveniently expressed as the number of moles) and the new quantity

*μ*is the (chemical) potential of the

_{i}*i*th chemical species. The chemical potential is related to the energy by the equation

where the subscript means that μ_{i} represents the rate of change of energy with respect to the mass of the *i*th component in this phase, the masses of all other components being held constant along with entropy and volume.

In a heterogeneous system composed of several homogeneous phases the fundamental equilibrium condition leads to the requirement that temperature, pressure, and the chemical potential of each independent chemical component must have the same values throughout the system. From these general conditions Gibbs derived the phase rule, that cornerstone of physical chemistry, which specifies the number of independent variations δin a system of *r* coexistent phases having *n* independent chemical components:

Gibbs also showed how to obtain the specific conditions for equilibrium when chemical reactions could take place in the system. Instead of one’s attention being restricted to a set of independent chemical components, all the relevant chemical species are considered. Suppose, for example, that a reaction of the type

can occur; here the *a _{i}’s* are integers, the stoichiometric coefficients, and the

*A*’s stand for the chemical symbols of the reacting substances. (An illustration would be the reaction H

_{i}_{2}+ Cl

_{2}−2HCl = 0, where

*a*

_{1}= 1,

*a*

_{2}= 1,

*a*

_{3}= −2 and the corresponding

*A*are respectively H

_{i}_{2}, Cl

_{2}, and HCl.) The equilibrium condition that Gibbs derived for such a reaction has the simple form

obtained by replacing the chemical symbols *A _{i}* with the chemical potential μ

_{i}of the corresponding substance in the reaction equation (10). Since the potentials could in principle be determined from experimental data, the equilibrium conditions were really established by equation (11).

The requirement that the energy have a minimum and not just a stationary value at equilibrium states. This stability depends ultimately on the second law of thermodynamics and manifests itself in the unique sign of certain properties of all substances; the heat capacity at constant volume, for example, must be positive, and the isothermal derivative of pressure with respect to volume must be negative for any substance. The most interesting aspect of Gibbs’s investigation of stability was his theory of critical phases, those situations where the distinction between coexistent phases vanishes and the stability is of a lower order than that usually found.

Gibbs’s memoir showed how the general theory of thermodynamic equilibrium could be applied to phenomena as varied as the dissolving of a crystal in a liquid, the temperature dependence of the electromotive force of an electrochemical cell, and the heat absorbed when the surface of discontinuity between two fluids is increased. But even more important than the particular results he obtained was his introduction of the general method and concepts with which all applications of thermodynamics could be handled.

Although Maxwell responded immediately to Gibbs’s work and influenced a number of his English colleagues to apply Gibbs’s results if not his methods, it took a little longer for Continental scientists to appreciate Gibbs. Such figures as Hermann von Helmholtz and Max Planck independently developed thermodynamic methods for chemical and electrochemical problems in the 1880’s quite unaware of Gibbs’s prior work. This situation changed gradually—Ludwig Boltzmann referred to Gibbs in 1883—but it was Wilhelm Ostwald’s German translation of Gibbs’s papers in 1892 that made his ideas more readily available to German chemists.

Gibbs wrote no other major works on thermodynamics itself, restricting himself to a few small applications and developments of his extensive memoir. He rejected all suggestions that he write a treatise that would make his ideas easier to grasp. Even Lord Rayleigh thought the original paper “too condensed and too difficult for most, I might say all, readers.”^{11} Gibbs responded by saying that in his own view the memoir was instead “*too long*” and showed a lack of “sense of the value of time, of [his] own or others, when [he] wrote it.”^{12}

During the 188’s Gibbs seems to have concentrated on optics and particularly on Maxwell’s electromagnetic theory of light. He was giving lectures
on Maxwell’s theory at Yale at least as early as 1885, and he published a series of papers in the *American Journal of Science* on double refraction and dispersion, that is, on the behavior of light as it passes through material media. Two aspects of Gibbs’s optical work are of more than technical interest. He emphasized that a theory of dispersion requires one to treat the local irregularities of the electric displacement due to the atomic constitution of the medium. Since he was writing before H. A. Lorentz’s theory of electrons (the first version of which did not appear until 1892), Gibbs had to make assumptions of a different kind from those of later theories, and he missed the essential contribution of the atomic structure of the medium—the frequency dependence of the dielectric constant. In the last two papers of this series, published in 1888 and 1889, Gibbs appeared as a defender of the electromagnetic theory of light against the latest versions of purely mechanical theories. These were based on special elastic ethers still being proposed by William Thomson. Gibbs showed that although such theories might account for the phenomena, they required rather artificial assumptions as to internal forces, while Maxwell’s theory was “not obliged to invent hypotheses.”^{13}

Gibbs’s reading Maxwell’s *Treatise on Electricity and Magnetism* led him to a study of quaternions, since Maxwell had used the quaternion notation to a limited extent in that work. Gibbs decided, however, that quaternions did not really provide the mathematical language appropriate for theoretical physics, and he worked out a simpler and more straightforward vector analysis. He wrote a pamphlet on this subject which he had printed in 1881 and 1884 for private distribution to his classes and to selected correspondents. No real publication of Gibbs’s version of vector analysis took place until 1901, when his student Edwin B. Wilson prepared a textbook of the subject based on Gibbs’s lectures. In 1891 Gibbs defended his use of vectors rather than quaternions against an attack by Tait, who was the great exponent of—and crusader for—William Rowan Hamilton’s quaternions. Gibbs more than held his own in the ensuing controversy over quaternions, debating both Tait and his disciples in the pages of *Nature*, on minor matters of notation and on the “deeper question of notions underlying that of notations.”^{14}

During the 1890’s the classical goal of the physicist—the explanation of natural phenomena in mechanical terms—was seriously questioned and sharply criticized. Among the most outspoken critics were such men as Ernst Mach, Pierre Duhem, and Wilhelm Ostwald, who rejected the concept of atomism along with that of mechanism. One of the spokesmen for the group who called themselves energeticists was Georg Helm, who in 1898 wrote a treatise on the historical development of energetics. In his book Helm claimed Gibbs as a fellow energeticist, pointing to Gibbs’s writings on thermodynamics as a sign that he was free of any prejudice in favor of atomistic mechanical explanations. To Helm, Gibbs’s thinking proceeded directly from the two laws of thermodynamics “without any hankering or yearning after mechanics”; his writings were “not decked out with molecular theories.”^{15} While it is true that Gibbs’s papers on thermodynamics were based only on the two laws and required no assumptions about the molecular structure of matter, it is equally true that Gibbs believed in and constructed a (statistical) mechanical explanation of thermodynamics itself. He certainly did not share the scientific values of the energeticists.

Gibbs had carefully studied the writings of Maxwell and Boltzmann on the kinetic theory of gases. This is evident from his obituary notice of Clausius, written in 1889, in which he commented perceptively on a peculiar feature of Clausius’ work in this field. Clausius never really accepted the statistical point of view. “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.”^{16} Maxwell had introduced statistical methods in his first paper on gases in 1860 and had emphasized the essentially statistical nature of the second law of thermodynamics with the help of the “Maxwell demon.” Boltzmann then developed all the essential features of a theoretical explanation of the second law based on a combination of mechanics and the laws of probability applied to a large assemblage of molecules in a gas. Gibbs had already paid his own respects to this fundamental insight in his memoir on heterogeneous equilibrium, when he argued that “the impossibility of an uncompensated decrease of entropy seems to be reduced to improbability.”^{17}

During the academic year 1889–1890 Gibbs announced “A short course on the a priori Deduction of Thermodynamic Principles from the Theory of Probabilities,”^{18} a subject on which he lectured repeatedly during the 1890’s. He referred to this subject in a letter to Lord Rayleigh in 1892:

Just now I am trying to get ready for publication something on thermodynamics from the

a prioripoint of view, or rather on “Statistical Mechanics” of which the principal interest would be in its application to thermodynamics—in the line therefore of the work of Maxwell and Boltzmann. I do not know that I shall have anything particularly new in substance, but shall be contented if I can so choose my standpoint (as seems to me possible) as to get a simpler view of the subject.^{19}

In fact Gibbs did not publish anything more than a very brief abstract on this subject (1884) until 1902, when his book *Elementary Principles in Statistical Mechanics Developed With Special Reference to the Rational Foundation of Thermodynamics* appeared as one of the Yale Bicentennial series.

Gibbs thought of his book as offering a more general approach to statistical mechanics than that used by Boltzmann or Maxwell. Individual results previously obtained by others could now assume their proper places in the logical structure Gibbs gave to the subject. He considered an ensemble of systems, that is, a large number of replicas of the physical system of interest—which might be anything from a molecule to a sample of gas or a crystal. The replicas were all identical in structure but differed in the values of their coordinates and momenta. The ensemble was then characterized by its probability density ρ in phase space, where ρ*dq _{1} ... dq_{N} dp_{1} ... dp_{N}* is the fractional number of systems in the ensemble whose coordinates

*{q*and momenta

_{i}}*{p*lie in the intervals

_{i}}*{q*and

_{i}, q_{i}+dq_{i}}*{p*respectively, at time

_{i}, p_{i}+dp_{i}}*t*. The phase point representing any individual system moves with time, and so the probability density ρ at any point in the 2

*N*-dimensional phase space varies in a way determined by the mechanical equations of motion of the system.

If the average behavior of a system in the ensemble was to describe the behavior of the actual physical system, then a physical system in equilibrium had to be described by a stationary ensemble, one whose probability density ρ was constant in time. Gibbs analyzed several such stationary ensembles but found the one he called “canonical” to be most useful. In a canonical ensemble the probability density ρ is given by the equation.

where E (*q _{1} ... p_{N}*) is the energy of the system having coordinates and momenta

*q*and ψ,Θ are constants in phase space. Gibbs showed that the energy of such an ensemble has a sharply peaked distribution, if the systems have many degrees of freedom; only a very small fraction of the systems in the canonical ensemble have energies appreciably different from the average.

_{1}... P_{N}The principal theme of Gibbs’s book is the analogy, as he describes it, between the average behavior of a canonical ensemble of systems and the behavior of a physical system obeying the laws of thermodynamics. When this analogy is worked out, the modulus Θ of the canonical distribution is proportional to the absolute temperature, with a universal proportionality constant *k*

The average energy Ē, where Ē satisfies the equation

is identified with the thermodynamic internal energy *U*. The second parameter of the distribution, ψ, is fixed by the condition that ρ is a probability density, so one has the equation

and ψ is also related to the energy U (or Ē) and the meanings of heat and work in ensemble.

a result Gibbs obtained by analyzing the meanings of heat and work in ensemble.

Gibbs was very much aware of the gaps in his statistical mechanics. He had supplied a “rational foundation” for thermodynamics in statistical mechanics to the extent that thermodynamic systems could be treated as if they were conservative mechanical systems with a finite number of degrees of freedom. He could not incorporate the phenomena of radiation that were of so much interest at the turn of the century, nor could he surmount the longstanding difficulties associated with the theorem of the equipartition of energy. For these reasons he disclaimed any attempts “to explain the mysteries of nature” and put his work forward as the statistical “branch of rational mechanics.”^{20} He was also dissatisfied with the effort he had made in the twelfth chapter of his book to explain the irreversibility of nature embodied in the second law. His argument in this chapter was almost completely verbal rather than mathematical, and his statements were carefully qualified. Gibbs’s manuscript notes suggest that he was still struggling with the problem of irreversibility and the nature of the entropy of systems not in equilibrium.

Despite these difficulties Gibbs’s work in statistical mechanics constituted a major advance. His methods were more general and more readily applicable than Boltzmann’s and eventually came to dominate the whole field. Gibbs did not live to see the real successes of statistical mechanics, for his fatal illness came within a year of the publication of his book.

## NOTES

1. Everett O. Waters, “Commentary Upon the Gibbs Monograph ‘On the Form of the Teeth of Wheels in Spur Gearing,’” in L.P. Wheeler, E.O. Waters, and S. W. Dudley, eds., *The Early Work of Willard Gibbs in Applied Mechanics* (New York, 1947), p.43.

2. William Thomson, “An Account of Carnot’s Theory of the Motive Power of Heat,” in *Transactions of the Royal Society of Edinburgh*, **16** (1849), 541.

3. Rudolf Clausius,”On the Moving Force of Heat and the Laws of Heat Which May Be Deduced Therefrom,” in *Annalen der physik*, **79** (1850), 368, 500. Trans. into English by T. Archer Hirst as *The Mechanical Theory of Heat* (London, 1867), p.45

4. Clausis, “On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat,” *in Annalen der Physik*, **93** (1854), 481. Tranklation, *The Mechanical Theory of Heat*, p. 111.

5. Clausius, “On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat,” in *Annalen der physik*, **125** (1865), 353. Translation, *The Mechanical Theory of Heart*, p. 365.

6. Carl Neumann, *Vorlesungen über die mechanische Theorie der Wärme* (Leipzig, 1875), p. ix.

7. The list is reprinted as app. IV in Lynde Phelps Wheeler, *Josiah Willard Gibbs* (New Haven, 1952).

8. James Clerk Maxwell to Peter Guthrie Tait, 13 Oct. 1874.

9. Gibbs, abstract of “Equilibrium of Heterogeneous Substances,” in *American Journal of Science* 3rd ser., **16** (1878). 441; Scientific Papers, 1. 354.

10.*Ibid*

11. Lord Rayleigh to J. W. Gibbs, 5 June 1892.

12. Gibbs to Lord Rayleigh, 27 June 1892.

13. Gibbs, “A Comparison of the Electric Theory of Light and Sir William Thomson’s Theory of a Quasi-Labile Ether,” in *American Journal of Science*, 3rd ser., **37** (1889), 144; *Scietific papers*, II, 264.

14. Gibbs, “On the Role of Quaternions in the Algebra of Vectors,” in *Nature***43** (1891), 511; *Scientific Papers*, 11, 155.

15. Georg Helm, *Die Energetik* (Leipzig, 1898), p. 146.

16. Gibbs, “Rudolf Julius Emanuel Clausius,” in *Proceedings of the American Academy of Arts and Sciences*, **16** (1889), 458; *Scientific Papers*, 11, 265.

17. Gibbs, “On the Equilibrium of Heterogeneous Substances,” in *Transactions of the Connecticut Academy of Arts and Sciences*, **3** (1875–1878), 108, 343;*Scientific Papers*, I. 167.

18. Henry A. Bumstead to Edwin B. Wilson, 22 Dec. 1915.

19. Gibbs to Lord Rayleigh, 27 June 1892.

20. Gibbs *Elementary Principles in Statistical Mechanics* (New Haven, 1902), pp. ix-x.

## BIBLIOGRAPHY

I. Original Works. Gibbs’s scientific papers are repr. in *The Scientific Papers of J. Willard Gibbs*, H. A. Bumstead and R. G. Van Name, eds., 2 vols. (New York, 1906; repr. 1961). Vol. I contains the papers on thermodynamics; all others are in vol. II. There is also his *Elementary Principles in Statistical Mechanics* (New Haven, 1902; repr. New York, 1960). His work on vector analysis is in Edwin B. Wilson, *Vector Analysis Founded Upon the Lectures of J. Willard Gibbs* (New York, 1901). Gibbs’s engineering work is reprinted in *The Early Work of Willard Gibbs* in *Applied Mechanics*, complied by Lynde P. Wheeler, Everett O. Waters, and Samuel W. Dudley (New York, 1947). See also Edwin B. Wilson, “The Last Unpublished Notes of J. Willard Gibbs,” in *Proceedings of the American Philosophical Society*, **105** (1961), 545–558.

Gibbs’s notes, correspondence, and MSS are collected in the Beinecke Rare Book and Manuscript Library of Yale University .

II. Secondary Literature. There is extensive treatment of Gibbs’s writings in F.G. Donnan and Arthur Haas, eds., *A Commentary on the Scientific Writings of J. Willard Gibbs*, 2 vols. (New Haven, 1936). The official biography is Lynde P. Wheeler, *Josiah Willard Gibbs. The History of a Great Mind* (New Haven, 1951; rev. ed., 1952). The appendixes include a bibliography of writings on Gibbs, a catalog of Gibbs’s scientific correspondence, his mailing lists for reprints Scientifc Papers, I, xi-xxvi, and in J.G. Crowther, the text of his first unpublished paper, and family information.

A more popularly written biography, valuable for period background and a poet’s insight, is Muriel Rukeyser, *Willard Gibbs* (New York, 1942; repr. 1964).

Shorter accounts may be found in Henry A. Bumstead, “Biographical Sketch,” repr. in Gibb’s *Scientifc Papers*, I, xi-xxvi, and in J. G. Crowther, *Famous American Men of Science* (New York, 1937). pp. 229–297.

Two recent articles are Martin J. Klein, “Gibbs on Clausius,” in Russell McCormmach, ed., *Historical Studies in the Physical Sciences*, I (Philadelphia, 1969), pp. 127–149; and Elizabeth W. Garber, “James Clerk Maxwell and Thermodynamics,” in *American Journal of Physics*, **37** (1969), 146–155.

Martin J. Klein

# Gibbs, Josiah Willard

# Gibbs, Josiah Willard

**AMERICAN MATHEMATICIAN AND PHYSICIST1839–1903**

Josiah Willard Gibbs was born in New Haven, Connecticut, on February 11, 1839, into a family of academics whose connection to higher education dates back to the 1600s. Gibbs entered Yale University (where his father served as librarian) in 1854, graduated in 1858 with distinction in mathematics and Latin, and continued his studies there, in 1863 earning the first Ph.D. in engineering in the United States. He was appointed tutor and taught Latin and natural philosophy until 1866, whereupon he left New Haven to study in Europe. He attended lectures by the eminent scientists of Paris, Berlin, and Heidelberg, returning home in 1869. He was appointed professor of mathematical physics at Yale in 1871, a post he held until his death in 1903.

In 1873 Gibbs published the first two of his revolutionary papers, which unveiled graphical methods for visualizing properties such as stability, equilibrium, and critical states for thermodynamic systems. One of his major accomplishments was showing the usefulness of the "entropy" quantity that the German mathematical physicist Rudolf Clausius had posited in 1865. This was followed in 1876 by his famous "On the Equilibrium of Heterogeneous Substances" paper, in which he introduced the field of chemical thermodynamics, advancing the concept of chemical potential. Gibbs's free energy *G* determines whether it is favorable that chemicals will react. Gibbs employed a completely general treatment, making almost no assumptions about the properties of the interacting substances. His celebrated phase rule allowed for the identification of the number of coexisting phases in a system.

When one aspect of Gibbs's work in thermodynamics, the phase rule, is considered, just how important the discovery of this theoretical system was becomes clear. The phase rule allows one to determine the number of coexisting phases of any system, no matter how complicated. The rule relates the number of independently changeable intensive variables, or degrees of freedom, *F,* the number of components (i.e., substances) *C,* and the number of phases in equilibrium *P,* as follows:

*F = C − P* + 2

Thus, in a one-component system (such as pure water), at temperature and pressures where one phase is present (*F* = 1 − 1 + 2 = 2), those variables can be varied independently (carving out an area of coexistence on a phase diagram). In order to keep two phases in coexistence (*F* = 1 − 2 + 2 = 1), like water and ice, varying the pressure necessitates a specific change in the temperature (which forms a coexistence line on a phase diagram). Where all three phases (ice, water, and vapor) coexist, there are no degrees of freedom, which translates to a single point, the triple point, on a phase diagram. In other words, there is only one temperature and pressure where all three phases of water are in equilibrium. More complicated systems have more intricate phase diagrams, but the phase rule allows one to, relatively, easily determine the phase properties of any system.

Due in part to his mathematically demanding and abstract style of writing, and also since his papers were published in the obscure *Transactions of the Connecticut Academy of Arts and Sciences,* Gibbs's work was not immediately embraced by the scientific community, with the notable exception of the Scottish physicist James Clerk Maxwell. Consequently, many of Gibbs's results were not discovered until years or decades later by scientists ignorant of his work.

Gibbs also wrote an influential book on statistical mechanics, which developed a molecular theory of thermodynamic properties from first principles, with a treatment that was general enough to accommodate **quantum mechanics** , a theory that had not even been imagined yet. He championed the now standard use of vector notation over William Rowan Hamilton's quaternion algebra and wrote several seminal papers on electromagnetism in the 1880s that supported Maxwell's theory.

see also Equilibrium; Maxwell, James Clerk; Thermodynamics.

*Michael J. Fosmire*

## Bibliography

Hastings, Charles S. (1909). "Josiah Willard Gibbs." *Biographical Memoirs of the National Academy of Sciences* 6:372–393.

Seeger, Raymond J. (1974). *J. Willard Gibbs.* New York: Pergamon.

# Josiah Willard Gibbs

# Josiah Willard Gibbs

Josiah Willard Gibbs (1839-1903) was an American mathematical physicist whose pioneer work in statistical mechanics laid the basis for the development of physical chemistry as a science.

When Josiah Willard Gibbs began his work, thermodynamics had become a true science, firmly based on recently formulated laws of the conservation of energy. These included the law that equated heat and energy and the law of the dissipation or degradation of energy (first and second laws of thermodynamics), which had been worked out mathematically.

Gibbs began with the known thermodynamic theory of homogeneous substances and worked out the theory of the thermodynamic properties of heterogeneous substances. It was this work which, some years later, provided the basic theory for the new branch of science known as physical chemistry. Gibbs's great contribution, "On the Equilibrium of Heterogeneous Substances," was published in the *Transactions of the Connecticut Academy of Arts and Sciences* in 1876 and 1878. Before the end of the 19th century it had been translated into French and German, and Gibbs was widely recognized as one of the greatest mathematical physicists since Isaac Newton. He was one of the few American physicists of that century to achieve an international reputation and the only one to make a theoretical contribution of fundamental importance.

## Early Life and Education

Josiah Willard Gibbs was born Feb. 11, 1839, in New Haven, Conn., of a distinguished and learned family. His father was professor of sacred literature in the Yale Divinity School for many years and a well-known linguist. Gibbs graduated from Yale College in 1858 with prizes in mathematics and Latin. He continued his studies at Yale, earning his doctorate of philosophy in 1863. Afterward he was appointed tutor in the college, where he taught Latin for 2 years and natural philosophy for a third year. From 1866 to 1869 he studied in Paris, Berlin, and Heidelberg, where his teachers were some of the world's most distinguished mathematicians and physicists. This period in Europe constituted his only residence outside of New Haven, for in 1871 he was appointed professor of mathematical physics in Yale College—the first such chair in an American college—and he served in that capacity for the rest of his life.

## Work in Thermodynamics

From the beginning of his professorship, Gibbs devoted himself to the development and presentation of his theory of thermodynamics. His first two scientific papers made an exhaustive study of geometrical methods of representing by diagram the thermodynamic properties of homogeneous substances. These early papers brought Gibbs to the attention of England's leading physicist, James Clerk Maxwell. Maxwell constructed a model illustrating a portion of the work and sent a plaster cast to Gibbs. After investigating homogeneous substances, Gibbs went on to his great work, "On the Equilibrium of Heterogeneous Substances." It was this thesis that contained the "phase rule," which has been of such great practical value in industrial chemistry.

## Later Work

Gibbs's subsequent work in thermodynamics was valuable but not on the same order as the earlier papers. During the 1880s his interests began to turn in other directions. He modified his earlier work on quaternions and geometric algebra into a system of vector analysis especially suited to the needs of mathematical physicists, and he developed his own theory of optics based on electricity rather than electromagnetism. His theory was built chiefly upon the hypothesis that light is a periodic disturbance propagated through media whose structures are more fine-grained than the wavelength of light. The theory was published in the *American Journal of Science between 1883 and 1889.* As in his paper on thermodynamics, Gibbs relied to an unusual degree upon mathematical logic and avoided all special hypotheses concerning the constitution of matter.

During the 1890s Gibbs published nothing at all. His last and perhaps his greatest contribution, *The Elementary Principles of Statistical Mechanics,* was published in 1902. The book, termed by one authority "a monument in the history of physics which marks the separation between the nineteenth and twentieth centuries," laid the foundation for a new branch of theoretical physics that ultimately developed into quantum mechanics.

Although Gibbs received many honors during his lifetime, including the Royal Society's Copley Medal and the Rumford Medal of the American Academy of Arts and Sciences, his immediate influence was limited to a small circle of advanced physicists. The esoteric nature of his work made him practically unintelligible to students. Shy and retiring by nature, he made no effort to attract followers or to communicate to a wider audience. The first impact of his work, both theoretical and applied, was in Europe, for America had neither physicists nor a chemical industry capable of taking advantage of the many insights his work provided.

Gibbs never married. He died on April 28, 1903, in New Haven.

## Further Reading

There are two book-length biographies of Gibbs: Muriel Rukeyser, *Willard Gibbs* (1942), studies Gibbs as a creative thinker from a poet's point of view, and Lynde P. Wheeler, *Josiah Willard Gibbs: The History of a Great Mind* (1951), deals best with the technical side of Gibbs's work. Solid studies of Gibbs are in James Gerald Crowther, *Famous American Men of Science* (1937); Bernard Jaffe, *Men of Science in America* (1954; rev. ed. 1958); and Mitchell Wilson, *American Science and Invention* (1954). □

# Josiah Willard Gibbs

# Josiah Willard Gibbs

**1839-1903**

**American Theoretical Physicist and Theoretical Chemist**

J. Willard Gibbs is regarded as one of the greatest American scientists of the nineteenth century and one of the founders of modern physical chemistry. His theoretical and mathematical treatments of physical and chemical processes involving heat and work developed the science of thermodynamics into one of the most useful tools available to physicists and chemists.

Gibbs, whose father was a professor of sacred literature at Yale University, grew up in a highly intellectual environment. He developed into a shy, somewhat frail young man, deeply involved in intellectual pursuits. He graduated from Yale, then, in 1863, received the first doctorate of engineering awarded in the United States. In that same year he became a tutor at Yale.

After Gibbs's parents died and he and his two sisters inherited their estate, the three traveled together to Europe in 1866. They stayed for three years, giving Gibbs the opportunity to attend the lectures of some of Europe's most outstanding mathematicians and physicists.

Returning to the United States, Gibbs continued his work with engineering problems. As a result of an investigation of steam engines, he became interested in the science of thermodynamics, which involves the study of heat and work and their interchange. This interest became the basis of most of his lifelong scientific endeavors.

In 1871 Gibbs was named professor of mathematical physics at Yale and continued to devote his research efforts to the theoretical treatment of thermodynamic phenomena. In 1872 he published a theoretical treatment of the thermodynamics of the adsorption of substances on surfaces, and in 1876 produced his best known work, *On the Equilibrium of Heterogeneous Substances,* a general theoretical treatment of thermodynamics.

In 1878 Gibbs published his phase rule in *Transactions of the Connecticut Academy,* a journal that was so obscure that his work went unnoticed for 20 years. The phase rule relates the
number of intensive quantities (e.g., temperature and pressure) and the number of chemical components in a system with the number of phases of a substance (e.g., liquid or gas) that may exist simultaneously. Gibbs's phase rule has found considerable application in the development and improvement of industrial processes.

Perhaps his most outstanding contribution was the introduction of the concept of chemical potential or free energy, now universally called Gibbs free energy in his honor. The Gibbs free energy relates the tendency of a physical or chemical system to simultaneously lower its energy and increase its disorder, or entropy, in a spontaneous natural process. Gibbs's approach permits the calculation of the change in free energy that occurs in such a process—for example, a chemical reaction—and, consequently, enables prediction as to whether and how rapidly the process may be expected to occur. The use of this concept revolutionized the scientific understanding of those chemical and physical processes that involve changes in heat and work. Since virtually all chemical processes and many physical ones involve such changes, his work has significantly impacted both the theoretical and empirical aspects of these sciences.

Another of Gibbs's major contributions was the development of statistical mechanics and its application to thermodynamic systems. Statistical mechanics has subsequently found extensive application in other fields, including quantum mechanics.

Although the importance of Gibbs's accomplishments was immediately recognized in Europe, recognition came more slowly in America. This was partially due to the more empirical bias of American science, but it may also have been the result of his withdrawn nature. He remained a bachelor, living with his sister's family throughout his life.

In 1881 Gibbs received the Rumford medal from the American Academy in Boston and, in 1901, the Copley medal from the Royal Society in England.

**J. WILLIAM MONCRIEF**

# Gibbs, Josiah Willard

**Gibbs, Josiah Willard** (1839–1903) US theoretical physicist and chemist. His application of thermodynamics to physical processes led to statistical mechanics. He devised the phase rule and developed vector analysis.

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