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a t a temperature very close to, if not identical with, the critical temperature. Since similar behavior seems to be found experimentally in other systems (5,l l ) , it would mean that these relations among the coefficients must hold regardless of the details of the intermolecular forces. Our experiments were designed to answer experimentally the question of whether the theoretically expected second-order transition can be found. The answer, at least as concerns the system studied, seems to be a clear negative. Xot the slightest evidence was found for any anomalous behavior near the critical point. The results then pose a difficult problem for the theorist,-the problem of explaining why the predicted second-order transition fails to appear. REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
EINBTEIS,A , : Ann. Physik 33, 1275 (1910). FOWLER, R. D., et a l . : Ind. Eng. Chem. 39, 375 (1947). HILDEBRAND, J. H., AND COCHRAN, D. R . F.: J. 4 m . Chem. SOC.71, 22 (1949). KEESOM,W. H.: Ann. Physik 56, 591 (1911). KRISHNAN, R. S.: Proc. Indian Acad. Sci. 2, 21 (1934); 6, 577 (1937). MASON,Y. G., AND MAASS,0.: Can. J. Research 26B, 592 (1918). MAYER,J. E.: J. Chem. Phys. 6, 67 (1937). MCMILLAK, W. G., AND MAYER,J. E.: J. Chem. Phys. 13, 276 (1945). ORNSTEIN,L. S., AND ZERNIKE,F.: Physik. 2. 27, 261 (1926); 19, 134 (1918) RICE,0. K : J. Phys. & Colloid Chem. 64, 1293 (1950). ROWSET,-4.: Snn. phys. 6, 5 (1936). SMOLUCHOWSKI, M.: Ann. Physik 26, 205 (1908). ZIMM,B. H . : J. Chem. Phys. 16, 1099 (1948).
T H E THEORY O F CRITICAL POINTS', L. TISZA Deportment of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts Received December 19, 1949 I. IhTFiODUCTION
The existence of critical points was discovered by Andrews (1) in connection with his classical investigation of the carbon dioxide isothermals. Andrews was mainly concerned with the nature of the liquid, vapor, and gaseous phases of matter. His main conclusion and at the same time the definitive solution of the question was that there is only a single fluid condition of matter which can split into two coexisting forms of different density, provided the temperature is below 1 Presented a t the Symposium on Critical Phenomena, which was held under the auspices of the Division of Physical and Inorganic Chemistry a t the 116th Meeting of the American Chemical Society, Atlantic City, Kew Jersey, September 22, 1949. * This work waa supported in part by the Signal Corps, the Air Materiel Command, and the Office of Naval Research.
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a certain critical value. Gases appear “permanent,” that is, resist liquefaction, if compressed at a temperature above the critical. This interpretation provided a powerful stimulus for reaching lower temperatures than were available at that time and generally cleared the ground for the development of low-temperature physics. Yet this was only one of the repercussions of the discovery of critical points. It may be noted in the first place that the term “critical” seems to have been dictated by an unusual intuition. Originally referring only to the possibility of liquefaction of a gas, it turned out that the critical points of the phase diagram were “critical” also in a more profound sense of the word. It was realized that critical points have a certain anomalous nature; they prove rather elusive to routine thermodynamic investigations. Equilibria under critical conditions are extremely slow to establish themselves and prove very sensitive to the minutest perturbances. All this concurs in making it hard to obtain reproducible results. Perhaps the most spectacular among critical phenomena is the enormous amount of light scattering, the so-called critical opalescence. In 1908 Smoluchowski (11) gave his famous interpretation of this phenomenon, tracing it to abnormally large molecular fluctuations. This was at the same time the key to the above-mentioned peculiarities of the critical point. In fact, the smallness of molecular fluctuations is a prerequisite for the use of thermodynamic methods in general, since otherwise there would be no assurance that the molar quantities are adequate substitutes for the molecular variables of which they are the averages. The standard method of statistical mechanics for computing fluctuations makes use of the method of canonical ensembles. By applying this to the critical point one arrives at the result that the fluctuations tend to infinity. In reality, the fluctuations are unusually large, but obviously, finite. This means that at the critical point both thermodynamic and statistical methods reach their limits of applicability. The significance of the breakdown of these methods of great generality is amplified by the remark that critical phenomena are by no means limited to the type found in one-component fluid systems. A possible generalization was mentioned in the first investigation of Andrew (1): one might expect a critical point with respect t o the solid-liquid equilibrium. A search for such a critical point has consistently led to negative results. In fact, cogent theoretical arguments have been advanced in favor of the view that such a critical point is not possible (e.g., 3). Quite analogous phenomena have been observed, however, in two-phase systems of liquids of limited miscibility. Most of the measurements on critical opalescence have been carried out at such points of critical mixing. A somewThat more hidden variant of critical phenomena is provided by ferromagnetic and ferroelectric Curie points and the so-called A-points which are found in a large number of solids (order-disorder transitions in alloys and molecular crystals). This analogy seems to have been first pointed out by Onsager and Kaufman (4,5 ) . Subsequently, Semenchenko (10) tried to substantiate this idea by reformulating Ehrenfest’s relation for second-order transitions for cases of ferromagnetism, ferroelectricity, and related phenomena.
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The question has been reopened by the author (12), who has shown that Ehrenfest’s approach is not suited for a unified theory of critical points and Apoints. According to Ehrenfest, the derivatives of the thermodynamic potential, say the specific heat, are discontinuous along the A-line, but they are presumably quite regular within the tlyo phases. On the other hand Onsager (4, 5 ) has shown that, a t least in one case where exact calculations were possible, the specific heat tends to infinity on both sides of the A-point. The unified thermodynamic theory of critical points and A-points (12) has been huilt on the idea of stability. There arises the possibility of singular points in the phase diagram in which the thermodynamic stability of the system is the lowest which is compatible with the principles of thermodynamics. These singular points can be identified without ad hoc assumptions with the critical points and A-points. The study of critical points is also of interest from the point of view of statistical mechanics. Vnder critical conditions the standard method of canonical ensembles leads to infinite results and needs modification. h modification of a very general type has been carried out by Klein and Tisaa (2). After a survey of the older theories of critical fluctuations in Section 11,the new method of calculation will be outlined in Section 111. 11. T H E THEORY O F FLUCTUATIONS
As indicated in the introduction, the central problem connected with critical points is the calculation of fluctuations. Hence we shall consider a short summary of the classical theory of fluctuations in general and a survey of the attempts to deal with critical fluctuations in particular. Consider a closed system in thermodynamic equilibrium within an adiabatic envelope and isolate a small subsystem (briefly a cell) from the large residual system (briefly the reservoir) by (i) adiabatic or (ii) diathermal walls. In case ( i )both the energy and the temperature of the cell are constant; in case (ii) the energy is free to pass betveen cell and reservoir; the temperature of the cell is equal to that of the reservoir. It is an important experimental fact that there is no observable difference in the behavior of the cell in the two cases. This means that the fluctuations of the energy, present in case (ii) alone, do not significantly alter the properties of the system. In the language of statistical mechanics the two cases correspond to the microcanonical and canonical ensemble, respectively. The smallness of the fluctuations assures the equivalence of these methods. The preceding discussion can be easily generalized by replacing the fluctuation of the energy by that of some other extensive variable, such as the number of particles, the volume, or some electric or magnetic quantity. One speaks in these cases of grand canonical, or more simply (generalized) canonical ensembles. -1quantitative discussion of the fluctuations is based on a fundamental relation of statistical mechanics according to which the probability $7’ of some fluctuation is given by 9- e W ’ k * (1) n-here W is the minimum vork required to bring about the fluctuation from equilibrium.
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Considering] e.g., fluctuations of volume at constant temperature, one has (see, e.g., reference 11):
W = $a(AV)z
(2)
with
where p is the isothermal compressibility. The higher-order terms in the fluctuations AV can be normally neglected in equation 2. One obtains for the relative fluctuations:
For an ideal gas, p = 1/P; hence expression 4 becomes 1/N when N is the number of molecules in the cell considered. Thus the fluctuation is very small under ordinary conditions. It is even smaller for liquids, because of their smaller compressibility. On the other hand, at the critical point both compressibility and fluctuations become infinite. The phenomenon of critical opalescence shows that the fluctuations are indeed very big, but they of course cannot be infinite and the problem is to refine the theory so as to give finite answers. A considerable number of theories have been proposed which were to achieve this end. The analysis of the calculation of the fluctuation shows clearly that the infinite result stems from the fact that the work W required to bring forth the fluctuation vanishes at the critical point. Any additional contribution to W , however negligible under normal conditions, would imply a nonvanishing W and a finite expression for the fluctuation. Hence most theories can be classified according to the additional term which they assume in computing the minimum work connected with the fluctuation. We shall briefly enumerate the main theories : (i) Smoluchowski (11) remarked that equation 2 is only the first term in an expansion in AV. If the expansion is continued up to the quartic terms, a nonvanishing value is found for W . (ii) The extensive investigations of Ornstein and Zernike ( 6 ) are based on the idea that the instantaneous values of the fluctuations of density in neighboring volume elements of a fluid are not independent. The direct energetic interaction gives rise to a term in the work W and also to a correlation of fluctuations in distant volume elements. The spread of correlations becomes very big a t the critical point and implies a finite value for the critical opalescence. (iii) Also, Rocard (8) emphasized the inhomogeneity of the fluid because of fluctuations and pointed out that the local pressure can no longer be assumed to depend on the local density alone, but will depend also on the iicurvature” V2p of the density. This dependence provides the required additional term in the work W . The additional pressure can be conceived to be of capillary origin. In
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fart, Rocard succeeded in developing on this basis a molecular theory of capillary forces (9). (iv) Yvon (13) called attention to the importance of gravitational forces near the critical point. In view of the extreme compressibility of the system the gravitational field is sufficient to bring forth noticeable gradients of density. Thus the gravitational work provides a contribution to W . It is important to realize that the above theories are by no means incompatible with each other and more than one factor may contribute a term to the work W . The clarification of the interrelation of these theories made the reopening of the question desirable. A further reason was that the above theories refer uniquely to the gas-liquid critical point, whereas critical phenomena have a more general scope, as was pointed out in the introduction. The theory of Ornstein and Zernike proved to be the most convenient point of departure for the more general “cellular method’’ to be discussed in Section IV. I t is noteworthy that the results of this theory contain both Ornstein and Zernike’s and Rocard’s results as special cases. In this general theory neither the gravitational effects nor the quartic terms of Smoluchowski have been taken into account. Although gravitational effects are no doubt important in the case envisaged by Yvon, they cannot be relevant for A-points in solids. Smoluchowski’stheory seems too special, inasmuch as spatial inhomogeneities of the system are not taken into account. It was also pointed out by Rocarda that the use of the quartic terms leads to a critical opalescence which is not proportional to the volume. 111. THE CELLULAR METHOD
From the point of view of physical intuition it is obvious that the fluctuation in a “cell” which is part of a closed system must be finite, e.g., the density can vary only between zero and a maximum which occurs if the entire mass of the system is concentrated in the cell in question. Briefly, the conservation of mass, energy, volume, or whatever other extensive quantity we may be considering is sufficient to assure a finite value for the fluctuation. The difficulty is only that the conventional theory of canonical ensembles does not provide the possibility of introducing the conservation principles in any formal way into the theory. Hence Klein and Tisza (2) have generalized the concept of a canonical ensemble by introducing the “cellular method.” Instead of concentrating attention on a subsystem (a single cell) and schematizing the rest into a “reservoir,” the whole system is divided into cells which are treated on an equal footing. These cells are chosen as identical in size and shape and arranged in a regular spatial array. If dealing with a fluid system, one can choose a simple cubic array of cubic cells. For a crystalline system, it will be natural to let the array have the symmetry of the crysta1;‘the cells can then consist either of unit cells of the crystal or of groups of coytiguous unit cells. Whereas in the old theory the states of the cells are statistically independent, 3
Personal communication
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in the new theory an interaction depending on the instantaneous states of contiguous cells has been introduced. This remedies the above-mentioned defect of the old theory and makes it possible to give a simple analytic expression to the conservation of extensive quantities. It leads also to a correlation between the fluctuations of various cells. This correlation may extend to distant cells, particularly at the critical point. The theory leads to finite values for the critical fluctuations and this result is general enough to include A-points in solids. In the latter case the fluctuating quantity may be the magnetization, the polarization, or the long-range order. In particular for fluid systems, one gets for the critical light scattering a result of the form:
Io
-1 A?
(1
);
+ cos26’) + c + d2
-1
sin
Here I l l 0 is the ratio of scattered to incident light of wave-length A observed in the direction 8; p is the compressibility, which becomes infinite at the critical point; c and d are two constants which cannot be calculated without making more specific assumptions. For c = 0, equation 5 reduces to the result of Ornstein and Zernike ( 6 ) ;for d = 0 to that of Rocard (8). Qualitatively one can understand the origin of these terms as follows: Rocard’s constant c stems from the fact that the pressure in a cell depends not only on the local density but also on the instantaneous density of the neighboring cells. On the other hand, the term containing d originates in the fact that the correlation of the fluctuations in different volume elements implies phase relations for the scattered light. This correction gives rise to a 1/A2 wave-length dependence of the scattering and an enhancement of the forward scattering at the critical point. Assuming c = 0, the critical scattering would still be infinite for 6’ = 0. Placzek (7) has shown that taking the finite volume of the scattering medium into account introduces a factor preserving the finiteness without modifying the result for experimentally realizable situations. The presence of Rocard’s c term renders of course the same service. IV. DISCUSSION AXD SUMMARY
The discussion of thermodynamic stability leads naturally to the possibility of critical points as boundary points between ranges of stability and instability. The latter states are not realized in nature, and the system escapes instability by breaking up into two or more phases. Critical points in this general sense include A-points and Curie points in crystals. These points are all “critical” with respect to the applicability of the standard thermodyhamic and statistical methods. Thus the method of canonical ensembles leads to infinite results for critical fluctuations. The so-called cellular method has been developed to cope with this situation. With appropriate special assumptions the results of the general theory
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reduce to those of the theories of Ornstein and Zernike and of Rocard, respectively. The general methods used so far are not capable of deciding a priori which of these limiting cases is closer to reality. The measurement of the dependence of critical opalescence on wave length and angle of scattering could throw some light on this issue. The experimental difficulties under critical conditions are very considerable and at present the evidence (see reference 2) is inconclusive. Finally, it might be noted that theoretical results obtained with canonical (or grand canonical) ensembles cannot be extended without further analysis to the critical point. REFERENCES (1) (2) (3) (4) (5) (6) (7) (8)
(9) (10) (11) (12)
(13)
ANDREWS, TH.:Phil. Trans. 169, 575 (1869). KLEIN,hl. J., AND TISZA,L.: Phys. Rev. 76, 1861 (1949). LANDAU, L.: Physik. Z. Sowjetunion 11, 545 (1937). ONSAOER, IARS:Phys. Rev. 66, 117 (1944). ONSAGER,LARS, AND KACFMAS, BRURIA : Physical Society Cambridge Conference Report, p. 137 (1947). ORNSTEIN, L., AND ZERNIKE,F.: Proc. Acad. Sci. Amsterdam 17, 793 (1914); 18, 1529 (1916); 19, 1321 (1917); Physik. Z . 19, 134 (1918); 27, 761 (1926). PLACZEK, G.: Physik. Z. 3 1 , 1052 (1930). ROCARD, Y.: J. phys. [7] 4, 165 (1933). ROCARD, Y: J. phys. 171 4, 533 (1933). QEMENCHENKO, V. K . : J. Phys. Chem. Acad. Kauk U.S.S.R. 21, No. 12 (1948). The author is indebted to D r . D . Atnck for calling his attention to this paper. SMOLUCHOWSKI, 31.V.:Ann. Physik 141 26, 205 (1908). TISZA,L . : Sational Research Council Symposium on Phase Transitions, held August 1948 a t Cornell University; R.L.E. M.I.T. Technical Report S o . 127. YVOK,J . : dctualites scientifiques et industrielles, Nos. 542, 543. Hermdn et Cie, Paris (1937).
SOLUBILITY MEASUREMEXTS IX THE CRITICrlL TEMPERATURE REGION. I',* D . ATACK
AND
W . G. SCHKEIDER
National Research Council, Ottawa, Ontario, Canada Received December 19, 1949 IXTRODUCTION
During the course of the last two decades a considerable amount of attention has been given to the nature of the liquid-vapor phase transition occurring when either single or multicomponent systems are taken along their critical isochores (1-6, 9, 10, 12-18). Presented at the Symposium on Critical Phenomena, which was held under the auspices of the Division of Physical and Inorganic Chemistry at the 116th Meeting of the American Chemical Society, Atlantic City, Sew Jersey, September 22, 1949. a Contribution KO.2239 from the Sational Research Council, Ottawa, Canada.
