MEANINGFUL TEACHING OF PHASE DIAGRAMS' NORMAN 0. SMITH Fordham University, New York, N. Y.
ITIS the writer's opinion aft,ertwenty years of teaching that the basic principles and utility of phase diagrams are rarely grasped at the college level and that the study of heterogeneous equilibria from the standpoint of the phase rule, treated often in college as one small facet of physical chemistry tends, in the student's e n d , to remain so for the rest of his iife without the realization of its immediate impact on many of the procedures with which he is concerned as a working chemist. In undergraduate texts the subject is treated, perhaps unavoidably, in a chapter by itself, and references to it elsewhere are usually lacking. In one widely used book, for instance, a t least seven chapters contain material interpretable in terms of the phase rule or phase diagrams but in only one of these chapters is the interpretation as such undertaken. This is entirely understandable and probably necessary if clarity is to be preserved, but it is not uncommon to find that the relevance of phase diagrams t o common laboratory operations has been missed. How many college graduates, for example, would dispute a statement that an impurity always lowers the melting point of a substance? How often is the relation of simple eutectic diagrams to the phenomenon of steam distillation realized? How often is it seen that the distrihution law determines the slope of the tie lines under a binodal curve in ternary liquid systems? It is the purpose of this article (1) to suggest a few aids in the teaching of phase diagrams, (2) to emphasize the connection between many seemingly unrelated diagrams and (3) to voint out some conclusions derivable from them'khich-have a direct bearing on everyday laboratory phenomena, Only those aspects of the subject which seem to the writer to require comment will be referred to. In no sense. therefore. is what follows to be regarded as comvrehensive. ONE-COMPONENT SYSTEMS
There is rarely difficulty in grasping the meaning of the familiar ~ressure-temnerature diagram for a one-component system and the application of the Clapeyron relation to the vaporization, sublimation and fusion curves. In this connection, however, it is helpful to recall the pressurevolume isotherms (1) ("Andrews isothermals") commonly used in discussing the liquefaction of gases, and to remind the student that, for a given temperature, the vapor pressure of a liquid is the height of the horizontal line of the corresponding Andrews isothermal, which will normally
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Presented before the South Jersey Section of the American Chemical Society, Penns Grove, N. J., January 15, 1957, and before the Division of Chemical Education a t the 132nd Meeting of the American Chemical Society, New York, September, 1957.
VOLUME 35, NO. 3, MARCH, 1958
have already beeu described. The continuous change from a vapor to a liquid ordinarily dealt with when considering continuity of state in connection with the Andrews isothermals can now be demonstrated on the pressure-temperature diagram by passing from the field for vapor to that for liquid around the end of the vaporization line thereby emphasizing the abrupt end of the latter a t a critical point. This gives an opportunity to remark that the critical phenomenon is observable only on passing through the critical point to or from a point on the vaporization curve. The usual experimental demonstration of this phenomenon involves heating a sealed glass tube containing liquid and vapor in equilibrium, but it is rarely stated that only when the relative amounts of liquid and vapor are initially such as to give an over-all density equal to the critical density will both liquid and vapor persist up to the critical temperature and so give the desired effect. That this condition must obtain is immediately evident from the fact that on the Andrews diagram only an isochor for a particular volume of system, namely that equal to the critical volume, will pass through the critical point. Incidentally the horizontal portions of the Andrews isothermals provide an early opportunity, if so desired, to introduce the rule known variously as the lever law, the proportioning rule or the law of mixtures, according to which (in this instance) the relative amounts of coexisting liquid and vapor for a given over-all volume can be readily calculated. Familiarity with this rule is of great assistance later ~vhen the idea of tie lines is encountered. BINARY SYSTEMS The phase behavior of binary systems requires a three-dimensional model for its com~leterevresentation in terms of the conventional variables temperature, pressure, and concentration, and it is only through a study of such models that complete understanding of heterogeneous equilibria is gained. With some sacrifice of the over-all viewuoint but considerable gain in clarity and simplicity it is common, however, to draw only isobaric or isothermal sections of these models. In discussing equilibria involving vapor both kinds of. section are commonly used, whereas in discussing other types only the isobar is usually of consequence. Nevertheless, it is helpful to realize that, qualitatively, the isobar is often a mirror image of the isotherm. Examples of this will be seen below. Let us consider a hypothetical "binary system" A-B in which there is immiscibility in all states of aggregation (solids A and B are not isomorphous, liquids A and B are immiscible and so are gapes A and B (!)). Our isobaric "phase diagram" (one atmosphere total 125
Figure 1. Isobar for lmmiscibility in i l l States of A g p g a tion
2. l s o b u for Complete Miscibility in All Statn of Agmegetion figure
pressure) is shown in Figure 1 if the normal melting points (T) and normal boiling points (T') are in the order T , < T, < T', < TI,. The rectangular character of the fields expresses the chief feature of this system, namely immiscibility. An isotherm for this system would similarly be so characterized. Consider, on the other hand, a binary system in which there is complete miscibility in all states of aggregation. (The system bromobenzene-chlorobenzene may be cited as an example.) If we ignore, temporarily, the possibility of azeotropism, which may be regarded as incidental to the main diecussion, the phase relations are given in the isobar Figure 2, where, again, T, < T, < T', < T',. The rectangular character of Figure 1 has disappeared. The juxtaposition of vaporization and melting equilibria and their obvious similarity is, regrettably, rarely shown in textbooks. The relation of the upper pair of curves to fractional distillation needs no emphasis. I t may be noted parenthetically that if one of the components be regarded as an impurity in the other then it is clear that (volatile) impurities can either raise or lower a boiling point. The lower pair of curves, similarly, form the basis for fractional crystallization and show that impurities can raise as well as lower a melting point. Furthermore, all mixtures of A and B w~llhave a melting range. It is well, here, to admit to ambiguity in the usage of the term "melting point" where melting ranges exist. If the terms melting point and freezing point are taken to be synonymous, then, of course, the melting point will not be the temperature a t which insipient melting point occurs, although there would seem to be much in favor of such a usage. Both lenticular areas are crossed by (two-point) tie lines, such as ab, to which the lever law is applicable for closed systems. (The horizontal axis, be it noted, measures not only the composition of every phase present but also the total composition of the systemthis is invariably a trouble spot in teaching the subject.) With reduction of total pressure the diagram would gradually alter by the approach of the two lenticular regions until they would overlap eventually and a three-phase equilibrium would be possible. (Actually in this approach, both areas would drop, but the upper one would drop much faster because of the far greater sensitivity of liquid-vapor phenomena to pressure changes.) One may predict with some confidence that an isotherm for the system (over a restricted temperature range) would have the appearance of an approximate mirror image of Figure 2 with the mirror 126
a t the base AB. For ideal behavior in all three states of aggregation(this is usually assumed anyway in the gaseous state) equations can be found for all of these curves. I t is worth noting that, with this restriction, the lower curve of the upper pair on the isobar is still curved although its counterpart in the isothermal mirror image is straight (if concentrations are expressed in mole fraction), in accordance with Raoult's law. Deviations from ideal behavior can, but do not necessarily produce maxima or minima in one or both pairs giving rise to "azeotropism" in vaporization and/or melting phenomena, although according to its etymology, the term should be confined to vaporization. It may be commented that the generally accepted statement that in distillation the vapor is richer in the more volatile component is not necessarily true when azeotropism is present unless the azeotrope itself is regarded as a component. Whether maxima or minima result is determined by the sign of the difference between the deviations in the two stages of aggregation involved in the phase change, and not by the sign of the deviations themselves ( 2 ) . If, for example, there are positive deviations in the solid state which are much greater than the positive deviations in the liquid, a minimum in the freezing point curve on the isobar will result. The same will be true if there are negative deviations in the solid state which are much less than the negative deviations in the liquid. A reversal of these conditions will give a maximum. It is not correct therefore to assume, as is commonly done, that a minimum in the freezing point curve necessarily implies positive deviations in the solid state and so a tendency to partial miscibility. The same considerations apply in principle to the liquid-vapor equilibria. Here, however, one can assume the vapor to be ideal so that only the deviation in the liquid is the determining quantity. In the foregoing discussion of complete miscibility i t has, of course, been implied that both components are volatile. I t is hardly necessary to say that if only one of these is appreciably volatile we have isotherm and isobar tvoes as in Figures 3 and 4. the line in Figure 3 being straight only for ideal systems.
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" A
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MOLE FRACTION
Figure 3. Isotherm for Only One Volatils (Ideal Behavior)
component
Only compon.nt
Figu=. Volatile 4. Isobar
One
Let us now arrive at a very common type of diagram by combining the two preceding categories to give a two-phase equilibrium in which there is complete miscibility in one phase, say a gas phase, but complete immiscibility in the other, say the liquid state. Both curved and rectangular features now appear. By assuming ideality in all binary phases it is possible to JOURNAL OF CHEMICAL EDUCATION
draw both an isotherm and an isobar from readily available or readily calculated data. It is suggested that having the student construct such a drawing can be very instructive. The isotherm, shown in Figure 5, can be drawn merely from a knowledge of the vapor pressures of pure A and pure B (PA, P i ) a t the given temperature. Point E, which represents the composition and total pressure of the vapor saturated with both (pure) liquids, has a mole fraction of B equal to PB/(PA P i ) , and a pressure equal to P> Pi. Any other point on either curve can be plotted readily as
+
+
+
A
MOLE F R A C T I O N
Pig"?* 5.
B
Isotherm forImmiscib1e Liquid.
MOLE FRACTION Figure 6.
B
1.ob.r £0. Irnmiscibl. Liquids
follows: The curve EPL, for example, represents vapors saturated with liquid B but also containing A. Any point R on it, of mole fraction of B equal to x, must have an ordinate, P, equal to the partial pressure of A, namely (1 - x,) P plus the partial pressure of B, namely Pi. From P = (1 - xl) P Pi it is seen that P = Pilx,. It is, therefore, a simple matter to plot the whole diagram for given values of PA and Pi. (Note that the position of the right-hand curve is quite independent of the nature of the left-hand component and vice versa.) Thus we have the genesis of a diagram type which is prominent in most phase diagram studies. It is safe to say that the analogy between the phase behavior just discussed and that of the familiar eutectic diagram passes quite unnoticed by showing Figure 3 in textbooks merely as the three horizontal lines through E, PI and Pi, respectively, in connection with a discussion of steam distillation. The isobar (Figure 6) can also be easily plotted, ignoring changes of vapor pressure with total pressure. The necessary data are the vapor pressures of both pure liquid components over a range of temperature. The immiscible liquid system water-chlorohenzene at a total pressure of one atmosphere may be chosen for illustration. The handbook gives the following values of vapor pressure: water (A) 100, 400, and 760 mm. a t 52, 83, and 100°C. respectively; chlorobenzene (B) 100, 400, and 760 mm. a t 71, 110, and 13Z°C. respectively. An equilibrium vapor for which the partial pressure of B, P,, is 400 mm. will have a mole fraction of B, x,, of 400/760 or 0.53. If saturated only with liquid B the temperature must be 110'. Similarly a vapor for which PAis 400 mm. (x, = 0.47) and saturated only with A must be at 83". If P, is 100 mm. xB = 0.13 and the temperature must be 71' for such a vapor saturated with B; if PAis 100 mm.
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VOLUME 35, NO. 3, MARCH, 1958
xB = 0.87 and the temperature must be 52' if saturated with A. Figure 6 is a plot of the data. Point E is a t xB = 0.30 and 91°, the vapor composition and boiling point of the constant boiling mixture for one atmosphere Again the curves are independent. Clearly, the bases of the calculation are ideal behavior in the phase in which there is complete miscibility, and the Clausius-Clapeyron equation for the heterogeneous liquid-vapor equilibria for both of the immiscible liquids. The horizontal line through E is conveniently described as a three-point tie line marking the juxtaposition of three areas crossed by two-point tie lines. We are now in a position to reason that the isobaric phase diagram for solid-liquid equilibria, in which there is complete miscibility in the liquid state but complete immiscibility in the solid state will be entirely analogous to Figure 6, as shown in Figure 7. Here, however, the Clausius-Clapeyron equations for both the solid and liquid forms are involved, but by combining them with Raoult's law in the well-known way (3) it can be readily shown that the left-hand curve follows the equation
where AH,, is the molar heat of fusion of A. An approximate form of this relation, arrived a t by confining x, to small values, is invariably dealt with in discussing molecular weight determinations by freezing point depression, hut the connection between the latter and ideal solubility can easily be missed. It is well to make clear, too, that the eutectic is marked by the three-point tie line through E and not merely by the point E. Inspection of Figure 7 gives opportunity to discuss the effect of imuurities on melting points and melting ranges, and to give the basis for the use in the laboratory of "mixed melting point" determinations, purification by partial melting, zonemelting (4, 5), freezing mixtures, etc. Incidentally, the use of an ice-sodium chloride mixture for a freez- T~ ing mixture does not provide the best example in this connection, for the salt forms a dihydrate a t these temperatures (6). It may A rieum 7. Isobar for Imrniscibe pointed out that a shesp ble Salids h u t Miscible Liquids melting uoint is usually, but n i t always, characteristic of a pure compound, for intimate mixtures of the eutectic composition, racemic mixtures, and solid solutions with a minimum melting point all melt sharply. Erasing one of the curves of Figure 7 and rotating the whole diagram through 90° gives the same diagram usually encountered as a solubility curve. Systems with a eutectic also raise an interesting question which has not been answered to everyone's satisfaction: Why should an intimate mixture of A and B begin to melt a t a temperature lower than the melting point of both pure components, even when their volatility is negligible?
I n other words, why should the melting behavior of solid A depend on the presence or absence of solid B? We now consider the effect on the phase diagram of the appearance of partial miscibility. The familiar parabola-like curve enclosing the miscibility gap and seen in the phenol-water system now makes its appearance. If the two components are partially miscible in the liquid state they will probably be immiscible in the solid state, whereas if they are partially miscible in the solid state they will probably be completely miscible in the liquid state. This is because solid miscibility requires greater similarity of the solid components than is required of the liquid components for liquid miscibility. The superposition of a miscibility gap on melting phenomena will give the isobar, Figure 8, when the partial miscibility is in the liquid. There are here two isobaric invariant points a t temperatures T , and T,. Unfortunately, there seems to be no generally accepted name for the second of these. (Metallurgists have, on occasion, referred to such a point as a monotectic, but this term really refers to a special kind of eutectic-one in which the invariant liquid composition nearly coincides with that of one
Figws 8. I s o b u for Partid M i s ~ i b i l i tin ~ the Liquid but lmmissibility in the Solid
Figure 9. h o b o . for Miscibility in the Liquid but Partid Miscibility in the Solid
of the solid phases.) The reality of Figure 8 can be brought out by showing that it describes the phase behavior of "melting under solvent," which refers to the phenomena sometimes encountered when attempts to recrystallize a solid by heating it with excess solvent show that the solid, rather than dissolving, suddenly appears to melt, giving two liquid phases. Figure 8 zlso relates to the laboratory technique known as "wet melting" in which one measures the melting point depression of a solid produced by deliberately covering it with a considerable excess of water. A compound with intramolecular hydrogen bonding (chelation) may frequently he distinguished from an isomer with intermolecular hydrogen bonding (association) by the fact that in the presence of water the latter has its melting point lowered to a far greater extent than the former (7, 8). When the partial miscibility is in the solid state and complete miscibility in the liquid either Figure 9 (eutectic type) or a similar one (peritectic type) results. These will not be discussed here (9). The analogy of Figure 9 to boiling point diagrams of partially miscible liquids (partial miscibility in the liquid but complete miscibility in.the gaseous state) is obvious. The appearance of binary solid compounds with 128
congruent or incongrueut meking points will not be detailed here except as the subject relates t o hydrates in equilibrium with water vapor. It is the writer's opinion that, for instance, the commouly drawn isotherm for a system consisting of a salt and water, with its step-wise character does absolutely nothing t o relate such a phase diagram to any other kinds. The
Figur.
10.
1~0th-m for water and Salt Whish Form. Hydrates
steps usually leave off in mid-air with no accompanying explanation, and the two-and three-point tie lines, common elsewhere, suddenly seem to have vanished. A representation as in Figure 10, however, for a system which forms two hydrates, H I and Hz,although losing some simplicity, gains in every other way. The immiscibility of the solids appears in the rectangular quality of most of the areas. The upper righehand portion derives from Figure 3, modified for non-ideal systems. There are three horizontal three-point tie l i e s . Such a diagram forms an excellent basis for de~cribingnot only efflorescence and hydration in the usual way, hut also deliquescence, as well as heterogeneous equilibria from the standpoint of the mass law. Deliquescence can occur when the partial pressure of water in the vapor above the hydrate is larger than that corresponding to the highest horizontal. The composition of the resultine laver of saturated solution is " given, of course, by that of point Y.
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Figure 11.
Isobar for the System Sodium Chloride-Wster at 240 (Schematic)
at-.
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I t may, perhaps, form an interesting conclusion to a study of binary systems to present an isobar for the system sodium chloride-water. Figure 11 is the schematic isobar for 240 atmospheres (6, 10, 11); on one diagram are included melting, boiling and critical phenomena, miscibility, immiscibility, eutectic and peritectic phase relations, and the boiling point of a saturated solution-all for a system of two familiar components. The dotted lines are only conjecture. The table gives the coordinates of the salient points on the graph. (For the condensed equilibria the coordinates given are actually those for a pressure of 1 atm., hut it is unlikely that they would be greatly different a t 240 atm.)
Point
Cornmsiti& (wt. O/o NaC1)
Temperalure (T.) Point
purity both qualitatively and quantitatively is worth mentioning. Moreover, this diagram relates to the problem of determining whether a given unknown solid, containing only A and B but of variable composition, is a mixture or a solid solution. If a mixture, the composition of a liquid saturated with it would be independent of the solid composition, but not so if a solid
Comuositi& Ternpew(wt. O/o ture NaC1) ("C.)
Figure 14.
TERNARY SYSTEMS
The systematization adopted above for binary systems is not conveniently extended to ternary systems because of the much larger number of possible combinations of phenomena. In any case, there is only time in a college course for a brief reference to ternary phase diagrams. It is perhaps regrettable, for instance, that liquid-vapor equilibria are never described, as the student is likely, sooner or later, to encounter the fractionation of a three-component liquid in the laboratory (12). Only a few of the more deserving practical aspects, however, will be discussed here. The isothermal behavior of two salts with a common ion and water is shown in Figure 12 for a temperature
Iaotherm for Salt (S)-Organic Liquid (0)-wetar (W)
solution. (It is assumed that, if a mixture, not enough solvent has been added to dissolve either solid phase completely.) Figure 13 shows the behavior represented in Figure 12, but a t two temperatures, T and T' (T' > T, normally), and might well be used to describe recrystallization between two temperatures. A composition of impure solid, x, after addition of solvent to bring the total composition to y, becomes homogeneous below T' (after filtering off insolubles), but, on cooling to T, throws down solid A, which is then filtered off. If mother liquor is completely removed one such treatment is sufficient for purification, a t least in theory. Clearly, this would be true only when A "zO SOLVENT and the impurity form no solid solution. The concluding illustrations concern ternary liquid syetems zhowing partial miscibility. The classical example chloroform-water-acetic acid, or a similar one, can he used as an opportunity to refer to the distribution law and process of extraction insofar as the direction of the tie lines under the biuodal curve ie concerned. Procedures such as "salting out" of an organic liquid and the precipitation of a soluble salt A 8 A X IMPURITY by addition of an organic liquid can be illnstrated by rigur.la. 1 ~ 0 t h ~for ~ - TWO F ~ W ~~ ~~ I~~~ . t h ~ ~ ~ n imeans ~ . t noft .Figure 14, which may be imagined t o have Salts with a Common Ion and ing R.cvtalliration Between arisen by the superposition of partial miscibihty on the Wate~ Two Tsmpo~atures salt-water behavior shown in Figure 13, but with a above the freezing point of water. The course of the water-soluble organic liquid replacing the "impurity". curves J K and MK is usually determined by the pheI n Figure 14, for the system salt (S) -organic liquid nomena of common ion effect and salt effect. It may (0) -water (W), the isothermal solubility curves of the he noted, in passing, that theory requires the extensions salt, AB and CD, are separated by a region of partial J K and MK both to enter either the triangle AKB or miscibility bounded by the binodal BKC. Liquid B the fan-shaped areas. Juet as the lever law applies to and C are in equilibrium with each other and also with the tie lines in the fan-shaped areas, so also there i? a solid salt. The "salting out" may be thought of as simple rule by which the relative amount? of solid follows: An appreciable quantity of the 0 present in A, solid B and liquid K can be calculated for total an aqueous solution F is obtained by adding salt until compositions lying within the triangle AKB. For the total composition, always on the line FS, lies wit,hm those interested in analytical chemistry the utility of the triangle BCS and then separating the 0-rich layer this diagram in connection with the recent technique of composition C. On the other hand, the precipitation of phase solubility analysis (IS, 14) for determining of a ~ o h b l esalt from it.s aqueous solution can he visualVOLUME 35, NO. 3, MARCH, 1958
ized by imagining liquid 0 to be added to a saturated aqueous solution of the salt, namely., A. The total composition follows A 0 and the precipitation of the desired salt is seen to occur immediately. Addition of sufficient 0 gives a total composition in the triangle BCS and a second liquid layer appears in addition to the salt-familiar observations t o the practicing chemist. LITERATURE CITED (1) See, for example, DANIELS, F., AND R. A. ALBERTY, "Phyaical Chemistry," John Wiley & Sans, h e . , New York, 1955, p. 16. (2) See, for example, HILDEBRAND, J. H., AND R. L. SCOTT, "The Solubility of Nanelectrolytes," 3rd ed., A.C.S. Monograph Series, Reinhold Publishing Corp., New York, 1950, p. 304. (3) Ref. (I), pp. 215-6.
(4) PFANN, W. G., J . Metah, 4 , Trans., 747 (1932). ( 5 ) CHRISTIAN, J. D., J. CnEM.EDUC., 333 32 (6) "International Critical Tables," Val. IV, 1928, p. 235. T.E. .,Y,E S , J . (7) SIDGWrCK, N, V,, W. J. SPURRELL, Chem. Soc., 1915, 1202. (8) BAKER,W., J. Chem. Soe., 1934,1684. (9) For a discussion of these see P n u r r o ~ ,C. F., AND S. H. MARON,"Fundamental Principles of Physical Chemistry," Revised ed., The Milcmillan Co., New York, 1951, pp. 410-12. Or see Ref. (IS), pp. 164-9. (10) KEEVIL,N. B., J. Am. Chem. Sac., 6 4 , 841 (1942). A,, AND H. LIANDER, Ada Chem. Scandinauica, 4, (11) OLANDER, 1437 (1950). (12) FINDMY,A., A. N. CAMPBELL, AND N. 0. SMITH,'#The Phase Rule and Its Applications," 9th ed., Dover Puhlioations, Inc., 1951, Chap. 15. (13) WEBB,T. J., Anal. Chem., 20, 100 (1948). (14) MADER,W. J., "Phase Solubility Analysis" i n WEISSBERGER, "Organic Analysis," Vol. 4, Interscience Publishers. Inc., 1954, p. 253.
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