Popular Science Monthly/Volume 74/June 1909/Josiah Willard Gibbs and his Relation to Modern Science II

by Fielding Hudson Garrison

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JOSIAH WILLARD GIBBS AND HIS RELATION TO MODERN SCIENCE. II

By FIELDING H. GARRISON, M.D.,

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

The Thermodynamic Potentials.^[1]—In 1869 the physicist F. Massieu communicated to the French Academy of Sciences the discovery of two algebraic functions from which all the thermodynamic properties of a fluid may be derived.^[2] These "fonctions characteristiques" of Massieu contain in latent form two of the four relations which Gibbs derived independently from his general thermodynamic equation and which have since been variously interpreted as the fundamental functions or thermodynamic potentials of heterogeneous chemical systems. Mathematically they are simplifications which dispense with the necessity of endless transformations of equations and formulæ, evolving, as Bryan says, "order out of chaos."^[3] As the foundations of thermodynamics are its two laws, so the potentials may be regarded as the coping stones of the edifice, and all recent progress in the science, as in the physics of gas mixtures, osmosis, elastic solids or electrolysis has been made with their aid. The four potentials are now interpreted as the "free energy" $(\psi )$ and the "modified available energy" or total thermodynamic potential $(\zeta )$ for constant temperature, and the intrinsic energy $(\epsilon )$ and heat function $(\chi )$ for constant entropy.^[4] Of these the first two are the most important, being the analogues of the Newtonian or gravitational potentials (potential energies) of mechanical systems, generalized, as Larmor says, "so as to include the temperature" and connoting thermal effects,^[5] just as the Maxwellian potentials connote effects of electromotive force. They represent that part of the energy of a system which is due to changes in its mass or structure rather than to thermal or molecular changes, and so can take part freely in physico-chemical transformations. For this reason the potential $\psi$, which is the difference between the total energy of a system and its bound (molecular) energy, was called the "free energy" of the system by Helmholtz, who rediscovered the principle independently, not knowing that Gibbs had forestalled his labors by at least six years. In lecturing on the subject during the later period of his life, Helmholtz, with his usual breadth of spirit, was inclined to assign complete priority to his predecessor,^[6] while both Gibbs and Helmholtz have acknowledged their indebtedness to Massieu.^[7]

Criteria of Equilibrium and Stability.—Gibbs's conditions for the complete equilibrium of an isolated homogeneous chemical substance are that its pressure, temperature and the chemical potentials of its components should be constant throughout the mass, since changes of pressure and temperature disturb mechanical and thermal equilibrium, while difference of potentials destroys stability and precipitates chemical change. For an isolated heterogeneous system, as an enclosed liquid and a gas in contact, the following maxima and minima are criteria of complete equilibrium: The system must have and maintain the greatest entropy consistent with constant energy; or for adiabatic systems (at constant entropy), the intrinsic energy $(\epsilon )$ or heat function $(\chi )$ should have minimum values for constant volume or pressure, respectively, but for isothermal systems (at constant temperature) the free energy potential $(\psi )$ or the thermodynamic potential $(\zeta )$ should have minimum values for constant volume or pressure. Any deviation from these maxima or minima will again disturb equilibrium and produce changes of physical or chemical state. The essential feature of spontaneous chemical change is, then, either constant increase of entropy in self-contained or adiabatic systems or a corresponding decrease of free (mechanically available) energy in systems at uniform temperature, or in Lord Kelvin's phrase, a general dissipation of energy in all irreversible phenomena. For each potential, with appropriate choice of coordinates, a solid model or relief map can be constructed, upon which the different minima of the potentials appear as depressions in the landscape. When the lowest depression or minimum has been reached, complete and permanent equilibrium is attained, and we have what Gibbs calls a "phase of dissipated energy," at which, as in a bar of metal or a block of granite, no further spontaneous changes of physical state are possible so long as the system remains isolated from external forces. In connection with his discussion of equilibrium we may note Gibbs's forethought in extending his equations to n dimensions, since for more than three components a three-dimensional model no longer suffices; his early introduction of the time element into the discussion of chemical reactions^[8] and his pages on "passive resistance to change,"^[9] which should be read by every chemist, since they are of the essence of his subject, especially in regard to carbon compounds or colloidal substances. In applying dynamic principles to chemical phenomena Gibbs, and after him Helmholtz, thought decrease of free energy at uniform temperature to be the most important condition for equilibrium, since it measures the actual work done and is thus the "force function" of mechanics with reversed sign.^[10] The electromotive force of a reversible galvanic cell turns out to be identical with the free energy of chemical decomposition in the cell,^[11] and in the field of biology free energy is of equal importance, for relative uniformity of temperature is as common in living processes as in the laboratory. Well did Boltzmann say that "the struggle for existence of living matter is a war for free energy," for when the free energy of a living body becomes a minimum its death is at hand.

The Phase Rule.^[12]—Any aspect of a chemical substance which is homogeneous in regard to physical state and percentage composition has been called by Gibbs a phase of the substance, the components of a phase being its constituents of independently variable concentration. The phase rule asserts that a homogeneous substance having n components is capable of only n + 1 independent phases, while a heterogeneous system of r coexistent phases, each of which has n independently variable components, is capable of $n+2-r$ degrees of freedom or variations in phase, of which not more than $n+2$ can coexist at the same pressure and temperature. Such systems, in Trevor's nomenclature, are spoken of as invariant, monovariant, divariant, etc., according to the number of possible variations of state. Thus water (${\ce {H2O}}$) has three independent phases, ice, liquid and steam, and is invariant, the three phases being in equilibrium at only one pressure and temperature, called the triple point, where the steam-line, ice-line and hoar-frost line meet. But when calcium carbonate (${\ce {CaCO3}}$), calcium oxide (${\ce {CaO}}$) and carbon dioxide (${\ce {CO2}}$) are in equilibrium, we have three coexistent phases formed of two components (${\ce {CaO, CO2}}$) and the system is monovariant. By this rule the chemist is able to predict the number of modifications of which a chemical substance is capable from observation of its physical properties alone, or the number of substances in a mixture from notation of the number of phases possible, or the strength of a saturated solution from its temperature and pressure. Many different proofs of the phase rule have been given by mathematicians and physicists from varied and independent points of view,^[13] and there is every indication that it is a complete and accurate statement of a general chemical law. Its practical significance remained for a long time undiscovered until the Dutch chemist Van der Waals took it up, and when, in 1884-6, his colleague, Bakhuis Roozeboom, found himself unable to explain certain puzzling phenomena connected with the equilibrium of gaseous hydrates and of double ammonium salts, van der Waals was able to direct his attention to Gibbs's theorem and showed him, by working out a special case, how thermodynamic methods might be applied to practical chemistry.^[14] From that time on Roozeboom became the devoted champion of the phase rule, which he compares to the ground plan of a gigantic building in which all the collected phenomena of chemical equilibrium can be stored in a convenient and comprehensive manner. "This structure," he adds with pride, "has since been completed, almost exclusively by the work of the laboratories of Leyden and Amsterdam."^[15] In fact, the investigations of Roozeboom and van't Hoff upon double salts, solid solutions and metals are among the most brilliant results of modern chemistry. It is in connection with the graphic study of chemicophysical changes by the phase rule that Gibbs's diagrams and surfaces have proved of greatest value. The triangular diagram which he originally proposed for this purpose^[16] has been so improved by Roozeboom^[17] that the study of the chemical changes of a heterogeneous system and the prediction of its possible degrees of freedom become for the skilled worker a simple and easy matter.

Scientific Applications of the Phase Rule.—The doctrine of phases gives the chemist a new way of looking at things, serving at once as a basis of classification and a guide in qualitative research, and in contradistinction to the older gravimetric chemistry which dealt with some one continuous state of an isolated substance, it inaugurates the chemistry of substances in contiguity. Here the phase rule bears the same relation to physical chemistry that the periodic law of Lothar Meyer and Mendelejeff does to inorganic chemistry. Although only qualitative in its application it gives the chemist a new fundamentum divisionis by components and phases, necessitating a revision of substances, of which many formerly recognized as compounds are now no longer listed as such, while many new compounds have been introduced.^[18] In analytical chemistry the phase rule has found its widest application in classifying our knowledge of the dissociation of solid substances such as alloys, solid solutions, cryohydrates, tartrates, basic double and racemic salts, or in the solution of such special problems as the changing solubilities of metallic hydrates or the distributions of a dissolved substance between two solvents which do not mix. For example, Roozeboom discovered by the phase rule that four different hydrates can be formed with ferric chloride, of which only two were known before his investigation,^[19] while van der Heide's studies of the double sulphate of potassium and magnesium (Schonite) revealed the possibility of at least fifteen heterogeneous modifications of phase.^[20] No less than thirty different ferric sulphates are now on record,^[21] and Bancroft has said that a general system of qualitative research is not possible until we have studied the properties of such multi-component systems. Tammann's researches on the equilibria subsisting between solids and their melted states indicate that nearly all such substances have more than one solid modification of phase,^[22] while van't Hoff has completely revolutionized our knowledge of the double salts and of geological formations. One of the most beautiful applications of the phase rule is found in van't Hoff's investigations of the oceanic salt deposits at Stassfurt,^[23] in which from a laboratory study of the equilibrium obtaining between the sulphates and chlorides of sodium, calcium, magnesium and potassium the great chemist was able to reconstruct the past history of the formation of the earth's crust from the primeval ocean, giving even the limits of time, the pressure and the probable temperature at which the water evaporated. The importance of such methods in mineralogy and geology is self-evident and clearly as extensive as the subjects themselves. In metallurgy the work of Roozeboom, Le Chatellier, Sorby, Stead and others on steels, bronzes, tins, alloys and ingots of metal generally have, with the aid of the microscope, given us most valuable knowledge of the continuous chemical changes and "diseases of metals" going on in these substances, which, without the guidance of the phase rule, were formerly investigated in an aimless and haphazard way, at enormous expense and waste of time. Investigations of the very complicated equilibria in such "solid solutions"^[24] as carbon-steels, nickel-steels, cobalt-steels, etc., have explained the causes of brittleness and crystalline structure in steel-rails through extended use, and how such rails can be renewed by prolonged heating at high temperatures. "The variations of the engineering properties, such as tensile strength, torsional resistance, ductility, etc., with varying concentration and varying heat treatment, is a subject which can only be worked out satisfactorily with the phase rule as a guide," says Bancroft, and he adds, "we do not yet know one half the properties of our structural metals." The establishment of the true constitution of Portland cement is another telling application of the phase rule and it is thought that it will give us "clearer ideas as to the strength of cements and the elasticity of clays." Lord Kelvin expressed the hope that some day the architect might be in effect a sanitary engineer,^[25] and Bancroft predicts that "the time will come in our engineering schools when the subject known as 'materials of engineering' will have to be taught by the chemist rather than by the engineer."^[26]

In agricultural chemistry the phase rule serves as a guide in the investigation of soils. "The soil is the stomach of the plant," being in effect a complex system in three phases, of which the liquid phase furnishes the nutrient solution to the plant. The bacteria, molds and enzymes in the soil make its relation to the plant a complicated and difficult problem,^[27] but the application of physical chemistry to its solution of Cameron, Bell, Briggs and other chemists in the United States Department of Agriculture is clearly in the right direction. Recently Bancroft has shown how the phase rule may be applied to photochemistry when the radiant energy of absorbable light such as ultraviolet is converted into chemical energy. The light acts as another variable requiring n + 3 phases in an invariant system while in general we may have as many additional degrees of freedom as there are kinds of light.^[28] The application of the phase rule to organic chemistry is difficult owing to "passive resistance to change." Most of the reactions in organic chemistry are reversible, i. e., proceed to equilibrium, and if sufficient time be allowed will reverse backwards like a Carnot cycle, to some approximation of their initial state. Many reactions with organic substances, however, seem to stop short of equilibrium, and the chemist, in working with colloids, ferments, gums, etc., is balked by certain passive forces, which do not, like friction or viscosity, merely retard chemical change, but actually prevent it. Even such explanations as the hypothesis of reversibility in infinite time or Duhem's theory of false equilibria and pseudoreversible reactions, do not entirely account for these mysterious phenomena, and it is probably through new methods of laboratory procedure that organic chemistry will ultimately pass into the hands of the physical chemists.

In physiological chemistry the doctrine of phases opens out a new perspective, a new qualitative way of envisaging problems which, approached quantitatively, are a severe task even for an Emil Fischer. Recently the Dutch physiologist Zwaardemaker has proposed the application of the phase rule and the second law to general physiology.^[29] "The task of an energetic histology," he says, "would be to give the number of phases and their relations, while an energetic physiology would determine the equilibria and reversible processes by direct experimentation."^[30] Zwaardemaker proposes to regard the human body at rest as a complicated system of coexistent phases in equilibrium, the metabolic, reproductive and other processes of which are irreversible. The animal cell he holds to be a system of heterogeneous phases, the equilibrium of which can be disturbed by experimental removal of the nucleus. The red blood corpuscles are probably divariant, four component systems of four phases, while the endothelial cells of the ventricle of the heart are examples of a monovariant system. The "pressure phosphenes" of the retina, luminous sensations produced by pressure on the eyeball, and consequently "eye strain," may be due, Zwaardemaker thinks, to displacements of equilibrium through disturbance of the thermodynamic potential. These speculations are, of course, tentative, but there are indications that physiological problems may be attacked in a way that has some show of success in that it is qualitative.

The Law of Critical State.—When we have two contiguous phases of a substance, as a liquefying gas or a vaporizing liquid, there is a point where the two become continuous. This is called the critical state at which the distinction between coexistent phases vanishes. Gibbs's law asserts that a critical phase of independently variable components is capable of n − 1 independent variations. This theorem is the basis of the brilliant work of van der Waals, Duhem, van Laar and Kamerlingh Onnes upon continuous gaseous and liquid states.

Osmosis and the Theory of Solutions.—Gibbs's work is remarkable throughout for his avowed or explicit intention to have "nothing to do with any theory of molecular constitution" as leading to strained and unnatural hypotheses, and the wisdom of his decision is seen in his earlier treatment of the equilibrium of osmotic forces. He bases his theory of osmosis upon the idea of a semi-permeable diaphragm or membrane, which he introduced into physics as a purely theoretical concept, leaving the actual facts about it, he says, "to be determined by experiment." This ideal membrane, which he supposes permeable to one component and impermeable to others, is the key to the theory of solutions, for, to the mathematician, it admits of the condition of reversibility which he finds in algebraic or chemical equations; to the physicist, a solvent in the act of breaking up or wedging its way through a dissolving substance is the dynamic analogue of a liquid forcing its way into a denser liquid through the membrane, while to the chemist, the assumption that the membrane is selective for certain substances only implies some special chemical affinity between these substances and the membrane itself. If two fluids of different composition or concentration, say water and alcohol, are separated by a semi-permeable membrane, the osmotic flow of the water into the alcohol is due to definite forces. These, in Gibbs's argument, are, not a difference in pressure, but a difference in temperature which disturbs thermal equilibrium or a difference in the chemical potentials of components which can pass the diaphragm, the condition for equilibrium being equalized temperature and equality of chemical potentials. "Even when the diaphragm is permeable to all the components without restriction," he says, "equality of pressure in the two fluids is not always necessary for equilibrium."^[31] While Gibbs did not attempt a definite theory of solutions, it is clear that he regarded osmosis as a chemical or thermodynamic phenomenon. Let us see how his mathematical theory agrees with the facts of recent investigations. The mathematician Cayley thought double-entry bookkeeping an example of a perfect science, because its theory and practise are in complete agreement, so that the detection of sources of error becomes simply a matter of expert skill. For similar reasons one of the principal aims of modern physical chemistry has been to arrive at an adequate theory of solutions as a guide in chemical and biological research. Such a theory has been proposed by van't Hoff, who, starting from Pfeffer's measurements of osmotic pressure, bases his argument upon the widely known equation which asserts that osmotic pressure in very dilute solutions obeys the laws of Boyle, Gay Lussac and Avogadro with the same physical constants that obtain in mixtures of dilute or ideal gases.^[32] Pushing this analogy with gases farther, van't Hoff implicitly denied that there is any specific attraction between the solvent and solute (dissolved substance) or that the alleged semi-permeable membrane plays any active part in osmosis, holding that "osmotic pressure," like the pressure exerted by rarefied gases, is a real initial pressure caused by a bombardment of the membrane by the dissolved molecules. Now van't Hoff's equation, which Gibbs anticipated for dilute solutions of gases in liquids, and of which van't Hoff, Lord Rayleigh^[33] and Gibbs^[34] have each given rigorous thermodynamic proofs, was found to be true to the laboratory measurements for extremely diluted solutions of sugar and other substances, but (as Lord Kelvin said ten years ago) "wildly far from the truth" for solutions of acids, bases and salts.^[35] Arrhenius, in his theory of electrolytic dissociation,^[36] has explained these discrepancies as "harmonies not understood," due to free dissociation of ions in water and to increase of molecular conductivity with dilution; but Lord Kelvin's objection has still some force to this day. Two schools of chemists have thus arisen, one of which seeks to approximate the laboratory facts about solutions to van't Hoff's dynamic analogy with the gas laws, the other holding that osmosis is bound up with an ascertained selective action of the semi-permeable membrane, osmosis and solution being both due to "chemical affinity." Most prominent among those who have opposed the view that real solutions behave like ideal gases, are Louis Kahlenberg and J. J. van Laar. The special service of Kahlenberg has been to discredit the molecular or dynamic analogy between gases and liquids and to emphasize the point made by Fitzgerald in 1896, that "chemical forces are of a far more complex nature than electrolysis."^[37] Accepting the contention of the van't Hoff school that the gas equation and the Arrhenius theory are only true for infinite dilution, Kahlenberg has turned a clever flank movement upon them by insisting that if liquids act like gases we should expect a solution of increased concentration to behave at least qualitatively as gases do on increase of pressure. As a matter of fact, although practically all gases act alike, different solutions do not, as a rule, and solutions of solids in liquids, or liquids in liquids, do not behave like solutions of gases in liquids or gases in gases. Furthermore, the Arrhenius theory does not agree with many facts about aqueous solutions, while it falls completely to the ground for solvents other than water. This does not mean that Kahlenberg opposes electrolysis or electrolytic dissociation as such, or that he would have us abandon hypotheses of such value before we have found better ones, but he insists that "the question why certain solutions, molten salts, etc., conduct electricity and others do not will probably not be answered until we can tell why a stick of silver conducts electricity and a stick of sulphur does not.^[38] Morse and others have shown that the van't Hoff equation and the Arrhenius theory are true for very small dilutions, that is for solutions so mathematically ideal that they are practically independent of the nature of the solvent and the solute, but the experiences of Kahlenberg have shown that they are not always true for actual solutions of reasonable concentration. Moreover, the fact that the solute in tenth-normal solutions acts like a gas by no means explains all the phenomena of solution. Kahlenberg's experiments with semi-permeable membranes^[39] show that such membranes, while passive for gases, are active or selective for different liquids, so that the initial movement and actual direction of the osmotic current are determined by the specific nature of the membrane itself and of the liquids bathing it. Semi-permeable membranes, therefore, exist as such, and although none are strictly ideal in Gibbs's sense, their true "semi-permeable" or selective character is indicated by Kahlenberg's discovery that in some cases true measurements of osmotic pressure can not be obtained unless the solution is stirred to increase chemical action. The semi-permeable membrane shows that osmotic pressure is not an initial force, but a secondary hydrostatic pressure due "to the same affinity which produces adhesion, imbibition, absorption, adsorption, solution and chemical action." But all these forces are reducible in the simple reasoning of thermodynamics to the difference in temperature (Carnot's principle) and differences of chemical potentials which promote chemical change. As to molecular bombardment, "osmotic pressure," said Fitzgerald, in 1896, "is more nearly related to Laplace's internal pressure in a liquid which depends upon intramolecular forces, than to a gaseous pressure which is practically independent of the forces acting between the molecules."^[40] Van Laar pictures a sugar solution of reasonable concentration as made up of crooked movements of molecules, slowly crowding upon one another, with no intervening spaces, totally different from the rapid billiard ball movements with wide repulsions that are supposed to obtain in diluted gases. Osmosis, in van Laar's theory depends not upon the molecules of the dissolved substance, but upon the solvent itself, which, having the higher chemical potential, moves toward the solute. To explain the phenomena of osmosis by appealing to an initial osmotic pressure, says van Laar,^[41] is like saying that an angry man's loud talk and unseemly gestures are due to his red face.^[42] Anger is the real cause of both. So the movement in osmosis, which produces a difference in hydrostatic pressure, depends initially upon differences of chemical and thermodynamic potentials. Beyond this we know absolutely nothing of the interaction between the solvent and the solute. Again Bancroft has shown that the pressure for finite solutions in osmosis varies with the heat of dilution, which again varies with the specific nature of the solvent and the solute.^[43] All this brings us back to Gibbs's fundamental position that osmotic pressure "is a function of the temperature and the n potentials.^[44] From this point of view, Graham's original doctrine, that osmosis is the conversion of chemical affinity into mechanical power,^[45] is at once true to the mathematical theory and the laboratory facts. If now we agree with Whetham that "osmotic phenomena are intrenched in the strongest part of the vast lines occupied by the science of thermodynamics," it is clearly due to the early pioneer work of Gibbs that this vantage ground was gained in the first instance, while the molecular theory of osmosis remains in the debatable land of controversy and a true theory of solutions is still far to seek.

(To be continued)

1. Tr. Connect. Acad., III., 144-52.
2. "J'appelle cette fonction fonction caracteristique du corps: en effet, lorsqu'elle est connue, on peut en tirer toutes les propriétés du corps que l'on considère dans la thermodynamique. . . Je rapellerai d'ailleurs qu'une fois la fonction caracteristique d'un corps déterminée, la theorie thermodynamique de ce corps est faite." F. Massieu, Compt. rend. Acad. d. sc, Paris, 1869, LXIX., 859, 1058.
3. Bryan, "Thermodynamics," Leipzig, 1907, 109.
4. If $\epsilon ,\tau ,\eta ,\rho$ and $v$ represent the energy, temperature, entropy, pressure and volume of a chemical or thermodynamic system, its thermodynamic potentials will be the intrinsic energy $\epsilon$ obtained by integrating Gibbs's fundamental equation, the free energy $\psi =\epsilon -\tau \eta$, the total thermodynamic potential or "modified" available energy $\zeta =\epsilon -\tau \eta +\rho v$ and the heat function $\chi =\epsilon +\rho v$.
5. Larmor, "Encycl. Britan.," 10th ed., XXVIII., 167.
6. "Deshalb hat Gibbs, der auch diese Form der Darstellung zuerat fand, die Function A das isotherme Potential genannt." Helmholtz, "Vorles. über theoret. Physik," Leipzig, 1903, VI., 269. See, also, the lecture in his biography by L. Koenigsberger, Braunschweig, 1903, II., 369: "In diesem Sinne hat Herr Gibbs die Grösse F. das isotherme Potential des Körpersystems genannt, ich selbst habe dafür den Namen der freien Energie vorgeschlagen, well dieselbe Arbeitsäquivalente darstellt, deren Ueberführung in andere Formen der Energie nicht denselben Einschränkungen unterliegt wie die der Wärme." For Lord Kelvin's relation to the discovery of the free energy potential see Proc. Roy. Soc. Lond., 1908, LXXXI, No. A 543, pp. xlvi-xlvii.
7. "M. Massieu appears to have been the first to solve the problem of representing all the properties of a body of invariable composition which are concerned in reversible processes by means of a single function." Gibbs, Am. J. Sc., 1878, 3. s., XVI., foot-note to p. 445.
8. Gibbs, loc. cit., 113.
9. Ibid., 111-3.
10. See Gibbs, Am. J. Sc, 1878, 3. s., XVI., 442. "The transition from the systems considered in ordinary mechanics to thermodynamic systems is most naturally made by this formula. . . the mechanical properties of a thermodynamic system maintained at constant temperature being such as might be imagined to belong to a purely mechanical system, and admitting of representation by a force function."
11. Gibbs, Tr. Connect. Acad., III., 520.
12. Ibid., 152-6.
13. Nernst, "Lehrb. d. theoret. Chemie," 2. Aufl., 564. Wind, Ztschr. f. phys. Chem., 1899, XXXI., 390. Kuenen, Proc. Roy. Soc. Edinb., 1899-1901, XXIII., 317; J. Phys. Chem., 1899, III., 69. Le Chatellier, Rev. gén. d. sc., 1899, 759, Saurel, J. Phys. Chem., 1901, V., 31, 401. Trevor, ibid., 1902, VI., 185. Wegscheider, Ztschr. f. phys. Chem., 1903, XLIII., 93, 113. Raveau, Compt. rend. Acad. d. sc., 1904, CXXXVIII., 621. Mueller, ibid., 1908, CXLVI., 866.
14. Roozeboom, "Die heterogenen Gleichgewichte," 1901, I., 7.
15. Ztschr. f. Elektrochem., 1907, 94.
16. Gibbs, Tr. Connect. Acad., III., 176.
17. Roozeboom, Ztschr. f. phys. Chem., 1894, XV., 143; Arch, néerl, 1895-6, XXIX., 71. Bancroft, J. Phys. Chem., 1896-7, I., 403. In 1891, Sir G. Stokes suggested the graphic representation of physical states of ternary alloys by means of an equilateral triangle which he derived independently from Maxwell's color diagram. (Proc. Roy. Soc. Lond., 1891, XLIX., 174.)
18. See Professor Bancroft's Journal of Physical Chemistry (passim), from which most of the results in this section are taken.
19. Roozeboom, Ztschr. f. phys. Chem., 1892, X., 477.
20. Van der Heide, ibid., 1893, XII., 416.
21. Cameron, J. Phys. Chem., 1907, XI., 641.
22. Tammann, "Kristallisieren und Schmelzen," Leipzig, 1903.
23. 78 Van't Hoff, "Lectures on Physical Chemistry," Chicago, 1907.
24. Solid solutions are solids dissolved in solids, and were first described by van't Hoff (Ztschr. f. phys. Chem., 1890, V., 322), who found that when certain solutions are frozen, the separated solid is not the pure solvent, but a mixture of the selvent and the solute, i. e., a solid solution, of which we have examples in the alums, glasses, colored and hyaline minerals, alloys and the "ice flowers" of the Antarctic regions.
25. Kelvin, "Popular Lectures," 1894, II., 210.
26. Bancroft, J. Phys. Chem., 1905, IX., 209.
27. Cameron, ibid., 1904, VIII., 642.
28. Bancroft, ibid., 1906, X., 721.
29. Zwaardemaker, "Ergebnisse d. Physiol.," Wiesbaden, 1906, V., 108.
30. Loc. cit., 154.
31. See his abstract in Am. J. Sc, 1878, 3. a., XVI.
32. Van't Hoff, Ztschr. f. phys. Chem., 1887, I., 481.
33. Nature, London, 1896-7, LV., 253.
34. Ibid., 461.
35. Ibid., 273.
36. Ztschr. f. phys. Chem., 1887, I., 631.
37. "That other than purely electrical forces are operative in solution is indicated by Helmholtz's investigations of electrical diffusion through fine tubes." Fitzgerald, Helmholtz lecture, Nature, 1895-6, LIII., 297.
38. Kahlenberg, Phil. Mag., 1905, 6. s., IX., 229.
39. Kahlenberg, J. Phys. Chem., 1896, X., 141-209. Recently Tammann has advanced the view that in ideally diluted solutions the solute acts like a gas, while in concentrated solutions there is a chemical interaction between the solvent and the solute, and such solutions behave more like the solvent under higher pressure. (Tammann, "Ueber die Beziehungen zwischen den inneren Kräften und Eigenschaften der Lösungen," Leipzig, 1903.)
40. Fitzgerald, Nature, London, 1895-6, LIII., 297.
41. Van Laar, "Sechs Vorträge über das thermodynamische Potential," Braunschweig, 1906, 3.
42. Ibid., 34.
43. Bancroft, J. Phys. Chem., 1906, X., 319-29.
44. Gibbs, loc. cit., 139.
45. Graham, Phil. Tr., 1854, 227.