An INTRODUCTION to the PHASE
RULE. PART I
H. G. DEMING University of Nebraska, Lincoln, Nebraska
T
HE phase rule has to do with any set of homogeneous bodies having one or more ingredients in common. When such bodies are placed in contact these ingredients may pass spontaneously from one body into another. In the end, a condition of equilibrium may be reached, in which the rate of transfer of any ingredient, from any body into any other, is exactly balanced by a transfer in the opposite direction. The phase rule indicates the general nature of the conditions that must be fulfilled for such an equilibrium to be possible. The homogeneous bodies with which we shall have to deal are called phases. To count as a separate phase, in a system of phases, a body must not only be homogeneous but also (as we shall presently see) must differ in its properties from other phases. Thus a handful of crystals of identical character would not count as that many separate phases, but as a single phase of a stated composition. Furthermore, each phase must possess at least one ingredient that i s exchangeable with at least m e other phase. If a bottle is partly filled with an aqueous solution, the liquid and the moist air above i t would count as two separate phases, since water passes from the one to the other by evaporation and recondensation; but we would not count the bottle as a third phase unless it should be found to yield some ingredient (perhaps alkali) to the solution, or to absorb some ingredient from the solution, in sufficient quantities to be appreciable in the use for which the solution is intended. The rate a t which any given ingredient will pass from any given phase into any adjacent phase will usually depend : (1) On the temperature. ( 2 ) On the concentration of the giwen ingredient i n the given phase. (3) On the environment of its particles in the given phase, namely, the concentrations of other ingredients. (4) Indirectly, on the pressure, since this helps to determine concentration. Thus, in a phase containing I ingredients or chemical individuals, each of some definite concentration, there are ordinarily I 2 variables, including temperature and pressure, that determine the rate of transfer of any given ingredient from that phase into any other. These variables are, of course, not all independent. In a system of P phases, if we have the same temperature and pressure throughout, there will evidently be IP 2 variables, not all independent, that determine
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rates of transfer of material across the phase boundaries, from any one phase into its neighbors. By letting some of these variables take zero values we may provide for the fact that not all the ingredients of the system may be present in all its phases. DERNlNG THE PHASE RULE
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Our problem is to determine how many of the IP 2 variables, which determine the rates of transfer of the material across phase boundaries, are independent. To do this we must see what relationships exist between them. We notice, first, that there is for every phase a definite phase equation, relating concentrations to the temperature and pressure. The general phase equation is f(P, T,cl, c,, . . .) = 0; for an ideal gas mixture this becomes P V = .Zn.RT, namely, P = 2 . R T (since n / V = c, for each gas in the mixture). If concentrations are expressed in mole-fractions or in partial pressures, instead of in moles per unit volume, new variables are introduced, but an equal number of restricting equations, defining these variables in terms of those already used. The reasoning, therefore, remains unchanged. We also need to consider the possibility of certain ingredients being converted into others by chemical reactions, within any phase or a t any phase boundary. Every such reaction reaches a definite equilibrium, with reactants and resultants in concentrations that are related as given by the ordinary expression for the equilibrium constant, Kc, within the range in which concentration is a measure of activity. So, if R diierent independent chemical reactions occur within the system there will be R different equations, relating the concentration of the reactants and resultants a t equilibrium. Knowing the equilibrium constant we may calculate the concentration of any chemical individual in any given phase, if the concentrations of the others are known. Of course we introduce a new variable when we use Kc as just described, but offset this by a new restriction relating K, to the temperature. (Though a change in pressure may disturb the equilibrium, K, itself is independent of the pressure.) To these reaction equi~ibriumequations must be added to one other, if ionic reactions are considered. This expresses the fact that the sum of the charges on the cations is equal to the sum of the charges on the anions, since the system as a whole (not necessarily any individual phase) is electrically neutral. Finally, we must remember that the phase rule is
interested in equilibrium conditions, in which the rate of transfer of any ingredient, across any phase interface, is equal to its rate of transfer in the opposite direction. From any given phase, in a system of P phases, any ingredient may be transferred into P - 1other phases, either directly or through intervening phases. We thus have P - 1 equations for each ingredient, stating that the rate of transfer in one direction is equal to the rate of transfer in the opposite direction, a t equilibrium; and if there are I different ingredients the number of such transfer epilibrium equations will be I ( P - 1). Now let us summarize: We have found that the rates of transfer of material across the phase boundaries are ordinarily determined by IP 2 variables, not all independent. As restrictions, namely, equations connecting the variables, we have found: P phase equations, of the formf(P, T , cl, cz, . . .) = 0 R reaction equilibrium equations (chemical restrictions), perhaps including one expressing electrical neutrality for the system as a whole. I ( P - 1) transfer equilibrium equations. The number of independent variables appearing in any equation is one less than the total number of variables; for if values of all but one are known the equation will ~ e r m i tthe one to be calculated. Thus bv subtracCng the total number of equations from thk total number of variables we may find the number of independent uariables, otherwise called the number of degrees of freedom or the wariance.* Representing this by F we find
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F=(IP+2)-[P+R+I(P-1)]=I-R-P+2
This is an algebraical expression of the phase rule. Now I - R evidently represents the number of chemical individuals. whose total weights are independently variable, in spite of chemical transformations within the system. Representing these independently mriable ingredients, usually called components, by C we have F = C - P + 2
as an alternative expression of the phase rule. RESTATING THE PHASE RULE
The equation just deduced has four chief implications: 1. The maximum number of phases possible, in any system of fihases in equilibrium,is at most two greater than the number of components. (If there were more phases than this, F would be negative, namely, the number of equations to be satisfied would exceed the number of variables. This would be possible only in the very improbable event that an exchange of a given component between two phases might reestablish equilibrium, not only for that component but for some other.)
* The term degrees of freedom is commonly used but is less desirable, since it is employed with an e n t i ~ ~ different y meaning in the theory of the energy states of molecules.
2. This maximum number of phases is possible only at a definite temperature, a definite pressure, and a definite concentration of each component in each phase. (This must be true, since when P has its maximum value of C 2 then F = 0, namely, none of the variables can be varied, independently of the rest, and still maintain equilibrium.) 3. When less than the maximum possible number of phases are present, the conditions of eguilibrium are not so rigidly prescribed. Instead we have mobile equilibrium, in which one or more of the variables that determine rates of exchange of material between phases may be varied (within limits), thus causing some compensating exchange of material, which presently leads to a new condition of equilibrium. 4. For aery phase that is lacking, to complete the maximum number, some one of the nariables (temperature, pressure, or concentration of a selected component in a selected phase) may be arbitrarily chosen or fixed, or arbitrarily mried, within at least a small range. In the new equilibrium thus induced the same number of phases will be present as before, though altered in composition.
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EXTRA VARIABLES OR RESTRICTIONS
In what precedes we have assumed that the rate of transfer of material across phase boundaries is completely determined by the concentrations of the several ingredients in the different phases, and by the temperature and pressure. Actually, any factor that alters the potential energy of the particles in any phase with respect to those in any other phase will alter the distribution of the particles between the phases a t equilibrium. I n addition to temperature and pressure, in special instances, we might need to consider such things as differences in electrical potential, magnetic field strength, o r surface tension-in brief, any of the capacity factors of potential energy. The "2" in our statement of the phase rule would then need to be replaced by some number that would take account of the increased number of variables. For example, when electrons are transferred between an electrode and the ions in a solution, the electrons are a component of the system, and the potential of t h e electrode appears as an extra variable, hence we replace 2 by 3 in the equation expressing the phase rule. T h e reader may now pause to consider what variables must be fixed, to establish the potential of a hydrogen electrode, and whether this conclusion is in harmony with the phase rule, as just extended. The membrane potential, in the Donnan equilibrium, will furnish further opportunity for meditation. We would have extra variables, too, if the temperature and pressure, instead of being uniform for the whole system of phases were to vary from one phase to, another. We would have extra restrictions if definitediierences in temperature. pressure. or concentration were to be establiihed bet&& the different phases, or^ definite ratios of concentration. The variance, in any case, is the total number of variables minus the totaC
number of conditions to be satisfied. The reader will perceive bow much broader the phase rule really is than is implied by the algebraical equation commonly used in stating it. THE PHASE RULE AND THE PRINCIPLE OF LE CHATELIER
We can see that the pbase rule is an extension of the principle of Le Chatelier, in the special case of homogeneous bodies (phases) in equilibrium. If this equilibrium is disturbed by altering the temperature or pressure or by arbitrarily altering the composition of any phase, a spontaneous adjustment occurs, by chemical reactions within the phases or by transfers of material from one phase into another, until equilibrium is reestablished. The principle of Le Chatelier states that this spontaneous adjustment may include an adjustment of the altered temperature, pressure, or concentration, i n the direction of their former values. The phase rule determines how many variables may be altered arbitrarily and independently, with the number of chemical individuals and phases that are actually present, without causing any phase to disappear or any new phase to be produced.
first, introduces any new variables into the rates of transfer of material across phase boundaries, if provided we assume that these rates are not influenced by the size of a crystal nor by the proportion of comers and edges to surface area. In extreme cases this assumption is not justified; a miwocrystalline precipitate is actually slightly more soluble than the same material in coarse crystals. HOW TO COUNT COMPONENTS
Since difficultiesin applying the phase rule are usually due to doubt concerning the number of components, let us see how components are chosen or counted. As general rules: 1. The composition of every phase i n the system, and hence that of the system as a whole, must be expressible in terms of the chosen components, though zero values are permitted for some of them, in certain phases. Moreover, one hundred per cent. of every phase must be accounted for. 2. It is most convenient to choose simple components, for these are most likely to permit all phase compositions to be expressed without the use of negative concentrations. 3. If two ingredients m c h e m k l indim'duals, i n the AVOIDING A PITFALL system of phases, are interconvertible in a chemical We must now emphasize a point that textbooks com- transformation, within any phase, only one of them is monly neglect, namely, that in making adjustments of counted as a component. concentration, to reestablish equilibrium, we are as4 . In more complicated chemical transformations, sumed to have a t our disposal an unlimited external since C = I - R, one may count chemical indiwiduuls, supply of each of the components, from which material namely, molecular and ionic species, and subtract from may be brought into the system, and to which material this the number of chemical restrictionr expressed by may be removed. Otherwise, the components are not indepadent chemical equations*-with an additional really independent but are subject to the additional restriction, in ionic systems, due to the need for elecrestriction that the total mass of a given component, trical neutrality in the system as a whole. within the system, shall be constant or else definitely 5. Selecting or counting components as just described, related to the total mass of one or more others. we assume that all the components chosen may be indeTo restrict each component to the mass that it had in pendently introduced into the system from an ouLsidz the beginning we would need C new restrictions, of the source. By contrast, molecular or ionic species proform ZVc (for each component, in all the phases) = duced by chemical reactions within the system, and const. In so doing we introduce the volume, V , of each not independently available from an outside source, phase as a new variable (namely, P new variables). are not counted as components. With P new variables and C new restrictions the variance is changed from F = (C - P) 2 to F = (C AN EXAMPLE P) 2 (P - C) = 2. In brief, whenever we lack an unlimited external supply of the components, the As an example, consider an aqueous solution convariance, namely, the number of variables that may be taining the ions K t , Naf, C1-, Br-, together with altered through a t least a small range without altering simple and complex water molecules. We rule out the the number of phases, is always two. complex water molecules a t once, since these are If arbitrary variation of the temperature and pressure interconvertible with simple molecules by reactions in is carried far enough, certain phases may disappear, the liquid phase. The unassociated water molecules until the system a t last may be reduced to a single and the four kinds of ions represent five chemical pbase. In this process we lose a corresponding num- individuals. Subtracting the restriction for electrical ber of restrictions (phase equations) but an equal neutrality, we conclude that this is a four-component number of variables (phase volumes) hence the vari- system. Which particular four components to select, to avoid negative numbers in expressing phase comance remains unaltered, namely, two. We can now see why bodies of identical composition positions, cannot be told in advance of a knowled~e - of (as in a handful of crystals of some one kind) are not *Chemical equations are said to be independent if none of counted as separate phases. None of them, after the them may be obtained from the others by numerical operations.
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what phases are actually present, for example, as solid salts, in equilibrium with the solution. The number of components is not changed if we consider that the molecules NaC1, NaBr, KC1, and KBr are formed from the ions. This would give us four new individuals, but four new restrictions, due to chemical equations representing the formation of the molecules from the ions. The reaction NaU f KBr
NaBr
+ KC1
need not be considered, for it is not independent of the reactions by which these four salts are separately formed from their ions. If we consider that the water in this system is slightly ionized,
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9u.n + --*u.n+r , nu-
we have two new chemical individuals (ions) ; bence subtracting a restriction for the chemical equation we and OH- are get one extra component, provided actually made to vary independently within the liquid ~ h a s eby , adding an acid or alkali or because of a tion (hydrolysis). However, if both ions originate from water. and no hvdrolvsis . occurs,. they. are not independently variable, but must be present in equal concentrations. Neither of the ions would then count as a component, and we would still have a four-component system. If we begin to wonder whether an extra component is indicated by the presence of heavy water in the system, in the ordinary proportion of about 1 in 5000, the answer depends on the ends to be served. The important point is not that the proportion of heavy water is ordinarily too small to be noticed. The real test is to observe whether its activity (which determines rate of transfer across pbase boundaries or rate of participation in chemical reactions) is sufficientlydiierent from that of ordinary water to give i t a significantly different distribution between the different phases. If so, i t would need to count as an additional component. For isotopic variants of other elements the same principle applies.
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equation, and one restriction expressed by the equation, we conclude that this is a two-component system, provided both components (say NHCl and NHa) are independently introduced into the system from an outside source. But if only ammonium chloride is thus introduced, the ammonia and hydrogen chloride being derived from this by dissociation, we have a one-component system. Having decided how many phases and how many components are present, the variance becomes apparent. One unit of variance may he used in arbitrarily selecting the temperature. If another then remains it may be used to specify the pressure or the concentration of one of the ingredients of the vapor phase; or if two units of variance remain, after specifying the temperature, the concentrations of both the ingredients of the vapor phase may be specified. The reader may work this outfor the four cases that are possible, with condia svstem of one or two Dhases tions ~- so chosen as to " , and one or two components. In each case determine what things may be specified in order that the pressure exerted by the gaseous phase may assume some definite value, not subject to specification in advance. ~~~
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THE DISSOCIATION OF CALCIUM CARBONATE
When calcium carbonate is heated, in a closed vessel, we have dissociation, CaC08
= CaO + C0a
This is a two-component system, provided two of the three chemical individuals here appearing are independently introduced into the system from an outside source. Experience shows that this system, unlike the previous one, exerts a definite pressure a t each temperature, even though carbon dioxide (within limits) is introduced into the system from an outside source. The system therefore has one unit of variance (used in specifying the temperature). One unit of variance, with two components, indicates three phases. Thus in addition to the vapor phase we must have two separate and distinct solid phases. Without this proof, based on the phase rule, we might well have assumed an intermingling of calcium carbonate and calcium APPLICATION TO THE DISSOCIATION oxide to form a single solid phase, having a composition OF AMMONIUM CHLORIDE that varies with the temperature. Actually, when crystalline calcium carbonate disConsider a system consisting of solid ammonium sociates, dissociation begins a t the corners and edges of chloride, in equilibrium with a vapor phase in which the the crystals, since each carbonate ion here possesses salt is partly dissociated: less than the normal number of positively charged NH4C1S NHa f HCI neighbors, and is consequently less firmly bound. The There will be two phases (a solid and a vapor), provided second solid phase (calcium oxide) is then formed by the quantity of solid taken is so great that it is not com- rearrangement of the particles remaining after removal pletely vaporized, in the volume offered to it. (There of the carbon dioxide. At moderate temperatures an can never be more than one vapor phase in any system, adjustment of this sort sometimes occurs only in the presence of a catalyzer or solvent; in its absence a since gases or vapors intermingle freely.) With three chemical individuals appearing in the condition of equilibrium is not attained.
