An INTRODUCTION to the PHASE RULE. PART I1 H. G. DEMING University of Nebraska, Lincoln, Nebraska
SUCCESSIVE STATES OR APPROXIMATE EQUILIBRIUM
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N IMPORTANT extension of the phase rule is to the successive states of approximate equilibrium that are observed when some of the transformations and transferences within a system promptly attain equilibrium, whereas others progress slowly in one direction and never attain equilibrium. The substances concerned in a chemical transformation that is negligibly slow may be regarded as independent components; otherwise expressed, the chemical equation expressing equilibrium for such a transformation no longer appears as a restriction, hence the variance of the system is inaeased by one unit. Similarly if the transfer of any material across a phase interface is negligibly slow, the corresponding equation expressing equality of rates of transfer in opposite directions disappears as a restriction and the variance is increased by one unit. Such an extension of the phase rule merely asserts that if any of the adjustments (transformations or transfers) that tend toward establishing equilibrium happens to be extremely slow, the other adjustments will make nearly the same progress that they would in the complete absence of the slow adjustment. The phase may even be applied to cases in which there is a slow diffusion of some material within a nonhomo~eneousnhase. For examnle. with an aoueous solution in an open beaker, the space immediatelv above the surface of the liquid may be considereh saturated with the water vapor and in equilibrium with the liquid, even though the equilibrium is slowly being the diffusion of water vapor into the surdisturbed rounding atmosphere.
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assume that a vapor phase is present, even in systems in which the vapor pressure is not large enough to be measured. Nevertheless, by applying a sufficiently large external pressure we may cause a vapor phase to disappear completely, by being condensed to a liquid or solid. Diminishing the number of phases by one gives an extra unit of variance, which may be used in specifying the pressure, in the range above that a t which the vapor phase disappears. The phase rule does not indicate the nature of the phases that are present when equilibrium is attained, but merely how many phases must be present with the variance observed, or what variance will account for the observed number of phases. Furthermore, i t does not reveal the range of any variable. For example, if a system has a variance of one we may arbitrarily alter the temperature through a t least a small range. This alteration in conditions will induce chemical reactions or exchanges of material between the different phases, or both, until each phase has been so altered in composition that i t is again in equilibrium with other phases, at the new temperature. The phase rule does not state whether the ~ermissiblealteration in temperature is large or small. If i t happens to be only a few degrees that particular phase-set may be overlooked in a hasty experimental survey,
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APPARENT EXCEPTIONS TO THE PHASE RULE
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The phase rule appears to fail whenever phases become identical at the limit of a certain range of perature pressure, or at the limit of a certain range of composition, Thus a two-component system(e.g,,water and phenol), a t the critical solution temperature, would appear to represent two phases (liquid and vapor) in SOME COMMON DIFFICULTIES Is the air that is intermingled with the water vapor equilibrium. The variance should therefore be two. above a solution contained in a beaker to be considered Actually, with that particular phase-set, the temperaas contributing new components to the system? ture is invariable. If we raise i t the liquid phase disStrictly speaking, yes. But each of these components appears by evaporation; if we lower it the liquid phase (N,.-, 0, ~- A.. and so forth) is ordinarilv nresent under a separates into two liquid phases. The difficulty disappears if we observe that a t all definite partial pressure, the sum of the partial press u r e being ordinarily, nearly enough, one atmosphere. temperatures up to the critical solution temperature In specifying the partial pressures we use the extra we have two liquid phases and a vapor phase. The one variance that we gain by having more components, unit of variance thus indicated may be used in specifyhence the net variance and the number of phases re- ing that the concentration of either component in the mains unaltered. For this reason the atmosphere one liquid phase shall bear any chosen ratio to its concentration in the other. The temperature is then inmay commonly be neglected. The pressure, in applications of the phase rule, is that variable. At the critical solution temperature the set up by the vapor phase, if present. We may often chosen ratio of concentrations is a ratio of 1: 1, for 260 \
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definite compound, a mixture of two compounds, or a solid of continuously variable composition (for example, a hydrous precipitate or a solid solution). Neglect of the phase rule, in such cases, has resulted in numerous materials being reported as definite compounds, when they were really mixtures or variants. Consider two soluble salts, A and B, reacting in an aqueous solution to form a precipitate, representable by a reasonably simple formula. Is the precipitate really a definite compound or is its simple composition merely the result of the intermingling of two or more separate substances in just the right proportions, because orecioitation took olace from a solution havine just the right temperature and composition? As components we may count A, B, HzO, and OHor HaO+, namely, four components, since the total MELTING POINTS AND TRANSFORMATION POINTS amount of each of these, in the system as a whole, may Now for a few applications, in wh ch we shall empha- be varied independently of the total amount of any sizk some points that the textbooks commonly slight. other. This assumes, of course, that hydroxyl-ion or Consider, first, melting points and transformation hydronium-ion shall be introduced from an outside points. It is commonly taken for granted that a defi- source, and shall not be entirely derived from water. nite chemical compound will melt a t a definite tem- Otherwise the system would be regarded as one of perature. This would seem to suggest that we have three components. Let us first assume that the precipitate is really a to deal with an invariant system. But do we? single phase (whether of variable or fixed composition). There are two cases to consider. I . A Solid, Melting (with or without Decomposition) We then have three phases, including the solution to Produce a Liquid and a Vapor.-If the solid is one of phase and the vapor phase. In a four-component sysC components, the presence of three phases, in eqnilib- tem this indicates three units of variance. Two of these are used in selecting a definite temperature and a rium a t the melting point, indicates a variance of C-1. Ordinarily, this number of units of variance are used definite degree of acidity or alkalinity, and the third in in specifying that the composition of the liquid phase varying the ratio of the concentrations of A to B , within shall be the same as that of the solid phase.* This the solution. If the composition of the solid phase recan be true only if no appreciable quantity of any com- mains constant in spite of an appreciable alteration in ponent escapes, selectively, into the vapor phase. In the composition of the solution, the solid phase is to other words, the melting point, with the phases' just be regarded as a definite compound. If the compoconsidered, is fixed and definite only when the vapor sition of the solid phase varies continuously with the phase is of negligible mass or of the same composition composition of the solution, the solid phase is to be reas the other two phases. garded as of variable composition. 2. A Solid, Melting or Being Transformed, with It is easy to see that a t some definite A / B ratio (soluDecomposition, to Produce a Liquid, another Solid, and tion phase), not predictable in advance, two solid phases a Vapor (or, Conceiwably, Two Liquids and a Vapor).may be present. The system would then have a With C components and four phases we have C-2 units variance of two, completely used in specifying the of variance. In the special case in which only two com- temperature and alkalinity (or acidity). Any deparponents are present, the system is inwariant, namely, ture from that composition of the solution, a t the chosen the transformation temperature is constant, whether temperature and alkalinity, would cause one of the two the vapor phase is or not. We encounter this solids composing the precipitate to disappear, provided case whenever the transition temperature of a hydrate sufficient time were allowed for the system to reach is used as a means for securing a constant temperature. complete equilibrium. With more than two components the transformation ONE-COMPONENT DIAGRAMS temperature is constant only when certain conditions We turn now to graphical representations of systems are met with regard to the composition of the solid in equilibrium. A one-component system must have phases. a t least one phase, hence a variance of not to exceed CHARACTERIZING A PRECIPITATE two. It is therefore representable by a plot of temAn important application of the phase rule is in deperature against pressure (assuming these to be the termining whether a precipitate is to be regarded as a two independent variables). Such a diagram will show 'We need only C-i components, completely to specify the a number of areas, separated by curved lines. Within composition of any phase, since each concentration corresponds any area the temperature and pressure are indeto a definite mole-fraction, r = 5 , and the mole fractions are pendently variable, thus indicating a variance of two, zc related by the restriction Zx (for any phase) = 1. and hence a single phase, characteristic of the area. either component (and hence, almost necessarily, for both components). A similar dimculty is encountered whenever two components form a continuous series of solid solutions, with a minimum melting temperature (the system mercuric bromide-mercuric iodide, for example). At the minimum melting temperature we have a solid solution, liquid solution, and vapor, in equilibrium. This would indicate a variance of one; yet the minimum temperature is definite and invariable. The correct approach is to observe that a t all temperatures, in descending toward the minimum temperature, we have the phases just indicated and a variance of one; and we use this unit of variance in specifying that our temperature shall be a minimum.
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Boundary lines between two areas represent two phases in equilibrium and a variance of one. Points a t which three regions or three boundary lines meet represent three phases in equilibrium, a t an invariant temperature and pressure.
Solid phases of invariable composition:
A phase sometimes fails to appear, even under conditions under which i t would be stable. Rhombic sulfur, for example, may persist even a t temperatures under which its conversion into monoclinic sulfur is quite possible. Whenever this happens the area proper to the missing form disappears from the diagram, and three adjacent areas become extended until they meet within the disappearing area (see diagrams for sulfur or phosphorus, in textbooks). This represents a case of metastable equilibrium, which can persist only so long as the missing form is definitely excluded, or so long as transformation into this form is negligibly slow. TWO-COMPONENT DIAGRAMS
One or more solid phases of wariable composition:
FIG~RE 1.-Two COMPONENT SYSTEMS Curves and vertical lines represent the composition of the solid and liquid phases separating from a mixture of two components, A and B, a t different temperatures. Such diagrams may be viewed as showing the melting points of the solid phases, in the presence of the indicatedliquid phases; the freezing @kts of the liquids in the presence of the indicated solids; or the solubility of each component, a t the indicated temperature, in the other component. It is imDortant t o keep these three interpretations in mind, in &arching for d& in the literature. (1) The two components form no compounds or solid solutions. 2 Two liauid nhases exist. in eauilibrium. within a ce&n range of temperature. (3) he-two components form a compound, which is stable up to its melting point. (4) The two components form a compound, which is decomposed a t a definite transition temperature, pure B being separated as a solid phase. (5) A compound is formed, which exists in two different crystalline forms, the less stable form possessing the lower melting point, the higher vapor pressure, and the greater solubility in all solvents. (6) Two distinct com~ounds are formed. one of them be&g largely decomposhd in the liquid phase, into the components A and B (evidenced by flattened top t o curve). (7) Component A dissolves in B t o form a solid solution, stable through a limited range of composition. (8) ComDonent B dissolves in A t o form a solid solution: the two components also form a compound. (9) Each component dissdves in the other t o form a solid solution. (10) The two components form a continuous series of solid solutions. with a minimum meltine ~ o i n t . (11) A continuous series of solid solutions, of intermidiate meltkg points. (12) A cantinuous series of solid solutions, with a maximum melting
In two-component systems, with three phases present, we have a variance of one, which we ordinarily use in specifying the temperature. All other variables are then beyond our control; that is, for each temperature there is a definite concentration of each component in eachphase. We may therefore plot temperature against composition (in percentages or mole percentages of either component) and. obtain definite curves, as in Figure 1. Textbooks discuss the different cases shown in this figure, but commonly neglect to show how such diagrams may be subdivided into stable and metastable areas, as an important clue to the interpretation of the diagrams. Small circles, in Figure 1, represent melting points of solid phases of invariable composition (elements or compounds). The composition of any such phase is represented by a vertical straight line, extending downward from the melting point to indefinitely low temperatures. The uppermost curve or set of curves on each diagram represents the composition of a liguid phase. Other curves represent the composition of solid phases of wariable composition (solid solutions). To subdivide any diagram into stable and metastable regions, make sure that all the solid and liquid phases have been represented in the diagram, then draw a horizontal line through each eutectic point, E, each eutectoid point, E', and each transformation point, T', until the next adjoining lines on left and right, representing solid phases, are intercepted. Distinguish between case 2, in which two liquid phases are in equilibrium with each other, and case 3, in which either of two liquid phases is in equilibrium with a definite compound, represented by a vertical line. The former case may be recognized a t a glance by the fact that the point of inflection and the eutectic point, E, are a t the same level (namely, a t the same temperature). A vapor phase is assumed to be present, in all the cases shown by the figure, though its composition is not shown. The areas above the curves that represent liquid solution are stable areas, S. Other stable areas appear a t the right or left of curves representing solid solutions. The remaining areas are metastable areas, M. When we have marked out the limits of the different stable and metastable areas we are ready to interpret the
diagram. We may prepare a stable solid or liquid phase having a composition represented by any point within a stable area; by contrast, in a metastable area, such as the area MI of Figure 2, a single phase, of a specified temperature and composition, W, will separate or tend to separate into two phases of compositions given by points X, Y, a t opposite ends of a horizontal tieline passing through the point W. The relative amounts of the phases X and Yare as the more distant segments of the tie-line; that is, WY represents the relative amount of the phase of composition X, and WX the relative amount of the phase of composition Y. These relative amounts are in weights or moles, according as the diagram happens to be constructed in per cent. of weights or in per cent. of moles. As an exercise, the reader is invited to describe what will happen when mixtures containing twenty, forty, sixty, and eighty per cent. A , in each of the cases shown in Figure 1, is slowly cooled from high temperatures or heated from low temperatures. The change in temperature will represent movement along a vertical line passing through each stable region, but interrupted in each metastable region, since the given phase is there resolved into two phases, whose compositions, and hence whose relative amounts, may or may not alter with changing temperature. The reader should also describe what happens when B is gradually added to A, without limit, a t each of several different temperatures. Finally, indicate what phases and what relative amounts of each will separate from the five mixtures having compositions and temperatures represented by crosses, in Figure 2. In all the two-component systems here considered, the vapor phase is assumed to be present, though it is not shown on the diagrams. Thus with two condensed phases (liquids or solids) we have a total of three phases, hence one unit of variance, used in specifying the temperature, within the range in which the indicated phases are in equilibrium with one another.
3. A solution, on being cooled, separates a solid phase, then two solid phases (at a lower temperature), and perhaps ultimately three solid phases. This case is shown in Figure 4. The contour lines represent the temperatures a t which liquids of different compositions begin to deposit solid. In diagrams representing systems of this sort look for analogs of features shown on a
geographical contour map: hilltops, valleys between hills, direction of slope down a valley, an occasional pass or saddle between two hills, two valleys merging, three valleys meeting in a pit. These all have important physical interpretations. If the solution whose composition is represented by M, Figure 4, is cooled i t will begin to solidify a t 800°C. The solid separating will have the composition A a t the hilltop dominating that particular region of the chart. As more and more of this solid separates the composition of the remaining
THREE-COMPONENT SYSTEMS Three-component systems are represented by triangular diagrams, the interpretation of which is given in the textbooks. The tie-lines in a triangular diagram are no longer horizontal, since temperature (assumed uniform for all the phases) no longer appears as an ordinate. In the textbooks or the International Critical Tables (index a t the end of this article) the reader should look for examples of the following cases: 1. A mixture of three liquids 'shows one or more metastable regions. 2. A mixture of three liquids shows three metastable regions, overlapping to form a doubly metastable region (LMN, Figure 3). Any mixture having a composition represented by a point within this region will separate into three liquid phases, having compositions represented by points L, M, and N. The reader should show that this is a univariant system, the one unit of variance being used in s ~ e c i f v i nthe ~ temperature for which the diagram is t o b e cons
