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Biographical Memoirs: Volume 60

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Biographical Memoirs: Volume 60 LARS ONSAGER November 27, 1903-October 5, 1976 Courtesy, Yale Picture Collection BY H. CHRISTOPHER LONGUET-HIGGINS AND MICHAEL E. FISHER1 ONE DAY IN 1925 Pieter Debye was sitting in his office at the Eidgenössische Technische Hochschule in Zürich when a visitor from Norway was announced. In came a tall young man, who walked silently across the room, bent over the desk, and said solemnly: ''Professor Debye, your theory of electrolytes is incorrect.'' Whereupon Debye, after begging the stranger to sit down and inviting him to discuss his objections, offered him an assistantship for the following year. The young man's name was Lars Onsager.2 Forty-three years later Onsager was awarded the Nobel Prize in Chemistry for the "discovery of the reciprocal relations bearing his name, which are fundamental for the thermodynamics of irreversible processes." A group of physicists and chemists at Cornell had written of him: "We believe that his work is unique for its penetration, breadth, and influence in the development of theoretical and experimental studies of condensed matter. He is surely one of the outstanding physicists of this century." 1   This essay originally appeared in the Biographical Memoirs of Fellows of the Royal Society, London, vol. 24 (London: Royal Society, November 1978): 443-71. The text presented here corrects the original in various places, contains a few new references, and includes a completed bibliography of Onsager's published work. 2   T. J. Murphy and E. G. D. Cohen, "The Motion of Ions in Solution," BioSystems 8(1977): 255-60.

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Biographical Memoirs: Volume 60 NORWAY (1903-1925) Lars was born in Oslo on 27 November, 1903, to Erling and Ingrid Kirkeby Onsager. Erling was a barrister, and it is said that the family had interests in the steel industry, though Lars was later at pains to deny that his father was "a steel tycoon." His early education was liberal; his friends found him, in later life, extraordinarily well read in classical literature and philosophy and admired his taste in music and the fine arts. He attended high school in Oslo and at an early age familiarized himself with Norwegian literature, including the verse epics he loved to recite to his family and friends in later life, both in the original and in his own English translations. In 1920 Lars was admitted to the Norges Tekniske Hogskole in Trondheim to study chemical engineering in preparation for a technical career. But his inclinations were mainly intellectual; he had already bought a copy of Whittaker and Watson's classic monograph, Modern Analysis,3 and he worked through most of the (notoriously difficult) examples in his spare time. This early discipline equipped him for some of his most spectacular later achievements, notably his famous solution of the Ising problem in two dimensions. In other ways as well, Onsager's time as a student at the Norwegian Institute of Technology was prophetic of his later scientific work. As a freshman chemist he was introduced to the current theory of electrolyte solutions, according to which the properties of an electrolyte should be additive, not just over molecules, but even over the constituent ions. "In spite of some idealization," he declared later, "it sufficed for a great many purposes; it eased many tasks no end, and we were eternally grateful for that. However, very soon the journals rather than the textbooks taught 3   E. T. Whittaker and G. N. Watson, A course of modern analysis (Cambridge: Cambridge University Press, 1902), 4th ed., 1927.

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Biographical Memoirs: Volume 60 me about numerous observations which did not quite fit into the picture, and of tentative explanations for the discrepancies. . . . Suspicion centered on the long-range forces between the ions." (1969,1) This passing reference to "the journals" shows that Onsager was already exercising an independence of mind that later blossomed into a deep scientific originality. When, in 1923, Debye and Hückel published their new theory of electrolyte solutions,4 Onsager was quick to master their ideas and to detect a flaw in their account of electrolytic conduction and diffusion. It was his own ideas about these processes that ultimately led him to the reciprocal relations that now bear his name; but a parallel influence on his thinking was the experimental work of C. N. Riiber5 on the kinetics of tautomerism, which he had already begun to consider in 1924 in the light of the principle of microscopic reversibility. Onsager saw that this principle would supply a sufficient condition for detailed balancing to prevail—for there to be no chemical "circulation" when three or more tautomers were present under equilibrium conditions. The same principle might, he suspected, be brought to bear on the relative rates of other naturally occurring processes. In his five years at Trondheim, Onsager not only acquired the mathematical skill that he later put to such impressive use and an interest in electrolytes to which his attention was to return continually throughout his life, he also developed a deep appreciation of the relation of theory to experiment and of the duty of a theorist to propose experimental tests of his ideas. It is at least likely that his later interests in ther 4   P. W. Debye and E. Hückel, "Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen," Phys. Z. 24(1923): 185-206; "Zur Theorie der Elektrolyte. II. Das Grenzgesetz für die elektrische Leitfähigkeit," pp. 305-25. 5   C. N. Riiber, "Über Mutarotation I. Mitteilung", Chem. Ber. 55B(1922): 3132-43; and "Über Mutarotation II. Metteilung," Chem. Ber. 56B(1923): 2185-94.

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Biographical Memoirs: Volume 60 mal diffusion (1939,2,3,4; 1940,1), in colloidal solutions (1942,1; 1949,1), and in turbulence (1945,2; 1949,3) were also engendered by his training as a chemical engineer. At any rate, when he finally graduated with a Ch.E. in 1925, a most formidable intellect stepped onto the scientific scene. ZÜRICH (1926-1928) It was Onsager's interest in electrolytes that first took him to Zürich in 1925. Debye and his assistant Erich Hückel had put forward a new theory of electrolyte solutions founded on the idea that the electrostatic field of a dissolved ion is screened by an "atmosphere" of opposite net charge, the effective screening distance being inversely proportional to the square root of the ionic strength c, defined as where ci is the concentration of ion i, and zi is its electric charge in elementary units. The activity coefficient fi of any ion—its thermodynamic activity divided by ci—could then be calculated, for small c, from the equation in which Ai was a function of the charge zi, the temperature T, and the dielectric constant D of the solvent. The theory was quantitatively successful in accounting for the thermodynamic properties of dilute salt solutions, but Debye and Hückel's extension of their theory to deal with the conductivities of electrolytes ran into difficulties. Their equation

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Biographical Memoirs: Volume 60 for the molar conductivity Λ appeared to be correct in form, but the calculated value of Λ1, also dependent on T and D, differed considerably from the experimental value. This was puzzling because the most important physical effects had, it seemed, been allowed for: an "electrophoretic effect" in which the counter-ions in the atmosphere pull the solvent in the wrong direction, making it necessary for the central ion to "swim upstream"; and a ''relaxation effect" in which the ion is held back by the net attraction of the atmosphere itself. Accepting the general correctness of the physical picture, Onsager himself had explored its implications and in 1923 produced, as he himself put it, 'a modest but firm result': "The relaxation effect ought to reduce the mobilities of anion and cation in equal proportion. Much to my surprise, the results of Debye and Hückel did not satisfy that relation, nor the requirement that wherever an ion of type A is 10 Å west of a B, there is a B 10 Å east of that A. Clearly something essential had been left out in the derivation of such unsymmetrical results." (1969,1) By the time he visited Debye in his office in 1925, Onsager had pinpointed the origin of the discrepancy. Debye and Hiickel had evaluated Λ1 by assuming one particular ion to move uniformly in a straight line but allowing the other ions to undergo Brownian motion subject to the fields of their distorted atmospheres. All that was required was to relax the constraint that the central ion move uniformly, and to allow it to move as freely as its neighbors—subject, of course, to their influence and that of the external field. In this way the desired symmetry could be restored, and the calculated value of Λ1 brought into good agreement with experiment. Debye must have been deeply impressed by this astonishing insight. It is no wonder that he proclaimed his young critic a genius. In April 1926 he demonstrated his sincerity

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Biographical Memoirs: Volume 60 by taking Onsager on as a research assistant and in 1927 by promoting his ideas at a discussion meeting of the Faraday Society in Oxford. (1927,2) The Onsager Limiting Law, as it came to be called, was first put forward in the second of a pair of papers in the Physikalische Zeitschrift (1926,1; 1927,1). The evident thoroughness and maturity of these papers doubtless owed much to the influence of Debye himself. Onsager would have been the first to admit that he stood on Debye's shoulders both in his work on electrolytes and in his later investigations of the dielectric constants of polar liquids and solutions of polar molecules. During the next few years the meticulous experimental work of Shedlovsky6 confirmed the Limiting Law to a high degree of accuracy, and Onsager was to continue to develop his theory of electrolytic transport—especially in collaboration with Raymond M. Fuoss—for many years after that. But in the meantime his thoughts were beginning to turn to problems of greater generality and to horizons of wider scope. JOHNS HOPKINS UNIVERSITY (1928) In 1928 Onsager emigrated to the United States and was appointed an associate in chemistry at The Johns Hopkins University. The appointment was brief. In the words of Robert H. Cole, who was one of Onsager's associates for more than forty years: "They made the mistake there of assigning Onsager to the basic Chemistry I, II course. He just couldn't think at the level of a freshman. Frankly, he was fired."7 Onsager's difficulties in communicating with weaker intellects were acute and remained so throughout his life. It 6   T. Shedlovsky, "The electrolytic conductivity of some uni-univalent electrolytes in water at 25º," J. Am. Chem. Soc. 54(1932): 1411-28. 7   J. F. Barry, Jr., "Lars Onsager: The greatest theoretical chemist," Brown Univ. Alum. Mo. (November 1976): 2.

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Biographical Memoirs: Volume 60 may seem strange that a man who could see so deeply into physical reality should have been so conspicuously lacking in imagination when it came to reading other minds, but almost everyone who met him became immediately aware of this disability. "I won't say he was the world's worst lecturer," Professor Cole continues, "but he was certainly in contention. He was difficult to understand anyway, but he also had the habit of lecturing when his back was to the students and he was writing on the blackboard. To compound matters, he was a big man, and students had to peer round him just to try and see what was being written." Onsager's problems in communicating with lesser mortals were certainly not due to impatience or arrogance. The theoretical chemist Julian Gibbs, of Brown University, who got to know him some years later describes Onsager as a "very, very friendly man" who would always assume that his listeners were as advanced in their thinking as he was. "He assumed that if he knew it, others in the field automatically knew it," whatever the subject under discussion. Yet it was not only students who found Onsager difficult to understand; his colleagues had the same difficulty. Oliver Penrose, who worked with him as a postdoctoral associate, recalls a lecture to the Kapitza Club in Cambridge, many years later, at which Onsager was explaining his joint work with Bruria Kaufman on the Ising lattice. He had been warned that non-theoreticians would be present and that he should phrase his talk in not too technical language. He plunged, nevertheless, into the mathematics of spinor algebras. After about twenty minutes, one of the many experimentalists in the audience had the courage to ask him what a spinor was. Onsager replied, thoughtfully: "A spinor—no, a set of spinors—is a set of matrices isomorphic to the orthogonal group." With that he gave the famous Onsager grin,

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Biographical Memoirs: Volume 60 twinkled his Nordic blue eyes at the bewildered faces around him, and continued the lecture as if nothing had happened. "In private discussion," wrote Cyril Domb, "it was much easier to communicate with Onsager provided you were courageous enough to persist in questioning when you did not understand. He would drop the level one stage at a time until the gap could be bridged."8 In their introduction to the proceedings of a conference held in his honour in 1962, Shedlovsky and Montroll said in his defense: "Whether deserved or not, Onsager has the reputation of being verbally obscure, or at least enigmatic. However, those who know him well will testify that he is clarity itself and often responds at great length, if the question presented to him refers to Norse mythology, gardening, the more subtle aspects of Kriegspiel (a form of blindfold chess involving two opponents and a referee) and even encyclopedic facts about organic chemistry."9 And everyone who actually worked with him testified to his unfailing generosity with his time and ideas. Onsager's interest in Kriegspiel, incidentally, is confirmed by Penrose: "I never played this with him although I did play a game of chess with him once. He was a good player at chess, but very slow. At Kriegspiel he had worked out how to force the win with a king and two bishops—it may even have been king, bishop, and knight—against king alone; something which I had not believed could be done until he showed me." Onsager regarded chess, so he said, as too much like real problem-solving to spend much time on it. When he wanted to unwind from his work he would play solitaire, and bridge was a good relaxation in company. 8   C. Domb, C., Obituary, Nature, Lond. 264(1976): 819. 9   T. Shedlovsky and E. Montroll, Introduction, Proc. Conf. Irreversible Thermodynamics and the Statistical Mechanics of Phase Transitions (Onsager Symposium), J. Math. Phys. 4,2(1963).

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Biographical Memoirs: Volume 60 BROWN UNIVERSITY (1928-1933) So Onsager had to move. Fortunately, an opening appeared at Brown, where Charles A. Kraus was chairman of the Chemistry Department. The two men were very different—Onsager the young high-powered theorist and Kraus the hard-headed experimentalist. "But Kraus," reports Cole, "knew that Onsager would be good for Brown, and he signed him up as a research instructor. . . . A look at the University catalogues for the Onsager years at Brown reveals that he was listed at the bottom of the page simply as 'Mr. Onsager.' The fact was that he had no Ph.D. He did all his work at Brown that led to his Nobel Prize without the 'advantage' of a Ph.D." A lesser scientist might have been discouraged by the intellectual isolation in which Onsager must have found himself during those five years. The problems on which he was working and the ideas he was developing can hardly have appealed to his departmental colleagues. Speaking in 1973 of his now classic work on irreversible processes, which appeared in 1931, he said: "It wasn't doubted, but completely ignored." "It was not until after the Second World War," confirms Stig Claesson, commenting on the length of time before the full import of Onsager's ideas was recognized, "that it attracted great attention. The man was really ahead of his time.''10 As chairman of the department Kraus was always after Onsager to do an experiment of some sort rather than spending all his time on theoretical work. One day Onsager told him he had decided to try an experiment on the separation of isotopes by thermal diffusion. "Fine," said Kraus, and was 10   S. Claesson, "The Nobel Prize for Chemistry" (presentation speech), Les Prix Nobel en 1968, p. 42 (Stockholm: Imprimerie Royale P. A. Norstedt & Söner, 1972). Also in: Nobel Lectures—Chemistry 1963-1970, p. 269 (Amsterdam: Elsevier).

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Biographical Memoirs: Volume 60 doubly pleased when Lars told him that the only equipment he would need was a long tube. But his encouragement was quickly withdrawn when Onsager explained that the tube must be made of platinum and would have to stretch from the basement to the third floor of the chemistry building. Kraus never pestered him again about doing an experiment, which "was too bad," writes Julian Gibbs, "because no one succeeded in conducting this experiment until more than a decade later, when it was needed as part of the Manhattan Project for the atomic bomb."11 Onsager's pedagogic endeavours at Brown were hardly more effective than at Johns Hopkins, but they resulted in one major conversion. John F. Ryan, a Brown alumnus, recalling Onsager's "Sadistical Mechanics" course, as it was known, says that in the year he took the course a New Boy was attending it. Early in the second lecture Lars wrote a typically complicated equation on the blackboard, and turned to his audience with a hopeful "You see?" The gloomy silence that followed was broken by the New Boy: "But shouldn't it be 'times unit vector?"' Lars wheeled round to the board, shouted ''Yah! Yah!" scribbled in the unit vector symbol and beamed at the class and the world in general. He had found a disciple who understood him, and the rest of the course was a duet between the two, witnessed with irritated incomprehension by the rest of the students. The New Boy was Ray Fuoss, who took his Ph.D. with Onsager and became his first co-author (1932,1). Together again at Yale, he and Onsager collaborated on many joint papers during the next thirty-five years (1932,1; 1957,3; 1958,1; 1961,2; 1962,1; 1963,1,2; 1964,2; 1965,2). While he was at Brown struggling to meet the demands of the educational system, Onsager's mind was preoccupied 11   Barry, Brown Univ. Alum. Mo. (1976): 2.

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Biographical Memoirs: Volume 60 APPOINTMENTS 1926-1928 Research Assistant, Eidgenössische Technische Hochschule, Zürich, Switzerland 1928 Associate in Chemistry, Johns Hopkins University 1928-1933 Instructor in Chemistry, Brown University 1933-1934 Sterling and Gibbs Fellow, Yale University 1934-1940 Assistant Professor of Chemistry, Yale University 1940-1945 Associate Professor of Chemistry, Yale University 1945-1973 J. Willard Gibbs Professor of Theoretical Chemistry, Yale University 1951-1952 Fulbright Scholar, Cambridge University 1961 Visiting Professor, University of California, San Diego 1967-1968 Visiting Professor, Rockefeller University 1968 Visiting Professor, University of Göttingen 1970 Visiting Professor, University of Leiden 1972-1976 Distinguished University Professor, University of Miami, Coral Gables DEGREES 1925 Ch.E., Norges Tekniske Hogskole, Trondheim, Norway 1935 Ph.D., Yale University HONORARY DEGREES 1954 D.Sc., Harvard University 1960 Dr Technicae, Norges Tekniske Hogskole 1962 D.Sc., Brown University D.Sc., Rensselaer Polytechnic Institute Dr Naturwissenschaften, Rheinisch-Westfälisch Technische Hochschule, Aachen 1968 D.Sc., The University of Chicago 1969 D.Sc., Ohio State University 1970 Sc.D., Cambridge University 1971 D.Sc., Oxford University MEDALS AND PRIZES 1953 Rumford Gold Medal, American Academy of Arts and Sciences

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Biographical Memoirs: Volume 60 1958 Lorentz Medal, Royal Netherlands Academy of Sciences 1962 G. N. Lewis Medal, American Chemical Society, California Section J. G. Kirkwood Medal, American Chemical Society, New Haven Section J. W. Gibbs Medal, American Chemical Society, Chicago Section 1964 T. W. Richards Medal, American Chemical Society, Northeastern Section 1965 P W. Debye Award in Physical Chemistry, American Chemical Society 1966 Belfer Award in Pure Science, Yeshiva University 1968 Nobel Prize in Chemistry President's National Medal of Science ACADEMIC AFFILIATIONS AND DISTINCTIONS FELLOW American Physical Society (1933) New York Academy of Sciences (1942) National Academy of Sciences (1947) American Academy of Arts and Sciences (1953) MEMBER American Physical Society (December 1928) Sigma Xi Fraternity, Brown University Chapter (1929) Connecticut Academy of Arts and Sciences (1940) Royal Norwegian Academy of Science (1953) Royal Swedish Academy of Science (1957) Norwegian Academy of Technical Science (1958) Royal Netherlands Academy of Science (1958) American Philosophical Society (1959) American Chemical Society (1962) Deutsche Bunsen Gesellschaft (1969) FOREIGN MEMBER Norwegian Academy of Sciences and Letters, Oslo (1938) Royal Society of Sciences of Uppsala (1963) Royal Society (1975)

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Biographical Memoirs: Volume 60 HONORARY MEMBER Norwegian Chemical Society (1947) HONORARY FELLOW Institute of Physics, U. K. (1974) ASSOCIATE Neurosciences Research Program, U.S.A.

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Biographical Memoirs: Volume 60 BIBLIOGRAPHY 1926 Zur Theorie der Electrolyte. I. Phys. Z. 27: 388-92. 1927 Zur Theorie der Electrolyte. II. Phys. Z. 28: 277-98. Report on a revision of the conductivity theory. Trans. Faraday Soc. 23: 341-49, 356. 1928 Activity coefficients and mass-action law in electrolytes. J. Phys. Chem. 32: 1461-66. 1929 Simultane irreversible processor. (Abstract). Beret. 18d. Skand. NatForsk-Møde, Copenhagen, pp. 440-41. 1931 Reciprocal relations in irreversible processes. I. Phys. Rev. 37: 405-26. Reciprocal relations in irreversible processes. II. Phys. Rev. 38: 2265-279. 1932 With R. M. Fuoss. Irreversible processes in electrolytes. J. Phys. Chem. 36: 2689-778. Viscosity and particle shape in colloid solutions. (Abstract). Phys. Rev. 40: 1028. 1933 Theories of concentrated electrolytes. Chem. Rev. 13: 73-89. 1934 With N. N. T. Samaras. The surface tension of Debye-Hückel electrolytes. J. Chem. Phys. 2: 528-36. Deviations from Ohm's law in weak electrolytes. J. Chem. Phys. 2: 599-615.

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Biographical Memoirs: Volume 60 1935 Solutions of the Mathieu equation of period 4π and certain related functions. Ph.D. thesis, Department of Chemistry, Yale University. 1936 Electric moments of molecules in liquids. J. Am. Chem. Soc. 58: 1486-93. 1938 Initial recombination of ions. Phys. Rev. 54: 554-7. 1939 Electrostatic interaction of molecules. J. Phys. Chem. 43: 189-96. With W. H. Furry and R. Clark Jones. On the theory of isotope separation by thermal diffusion. Phys. Rev. 55: 1083-95. Separation of gas (isotope) mixtures by irreversible processes. (Abstract). Phys. Rev. 55: 1136-37. With W. W. Watson. Turbulence in convection in gases between concentric vertical cylinders. Phys. Rev. 56: 474-77. 1940 Separation of isotopes by thermal diffusion. (Abstract). Phys. Rev. 57: 562. 1942 Anisotropic solutions of colloids. (Abstract). Phys. Rev. 62: 558. Crystal statistics. (Abstract). Phys. Rev. 62: 559. 1944 Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65: 117-49. 1945 Theories and problems of liquid diffusion. Ann. N.Y Acad. Sci. 46: 241-65. The distribution of energy in turbulence. (Abstract). Phys. Rev. 68: 286.

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Biographical Memoirs: Volume 60 1947 With B. Kaufman. Transition points. Rep. Int. Conf. on Fundamental Particles and Low Temperatures, Cambridge, July 1946, vol. 2, p. 137. London: The Physical Society. 1948 With J. E. Robinson. De Haas-Van Alphen effect in zinc. (Abstract). Phys. Rev. 74: 1235. 1949 Effects of shape on the interaction of colloidal particles. Ann. N.Y. Acad. Sci. 51: 627-59. With B. Kaufman. Crystal statistics. III. Short-range order in a binary Ising lattice. Phys. Rev. 76: 1244-52. Statistical hydrodynamics. Nuovo Cim. Suppl. (9) 6: 279-87; see also 249, 261. With W W. Watson and A. Zucker. Apparatus for isotope separation by thermal diffusion. Rev. Scient. Instrum. 20: 924-27. 1952 Interpretation of the de Haas-Van Alphen effect. Phil. Mag. (7) 43: 1006-8. With L. J. Gosting. General theory for the Gouy diffusion method. J. Am. Chem. Soc. 74: 6066-74. Kinetic theory and statistical mechanics. Lecture notes of a course of the same title given at Yale University by Lars Onsager, compiled by Don E. Harrison, Jr. Unpublished manuscript in Kline Library, Yale University, New Haven. 1953 With S. Machlup. Fluctuations and irreversible processes. Phys. Rev. 91: 1505-12. With S. Machlup. Fluctuations and irreversible processes. II. Systems with kinetic energy. Phys. Rev. 91: 1512-15. Diamagnetism in metals. Proc. Int. Conf. Theoretical Physics, Kyoto and Tokyo, September 1953, pp. 669-75. Introductory talk [on liquid helium], pp. 877-80. Tokyo: Science Council of Japan.

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Biographical Memoirs: Volume 60 1955 With R. M. Fuoss. Conductance of strong electrolytes at finite dilutions. Proc. Natl. Acad. Sci. USA. 41: 274-83. 1956 With O. Penrose. Bose-Einstein condensation and liquid helium. Phys. Rev. 104: 576-84. 1957 With S. K. Kim. Wien effect in simple strong electrolytes. J. Phys. Chem. 61: 198-215. With S. K. Kim. The relaxation effect in mixed strong electrolytes. J. Phys. Chem. 61: 215-29. With R. M. Fuoss. Conductance of unassociated electrolytes. J. Phys. Chem. 61: 668-82. 1958 With R. M. Fuoss. The kinetic term in electrolytic conductance. J. Phys. Chem. 62: 1339-40. With J. L. Lebowitz. Low temperature fluctuations. Proc. Fifth Int. Conf. Low Temperature Physics and Chemistry, Madison, Wisconsin, Aug. 1957, p. 119. Madison: University of Wisconsin Press. Many-electron wave function. (Abstract). Bull. Am. Phys. Soc., Series 11, 3: 146. 1960 With M. Dupuis and R. Mazo. Surface-specific heat of an isotropic solid at low temperatures. J. Chem. Phys. 33: 1452-61. With D. R. Whitman, M. Saunders, and H. E. Dubb. Proton magnetic resonance spectrum of propane. J. Chem. Phys. 32: 67-71. With M. Dupuis. Electrical properties of ice. Re. Scu. Int. Fis. ''Enrico Fermi," Corso X, Varenna, 1959, pp. 294-315. Bologna: Nicolà Zanichelli (Supplement to Nuovo Cimento). 1961 Magnetic flux through a superconducting ring. Phys. Rev. Lett. 7: 50. With R. M. Fuoss. Thermodynamic potentials of symmetrical electrolytes. Proc. Natl. Acad. Sci. USA 47: 818-25.

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Biographical Memoirs: Volume 60 Statistical mechanics course. (Lecture notes from L. Onsager's Statistical Mechanics, Yale University, taken by Robert Hill). Unpublished manuscript in Kline Memorial Library, Yale University, New Haven. 1962 With R. M. Fuoss. The conductance of symmetrical electrolytes. I. Potential of total force. J. Phys. Chem. 66: 1722-26. The electrical properties of ice. Vortex 23: 138-41. With M. Dupuis. The electrical properties of ice. Electrolytes, Proc. Int. Symp. Trieste, Yugoslavia, 1959, pp. 27-46. Oxford: Pergamon Press. 1963 With R. M. Fuoss. The conductance of symmetrical electrolytes. II. The relaxation field. J. Phys. Chem. 67: 621-28. With R. M. Fuoss. The conductance of symmetrical electrolytes. III. Electrophoresis. J. Phys. Chem. 67: 628-32. With L. K. Runnels. Mechanism for self-diffusion in ice. Proc. Natl. Acad. Sci. USA 50: 208-10. Helium II. Proc. Symp. on the many-body problem, Stevens Institute of Technology, Hoboken, New Jersey, January 28-29, 1957, pp. 457-464. New York: Interscience. 1964 A correction to the Poisson-Boltzmann equation for unsymmetrical electrolytes. J. Am. Chem. Soc. 86: 3421-23. With R. M. Fuoss. The conductance of symmetrical electrolytes. IV. Hydrodynamic and osmotic terms in the relaxation field. J. Phys. Chem. 68: 1-8. 1965 Electrons in liquids. In: Modern quantum chemistry. Istanbul Lectures 1964, ed. O. Sinanoglu, pt. 2, pp. 123-28. New York: Academic Press. Electrons in metals. In: Modern quantum chemistry. Istanbul Lectures 1964, ed. O. Sinanoglu, pt. 2, pp. 265-78. New York: Academic Press. With C. T. Liu. Zur Theorie des Wieneffekts in schwachen Elektrolyten. Z. Phys. Chem. (Leipzig) 228: 428-32.

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Biographical Memoirs: Volume 60 With R. M. Fuoss and J. F. Skinner. The conductance of symmetrical electrolytes. V. The conductance equation. J. Phys. Chem. 69: 2581-94. 1966 With L. Mittag and M. J. Stephen. Integrals in the theory of electron correlations. Ann. Phys. (Leipzig) 7. Folge 18: 71-77. 1967 Ferroelectricity of ice? Proc. Symp. on Ferroelectricity, Warren, Michigan, Sept. 1966, ed. Edward F. Weller, pp. 16-19. Amsterdam: Elsevier. Thermodynamics and some molecular aspects of biology. In: The neurosciences. A study program, eds. G. C. Quarton et al., p. 75. New York: The Rockefeller University Press. 1968 With S. W. Provencher. Relaxation effects in associating electrolytes. J. Am. Chem. Soc. 90: 3134-40. 1969 The motion of ions: principles and concepts. Les Prix Nobel en 1968, pp. 169-82. Stockholm: Norstedt & Söner. Also in Science 166:1359-64. With L. K. Runnels. Diffusion and relaxation phenomena in ice. J. Chem. Phys. 50: 1089-1103. Protonic semiconductors. In: Physics of ice, Proc. 3rd Int. Symp., Munich, 1968, eds. Nikolaus Riehl et al., pp. 363-68. New York: Plenum Press. 1970 Possible mechanisms of ion transit. Physical principles of biological membranes. In: Proc. Coral Gables Conf., 1968, eds. F. Snell et al., p. 137. New York: Gordon and Breach. 1971 The Ising model in two dimensions. In: Critical phenomena in alloys, magnets and superconductors (Report on the Battelle Symposium), eds. R. E. Mills et al., pp. 3-12. New York: McGraw-Hill.

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Biographical Memoirs: Volume 60 1974 Interpretation of dynamic and equilibrium properties of water. In: Structure of water and aqueous solutions, Proc. Int. Symp. Marburg, 1973, ed. Werner Luck, pp. 1-7. Weinheim: Verlag Chemie. (a) Life in the early days, pp. 1-14; (b) with Edmond Drauglis. The effect of wall charge on the capillary rise of electrolytes, pp. 167-200; (c) with Tag Young Moon. Surface specific heat of crystals. I., pp. 227-79. In: Quantum statistical mechanics in the natural sciences, Coral Gables Conf., 1973, eds. S. L. Mintz and S. M. Widmayer. New York: Plenum Press. With A. M. Stewart. Asymptotic forms for luminescent intensity due to donor-acceptor pair recombination. J. Phys. C. 7: 645-48. With Mou-Shan Chen, Jill C. Bonner, and J. F. Nagle. Hopping of ions in ice. J. Chem. Phys. 60: 405-19. 1975 With J. McCauley, Jr. Electrons and vortex lines in He II. I. Brownian motion theory of capture and escape. J. Phys. A 8: 203-13. With J. McCauley, Jr. Electrons and vortex lines in He II. II. Theoretical analysis of capture and release experiments. J. Phys. A 8: 882-90. 1977 With Shoon K. Kim. The integral representation of the relaxation effects in mixed strong electrolytes in the limiting law region. J. Phys. Chem. 81: 1211-12. With Mou-Shan Chen. The generalized conductance equation. J. Phys. Chem. 81: 2017-21. With J. B. Hubbard, W. M. van Beek, and M. Mandel. Kinetic polarization deficiency in electrolyte solutions. Proc. Natl. Acad. Sci. USA 74: 401-4. With J. Hubbard. Dielectric dispersion and dielectric friction in electrolyte solutions. J. Chem. Phys. 67: 4850-57. 1978 With David L. Staebler and Sergio Mascarenhas. Electrical effects during condensation and phase transitions of ice. J. Chem. Phys. 68: 3823-28.

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