JOURNAL O F CHEMICAL EDUCATION
AN INTERPRETATION OF THE PHASE RULE C. A. HOLLINGSWORTH University of Pittsburgh, Pittsburgh, Pennsylvania
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phase mle can be interpreted as a statement of the invan'ance of the number of independent variables of a system in the following sense: The number of i n d e pendent variables necessary to describe a system (with respect to both its extensive and intensive properties) i s independent of the number of phases present. 0. Redlich' has taken tbis interpretation as a basic assumption and hasused it to derive the phase rule without use of the second law. However, since the invariance of the number of independent variables is not at all obvious directly, and since it can be derived by use of the second law-that is, by deriving the phase rule in the usual m a n n e ~ iwould t seem preferable to think of this invariance as a principal equivalent to the phase rule rather than a basic assumption to be used to deduce the phase rule. That the principle of the invariance of the number of independent variables cannot be accepted directly can be shown by the following example: Consider a one-component, one-phase system. The total number of independent variables (extensive and intensive) is three, and these can be taken as the mass, temperature, and pressure. Now suppose that the state of the system is varied so that a second phase appears. How could we know (if we had no phase rule) that tbis two-phase system would not have four independent variables: the mass of the first phase, the mass of the second phase, the temperature, and the pressure? There is no scientific principle that tells us that this cannot be tme--except the phase rule, or some other relationship such as the Clapeyron equation which also depends upon the second law. As an interpretation of the phase rule the invariance principle is very suitable for presentation to students of elementary physical chemistry. It gives the student ' REDLICH, Omo, J. CHEM.EDUC., 22, 265 (1945).
a clear picture of the meaning of the phase rule, especially emphasizing the fact that the degree of freedom is a variable which is independent of the masses of the phases. It is easy to visualize and apply a principle that means in effect that for each additional phase present there is one additonal independent extensive variable (the mass of that phase), and, therefore, there is one less independent intensive variable (degree of freedom). From this point of view the phase rule takes on a simple significance and thus becomes more than a mathematical equation t o be memorized. The simple examples given later will illustrate this point. Before considering these examples we will define carefully some important terms and then show that the principle of the invariance of the number of independent variables is indeed a correct interpretation of the phase rule. DEFINITIONS
Before the phase rule can be discussed from any view point it is necessary t o make a clear distinction between intensive and extensive variables, and between chemical species and independent components, and to define precisely the degree of freedom. Although definitions of these quantities are easily found in textbooks and other places (ref. I), it seems desirable to include them here with some discussion. Suppose we have two identical systems, and that some property Q of each system has the value q. Now consider the two systems to be one system of mass twice that of each of the original systems. If the value of Q for the combined system is 29, the property Q is an extensive property (variable). Examples are volume, entropy, internal energy, etc. If the value of Q for the combined system is q, the property Q is an intensive property (variable). Exam-
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We must show that F P is the total number of independent variables when P phases are present. The total mass of a phase is an independent (extenThe numher of independent components of a system in a given state is the smallest number of chemical sive) variahle since the mass of a phase can be varied species that would be required to prepare the system independently, say, by mechanically removing a porin the given state with masses of the phases in any tion of that phase or adding to that phase from another arbitrary proportions. All independent chemical re- identical system. The system can have no other inactions which are at equilihrium in the system must he dependent extensive variables if the temperature and considered to reduce the required number of chemical pressure are used as variables since any other extensive variable is determined by the mass of each phase, the species to the smallest possible value. That one would not he sufficiently precise by defining temperature, and the pressure. By definition F is the number of independent components simply as the the number of independent variahles that remain after smallest numher of chemical species that are required the masses of the individual phases have been elimito prepare the system can be illustrated by the following nated as variahles. Thus, it follows that F P is the total number of independent variables when P example: Consider a system composed of one mol of CaCOa phases are present, and the phase rule states that thisis (solid), one mol of CaO (solid), and one mol of C02(gas) constant (independent of P ) and equal to C 2. all in equilibrium a t some temperature, say, 500°C. This is a three-phase, two-component system. How- EXAMPLES (1) One-component. Consider a one-component sysever, this special system, with its CaO and COr phases in this particular ratio, could have been prepared from tem, such as pure water. When the system exists as one chemical species-CaC03. Tbus it is necessary one phase we know that three independent variables to consider systems in which the masses of the phases are necessary to describe the system with respect to are in other ratios. Sometimes this difficulty is pre- its extensive and intensive properties-for example, sumed t o be overcome by requiring that each phase be the mass, temperature, and pressure. According to prepared separately. However, according to such a the phase rule this number of independent variahles is definition it is not clear that the three-phase system independent of the number of phases. As long as we consider all of the independent variCaC08(solid) - CaO (solid) COz(gas) is not a threecomponent system. ables of this one-phase system, we can use one, two, The number of independent components may depend or three extensive variables as our independent variupon the state of the system. For example, the one- ables. Three extensive variables could be mass, component, one-phase system CaC08 is a special case volume, and internal energy. However, if we ignore of a system which is usually a two-component system. the mass in order to count the degrees of freedom, the However, it should he noted that if the solubility of volume and internal energy no longer have meanings, CaO and Cot in CaC08 cannot be neglected, this one- and we must use intensive properties such as molar phase system will seldom exist a t the exact composi- volume and molar internal energy, i. e., volume and tion of CaCOZand will, therefore, he a two-component energy per mol. Thus, the degrees of freedom of this system. one-phase system is always two, regardless of what The degree qi ,freedom is a property (variahle) of the variables we use to describe the system. system which is independent of the masses of the phases. When this system exists as two phases in equilibrium The degree of freedom is an intensive variable. How- (such as liquid and vapor), there are two independent ever, some intensive variables cannot be degrees of extensive variables: the mass of the vapor and the freedom according to the definitions given here. For mass of the liquid. This leaves only one intensive example, the average density (ratio of total weight to variahle as a degree of freedom. When the system is vapor, liquid, and solid in equihhtotal volume) of a heterogeneous system is not a degree of freedom since its value depends upon the rium there are three extensive variahles and no degrees of freedom. relative masses of the phases. ) Two-component. Consider the two-component system KaC1 and water. The total number of indeINVARIANCE OF THE NUMBER OF INDEPENDENT pendent variables is four: the mass of the water, the VARIABLES mass of the NaCI, the temperature, and the pressure, The ordinary phase rule can be written in the form for example. The phase rule tells us that this number four is independent of the number of phases. Tbus, if there are three phases present (say, water vapor, ice, where F is the number of degrees of freedom, P is the and solution) the mass of each phase can be counted number of phases, and C is the numher of independent as an independent extensive variahle leaving only one components. If the system can exist as one phase we degree of freedom as the remaining independent variknow that the numher of independent variables is able. This degree of freedom can be taken as the C 2, namely the mass of each component and two temperature, or the pressure, or the concentration of other variahles such as the temperature and pressure. the solution, or the density of the vapor, etc. ples are temperature, pressure, concentration, density,
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