On the Phase Rule OTTO REDLICH The State College of Wushington, Pullman, Washington CIENTIFIC ideas, on their journey through the textbooks, usually gain in precision. In general the presentation becomes stricter, clearer, and a t the same time simpler. Sometimes the opposite happens. In recent discussions of the phase rule the number of degrees of freedom is called the number of variables which must be specified for a complete definition of the system. The unprejudiced reader will have some difficulty in understanding that the "complete definition of the system" extends only to the intensive properties. Actually the phase rule is based just on the distinction between extensive and intensive properties. It can be derived by simply counting the number of independent intensive variables.' A slightly more precise definition of the terms "intensive" and "extensive" properties is desirable. If a homogeneous system consists of two identical parts, its intensive properties have the same value for the total system and its parts; the value of an extensive property of the total system is twice its value for either part. We talk of an intensive property of a heterogeneous system only if it is defined for each phase and has the same value for each phase. Any sum of extensive properties of the phases is an extensive property of the system. These definitions, while perhaps not explicitly specified before, are in accord with general usage. Thus, temperature, pressure, concentration, dielectric constant, extinction coefficient, molal volume, density are intensive properties, but the average density or the average specific heat of a heterogeneous system is not called so. Likewise, extensive properties are the weight, volume, energy, heat capacity, or surface area of a homogeneous or heterogeneous system. But artificial constructions like the product of the weights of two phases divided by the weight of a third phase are not included in the present definition of extensive properties although they are in the traditional definition. The present or some equivalent definitions, which are only more precise expressions of actual usage, are indispensable for a correct statement of the phase rule. For didactic reasons i t is advisable to stress the obvious fact that a system has many more properties than the intensive and extensive ones. Thesquare root of the weight of a system is a property which is neither intensive nor extensive. On the other hand, all properties can be expressed by means of intensive and extensive properties and therefore attention may properly be restricted to these.
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1 The following derivation was outlined earlier in a somewhat differentform, cf. 0.REDLICE, Z. anorg. dlgcm. Chenz., 174,218 (1928).
If surface phenomena are disregarded, all properties of a single phase can be expressed by means of intensive properties and a single extensive property, for instance its weight. A system consisting of P phases, therefore, is completely described by a number of intensive properties and P extensive properties (however, see Note 2). The total number of variables, determining the equilibrium state of a system, can be immediately deduced from a construction of the system. An example will furnish the best explanation. We take 100 kg. of water, 5 kg. of nitric acid, 10 kg. of sulfuric acid, and 1 kg. of benzene, all substances in their normal states a t 7'C., pour them into a platinum box of a volume of 135 liters, insulate the box from its environment, and leave i t to itself. This prescription specifies the amounts of the four independent components, their total energy content, and the volume. These six variables are independent, i. e., if any of the six specified values is changed the result will be a different system. The phase rule will be derived from the basic assumption that this description completely defines the system in its final equilibrinm state. This means, that the properties of the system in equilibrium will be the same, whatever the order of mixing the component (but see Note I ) or the original distribution of the total energy among the components or the shape of the box or the elevation above sea level, etc., may be. Thus i t is assumed that the system in equilibrium is described by 2 variables where Cis the number of inprecisely C dependent components. Since the system also can be described by P extensive and a number of intensive variables, there must be F = C 2 - P independent intensive properties. There cannot be more, because P of the C 2 independent variables are necessarily extensive. The number of degrees of freedom F is the number of intensive variables which can be arbitrarily changed while P phases are present. It is believed that this derivation, by simplifying the arithmetics of the proof, directs the attention to the fundamental concepts. A few details are discussed in the following notes. Note 1. The derivation demonstrates that the phase rule is valid precisely to the extent the basic assumption holds. This assumption implies that, after the system has been built up, a reaction either does not occur a t all or proceeds to the equilibrinm. The phase rule applies to both cases but the resulting systems are different. It is well known that the term "eqnilibrium" has a meaning only with respect to specified reactions. In the foregoing example nitration and sulfonation of benzene should be explicitly excluded. Note 2. The derivation shows immediately the sia-
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nificance of the term "independent components" in accord with the usual definition. Nothing need be said with respect to thefact that the number of independent components depends on the reactions regarding which equilibrium is established (hydrogen, oxygen, and water as two or three components). But a special case, thoroughly discussed in the earlier literature, should be mentioned. If calcium carbonate undergoes a partial dissociation, the amounts of calcium oxide and carbon dioxide are stoichiometrically equivalent. If the system is built up by means of one component (calcium carbonate), the amounts of the phases are not independent. To exclude this case, one usually and correctly states that the independent components must suffice for the construction of each phase separately.
Note 3. The areas of interfaces between the phases are extensive quantities. Plane interfaces, therefore, can be taken into consideration without any change of the phase rule. If, however, the curvature of an interface is appreciable, a new intensive property appears, i. e., the number of degrees of freedom is increased. Note 4 . In the usual derivation, phases which do not contain all components cause an unpleasant complication. Some authors attempted to circumvent this difficulty by means of the entirely unjustified fiction that all components are present in all phases. The present derivation is free of any difficulty of this kind. Note 5 . No use has been made of the second law. This is the essential difference between the usual derivation and the present one.
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