Continuity of Chemical Transformations of Matter

N. S. Kurnakov.

The first decades of the twentieth century are characterized by profound changes in views on the fundamental concepts of the structure of matter. The enormous quantity of factual material accumulated by new methods of research in physics and chemistry compels us to take a critical attitude toward conceptions that had seemed to be established quite unshakably.

But under such conditions, in order properly to evaluate the achievements of the present, it is useful to look back also at the history of the past.

On November 6, 1922, one hundred years had passed since the death of the famous founder of chemical mechanics and of the doctrine of equilibria, Claude-Louis Berthollet (1748–1822). It is very interesting to trace the development of Berthollet’s ideas, which were of such enormous importance for the study of chemical transformations in general. It must undoubtedly be acknowledged that a large amount of factual data for judging the essence of chemical equilibria was obtained by him in the course of investigations of various technical questions—for example, the extraction of saltpeter from soil and the formation of soda in the natron lakes of Egypt, which he studied as a participant in Bonaparte’s brilliant but ephemeral expedition to the land of the pharaohs. The origin of soda in nature gave Berthollet a vivid confirmation of his fundamental idea that exchange decomposition, in this case of carbonate salts with sodium chloride or sulfate, is the result of a mobile equilibrium of two reciprocal reactions proceeding in two opposite directions1.

From the idea of mobile equilibrium there follow two fundamental propositions of chemical mechanics: 1) the existence of a limit of exchange decomposition, or equilibrium between two reversible transformations, and 2) the law of action of the masses of substances reacting with one another.

In his views Berthollet was far ahead of his time; the scientific ground had not yet been sufficiently prepared. Some of his propositions received their further development only many decades later. Thus, for example, the mathematical formulation of the law of mass action was given only in 1867, in the classical work of the Norwegian scientists Guldberg and Waage.

All the more must we marvel at Berthollet’s brilliant foresight: how could this physician by profession, who had received no special mathematical education, penetrate so deeply into the essence and properties of the equilibria of mechanics and transfer them to chemical systems.

Of course, in this direction there operated above all the inspired atmosphere of Lavoisier’s laboratory (in the Arsenal), in which the young Berthollet found himself from the first years of his stay in Paris. Here the flower of the learned world of France and of all Europe gathered; here Berthollet met two mathematicians of genius—Monge and Laplace—who later became his closest friends and comrades in scientific and public work.

The creator of descriptive geometry—Gaspard Monge—himself pursued chemistry and its applications rather intensively. This is attested by a number of his works, for example, an independent experimental investigation of the decomposition of water into its constituent parts (1783), and the study of methods for extracting saltpeter for the production of gunpowder. Together with Berthollet he discovered fulminating silver and published an instruction for metalworkers on the preparation of steel (Avis aux ouvriers en fer sur la fabrication de l’acier, 1794).

Monge, together with Lavoisier, Guyton de Morveau, Berthollet, and Fourcroy, was a member of the editorial board of the first chemical journal, Annales de Chimie, which was the ancestor of the well-known Annales de Chimie et de Physique1.

Berthollet’s friendship with Monge began as early as 1780, at the time of their election to the Academy, and continued, despite the terrible trials of that turbulent epoch, for more than a third of a century. Napoleon entrusted Monge and Berthollet with the organization of the scientific part of his expe-

sur les lois de l’affinité, par le citoyen Berthollet, an IX (1801); cf. the German translation in Ostwald’s Klassiker d. exakten Wissenschaften (1896), No. 74.

Berthollet’s data concerning the formation of soda in nature were confirmed in subsequent observations by Hilgard (Ber. d. Deutsch. Chem. Ges. 25, 3226), Melikov (Zhurn. R. Khim. Obshch. 28, 307, 551) and Tanatar (ibid. 28, 327, 376).

...expedition to Egypt. In this campaign the names of the two scholars were constantly associated with one another; Monge–Berthollet seemed to designate one and the same person. Arago, in his biography of Monge¹), reports that the soldiers who took part in the Egyptian expedition argued heatedly among themselves: some, who had seen Berthollet, said that Monge–Berthollet had fair, flowing hair, while others—who knew only Monge—asserted with no less certainty that Monge–Berthollet was a strong brunette and wore a wig with a long queue.

No less close relations existed between Berthollet and Laplace. In Duhem’s excellent essay “Une science nouvelle — la chimie physique”²), we find the following picture of the scientific movement at the beginning of the last century: “It was a time when the new chemistry, which had grown out of Lavoisier’s experiments, was spreading everywhere its fundamental ideas and terminology, while the system of molecular forces—attractive and repulsive—had as its lawgiver Laplace, who applied exact mathematical methods in this field. Between Lavoisier’s chemistry and Laplace’s physical mechanics there arose a fruitful rapprochement. This combination of two different doctrines was the work of Berthollet. At Arcueil near Paris, in a laboratory situated in a garden next to Laplace’s house, Berthollet continued his investigations on affinity. Under the influence of, and in collaboration with, Laplace, Berthollet wrote his Essay on Chemical Statics, while Laplace, in turn, when preparing the Exposition of the System of the World, made extensive use of Berthollet’s work.”

It should be noted that, likewise at Arcueil, every two weeks there were meetings of the first Physico-Chemical Society, the Société d’Arcueil, which numbered among its members Laplace, Berthollet, Biot, Gay-Lussac, Thenard, de Candolle, and other scholars who enjoyed wide renown. During the decade of its existence (1807–1817) this society published 3 volumes of original memoirs.

Such collaboration of chemists with geometers must be regarded as highly remarkable. In this way a favorable environment was created for the penetration of mathematical and physico-mechanical methods of thought into the chemical domain.

Berthollet’s mobile equilibrium is nothing other than the application of the geometrical idea of continuity to chemical transformations. The continuity of changes of geometrical figures characterizes the most general transformations of space, which are the subject of the geometry of position (Analysis situs, or topology).

¹) François Arago: Œuvres complètes, t. II, p. 532.
²) P. Duhem, Revue philomatique de Bordeaux et au Sud-Ouest, 2 année, No. 5–6 (1899).

The introduction of the idea of continuity (continuum) into science goes back to Leibniz (1684–1686). Monge and his school had it in mind in geometrical investigations, but the conscious and systematic application of the principle of continuity (principe de continuité) was carried out with great success by Monge’s pupil—the famous Poncelet—in his fundamental work on the projective properties of figures1. The essence of this principle, as conceived by Poncelet, consists in the fact that, under a continuous change of the parameters determining transformations of geometrical figures, there may exist properties of them that remain constant and real even in those cases when the figures themselves lose their real form and become imaginary. A characteristic example may be the common chord of intersection of two circles, or their so-called radical axis, which remains a real straight line for all real values of the parameters in the equations of the circles, even when these circles become imaginary.

In application to chemical equilibria the principle of continuity may be formulated as follows. In the functions determining the states of an equilibrium system, there exist continuous relations between the variable values of the various factors of equilibrium—temperature, pressure, concentration of the substances or components making up the system—and the measurable properties of homogeneous bodies or phases participating in the equilibrium: electrical conductivity, elasticity of vapors, etc. Thus, for example, successive changes in the composition of liquid and solid solutions correspond to continuous changes in their properties—specific gravity, electrical conductivity, etc.

Berthollet did not in essence make any distinction between physical and chemical processes; proceeding from general views on equilibrium, he transferred the concept of continuity also to chemical transformations of matter and asserted that the relations in which bodies enter into chemical compounds are not constant, but change together with the conditions determining the act of interaction.

In support of his view Berthollet cited the existence of homogeneous liquid solutions, glasses, slags, mineral compounds, etc.

Against these logical conclusions of the theory there rose the well-known French scholar Proust (J. L. Proust), who, on the basis of precise analytical data, proved that the weights of the constituent parts forming a compound are in a strictly constant ratio, independent of the conditions of interaction of the bodies. Proust regarded this feature as a characteristic property of true chemical compounds (combinaisons réelles). In the history of chemistry it is commonly considered that the dispute between the two—

...its opponents ended in the victory of Proust. In science the law of constant composition was established, predetermining the main direction of chemists’ work for almost an entire century. Chemistry separated from physics, turned to the study of discontinuous (discrete) transformations of matter, and set out on an independent path, where it achieved those brilliant results in the field of atomic theory that characterize its present state.

But there is no doubt that Proust’s victory was only temporary. At the beginning of the last century the methods for studying substances of variable composition had not yet been developed. Now, a century later, we are engaged in solving the very questions that troubled the contemporaries of Proust and Berthollet, but armed with improved methods of theoretical and experimental investigation. A new branch of general chemistry—physicochemical analysis—gives us the possibility of systematically studying those regions that had already been indicated by Berthollet, but for a long time remained wholly inaccessible to the ordinary methods of chemical observation.

The basic task of physicochemical analysis is the measurement of properties under a successive change in the composition of an equilibrium system, the result of which is the graphical construction of a “composition–property” diagram. Thus there arises a geometrical method for investigating chemical transformations. We obtain an exact geometrical model of the complex function that must represent the dependence among temperature, volume, concentration, and other physical and chemical factors determining the state of the system.

The “composition–property” diagram is a closed complex of surfaces, lines, and points. By this name, in the geometry of position (Analysis situs) or topology, one denotes figures that bound part of space or of a plane. Topology has as its subject the consideration of geometrical figures independently of the form of curves or surfaces. It seeks out such relations of these figures as remain unchanged, or invariant, under the most general continuous transformations of space.

In exactly the same way, physicochemical analysis, which may be called topological chemistry, makes no distinction between physical and chemical changes of states and establishes those properties of the elements of the geometrical complex of the diagram to which there correspond constant, or invariant, relations of composition under all transformations of the given system.

Despite the fact that chemistry in the nineteenth century seemed to have turned into a science of ruptures in the continuity of matter, many investigators of chemical equilibria instinctively made use of Berthollet’s idea of continuity. With complete definiteness Academician D. P. Konovalov expressed himself in this direction in his speech

“On Chemical Affinity,” delivered at the general assembly of the Tenth Congress of Russian Naturalists and Physicians in 18981.

He says: “The act of chemical transformation is caused by interactions subject to the law of continuity. We inevitably fall into contradiction if, in our conceptions of affinity, we confine ourselves to the realm of fixed proportions... Undoubtedly, the inherited boundaries of chemistry, which enclosed it within the domain of constant quantities, prove to be narrow.”

At the time when these words were spoken, new methods of investigation had already arisen and were being perfected, and these made it possible to broaden considerably the limits of our chemical horizon.

The methods of physicochemical analysis, although belonging to the most recent period, were in individual cases applied in deepest antiquity. Thus, the first application of one of the methods of physicochemical analysis—namely, the measurement of the specific gravity of a gold–silver system—was made by the great Greek geometer Archimedes in the third century before our era, in order to solve the well-known problem of the composition of the crown of the Syracusan king Hiero the Younger. At the present time we may say that Archimedes’ correct solution of this problem is based on the existence of a linear dependence between the specific volumes and the composition of a continuous series of solid solutions, or isomorphous mixtures, of which the solidified alloys of gold and silver consist. It should be noted that an exact determination of the indicated linear dependence, foreseen by the genius of Archimedes, was made by Matthiessen only at the end of the fifties of the last century.

Nevertheless, the modern period in the development of physicochemical analysis begins only in 1873–1878, when Gibbs’s classical memoirs (W. Gibbs) on the equilibria of chemical systems were published in the Proceedings of the Connecticut Academy (in North America).

Here, for the first time, new concepts of phases and components were established, which exerted a great influence on the study of chemical equilibria. These concepts brought simplicity and unity into the complex field of chemical transformations and served as the basis for a rational classification of equilibrium systems. Gibbs’s ideas for a long time found no application, until Van der Waals and Roozeboom, together with their students, demonstrated their fruitfulness in a whole series of experimentally developed examples.

But all these abstract investigations would have remained the property of a few physicochemical laboratories, had not the needs of metallurgical technology compelled broader circles to turn their attention to the young scientific discipline.

The modern doctrine of metallic alloys was born in the atmosphere of steelmaking plants—the Zlatoust, Obukhov, and Creusot works—thanks to the labors of Anosov, Chernov, and Osmond.

“If Sorby was the pioneer of metallography, and Chernov its father, then Osmond carried its torch, because he, more than anyone else, was our guiding star,” writes the American Sauveur in the preface to his course The Metallography of Iron and Steel.

To this characterization it should be added that the first investigator to use, as early as 1831, the microscope for studying the structure of polished and acid-etched steel was the Russian mining engineer Pavel Petrovich Anosov, head of the Zlatoust works in the Urals. According to the researches of N. T. Belyaev1, this historical fact must be regarded as fully established. P. P. Anosov was the first to employ that consecutive combination of procedures which at the present time bears the name of the micrographic method and forms the basis of metallography. The use of the microscope by Anosov at the Zlatoust works for determining the properties of damask-steel blades was undertaken more than 30 years before the Englishman Sorby, who in 1863 began the systematic study of sections of rocks and metals in transmitted and reflected light by means of the microscope.

Although with great delay, we are nevertheless obliged to render due honor to the works of the Russian pioneer in the field of metallic micrography.

Another of our compatriots, Professor D. K. Chernov, in 1868 pointed out the existence of characteristic temperature points at which mutual transformations of the structural elements of carbon iron take place; and the French engineer Osmond applied Le Chatelier’s thermoelectric pyrometer for their accurate measurement. Thus the beginning was laid for thermal analysis, which received further development in the works of van ’t Hoff, Roozeboom, Le Chatelier, Roberts-Austen, Heycock with Neville, Howe, Tammann, and other investigators.

The combination of the microscope with the pyrometer proved, in Osmond’s hands, extremely fruitful and quickly opened for metallography a broad field of application. The melting-point (solubility) curves of thermal analysis, indicating the relation between the temperature \((t)\) and the composition \((x)\) of a system, became the typical model of “composition–property” diagrams.

Beginning in 1906, in the chemical laboratories of the Mining and Polytechnic Institutes in Petersburg, a number of new properties were successively introduced into the circle of observations—electrical conductivity, hardness,

viscosity—which made it possible to establish extraordinarily subtle differences in the transformations of substances of the most diverse origin.

We now have more than twenty measurable properties that can serve for the corresponding geometrical constructions. These diverse properties may be divided, according to their mutual relations, into several categories indicating the principal branches of the methodology of physico-chemical analysis: thermal, electrical, volumetric analyses, determination of viscosity, elasticity of vapors, etc.

From the materials of a special field of metallography there gradually arose the enormous edifice of physico-chemical analysis. With a large quantity of numerical material, indispensable for constructing a “composition–property” diagram, systematic observations often become beyond the capacity of a single person and require the participation of several experimenters working according to a definite plan. The need to pass over to collective work became more and more perceptible.

It is therefore understandable that, in various countries, special institutions began to be established with the aim of applying the new methodology in particular branches of knowledge; among them one should mention the Alloys Research Committee in England, the Carnegie Geophysical Laboratory in Washington (North America), and the Union for the Scientific Study of German Potash-Salt Deposits. In our country, on the initiative of the Commission for the Study of Natural Productive Forces under the Academy of Sciences (KEPS), the Karabugaz Committee arose in 1916, subsequently merging with the Institute of Physico-Chemical Analysis.

For judging the form of chemical diagrams, Figs. 1 and 2 may serve, depicting Monge’s horizontal projections for spatial complexes belonging to various kinds of equilibrium.

In Fig. 1 there is presented the melting diagram of a ternary system in which the components are three of the most widespread substances in nature: lime, alumina, and silica. This major work by staff members of the Geophysical Laboratory in Washington—Shepherd, Rankin, and Wright1—required many years of persistent labor; here all the latest improvements in the determination of high temperatures and in the optical characterization of mineral substances were applied. To compile the diagram it was necessary to measure more than 500 points and to make up to 5,000 separate thermal and optical determinations.

The components, or constituent substances, of the system are placed at the vertices of an equilateral triangle, proposed by Gibbs for such representations. The three sides of the triangle correspond to the binary systems formed by combinations of the constituent substances taken two at a time.

Points inside the triangle belong to ternary melts, and the content of the components is determined by the length of three lines drawn from the given point parallel to the sides of the triangle. Their sum is a constant quantity; taking the latter as 100, one obtains a geometric representation of the coordinates of the percentage composition of the ternary system.

Fig. 1.

To each substance crystallizing upon cooling of the molten mass there corresponds a definite field of crystallization, bounded by lines that intersect at points where the joint separation of the corresponding number of solid phases takes place. As many as fifteen such fields have been found in the system under study. Many of them belong to minerals occurring in nature—anorthite, sillimanite, corundum, and various varieties of silica.

A mere glance at this diagram already indicates its resemblance to a geographical map. Indeed, to the latitudes and longitudes of points on the earth’s surface there correspond, on the chemical map, the coordinates of composition; to the elevations of a geographical survey—the temperatures of equilibrium systems. Topographical contours, or lines of equal elevation, find their analogues in the isotherms of chemical leveling, represented in Fig. 1 by thin curves. The analogy in the final results of measurement is quite complete, but how different are the instruments of geodetic and chemical surveys! The role of the theodolite is performed by the pyrometer, and that of the level by the thermostat.

Of great interest are the relief forms of the chemical surface. First of all, we note the summits of mountains and plateaus; these highest or singular points, marked by black circles, characterize definite compounds that obey the law of multiple proportions and melt without decomposition. Among such substances is one of the most important feldspar minerals—anorthite \((\mathrm{CaO}.\mathrm{Al}_2\mathrm{O}_3 \cdot 2\mathrm{SiO}_2)\), which melts without decomposition at \(1550^\circ\). If one takes into account the amount of energy and labor involved, then the attainment of chemical summits is perhaps no less complex and difficult a task than the ascent of the highest mountains of the earth’s surface.

The depressions, or basins, of the chemical country are of enormous significance. Here, at the intersection of three boundary lines of fields, are located the so-called eutectic points, e.g. Nos. 21, 23, 25, belonging to ternary alloys with the lowest melting temperatures. Such bodies form the principal mass of slags obtained in the smelting of pig iron from iron ores in blast furnaces, annually in the amount of several billions of poods. The determination of the composition of the most fusible slags, in order to reduce the expenditure of heat energy in the blast-furnace process, has long constituted an urgent problem of metallurgy. The diagram of the Geophysical Laboratory gives us perfectly precise indications in this respect.

The zone of the lowest melting temperatures approximately corresponds to the line connecting the binary eutectic \(A\) \((1436^\circ)\) with the spinel eutectic \(P\) (m. p. \(1335^\circ\)). The ternary fusible eutectic points, marked in Fig. 1 by the numbers 21 and 23, are very important. The first of them, with a melting temperature of \(1165^\circ\), determines the joint separation of three minerals: tridymite, anorthite, and pseudowollastonite. It belongs to the acidic slags of charcoal blast furnaces.

The second point (No. 23), characterized by the simultaneous crystallization of pseudowollastonite \((\alpha\text{-}\mathrm{CaSiO}_3)\), akermanite and \((3\mathrm{CaO}.2\mathrm{SiO}_2)\) and gehlenite \((2\mathrm{CaO}.\mathrm{Al}_2\mathrm{O}_3.\mathrm{SiO}_2)\) silicates, has a higher melting temperature—\(1310^\circ\)—and is characteristic of the basic blast-furnace slags of coke-fired furnaces. The position of point No. 23 agrees with the data

industrial practice and with the determinations of the well-known Swedish metallurgist Ockerman, made by an entirely different method.

A considerable amount of time and labor was spent on a detailed study of the structure of the region adjoining the crystallization fields of calcium orthosilicate (\(\mathrm{Ca_2SiO_4}\)) and free lime (\(\mathrm{CaO}\)). The results obtained here clarify to a significant degree the methods of manufacture of such an important building material as Portland cement. There is no doubt that Washington’s diagram will for a long time remain a most valuable aid for chemists, mineralogists, and metallurgists in considering various processes of nature and technology in which calcium and aluminum silicates take part.

Another drawing (Fig. 2) concerns reactions in aqueous solutions. This diagram is one of the links in the chain of investigations undertaken by our Commission of Productive Forces (KEPS) for the study of salt lakes and limans scattered over the enormous territory of the Aral-Caspian and Black Sea basins.

The well-known work of van ’t Hoff and his collaborators on the study of the Stassfurt deposits had already paved the way for determining complex salt equilibria. For the understanding of salt lakes it was necessary to carry out a systematic investigation of the separation of solid salts formed in the exchange decomposition of the reciprocal system: sodium chloride–magnesium sulfate. Such observations were carried out with the participation of S. F. Zhemchuzhnyi in the chemical laboratories of the Polytechnic Institute and the Academy of Sciences.

Fig. 2 depicts the results of our determinations of solubility at \(25^\circ\) by means of the so-called four-axis diagram, which represents the horizontal projection (according to Monge’s system) of the lower half of a regular octahedron placed on its apex. Along three mutually perpendicular lines (dashed in the drawing), which are the horizontal projections of the edges of the octahedron, there are laid off from the zero point the contents of four salts—sodium chloride and sulfate, magnesium chloride and sulfate—which constitute the reciprocal system. To determine the composition of a given solution, it is sufficient to plot the contents of three salts; the amount of the fourth is found as a function of the first three. The adopted scale of concentrations expresses the number of molecules of salt per 1000 molecules of water in the solution.

In the present case the isotherm at \(25^\circ\) (Fig. 2) represents a closed topological complex bounded by 11 surfaces, of which seven belong to fields of equilibrium indicating the limits of crystallization of the corresponding number of salts. All these substances are deposited during the evaporation of sea water and are found in Russian salt lakes. The greatest importance belongs, of course, to the field of crystallization of the most typical and widespread salt-like substance—sodium chloride, or common salt. The evaporation of each

of the brine lying in the region of this field there corresponds a special path of crystallization, indicated by a ray that issues from the pole of the field. Thus, for example, the boundary point XVI is so called, representing the composition of a solution of pure sodium chloride saturated at 25°.

Fig. 2.

The diagram in Fig. 2 shows a bundle of such rays belonging to four brines characteristic of Russia1. Thus we are able to determine precisely the structure of the sodium-chloride field and to predict the phenomena connected with obtaining this substance by evaporation of seawater and of the brines of various lakes.

The chemical expeditions sent in 1916–1917 by the Academy of Sciences to the Crimean lakes yielded material entirely consistent with the study of the equilibrium diagram. For example, the characteristic stages of the brines of the Perekop magnesium lakes are well expressed by the boundary line (II—XVI) of the contour of the sodium chloride field.

For the bitter lakes of the Aral-Caspian basin, the regions of crystallization of decahydrate and anhydrous sodium sulfate salt are highly characteristic. A typical representative of such formations is the Kara-Bogaz Gulf, which is undoubtedly the greatest deposit of Glauber’s salt in the world. The forces of Russian investigators and industrialists must be devoted to the study and utilization of this natural wealth. If the surface brine of Kara-Bogaz is placed on the solubility diagram (Fig. 2), it proves to be unsaturated both with respect to sodium chloride and to sodium sulfate. From the geometric construction it follows that, under isothermal evaporation, the composition of such a brine enters the field of common salt, the deposition of which we do in fact observe in many self-sedimented lakes that were once connected with Kara-Bogaz but subsequently became separated from it. In the gulf itself, owing to the constant inflow of water, the brine never separates out sodium chloride; but when the temperature is lowered to 0°, the equilibrium shifts into the field of crystallization of Glauber’s salt.

Laboratory measurements showed that the precipitation of the latter substance from the upper layers of the Kara-Bogaz brine begins to occur at about 5.5°. It must be acknowledged with deep satisfaction that these conclusions from the laboratory experiment received full confirmation in work carried out in the gulf itself. Last year, the Academy of Sciences and the Main Mining Administration organized the Kara-Bogaz expedition, under the direction of N. I. Podkopaev, for the purpose of making a year’s series of hydrometeorological and chemical observations. The measurements showed, for the beginning of the separation and dissolution of Glauber’s salt, a water temperature in the gulf of 5.4–6.0° (20 November 1921–10 March 1922), differing very little from the value observed in the laboratory.

Thus it becomes possible to depict the general picture of salt transformations, to determine precisely the conditions for the crystallization of the various salts, and to find the limits of their stable state. The diagram in Fig. 2 indicates the correct path toward understanding the genesis of salt deposits in nature and provides technology with a reliable means for separating individual substances in the pure state.

Without the graphical constructions of descriptive geometry, the study of chemical equilibrium systems, especially with a large number of components, becomes impossible. One may say that the shades of Monge and Berthelot are invisibly present in the modern work of chemists and encourage them toward further progress.

Interest in the equilibria of the reciprocal system sodium chloride–magnesium sulfate is indicated by the following circumstance as well. In recent years (in the interval 1915–1921), a period of interruption in scientific relations, the named system was subjected to entirely independent experimental study, besides in the USSR1, in three other places on the globe: by d’Ans in Germany2, by Blasdale in North America3, and by Takegami in Japan4. As far as can be judged from the brief abstracts, the results obtained are in general mutually similar, but there are also discrepancies. This depends, in all probability, on the retardations in transformations characteristic of the sulfate compounds of magnesium. The near future must decide whose results will prove more accurate.

The examples cited give an idea of the applications of physicochemical analysis. The number of these applications in various branches of natural science and technology is very great and increases rapidly with time.

At first glance it may appear that the new discipline has chiefly an applied, practical character. But its methods penetrate far beyond the usual techniques of chemical investigations and open up unknown, virgin regions. Here new concepts are created and paths are prepared for the solution of fundamental questions concerning the transformations of matter.

Among such important scientific acquisitions undoubtedly belongs the concept of a solid solution, as a homogeneous solid complex of variable composition, introduced by van ’t Hoff in the nineties of the last century. The name itself already indicates the analogy between the phenomena of solution in the solid and liquid states. We must conclude that the capacity for mutual solubility is a general property of solid crystalline bodies, and is not limited, as was formerly thought, to substances of analogous composition.

As early as 1884, Mallard asserted that the crystal lattice of all bodies is cubic or close to cubic, and according to Fedorov all crystals can be reduced to two limiting types—pseudohexagonal or pseudotetragonal. Therefore, the possibility of mutual alteration and subordination of lattices in the act of solution

determines the joint crystallization of substances of unlike chemical nature.

In essence, solid phases with variable composition have already been known for a long time. The names “Bronze Age,” “age of iron and steel” correspond to solid solutions of tin in copper, and of nickel and carbon in iron. But experimental proofs of the wide occurrence of the phenomena of mutual solubility in the solid state belong to recent times and are closely connected with the application of sensitive methods of metallography and physicochemical analysis—electrical conductivity, hardness, cooling curves, and so on.

Solid substances of variable composition are especially characteristic of metallic alloys, silicates, and many mineral and organic bodies. These include Vernadsky’s dissociation systems, Fersman’s mutabile compounds, zeolites, and crystalline proteins. If the conditions of the concept of homogeneity are broadened by introducing various degrees of subdivision of a substance, then molecularly dispersed solid solutions gradually pass into the immense region of more coarsely dispersed colloids of various origins.

Experience establishes with complete clarity that many definite compounds of metals, for example the argentides and cadmides of magnesium,

Fig. 3.          Fig. 4.

aurides of zinc and cadmium, are capable of forming solid solutions with an excess of their components. The composition of the solid phase of such substances changes within broad limits while the complete homogeneity of the substance is preserved, but on the “composition–property” diagram of the given phase there appears a special or singular point, which lies at the intersection of the separate branches of the diagram shown in Figs. 3 and 4. The composition of the singular point remains unchanged, or invariant, for all properties and serves as the characteristic of a definite compound subject to Dalton’s law of multiple proportions.

The variability of the composition of solid solutions as a function of temperature and other equilibrium factors gives us full confirmation

correctness of Berthollet’s views on chemical transformations. Diagrams of physico-chemical analysis establish beyond doubt the existence of a close connection between changes of position, or geometrical transformations of space, on the one hand, and changes of state or chemical transformations of a substance, on the other. At the same time the principle of continuity is revived once more in chemistry and becomes our faithful guide.

It makes it possible to find connections and mutual transitions in those cases where apparent breaks are observed in the continuity of a geometrical diagram.

As shown in Fig. 5, two separate branches ($a$ and $b$) prove to belong to one and the same fusibility curve ($t$—temperature, $X$—composition), even under those conditions when these branches are connected by means of a (dotted) segment that cannot in fact be realized. Such serpentine segments are characteristic of the formation of surfaces of separation in a homogeneous medium, for example, when a liquid separates into two layers (Fig. 5).

Fig. 5.

By the classical works of Van der Waals and his school, the continuity of the gaseous and liquid states has been established for systems composed of one or two components. Thanks to this, a whole series of such phenomena as the mutual solubility of liquids are brought under one common point of view and become fully intelligible.

At the present time we can go further. On the one hand, Poynting, Planck, Ostwald, Kamerlingh-Onnes, and other investigators have made probable the existence of relations of continuity in the transition of a liquid homogeneous body into a crystalline one and back again, i.e. for the processes of melting and crystallization. On the other hand, the experimentally demonstrated analogy between the properties of liquid and solid solutions creates the possibility of applying the principle of continuity to the chemical transformations of all substances, irrespective of their physical state. The variable composition of a solid phase, owing to the existence of solid solutions, appears merely as a consequence of the principle just named.

It should be noted that solid bodies of constant composition, used in chemical analysis, are considered the chief support of the law of multiple proportions and of atomistic views. Therefore the variability of composition, as a general property of solid homogeneous substances, involuntarily

caused anxiety for the accuracy and for the fate of the fundamental law of chemistry, with all the consequences following from this. The idea of continuity seemed incompatible with the views of Proust–Dalton. But these fears do not correspond to the actual state of things. At the present time, the totality of the data of physicochemical analysis makes it possible to assert with complete confidence that both sides are right in their assertions, but that Berthollet’s point of view is the more general one.

We must regard solutions and substances of variable composition, or solvates, as the fundamental type of chemical transformations. Strange as it may seem at first glance, it is precisely the principle of continuity that is henceforth destined to defend the inviolability of the law of constant composition and to give an exact geometrical characterization of discontinuities in the formation of definite chemical compounds. Indeed, it is not the composition of a solid substance that characterizes a definite compound, since it is in general variable, but rather the constant composition of a singular or invariant point on the diagrams of the properties of a solid substance. For this reason the properties of singular points acquire especially important significance. At such points the homogeneity of a solid or liquid substance undergoes no violations whatever, but on the “composition–property” diagram there is observed the characteristic intersection of two branches at an angle, as is seen in Figs. 3 and 4.

Proceeding from the parallelism between chemical transformations and geometrical transformations, we may admit, at singular points, the existence of an unstable material node (Fig. 3), characteristic of algebraic curves of the 3rd and higher orders, or of an isolated node (point anguleux of the French geometers), which is encountered in exponential and logarithmic curves. Such a nodal point is presented in Fig. 4. Under such assumptions both branches prove to belong to one and the same continuous curve, which characterizes the given solid or liquid phase.

A curious type of singular points has recently been established in the chemical laboratory of the Academy of Sciences by the investigations of T. F. Genke, on the solubility isotherms at 25° and 80° of sodium chloride in the ternary system: hydrochloric acid—sodium oxide—water. The singular isotherm shown in Fig. 6 consists of two intersecting branches which undoubtedly belong to one curve. A series of such curves, corresponding to different temperatures, gives us for the first time a conception of a singular fold in the chemical space of the diagram—a fold accompanying the formation of a typical undissociated salt in the reaction of neutralization of an acid by a base.

In analytic geometry it is proved that a necessary condition for finding a singular point is the equality to zero of the discriminant, or of a function composed of the coefficients

equation of the curve. This function remains unchanged under geometric transformations and, therefore, is also called an invariant of the curve. Thus singular points, or geometric invariants, characterize the composition of certain compounds which, in turn, represent chemical invariants, since they remain unchanged during transformations of an equilibrium system.

Fig. 6.

The correspondence between geometric and chemical invariants must be considered remarkable; it lies at the foundation of the chemical diagram and determines its entire structure. Under such conditions, matter and space, one may say, mutually penetrate one another; they seem inseparable.

Among continuously proceeding transformations of the type of solutions, the process of formation of definite compounds is the prototype of the creation of new, immutable forms of matter. At the sight of the singular nodes and folds of the chemical diagram, one involuntarily recalls the profound thought expressed by the well-known English mathematician and philosopher Clifford1: “matter is a fold in our space.” Apparently, Plato and Descartes were right in identifying both fundamental concepts. This question belongs to the theory of knowledge and awaits its consideration. Chemistry here moves toward the grand evolution of modern ideas about matter and space, and the decisive word will belong to it.

The friendship of Monge and Berthollet may be regarded as symbolic. In their persons geometry and chemistry entered into an alliance which proved fruitful and, with the passage of time, grows ever stronger and develops. It not only helps us in the struggle to master nature, but also indicates new paths for resolving the fundamental problems concerning the nature of matter.

1. K. Pearson, The Grammar of Science, p. 316.—W. Clifford, The Common Sense of the Exact Sciences, pp. 256–258. ↩↩↩↩↩↩↩↩↩

2. d’Ans, Zeitschrift «Kali», 1915. The work was carried out by commission of the Union for the Scientific Study of German Potash Deposits. ↩

3. Blasdale, Journ. Ind. Eng. Chemistry (1920), 164; J. Chem. Soc. Abstracts, 1920, 227. ↩

4. Shiro Takegami, J. Tokyo Chem. Soc. 41 (1920), 831; J. Chem. Soc. Abstracts, 1921, 30. ↩