Joseph Mindel Lincoln Laboratory,' MIT Lexington, Massachusetts
Gibbs' Phase Rule and Euler's Formula
In a series of remarkable papers on thermodynamics ( I ) , J. Willard Gibbs devoted approximately four pages to deducing, almost in passing, the generalization to which he gave no name, hut which is now known as the phase rule. In his own words, "the number of independent variations of which a system of coexistent phases is capable is n 2 - r, where T denotes the number of phases, and n the number of independently variable components in the whole system (Z)." Written in the form, f = n - r 2, f designates the variance or number of degrees of freedom of the system. Before Gibbs, Euler developed a formula for simple polyhedra which may be written V = E - F 2, where the symbols denote, respectively, the numbers of vertices, edges, and faces of the polygons composing the surface of a polyhedron. The similarity between the two expressions has been termed, on the one hand, a purely formal one based on similarity of algebraic groupings (3); on the other band, the two equations have been considered mathematically identical (4) or synonymous (5). Although the similarity in form of the two expressions is not coincidental, the respective terms can not be equated on the basis of algebraic identity. The nature of the relationship may be deduced by considering the origin of the phase rule in Gibbs' thermodynamic reasoning. Any paraphrase of Gihbs' proof inevitably loses the elegance and belittles the comprehensiveness of his method. The derivation is an integral part of his application of the laws of thermodynamics to systems in equilibrium. An aspect of his work that is of particular significance is his reliance on geometrical illustrations, rather than on mechanical models, as symbols and aids in visualizing thermodynamic processes.
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' Operated with support from the U. S. Army, Navy, and Air Force.
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Gibbs extended the familiar volume-pressure diagram to three dimensions, creating what he called the "thermodynamic ~ u r f a c e . " ~To descrlbe a system in equilibrium, he preferred the three functions, volume, entropy, and energy to pressure, volume, and temperature, because the latter three can be deduced when the former are known,3but the reverse is not generally true. The thermodynamic surface is constructed by plotting, in rectangular coordinates, the points corresponding to each set of values of the three functions selected. As a surface, it has the following features: Points, the number of which may he as large as desired, each corresponding to one set of conditions under which the system is in equilibrium; Lines, whose projections on the planes of the coordinate axes are isentropics, isotherms, etc., depending on the functions chosen; enclosed Areas of the surface, formed by intersecting lines. This is the primitive surface, which represents only homogeneous systems. The derived surface which represeuts heterogeneous systems can he formed from it, however. Gibbs showed (6) that if two different physical states can exist permanently in contact with each other, the points representing these states in the thermodynamic surface have a common tangent plane. In a volumeentropy-energy surface each such plane indicates a fixed temperature and a fixed pressure with magnitude given by the slope of the plane. If temperature and pressure are now varied, the two points of the primitive surface and the tangent plane will change position. The envelope of the successive positions of the plane constitutes the derived surface, which describes the relationship of the various states of a heterogeneous system. As a consequence of its origin from the primitive surface, it has the same general features: i.e., 8 Gibbs claimed priority neither for the idea of solid diagrams nor for the use of entropy as a function in developing them. a From the equation of state, dE = TdS - PdV, P = -(aE/aV)s and T = ( a E I a 8 ) ~ .
points, lines and enclosed areas.4 The derived thermodynamic surface thus developed allows for heterogeneity in the number of phases but is limited to onecomponent systems. A multi-component system of non-uniform composition can not be defined by specifying volume, entropy, and energy; pressure, volume, and temperature; or any other three thermodynamic variables. It is necessary to introduce new variables that measure the quantity of each chemical constituent; e.g., m,, mn,m a .. . m, denoting the number of moles of constituents 1, 2, 3 . . . i. Hence it follows in general that E = E(S, V , m,, ma,m ~ ... m J (other functions which depend on many variables are similar). Thus the equilibrium states of the system can not be represented by a threedimensional surface. I t is possible, however, to represent a multi-component system by a series of such surfaces, each of which has a composition that varies for each successive surface -for example, a volume-entropy-energy surface for a system of fixed composition. The process may be considered to be the projection of a multi-dimensional figure on three-dimensional space. I t is analogous to the projection of a three-dimensional figure on a twodimensional area, producing, for instance, a series of isotherms on a pressure-concentration diagram. The entire series of derived thermodynamic surfaces, however large in number, represents one system in equilibrium. In general, temperature, pressure and chemical potential of each of the components in coexisting phases in any part of the system must be equal, while the total volume, entropy and energy must be equal to the sums of the volumes, entropies and energies of the parts. It is not, necessary to describe in detail the series of surfaces, but only to indicate how the number of components and phases are to be summed over the whole system in equilibrium. The total number of components in the system is the sum of the number of components in the first surface of the series and the new and independently variable constituents that appear in the successive surfaces of the series. Similarly, the total number of phases is the sum of the number of phases in the first surface and the number of new phases, differing in composition or physical shate, that appear in successive surfaces. Since we are not concerned with the magnitudes of the thermodynamic functions but with the relationships arising out of the characteristics of the thermodynamic surface, we may choose any from the series of surfaces without examining all possible ones. Since each describes a state of the system in which n components are distributed among r phases, it is only necessary to apply the proper method of summation. Returning then to the features of the thermodynamic surface, we consider the meaning of the points, lines and enclosed areas of the surface. In general, each enclosed area of the surface defines the range of states in which that portion of the system in equilibrium may exist, limited by the enclosing lines James Clerk Maxwell recognized the originality and impora n c e of Gibbe' papers. In fact, his enthusiasm was such that he made a model of a thermodynamic surface and sent a plaster cast of it to Gibbs. For a photograph of Maxwell's model, see CROWTBER, .T. G., "Famous American Men of Science," W. W. Norton and Co., New York, 1937, p. 264.
that fix the boundary conditions. Each such area thus represents the number of variations of phase per component. Since in the system as a whole all r phases are in equilibrium, only r - 1 need be specified or counted. Taken together, therefore, the enclosed areas correspond to the quantity n(r - 1 ) . Each line, as the boundary between enclosed areas, represents a series of pairs of coexistent states, and is a measure of the distribution of components among the phases. To define the composition, only n - 1 components need be specified; the total number of lines corresponds to the quantity r(n - 1). See Table 1. Table 1
Comparison of Thermodynamic with Topological lnter~retotionsof a Surface
Thermodynamic interpretation Enclosed areas Lines Points
n(r
- 1)
~ (n 1)
All possible variations of the swtem
Topological interpretation Faces (F) Edges (E) Vertices ( V )
Then, since each point represents one possible set of conditions under which the system exists, the question of the number of possible variations in a system consisting of r phases and n components may be re-worded as follows: How many points correspond to the network of lines and enclosed areas on the thermodynamic surface? The relationship V = E - F 2 was observed by Descartes in 1640 and re-discovered in 1752 by Eulerthe formula bears his name. Topology is not concerned with the comparison of lengths and angles of geometric figures, nor does it involve the concepts of straight line and plane. It deals only with the continuous connectedness between the points of a figure. Thus the cube and the tetrahedron are topologically equivalent, and Euler's formula applies to both. Since straight lines and planes are not essential characteristics, the cube and the tetrahedron have the same topological properties as the sphere. This in fact provides a definition of a simple polyhedron: one whose surface can be deformed continuously, without tearing or pasting, into the surface of a sphere; The class includes all the regular and many irregular polyhedra. Euler's formula does not hold, however, for a rectangular solid that has a rectangular hole with beveled edges running through it. A count of its vertices, edges and faces yields t,he result V = E - F. Such a figure is not a simple polyhedron for its surface can not be continuously deformed into the surface of a sphere; but it can be deformed into a torus, or doughnut, whose topological properties it shares. For a rectangular figure with two rectangular holes (or its equivalent, a two-holed doughnut), the counting process results in the relationship, V = E - F - 2. It is evident that the Euler formula can be generalized: V =E-F 2 - 2p, where p is called the genus of the solid and is equal to the number of "holes." The thermodynamic surface may now be interpreted in terms of its topological properties. As its features suggest, it is equivalent to the surface of a polyhedron. I t can be shown that its genus is p = 0; i.e., there are no holes.
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Assume that there is a hole in the thermodynamic surface. The size, shape and direction are irrelevant; the only requirement is that it must run continuously through the surface. Where it intersects the surface, there will be a closed curve, outlining in effect a missing section of the surface. In any other portion of the surface, an infinitesimal change in one of the thermodynamic functions produces corresponding and reversible changes in the others, and hence in the state of the system as determined by the contours of the surface. The edge of the hole, however, is a discontinuity. Within the region bounded by the closed curve, this reversihility of infinitesimal change is absent. In a physical sense, the values of the thermodynamic functions corresponding to the points in the missing section of the surface do not represent equilihrium states of the system. By its origin and construction, the surface contains only points representing equilibrium states; hence there can be no holes in the thermodynamic surface. It should be noted that an indentation in the surface, regardless of its depth or the slope of its walls, does not have the same effect as a hole. For topologically, the solid bounded by the indented surface and the planes of the coordinate axes can still be deformed continuously into a sphere; and physically, the criterion of continuous and reversible changes in the system due to infinitesimal changes in condition is still satisfied. The thermodynamic surface is thus topologically equivalent to the surface of a simple polyhedron. The enclosed areas, lines and points are then seen to be, respectively, the faces, edges, and vertices of the polygons composing the surface. For a comparison, see Table 1. The question of the number of points corresponding to the network of lines and enclosed areas of the thermodynamic surface is now transformed into the topological question of the relation between the number of vertices and the number of edges and faces of a simple polyhedron. The relation is defined by Euler's formula, V=E-F+2. Hence, consideration of Gibhs' thermodynamic surface in terms of its topological equivalent yields the immediate result that the number of possible variations of a system consisting of r phases and n components is or, from Table 1,
If, however, other potentials-such as gravitational, electrical, magnetic or those due to light-also act on the system, then these must be taken into account. Their effect will he in general to increase the number of possible independent variations of which the system is capable; that is, the number of degrees of freedom. This may he expressed algebraically as f = n - r 2 k , where k represents the number of independently variable conditions, other than tem~eratureand vressure. The similarity to the generalized Euler relation must be viewed with caution. As in the case of the other terms in the two expressions, k can not be equated, in a purely formal way, with p. (The factor of 2 is not. significant; it can be eliminated by modifying . . the definition of p.) An obvious difficulty arises from the difference in sieu. - , which can not be elimiuated bv re-definine either p or 1c. For if we think of the genus of a geometrical figure as equal to the number of "holes," then p 2 0. That is, the difference in sign cannot be reconciled by considering p to have a negative value. As for lc, its physical significance requires it to be a positive integer. More decisive, however, is the fact that the thermodynamic surface cannot have holes. Hence the generalized Euler equation is applicable only in the specific case for which p = 0, when it reduces to the formula for simple polyhedra. Thus, while the similarity in algebraic form of the two expressions is suggestive, an essential relationship can be established only in the case for which their thermodynamic and topological meanings coincide.
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Summary
1. The thermodynamic surface, constructed by plotting in rectangular coordinates the points corresponding to each set of values of three thermodynamic functions of a system in equilibrium, is topologically equivalent to a simple polyhedron. Sub~t~itution of the algebraic expressions corresponding to the points, lines, and enclosed areas of the surface into the Euler formula for simple polyhedra leads directly to the usual statement of the phase rule. 2. A similar relationship between the generalized statements of the Euler formula and the phase rule can not be established because the thermodynamic surface is restricted by its physical meaning to a specific topological class. Acknowledgment
and hence
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The generalized Euler formula, V = E - F 2 - 2 p , suggests the possibility of an analogous formulation of the phase rule. A similar one does, in fact, exist. In developing the thermodynamics of heterogeneous equilibria, Gibbs first considered the simplest case: that in which the system was acted upon only by changes in temperature and pressure. The usual expression of the phase rule arises from this assumptionthe number two referring to temperature and pressure which, together with the chemical potentials of the n components distributed among r phases, completely establish the system. 514
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The author wishes to express appreciation to Prof. J. A. Beattie and to Dr. 71'. W. Harvey for valuable suggestions. Literature Cited (1) GIBES,J. W., Trans. C a n . Acad., Vol. 2 , 309-42, 382-404 (1873)3, 108-248(1875-6), 343-524(1877-8). (2) GIBBS,J. W., "The Collected Works of J. Willard Gibbs," Yale University Press, New Haven, Conn., 1957, Vol. 1, pp. 96-7, 359. (3) K ~ o c a ~ oM., A,, Imest. Selctora Fig.-Khim. Analiza, Inst. Obshch. i Neorgan. Khim., Akad. Nauk SSSR, 19, 82-8 (1949). (4) RODEL,R., 2.Elekt~ochem.,35, 54 (1929). (5) LEVIN,I., J. CAEM. EDUC.,23, 183-5 (1946) (6) GIBES,"Collected Works," Vol. 1, 35-8.
