- Josiah Willard Gibbs (1839–1903), a Yale scientist and the first recipient of a US engineering PhD, helped transform thermodynamics by linking physical and chemical changes to concepts such as entropy and free energy, laying foundations for modern physical chemistry.
- His landmark 1876–78 work On the Equilibrium of Heterogeneous Substances introduced the concept of a ’phase’ and presented the equation later known as the phase rule, expressed today as F = C – P + 2, which relates the number of components, phases and degrees of freedom in a system.
- Despite its importance, Gibbs’ work was initially overlooked because of its highly mathematical and abstract style. Even leading scientists found it difficult to understand, although James Clerk Maxwell became a key advocate and helped promote Gibbs’ ideas.
- By the end of the 19th century, translations and growing recognition established Gibbs’ reputation. The phase rule became a fundamental tool across chemistry, metallurgy, mineralogy and petrology, while Gibbs himself was honoured with awards, memorials and lasting scientific recognition.
The originality, power and beauty of Gibbs’ work in the domain of thermodynamics have never been surpassed… The generality and abstract nature of the reasoning have, however, made the understanding of his methods and results a difficult task for many students of science. This has been particularly true of students of chemistry, who in general are deficient in mathematical training …
F G Donnan & A Haas:Commentary on the Scientific Writings of J Willard Gibbs, Vol. 1 (Preface), Yale University Press, 1936.
The originality, power and beauty of Gibbs’ work in the domain of thermodynamics have never been surpassed… The generality and abstract nature of the reasoning have, however, made the understanding of his methods and results a difficult task for many students of science. This has been particularly true of students of chemistry, who in general are deficient in mathematical training …
F G Donnan & A Haas:Commentary on the Scientific Writings of J Willard Gibbs, Vol. 1 (Preface), Yale University Press, 1936.
Through centuries of trial and error, many artisans – including smiths, potters, dyers and distillers – have tried to control chemical transformations by adjusting the physical conditions which affect them. After chemistry had evolved into an academic discipline such investigations were pursued more systematically, and in the late 1800s a new hybrid subject – physical chemistry – began establishing theoretical guidelines for them.
150 years ago, the publication of On the Equilibrium of Heterogeneous Substances by Josiah Willard Gibbs contributed to this development, by introducing a simple equation with wide applicability – later to become known as ‘the phase rule’. Initially, most of his contemporaries ignored it, but its significance gradually became recognised. Today, the story of its discovery (and its delayed acceptance) remains instructive.
Gibbs was born in New Haven in the US state of Connecticut on 11 February 1839. His father, Josiah Willard Gibbs senior – then professor of sacred literature at Yale College (as the modern university was then) – had studied mathematics and science before specialising in biblical languages. His mother, Mary Anna Van Cleve Gibbs – the daughter of Princeton’s chemistry professor – took an active interest in botany and ornithology.
Gibbs at Yale and in Europe
At Yale, young Willard’s studies were interrupted by illness, and for some time tuberculosis was suspected. But he graduated in 1859, and in 1863 earned the US’s first engineering PhD with a thesis on the geometry of gear wheels. Remaining at Yale as a tutor, he patented an innovative railway carriage brake in 1866.
By 1861, the deaths of both his parents had left Gibbs responsible for the welfare of his unmarried sisters – Anna, Julia and Emily – and for the home all four siblings still shared. A tranquil life in this quiet academic community might have appeared to lie ahead of the family, but in 1866 he decided to rent out the house and undertake an extended tour of Europe with Anna and Julia (Emily had died in 1864.)
In Paris, Gibbs pursued advanced studies in mathematics and physics. During the winter his health broke down, and as tuberculosis was again suspected the travellers migrated to the Riviera. On their return to Paris, however, his doctor found no further signs of the disease and the party proceeded to Berlin University as planned. There, they were joined by Addison Van Name – Yale’s librarian and Julia’s fiancé.
The couple married in Berlin, and returned home (via the Riviera) while Anna and Gibbs continued their tour. His Berlin notes reveal wide-ranging interests in mathematics, physics and engineering. Fewer records survive from his stay in Heidelberg, although he certainly had access to advanced teaching there.
After returning to New Haven in 1869, Gibbs – together with his sisters and brother-in-law – reoccupied the old family home. He resumed teaching, and in 1871 Yale awarded him the title of professor – though for years it could not afford to pay him a professorial salary. Meanwhile, another project was occupying his thoughts.
Thermodynamics before Gibbs
The Scots engineer James Watt had invented a ‘governor’ that kept steam engines running at constant speed. When they ran too fast, a spinning assembly – powered from the drive shaft – opened a valve to release pressure from the cylinder. That slowed the mechanism, whereupon the valve closed and pressure rose again.
Watt’s governor worked sluggishly and tended to over-correct. Gibbs improved the device slightly – but this task may also have focused his attention on the issue of thermodynamic equilibrium. Yale graduates later recalled that in 1872 Gibbs began discussing the ideas of German physicist Rudolf Clausius in his lectures.
Clausius’ studies of the relationship between heat and mechanical work – plus the related contributions of Rumford, Carnot, Mayer, Joule, Thomson and Maxwell – cover far more scientific territory than this article can survey. To summarise briefly: by the time Gibbs entered the field, the first and second laws of thermodynamics had already been stated, and their significance recognised. But their wider implications remained imperfectly understood.
A critical turning point had come in 1865, when Clausius coined the word ‘entropy’ in order to pin a label on an issue which had concerned him for years – namely, the fact that, although energy cannot be created or destroyed, over time it inevitably becomes more dispersed, and therefore less available for use. 30 years later, this concept would inspire HG Wells’ bleak vision of a post-human future in The Time Machine.
Gibbs free energy
Meanwhile, thermodynamics remained a work in progress. Physicists still disagreed about various aspects of it, while many chemists simply ignored it. Gibbs helped to transform this situation, by showing how changes in physical state or chemical composition can be related to changes in entropy.
In 1873 Gibbs published two papers in Transactions of the Connecticut Academy of Arts and Sciences – ‘Graphical Methods in the Thermodynamics of Fluids’, and ‘A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces’. They contained many equations – but very few diagrams – and introduced a new concept, later to become known as Gibbs free energy.
His most consequential work, however, was the 300-page study ‘On the Equilibrium of Heterogeneous Substances’, published (in sections) between 1876 and 1878. It contributed another term to the debate:
In considering the different homogeneous bodies which can be formed out of any set of component substances, it is convenient to have a term which shall refer solely to the composition and thermodynamic state of any such body without regard to its size or form. The word phase has been chosen for this purpose.
’In considering the different homogeneous bodies which can be formed out of any set of component substances, it is convenient to have a term which shall refer solely to the composition and thermodynamic state of any such body without regard to its size or form. The word phase has been chosen for this purpose.’
As the Academy’sTransactions had a limited circulation, Gibbs sent off-prints to numerous eminent scientists. Few replied – though some later admitted that they had read the work, and found it incomprehensible! Scottish physicist James Clerk Maxwell praised it in the Proceedings of the Cambridge Philosophical Society. Yet even he had difficulty with Gibbs’ equations, and only became convinced after recalculating them from first principles himself.
Although Maxwell’s death in 1879 probably delayed the uptake of Gibbs’ ideas, his support did have some impact. In 1880, Cambridge University chemist Matthew Pattison Muir informed Gibbs:
I was told of your paper on the ‘Equilibrium of Heterogeneous Substances’ but being no mathematician … I did not venture to read it. But recently I found Clerk Maxwell’s translation of your paper into ordinary language in the Science Conference at South Kensington (1876). Will you allow me as a chemist to thank you most sincerely for the very wonderful work you have done for us who, without the aid of mathematics, must needs grope so much in the dark?
’I was told of your paper on the ‘Equilibrium of Heterogeneous Substances’ but being no mathematician … I did not venture to read it. But recently I found Clerk Maxwell’s translation of your paper into ordinary language in the Science Conference at South Kensington (1876). Will you allow me as a chemist to thank you most sincerely for the very wonderful work you have done for us who, without the aid of mathematics, must needs grope so much in the dark?’
Nevertheless, years passed before Europe’s emerging community of physical chemists recognised the full significance of Gibbs’ work. Wilhelm Ostwald in Germany, and Jacobus van ’t Hoff in the Netherlands – co-founders, in 1887, of the influential journal Zeitschrift für Physikalische Chemie – had both rediscovered some of Gibbs’ results before learning that he had anticipated them.
Introducing the phase rule
Perhaps the most consequential element of Gibbs’ 1876 treatise was his announcement of the equation later to become known as ‘the phase rule’. On p446 he states:
The number of independent variations of which a system of coexistent phases is capable is n + 2 – r, where r denotes the number of phases, and n the number of independently variable components in the whole system … When the number of phases exceeds the number of components by unity, the system is capable of a single variation of phase.
‘The number of independent variations of which a system of coexistent phases is capable is n + 2 – r, where r denotes the number of phases, and n the number of independently variable components in the whole system … When the number of phases exceeds the number of components by unity, the system is capable of a single variation of phase.’
In this context, a ‘phase’ is a homogenous state of matter (solid, liquid or gas), while a ‘component’ is a chemically distinct substance (compound, element or allotrope). In modern terms, Gibbs’ rule states that an enclosed system with C components and P phases must have F degrees of freedom (independently variable parameters, like temperature or pressure), such that F = C – P + 2. The equation tells a chemist how many variables – such as temperature or pressure – can be independently adjusted without changing the number of phases present in a system.
Gibbs presented this conclusion via a lengthy sequence of equations, unsupported by graphs or diagrams. When the English physicist Lord Rayleigh suggested that On the Equilibrium of Heterogeneous Substances was ‘too condensed and too dense’, Gibbs replied that, in his opinion, it was already too long!
Understanding the phase rule
Many students today – like most of Gibbs’ original correspondents – find the phase rule difficult to comprehend. Meanwhile, diagrams, which Gibbs would probably have thought superfluous, remain helpful to struggling beginners. Thus, the phase changes of a one-component system – water – are conveniently represented by this familiar graph.
At any point on a boundary line between two phases, C = 1 and P = 2, giving the system one degree of freedom there; the temperature and pressure cannot both be changed without the phase changing. At the triple point, C = 1 and P = 3, giving zero degrees of freedom – both temperature and pressure must be fixed for all three phases to exist together.
With more components, the diagrams become more complicated – particularly when components are interconvertible, like the allotropes of sulfur. Here, the phase diagram has three triple points: at each of them the system has zero degrees of freedom.
When two or more components of an enclosed system can interact chemically, the phase diagram becomes even more complex. But for the present, it is sufficient to note that operations such as combining hydrogen and oxygen to make water, or freezing water to make ice, involve changes in entropy levels as well as exchanges of energy – all of which must be strictly accounted for, as Gibbs recognised.
Maxwell’s models
Gibbs was a profound and wide-ranging thinker – he also earned himself a niche in mathematical history, through his contributions to the field of vector analysis. But even when his equations extended into three dimensions, he saw no need for pictorial representations of them. Initially, the three physical variables he charted were pressure, temperature and volume, though he later went on to track the relationships between volume, entropy and energy.
But while Gibbs believed that equations alone were sufficient to convey his meaning, his most eminent supporter disagreed. Maxwell therefore commissioned some three-dimensional plaster models to illustrate the relationship between Gibbs’ three variables. One which he sent to Gibbs is still preserved at Yale while another survives in Scotland’s National Museum.
However, Maxwell also developed an alternative method of representing Gibbs’ equations in two-dimensional form, through line drawings – now known as ‘Maxwell’s Surfaces’ – which Gibbs himself later endorsed.
Later recognition and lasting legacy
As the century ended, Gibbs’ ideas became more widely appreciated, as can be gathered by the eminence of the scientists who translated his work: a German translation of his principal works (by Ostwald) appeared in 1892, and a French version (by Henry le Châtelier) in 1899. In 1901, the UK’s Royal Society awarded Gibbs its Copley Medal, describing him as ‘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’.
Gibbs died in 1903, fondly remembered by family members, friends and former students. By 1910 he was considered notable enough to merit a biographical entry in the latest edition of EncyclopaediaBritannica. In the same year, the American Chemical Society inaugurated its Gibbs Award for Excellence in Chemistry – recipients have included Marie Curie, Linus Pauling and Robert Woodward.
In 1951 Gibbs’ biographer, Lynde Phelps Wheeler, declared: ‘This celebrated law [the phase rule] has found innumerable applications in metallurgy, mineralogy, and petrology as well as in theoretical chemistry.’ Since then, the rule’s applications have continued to increase, and Gibbs’ reputation has become further enhanced. In 1964 the International Astronomical Union named a lunar crater after him. And in 2005 his portrait appeared on a commemorative postage stamp – finally giving nationwide visibility to the man whom Albert Einstein had once declared to be the greatest scientific mind yet produced by the US.
Mike Sutton is a historian of science based in Newcastle, UK
