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INTRODUCTION TO T H E SYMPOSIUM ON CRITICAL PHENOMENA1 GENERAL COXSIDERATION OF CRITICAL PHENOMENA 0. K. RICE Department of Chemistry, University of North Carolina, Chapel Hill,North Carolina Received December 19, 194.9
In elementary accounts the critical temperature of a one-component system is defined as that temperature above which the two-phase equilibrium between liquid and vapor does not occur. Likewise, in many binary liquid systems there is stated to be a critical solution temperature, beyond which two liquid phases cannot coexist. However, in the past few decades there has accumulated much evidence which indicates that we should speak of a critical region rather than a critical point. A large number of very peculiar phenomena have been observed in the neighborhood of the critical point. Some of these may perhaps be explained as occurring because equilibrium has not been established; some, however, are without doubt true equilibrium phenomena. In spite of much effort, the situation is not yet entirely clarified, from either an experimental or a theoretical point of view. 1. ISOTHERMS KEAR THE CRITICAL TEMPERATURE
One of the problems which is not yet entirely settled has to do with isotherms in the neighborhood of the critical point. Various possibilities for a one-component system are illustrated in figure 1. The three parts of this figure are schematic diagrams constructed to illustrate three types of pressure-volume isotherms which appear to be possibilities. The diagram on the left shows the familiar type of isotherm, which is predicted by the van der Waals theory and which may be designated as the classical diagram. The coexistence curve, which indicates the region in which two phases can be seen, has a simple, approximately parabolic, shape. The middle diagram of figure 1 illustrates the prediction of Mayer and Harrison (4,7) on the basis of their statistical-mechanical theory of condensation. The coexistence curve, representing the region over which a meniscus can be seen, has a horizontal top, rather than being parabolic. However, the coexistence curve is extended, as is indicated by the broken curve. Inside this broken curve the isotherms are supposed to be horizontal, though no meniscus can be seen; and the isotherms do not show a break in slope, as in the ordinary type of condensation, but flatten out gradually as they approach the region where they are entirely horizontal. According t o this theory, a phase having any volume within the broken curve can be in equilibrium with a phase at the same temperature having any other volume within the broken curve. There are thus two critical temperatures: one, the temperature T, at the top of the solid coexistence curve Presented a t the Symposium on Critical Phenomena, which waa held under the auspices of the Division of Physical and Inorganic Chemistry a t the 116th Meeting of the American Chemical Society, Atlantic City, S e w Jersey, September 22, 1919.
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where the meniscus disappears, and the other, the temperature T,at the top of the broken curve, which is the temperature of the last isotherm having a horizontal portion. The right-hand diagram of figure 1 brings out the suggestion of the present author (11). The coexistence curve has a finite horizontal portion, which coincides in this volume range with an isotherm which also has a finite horizontal part. This is the temperature at which the meniscus disappears. Above this critical temperature, called Toin this case, no isotherm has a horizontal portion, but the isotherms do have a relatively flat portion, of gradually increasing slope as the temperature increases. No two phases can be in stable equilibrium with each other above T,. There is evidence that, at least in a number of cases, the classical diagram is not correct. However, the question as to which of the other two diagrams is to be preferred is not easy to settle. I have discussed the evidence elsewhere (12) and shall undertake here to give only a brief r6sum6 of some of the more im-
V
V
V
FIG.1. Possible types of isotherms for a one-component system. Pressure, p , as a funotion of molal volume, V .
portant and more recent experiments. Naldrett and Maass (9) made a study of the coexistence curve for ethylene and for ethyl ether by observation of the meniscus, and found that the meniscus disappeared at a fixed temperature over a range of volumes, so that the coexistence curve appeared to have a finite horizontal portion at the top. I t is true that the meniscus disappeared within the tube (rather than seeming to go out the top or the bottom) over only a portion of this range of compositions, but even this portion definitely included a finite range of compositions. Whether the isotherms which pass just above the horizontal part of the coexistence curve have, themselves, a horizontal portion as claimed by Mayer and Harrison, or whether they have at all points a finite (though near the critical temperature a very small) slope, is more difficult to decide experimentally. Data on ethylene by McIntosh, Dacey, and Maass (3, 5 ) are not entirely unequivocal (11). Data on carbon dioxide by Michels, Blaisse, and Michels (8) would lead one to believe that there is a last isotherm with a finite horizontal portion, and that above that temperature the isotherms have a relatively flat portion which has, however, a finite slope. Unfortunately, the meniscus was not observed in this case.
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Mayer and Harrison pointed to the persistence of density differences above the temperature at which the meniscus disappears, a phenomenon which has been frequently observed, as evidence that the isotherms have a horizontal portion above the last temperature at which the meniscus can be observed. Naldrett and Maass (9, p. 120) have recently concluded, however, that these density differences do not persist indefinitely, and from this we might infer that in a state of true equilibrium the system always has a definite uniform density at these temperatures. If this is the case, the isotherm cannot have a horizontal portion. This matter is, perhaps, not entirely settled at the presont time; however, it is well known that hysteresis readily occurs in the critical region, and one would have to be careful in any event in concluding that density differences, even though rather persistent, represent a state of true equilibrium. In the case of binary liquid systems there are possibilities which are quite analogous to those of one-component systems, as illustrated in figure 1. The three corresponding situations for two-component systems are shown in figure 2,
4
4
A xb B A xb 6 FIG.2 . Possible types of isotherms for a binary liquid system. Partial vapor pressure, pa, of component E , as a function of mole fraction,
zb,
of this component.
where the partial vapor pressure of one of the components is plotted against the composition, in a series of isotherms. These diagrams should be self-explanatory, in the light of the analogy with one-component systems. Usually the coexistence or solubility curve for two-component systems is plotted as a temperature us. composition plot, in which the compositions plotted are those of the two phases which are in mutual equilibrium at any given temperature. A temperature above (or below) which there is only one homogeneous phase is called a critical mixing point. A system will, of course, appear to be homogeneous if no meniscus can be seen, and so these curves will generally be determined by observation of the meniscus. If the coexistence curves plotted in figure 2 have a horizontal portion a t their top (or bottom, in the case of a lower critical point), the ordinary temperature us. composition coexistence curves will also have a horizontal portion at the top. The coexistence curves for binary liquid systems to be found in the literature are often quite flat at the top. Though it is impossible to prove from this kind of data that the curves are actually absolutely horizontal a t the top, I have the impression that in many cases more carefully performed experiments and greater
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purity of the components would result in accentuating the flatness. Some very careful work of Woodburn, Smith, and Tetewsky (16) showed that, in a number of organic binary mixtures, the temperature at which the mixture became homogeneous was practically independent of concentration over a range of concentrations. On the other hand, there do seem to be cases in which there is no reason to doubt the data and in which the top of the coexistence curve is rounded. It appears likely that both types of curves actually occur, and I have given a number of figures to illustrate both cases in a recent article (12); since this is readily available, it seems unnecessary to copy the figures here. Mr. R. W. Rowden, working in my laboratory, is studying the critical phenomena in the cyclohexane-aniline system. In this case he has found that the meniscus disappears near the center of the tube at compositions of 0.480 and 0.453 mole per cent aniline. This would indicate a horizontal portion in the coexistence curve of this system; however, the results are preliminary as yet. We also hope to measure vapor pressures in the critical region. Vapor pressure data and related data, available in the literature, are not sufficient to decide between the diagrams of figure 2. This I have also recently discussed with relevant figures (12). The scattering of light in the critical region is another property which can be used to obtain information on the basis of which one may decide between the types of diagram of figure 2. This will be discussed in Zimm’s paper in this Symposium (17). It may be remarked that Zimm has pointed out that it is useful to plot what is essentially the slope of the curves of figure 2 as a function of the composition, in order to bring out the differences between the various diagrams shown there. Zimm has given figures of this sort for the classical case, and for the type of diagram predicted by Mayer and his coworkers, and has remarked that the latter seems more inherently probable. In figure 3 I have shown the slope of the curves in the right-hand diagram of figure 2; this may be compared with Zimm’s figures l a and lb. It appears to me, however, t o be difficult to assess the inherent probability of this diagram without other considerations. 11. ASSOCIATION THEORY AND CRITICAL PHENOMENA
Mayer and Harrison came to the conclusion, as an inference from certain theoretical conclusions, that the middle diagram of figure 1 would be the correct one. McMillan and Mayer (G) extended these considerations to binary liquid systems, and believed that the middle diagram of figure 2 would very likely be the correct one in this case. However, objections can be raised regarding the range of validity of the mathematical treatment, and it seems quite possible that it would break down in the critical region (1 1). A simpler approach, which leads to the conclusion that the right-hand diagrams of figures l and 2 are correct, lies in the discussion of the molecular association in the vapor phase to form clusters of molecules, an association which may be considered as a kind of incipient condensation (11). Very similar considerations can be used in discussing the binary liquid systems (12); since the
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calculations are so completely analogous, me shall here confine ourselves to the study of the vapor-liquid equilibrium in EL one-component system. By a simple application of the principles of equilibrium, involving minimization of the free energy of the whole system, it may be shown that the concentration in the vapor phase of clusters containing s molecules is given by C. =
exp [ ( p - as
- ys2’a- c,)/kT]
(1)
Here p is the free energy of condensation, per molecule, of single molecules at unit concentration to form a very large drop; a is a coefficient originating as a Laplace multiplier in the minimization of the free energy, but shown to be equal to the free energy per molecule of a very large drop (or the liquid) minus the free energy per molecule of the vapor as it actually exists with all its clusters a t
A
xb
B
FIG.3. Slopes of a set of curves of the type indicated in the right-hand diagram of figure 2. These curves (though slightly different ordinates and abscissas are indicated) can be compared directly with figures l a and l b of Zimm’s article (17). T. is the temperature a t which the meniscus disappears.
the equilibrium concentration; y is proportional to the surface tension; and E* is a correction term which arises because the surface free energy cannot be expressed in the form ys2/3for very small drops, and which may be assumed to approach a constant value as s --t m . When a is large and positive the concentration of large clusters in the vapor is relatively very small, as may be seen by inspection of equation 1. As a decreases (the vapor being compressed at constant T ) ,the relative number of large clusters increases. However, if y is positive, it is seen that, even when a = 0, c a must decrease with s on account of the ys2’3term in the exponential. But once a has become ever so slightly negative, the term as will eventually predominate, as s -+ m , over the term ys2l3.It is thus seen that, as a crosses the value zero, very large clusters become stable quite suddenly. This corresponds to the ordinary process of condensation at a temperature below the critical temperature. According to the classical view of condensation, it mas generally accepted that
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the surface tension would vanish a t the critical point. Both Mayer and Harrison (4,7) and Rice (11, 12) assume that the temperature at which the surface tension vanishes will coincide with the temperature at which the meniscus disappears. I t is especially interesting to consider the behavior of equation 1 when y = 0. In this case the sudden stability of large clusters will not occur. I n fact, it may be shown that a will not become zero until c8 for s + m becomes infinite. This may be readily understood by the following considerations. Let us fix our attention on the reaction 2 clusters of size s F! 1 cluster of size 2s with s very large. If y is positive, such a coalescence of two clusters will occur with a lowering of the surface area, and hence with a lowering of the surface free energy. This tends to favor the coalescence. The coalescence is opposed, however, by the loss of entropy, since the freedom of motion of a pair of clusters is much less when they are combined than when they are separate. At a certain definite concentration the driving force toward coalescence will overcome the entropy effect. This is the point of condensation where a = 0. When y = 0, however, the driving force toward coalescence will not be present, and coalescence will not tend to occur until the free motion of the large clusters is entirely squeezed out at infinite concentration. I t will not be necessary to compress the system to zero physical volume before the coalescence occurs. The concentration cI should be measured in free volume, not ah total volume. Coalescence occurs when large clusters no longer have any room to move without hitting other large clusters. When the surface tension is small, even though still positive, we may expect a certain amount of wrinkling at the surface. Such wrinkling will result on an increase in surface entropy; therefore, if the concomitant increase in surface area does not result in too much increase in surface free energy, a noticeable deviation of the droplets from a purely spherical shape may be expected. This means that the clusters will themselves occupy effectively a larger volume than would otherwise be the case, and the condition of zero free volume will occur at a larger total volume than if the drops were spherical. As a matter of fact, when the surface tension is zero or very close thereto, the degree of wrinkling may become much greater than implied in the preceding paragraph, and the shape of a large cluster may become very complex indeed. Thus, two droplets with depressions and protuberances on them could unite to form a larger droplet containing holes. The final coalescence a t vanishing free volume would result in a spongy mass of liquid-like material surrounded and interpenetrated by gas-like material. It seems probable that the degree t o which the surface can extend itself will be limited by an eventual increase in the free energy if the radius of curvature becomes too small. Otherwise, we should expect the evolution of the surface to go on to the end, in which case clusters consisting of sheets and filaments would be stable rather than thicker clusters of the droplet type. However, a sheet or filament is really a different kind of phase, since it would differ radically in
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molecular energy and entropy from liquid in bulk, and so would be expected to become stable suddenly under certain conditions. This would mean that the surface tension would apparently drop to zero suddenly. Since there is no evidence that this happens we shall awume that we do not have to deal with sheets or filaments, and hence that the curvature of the surface of the droplets is in some way limited. There seems to be no reason why the surface tension should not become negative. The surface tension is a free energy, and may be considered as composed of an energy and an entropy term. Thus we write y = v - T u
(2)
where y is proportional to the surface tension, 9 is proportional to the surface energy (more accurately the surface enthalpy), and u is proportional to the surface entropy. Our view is that y vanishes when 9 = T u ;this occurs a t temperature T,. The term TUhas a larger temperature coefficient than 9 because of the factor T.If the liquid and the vapor actually became identical at T,, then y would vanish because 9 and u each became zero simultaneously. It seems much more likely that y vanishes because of fulfillment of the conditional equality. If this is the case, then it is obvious that we should expect T u to be greater than 9 and hence y to be negative at temperatures above T,. The term u contains a contribution due to wrinkling of the surface, but this has no effect except to lower Tobelow what it would otherwise be. It is clear that it would be impossible t o measure a negative surface tension, and very difficult even to measure a very small positive surface tension, when the wrinkling of the surface becomes important. With a negative y the surface would tend to expand, but the extent of the surface would be limited, if the views expressed in the preceding paragraph are correct, because increase in the extent of the surface would result in increase in the curvature of the surface, and it is assumed that this ultimately results in an increase in the surface free energy. Thus a state of equilibrium would be reached, in which further finite increase in the surface would be resisted, and so in a certain sense one could say that the surface tension had become zero. However, the actual free energy of the surface would still be negative, and the surface would resist even an infinitesimal decrease in area which was not accompanied by a change in average curvature. Since the area of a large cluster would, under conditions of negative surface tension, be limited only by the ultimate resistance against increase in average curvature, we should expect the the surface to become proportional to s rather than s*’~,and the term ys2I3in equation 1 would effectively disappear, being absorbed into the term as. In spite of this,, the idea of negative surface tension is of great importance in the discussion of the isotherms in the neighborhood of the critical temperature. Once the condition of zero free volume has been reached and the large clusters have coalesced, it is seen that the system is still in a state which is far from compact. Compression of the system will result in what might be called a turning inside-out of the spongy mass of liquid-like material. I t will change from a sys-
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tem which has just been formed from, and is just ready to revert to, a vapor with clusters therein, to a system which is just ready to form a liquid containing bubbles of vapor. We assume that this process of compression occurs over a definite range of volumes. For it seems probable that the coalescing droplets have enough wrinkling in their surfaces and sufficient bumps or “pseudopodia” so that the point of effective zero free volume occurs when the volume occupied by the liquid-like material of the clusters is less than half of the total volume. Likewise, in the reverse process of vaporization, coalescence of bubbles in the liquid will occur before the gas-like material of the bubble occupies half the total volume. There will thus be a range of volumes between the points when the clusters coalesce and the bubbles coalesce, the former having the larger volume and also obviously the larger surface, assuming as we have that the extent of surface is delimited by the average curvature of the surface. We now visualize the process of condensation in the critical region as the squeezing out of surface. At To, where the surface tension is zero, this is accomplished with no work and no increase in pressure. Above T,, where the surface tension is negative, the squeezing out of surface requires work, and the pressure along the isotherms increases as compression occurs. Thus we expect isotherms of the type shown in the right-hand diagram of figure 1. These arguments can be adapted to a critical solution region in a two-component system, by replacing surface tension by interfacial tension, and by using a semipermeable piston to compress one of the components. I n concluding this section we may remark that the association theory leads to the conclusion that there are large fluctuations in the density of the system in the neighborhood of the critical point. These are undoubtedly related to, but are probably not identical with, the ordinary type of critical fluctuations, which are considered from various points of view in the papers of Zimm (17) and Tisza (15) in this Symposium. I t seems likely that the effects of the curvature of the surface, which we believe to be responsible for stabilizing the surface at a certain surface area, do not become noticeable until the curvature is considerably smaller than the wave length of visible light. This implies variations in density over a range of distances of this order. On the other hand, the critical fluctuations which result in opalescence appear to approach the wave length of light. I t therefore seems likely that these critical fluctuations are superimposed upon the changes of density which we have considered. The compressibility is very large in the critical region. This is in large part the result of the cluster structure which we have described above and, in turn, allows fluctuations on a larger scale to be superimposed. 111. EFFECTS O F IMPURITIES ON CRITICAL PHENOMENA AND SOLUBILITY CURVES
Several of the papers of this Symposium deal with the effects of an added substance on the critical phenomena in the case of the vapor-liquid equilibrium. The paper of Atack and Schneider (1) is chiefly concerned with the use of the added substance to elucidate the critical phenomena of the one-component SYStem. The papers of Secoy (14) and Boyd (2) have more to do with the effect of the added substance on the coexistence curve. In the latter papers the critical
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point is treated from the classical point of view, a procedure which is probably reasonably adequate for the purpose in mind. Some theoretical considerations of the effect of a third substance on a twocomponent liquid system on a classical basis have been given by Prigogine (10). It seemed worthwhile to consider the effect of an impurity on critical temperature from the point of view of Section I1 of this paper. This has already been started (12), but more detail will be given here. In this section we confine ourselves to the effect on binary liquid systems of the addition of a small amount of impurity. Such an impurity can have two effects on the solubility curve of two partially miscible liquids: it can affect the chemical potentials of the principal constituents, which means that it affects the body of the solution; or it can affect the interfacial tension. To illustrate the possibilities, we shall consider the possible effects of two different kinds of impurities: ( I ) a substance which is only slightly soluble, but much more insoluble in one of the principal constituents than in the other, and is not absorbed at the interface, and (2) a substance which is nearly insoluble in both of the primary constituents, but is strongly absorbed a t the interface.
Impurity of type 1 We shall designate the two principal components, A and B , by the subscripts a and b and the impurity by subscript c. We shall designate the phase rich in component A by a prime ( I ) and that rich in B by a double prime (”). The chemical potentials, which are common to the two phases when in equilibrium, we shall designate simply by p’s without primes. The effect of an impurity on the equilibrium between the two phases is governed by Gibbs-Duhem relations, as follows: 2,” dp.
-4-
d
dpb f
2;
dr. = 0
(4)
where the x’s are mole fractions. Kow we can write, when x: and x; are small: dpo = RT d In xc = R T d In xf (5) These relations will hold even though a change in 2,’ or x; induces exchange of the other constituents between phases. For such an exchange will be proportional to a small change, dx: or dx:, and hence the extra effect on the partial pressure or fugacity of constituent C caused by this additional change in the solution will itself be proportional to dx: or dxf, and the total change in fugacity will thus be proportional to dx: or dx: so long as x: and xf are small. Substituting equation 5 in equations 3 and 4, and solving for dp, and dpb, we obtain:
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Let us now suppose that component C is much more soluble in component A and the phase ’ than in component B and the phase ”, so that dxf can be considered negligible. Then equations 6 and 7 reduce to:
-
Since xb xa xa xb must be positive, since x: > x: and x: > xi, we see that dp. is negative and dpb is positive. Suppose now that we had not allowed transfer of material between the phases while adding dx: to the one phase and dx: to the other. Since dx: is so small there would have been practically no change in d g and dp:. The decrease in dp. and the increase in dpb therefore arise from transfer of component A from phase ” to phase ’, or from transfer of component B from phase ’ to phase ”, or, most likely, both. Thus the effect of the impurity which is preferentially soluble in one of the principal substituents is to make the phases in equilibrium further apart in composition, that is, the branches of the coexistence curve are spread apart. This is in essential agreement with the conclusions of Prigogine. The changes in chemical potential with concentration are very small near the critical temperature. Therefore, in this region a small amount of impurity can be especially effective in spreading out the coexistence curve. In general one will expect the preferentially soluble substituent to be held away from the interface, where the disliked molecule is beginning to be present in appreciable concentration. Hence it should raise the interfacial tension. This is governed by the Gibbs equation, f f f
V
I
-
d r = -r,dp. - radpb Fodpo (10) where y is the surface tension and the ris represent the amounts absorbed a t the surface. Now the concentrations, C. and Cb, of the main constituents, when plotted as a function of the distance T along a line normal to the surface, should look somewhat as shown in figure 4. It is clear that, unless there is to be a region in which the total density is very different from what it is in bulk, a dividing surface drawn to make rozero will make J?b small. On the other hand, there can be a region of the order of 3 X 10-8 cm. near the boundary of phase in which the concentration of impurity is practically zero. Therefore, .’l may be of the order -3 X C: moles/cm.*, where C: is the concentration in moles per milliliter of component C in the phase in which its solubility is greater. It is clear that r. and rb will not approach 3 X Ct, respectively, CA or 3 X in absolute magnitude, unless, as we have noted, there is a region of order 3 X 10-8 cm. in which the overall density is very different from what it is in the bulk of the liquid phases. The Gibbs-Duhem equation may be written in either of the following forms:
++c:C: dpcdpc
0 = ci dr. 4- c: drb 0 = Cz dpc, C6 drb
+
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Adding these two equations:
0=
(c: 4-c.")dpG f (c;
+ c:)
dpb
+ (c: + c:)
dp,
(11)
Solving equation 11 for dp, and inserting the result in equation 10 we obtain: dr =
-[re
- rC(C+ C.")/(C:
+ C:)1 -[rb
dlr.
- ro (c: -k c:)/(c:
+ c:)]
dpb (12)
In view of our discussion of the mgnitudes of Fa,r b , and it can be seen that r. and rb are negligible, or, at least, relatively small in equation 12. Dropping I",, and rb, and restoring dpc by reverse use of equation 11, we see that equation 12 yields: dr E
- redp,
(13)
T
A
%b
FIG.4
B
FIG.5
FIG.4. Concentrations of components A and B as functions of t b distance from the dividing surface. FIG.6. Solubility curves for binary systems, with impurities. Curve I, no impurity; curve 11, preferentially soluble impurity; curve 111, surface-active impurity. We have already suggested (12) the applicability of equation 13 but with less justification than that giyen here. Since dp, = RT d In C. for small concentrations of compound C,we see that the estimate of l', as 3 X le8 C: gives: d r E3 X
RT dC:
(14)
We have already applied this equation to the case of sodium chloride with the phenol-water system. We shall now apply it in the case of water as an impurity in the cyclohexane-aniline system. The solubility of water in aniline is about 5 per cent a t room temperature and about 0.01-0.02 per cent in cyclohexane. Water is, therefore, the kind of impurity we are thinking about. What was estimated as 2 x 10" mole of water per milliliter of solution was accidentally introduced into a critical mixture of cyclohexane and aniline. This raised the mixing temperature 0.2' and caused the mixture to cease being a critical mixture. It is, therefore, clear that the critical temperature was raised by more than 0.2'. Using R = 8 X lo7 ergs per degree per mole, T = 300'K., and dC, = 3 X
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moles per milliliter, we obtain from equation 14: d-y = 0.0015 dyne/cm. This implies that the rate of change of interfacial tension is of the order of 0.01 dyne per centimeter per degree, which is about what is expected (about what is observed for the phenol-water system, for example). One more point deserves mention in connection with this type of impurity. A given percentage of the third component in the total mixture, if added to a two-phase system, should be more effective in raising the interfacial tension the less the amount of the phase in which the third component is more soluble. A coexistence curve with a horizontal section on top should, therefore, be converted into one in which the flat portion on top slopes down in the direction of the component in which the third component is soluble, as is indicated in figure 5 . An effect of this sort has been observed by Schreinemakers (13) with the phenolTyater system and sodium chloride as an impurity. I n this case the effect is still visible, even though he plots his curves for a constant ratio of sodium chloride to water, rather than for constant ratio of sodium chloride to water phenol. Water is more soluble in phenol than is sodium chloride. Hence addition of phenol makes the water layer richer in sodium chloride, and the critical temperature of phenol-rich mixtures (overall phenol concentration higher) is raised more than that of those richer in water.
+
Impurity of type 2 In this case we assume the solubility to e! so small that and f i b are not appreciably affected. The chief effect is a surface effect. re can be much larger in this case. r. and rb will have larger negative values than in the other case, because these constituents will be pushed away from the surface by the impurity. However, the effect of the two latter will not be comparable with that of rc; hence equation 13 holds again. rcis positive, so y is lowered. The interfacial tension of the mixture may be lowered below zero, at a temperaature below the critical temperature for the pure binary system. This means that there will be mutual emulsification. This will go on until so much of component C has been taken into the interface that its concentration in the body of the phases becomes so small that y has risen to zero. At this point emulsification will stop, and two layers will remain of net concentration closer together than the concentration at which they started. 9 smaller change of temperature than when the impurity was absent will now cause one of the layers to engulf the other. Thus the coexistence curve will be lowered a t this concentration. Presumably the temperature at which the one phase engulfs the other depends upon the relative amounts of the two phases, and it should be easier for one phase to be engulfed if the other is present in considerable excess. Thus a flat-topped coexistence curve is rounded off, as indicated in figure 5 . One question remains, Can the line of demarcation between the two phases be observed under the circumstances? If it cannot, this might readily lead to the belief that we can have constant vapor pressure over a range of concentrations of
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a homogeneous solution, when the system is not, in fact, homogeneous. In considering questions of this sort, however, it should be noted that the problem is actually somewhat more complicated than, perhaps, our sketchy account has indicated. The curvature of the surfaces of the emulsified droplets may have important effects on the surface tension and the equilibrium. Many problems in this field must await the future for solution. IV. SUMMARY
1. A general discussion is given of the various possibilities with respect t o liquid-vapor coexistence curves and the isotherms of one-component systems. A similar discussion is also given of the coexistence or mutual solubility curves in binary liquid systems, and the corresponding isotherms. There is a brief consideration of the relevant experimental data in each of these cases. 2. A brief review is given of the association theory of condensation, in which condensation is considered as a limiting case of the formation of clusters of molecules in the vapor phase. The application to critical phenomena is indicated, and it is shown that these considerations lead one to anticipate an isotherm (pressure us. volume in a one-component system) which has a finite horizontal portion at the critical temperature, with isotherms above the critical temperature having everywhere a finite slope. The importance of a correct evaluation of the free volume in which a cluster can move and the significance of the surface tension of large clusters are particularly stressed. 3. The effect of an impurity (small amount of a third component) on the critical phenomena in a binary liquid is considered. It is shorn that the effect of the third component on the interfacial tension of a two-phase system can play a very important role.
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