SIZE. I

Kentucky Color and Chemical Company, Inc., Louisville, Kentucky. Received November 10, 1946. I. INTRODUCTION. When particles of solids become ...
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LAWRENCE HARBURY

3. The results of an attempt t o demonstrate the transitory presence of pentavalent urahium were negative. 4. A mechanism consistent with the experimental results is suggested. REFERENCES (1) ABEL,E.: Z. Elektrochem. 43, 629 (1937). (2) B R ~ X S T EJ. D ,N.: J. Phys. Chem. 30, 777 (1926). (3) BUCHI,P. F.: Z. physik. Chem. 111,269 (1922). (4) CARTER, A. H., AND WEISS,J.: h o c . Roy. soc. (London) M74,351 (1940). (5) DITTRICH, C.: Z . physik. Chem. 29, 449 (1899). (6) FRANK, J., AND LIVINCSTON, R. s.: J. Chem. Phys. 9,184 (1941). (7) FURMAN, H. F., AND SCHOONOVER, I. C.: J. Am. Chem. SOC.63,2561 (1931). (8) HEIDT,L. J.: J. Phys. Chem. 46,624 (1942). (9) LEIGHTON, w. G., AND FORBES, G.s.: J. Am. Chem. SOC.62,3139 (1930). (10) LIVINGSTON, R . S. : J. Phys. Chem. 46, 1312 (1941). (11) LUNDELL, G. E., AND KNOWLES, H. B.: J. Am. Chem. SOC.47,2637 (1925). (12) MCBRADY, J. J.: Ph.D. thesis, University of Minnesota, 1944. (13) McCoy, H. N., A N D BUNZEL, H. €1.: J. Am. Chem. SOC.31, 367 (1909). (14) PARTON, H. N., AND GIBBONS, R . C.: Trans. Faraday SOC.36,542 (1939). (15) PITZER, E. C., GORDON, N. E., AND WILSON,D. A.: J. Am. Chem. SOC.68,67 (1936). (16) RABINOWITCH, E., AND LEHMANN, H. L.: Trans. Faraday Soc. 31,689 (1935). (17) ROLLEFSON, G. IC.: Chem. Rev. 17,425 (1935). (18) ROSCOE, H. E.: J. Chem. SOC.27,933 (1874). (19) THUNBERG, T.: Handbuch der biologischen Arbeitsmethoden,Teil I, Heft 7. Urban und Schwarzenberg, Berlin (1920). (20) WEISS,J.: Trans. Faraday SOC.34,455 (1938).

SOLUBILITY AND MELTING POINT AS FUNCTIONS OF PARTICLE SIZE. I LAWRENCE HARBURY'

Kentucky Color and Chemical Company, Inc., Louisville, Kentucky Received November 10, 1946 I. INTRODUCTION

When particles of solids become sufficiently small, the physicochemical properties of the solids become a function of particle size. Wide differences have become manifest in this field, not only with regard to the nature of this function, but even with respect to the experimental evidence and its interpretation. In view of .the theoretical as well as practical importance of this problem, it seemed worthwhile to reconsider various aspects thereof. This paper is restricted to theoretical and factual considerations of solubility and melting point as functions of particle size. 11. TWO BASIC LINES O F THOUGHT

Two main notions can be distinguished. On the one hand there is the idea particularly followed up by Kossel (9), who emphasized the atomistic or molecu1

Director of Chemical Research.

PHYSICAL PROPERTIES AS FUNCTIONS O F PARTICLE SIZE

191

lar way of stepwise formation (or breakdown, respectively) of a lattice. On the other hand a thermodynamic notion prevails, according to which it is held acceptable that t.he principle of continuity be applied to crystals in equilibrium with their vapor, melt, or solution. 111. APPROXIMATE THERMODYNAMIC RELATION FOR SOLUBILITY AS A FUNCTION

O F PARTICLE SIZE

After Thomson (17) had derived a formula for the vapor pressure of finely dispersed liquids as a function of the particle size of the droplets, an effort was made by Wi. Ostwald (11) to obtain a similar relation for the solubility of small crystals as a function of their size. This formula, corrected by Freundlich and Jones and simplified by Dundon and Mack, takes the following form (when allowance is made for some approximation) :

iRT In c/co = 2Mu/dr

(1)

Here i reflects the increase in number of particles in the solution due to electrolytic dissociation, with proper correction for ionic activity.2

A . Criticism of equation 1 Both the basis upon which formula 1 rests and the way the formula has been used have been the subjects of severe criticism. It may be stated, e.g., that already Gibbs in his Thermodynamics had warned against the assumption that processes of crystal growth or of crystal breakdown (by solution, melting, or evaporation) are of a completely continuous nature. It is not permissible to consider these transformations in the same way as the liquefaction of & gas, or the evaporation of a liquid, or the transition between an amorphous solid and a liquid. Kossel has elaborated this point, and illustrated the same by means of energy figures which, in many respects, are convincing. It may further be noted that in the case of very small particle sizes the surface tension and the specific gravity of the particles will change with their size. When a particle comprises only a few molecules, the distinction between specific free surface energy and lattice energy loses sense. One has to take into account, accordingly, that when the dimensions of small particles are approaching such minute size, this difference between both energy forms will amount to only a small fraction of the “normal value.” Again, in the case of highly soluble compounds, the values for c and co will be high; the dissolved substance no longer behaves in accordance with the osmotic laws which were used when formula 1 was derived. Finally, departures from a spherical shape, as discussed by Jones ( 8 ) ,may give rise to quite marked changes in c/co values. In this formula C O stands for the normal solubility, c for the greater solubility of t h e small particles (assumed t o be spheres with radius T ) , T for the absolute temperature, M for the molecular weight, d for the specific gravity, and u for the specific free surface energy of the solid particle versus the liquid.

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LAWRENCE HARBURY

B. Experimental d i f i u l t i e s There are also serious difficulties of an experimental nature. For example, a genuine influence of particle size on solubility starts to be large only a t values of r escaping ordinary microscopic observation. Conditions for a uniform particle size are almost never satisfied. Grinding gives rise to an uncontrolled frequency distribution of particle size, and to sweeping defects in the interior of the particles, as well as to changes in their surface. These defects entail substantial changes in energy levels. Again, in many investigations attempts are made to determine solubility as a function of particle size, after endeavoring in a conventional way t o separate the liquid from the solid particles. But already in 1903 JaffC (7) had found th& very small particles will readily pass through ordinary filters. This point having all too often been neglected, much too high values for c were published, and ridiculous values for c/co (as a function of particle size) found their way into textbooks. This point has again been stressed by Cohen and Blekkingh (3). Preferential adsorption of ions is another factor which formula 1 does not account for. Such adsorption may result in smaller particles ceasing to be unstable in the presence of larger ones. Up to the present, moreover, we have no reliable means of measuring the free surface energy of crystallized substances against their environment, a point well stressed by Saal and Blot (15). It is obvious that values for u obtained via formula 1 from “measurements” of c/co and r cannot be trusted. Last but not least, small amounts of “impurities” can substantially change the solubility.a Measurements made with chemically and physically pure substances are still the exception. ?

C. Theory of stepwise crystal growth also oversimplified Though we should beware of giving undue weight to some of the above-mentioned objections against formula 1 and its use, there can be no doubt that the notion of a stepwise (molecular) process of aggregation and orientation has more merits. It will be necessary to give more attention to this notion and its consequences in a later communication. We shall then see, however, that even this school of thought tends to represent’things in too simple a way, or was compelled to do so, in order to make calculations of an approximate nature possible.

D. The contrast between the notions of Section II should not be pushed too f a r In spite of all justified criticism it would go too far to deny any merit to formula 1, provided it is appropriately modified and used under well-defined conditions in order to serve as an approximation. As a matter of fact it is not permissible to lose sight of the existence of some real analogy of transformations in liquid-vapor systems to transitions between crystallized solids and their vapors or melts. Debye (4) has pointed out that a liquid can be considered as a quasicrystal. While, on the one hand, the molecules of amorphous substances in 3

Further attention will be given t o this point in a separate paper.

Q

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PHYSICAL PROPERTIES AS FUNCTIONS O F PARTICLE SIZE

some instances are quite tightly held in place, there are many instances in which the crystalline structure is not at all a guarantee of immobility. These facts have been too much neglected by Kossel. There are tendencies toward mobility, if necessary around crystal edges. A striking instance is the tendency of a damaged crystal toward recovery. In the case of ammonium oleate this process is even of such a nature that, after a pinnacle of a crystal is cut off, the rest stretches itself in such a way as t o restore the original shape. This occurs even without any new substance being supplied. For reasons of this nature it makes sense t o beware of exaggerating the contrast between the notions of a continuous process versus a structural stepwise one, as has been indicated above (Section 11). Rather, an attempt will be made in this and later communications to bridge over this contrast.

E. Introduction of the symbol u’, serving as a “catch-all” For this reason it is suggested that u of formula 1 be replaced by the symbol u’, in which all corrections are discounted that are required in view of the aforementioned objections and also of other^.^ We then get:

iRT In c/co

= 2Afa’/dr

TABLE 1 SALT

Potassium nitrate.. . . . . . . . . . . . . . . . . . . . . . . . . Potassium chlorate.. . . . . . . . . . . . . . . . . . . . . . . . Potassium dichromate.. . . . . . . . . . . . . . . . . . . . . Sodium sulfate decahydrate., . . . . . . . . . . . . . . .

Yld

48.1 52.6 109. 220.

cmax./co

ca. ca. ca. ca.

2.75 3.0 6.0 13.0

In cmax./co

+vq5

3.3 3.4 3.4 3.1

As some of these corrections are quite substantial, it should be noted that u’, though having the same dimensions as u, no longer represents the specific free surface energy of a macro crystal lattice.

F. For a variety of inorganic salts with rather complicated structure In cmaz./co is found proportional to the molar surface It has been found that, with proper precautions, quite a variety of substances can form strongly supersaturated solutions. Table 1 gives some instances for a group of inorganic salts with rather complicated molecular structure and a high solubility in water, as compared, e.g., with atomic lattices of pure metals. Its last column shows rather const’ant values for

(in which the denominator stands for the “molar surface”). Assuming i as about equal for these solutions (Noyes), this result is reasonably well in line with what could be expected from formula l a if it is further assumed that the values 4

See, for example, the end of Section IV.

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LAWRENCE HARBURY

of both u‘ and T (or a similar term, representative of the particle size), for which the finest crystalline particles of these salts are in equilibrium with their highly supersaturated solutions, are about the same for all these salts.5 IV. SIZE OF SMALL PARTICLES I N EQUILIBRIUM WITH UNDERCOOLED MELT

For reasons similar to those set forth above, we also think that for undercooled melts some significance should be attached to an expression of the type: v

ATIT, = 2a’/rQd (2) in which T , represents the normal melting point of the pure substance, and AT the range of undercooling which can be observed under appropriate conditions before a marked capacity of spontaneous formation of nuclei of crystallization (germs) in the melt makes itself fe1t.l Of the objections which can be made to relation 2 the following may be mentioned: Again the physical meaning of u’ is not quite definite; in many cases Q is not known; the value of AT cannot be sharply determined.7 Finally, d is not always independent of the representative value of r , the particle size. This objection may also be valid in regard to formula la. V. COMPARABILITY OF GERM FORMATION IN SUPERSATURATED SOLUTIONS AND UNDERCOOLED MELTS

The use of formulas la and 2 for cases of spontaneous crystallization (either in supersaturated solutions or in undercooled melts) has certain advantages inFor potassium nitrate the value of cmax./cg has been derived from a statement of Jaff6, according to which a very carefully prepared solution of 62.9 g. of potassium nitrate in 100 g . of water (after having been heated and cooled fifty t,imes) finally started crystallizing a t around 20.9”C.,when very gradually cooled from a relatively high temperature. For potassium chlorate a statement of Jaff6 has been used (7, page 588) to the effect that a solution saturated with this compound at 54”C., when repeatedly heated and cooled, eventually only started crystallizing below 20°C. A statement concerning potassium dichromate mentions in the same place a C/CO ratio of 5.0, which did not yet represent a maximum ratio of supersaturation. The same applied to Glauber’s salt (7, page 589), where a ratio C/CO = 11 was obtained, though in this case additional stabilization had been aimed at by an improved procedure of purification. For these reasons the values for cmax./c~ which the table mcntions for potassium dichromate and Glauber’s salt are the ratios just mentioned for C/CO increased by 20 per cent. Jaff6 did not pay particular attention t o these ratios, because he was more concerned about the existence of Ostwald’s metastable st,ate and applications of the ‘‘Stufenregel” of his teacher. 6 Q stands for thc normal heat of crystallization, and d for the density of the solid phase. I n the case of cubic instead of spherical particles, r becomes half the edge. For lamellae with a square base and thickness equal to one-fourth of the side, T equals the thickness of the lamella, etc. In view of the fact that, in general, maxima of rates of germ formation and of crystal growth need not coincide, the usc of formula 2 requires, inter diu,that the rate of germ formation have the upper hand. Conccrning spontaneous crystallization within the range T and T A T , see Haber ( 5 ) . 7 This is evident, since this term, in accordance with its definition, corresponds to the temperature range which has to be passed before a “large” number of nuclei is formed per unit of time. Though, in the case of many substances, this number rapidly rises when a certain temperature limit is reached, such a limit cannot be called a strictly sharp one.

-

PHYSICAL PROPERTIES AS FUNCTIONS O F PARTICLE SIZE

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sofar as, e.g., the factor of an undefined grinding effect (with its dubious frequency distribution of particle size and undefined consequences of serious “lattice defects”) does not present itself here. With carefully prepared strongly supersaturated solutions and widely undercooled melts, moreover, the smallest germs (the formation whereof is decisive for spontaneous crystallization) are so extremely tiny that even in solution the influence of a preferential ion adsorption will not easily assume serious proportions.8 In many cases it is permissible to assume the existence of a significant comparability between germ formation in supersaturated solutions and in undercooled melts. It is actually found for many substances, of which strongly supersaturated solutions can be obtained, that their melts also have quite a capacity for maintaining a condition of marked undercooling. In both fields it is found chat, among the molecules of the dissolved or molten substance, there are only a few having the properties required for a decisive contribution to the formation of (new) germs. VI. CALCIUM LACTATE, A TYPICAL INSTANCE

Calcium lactate easily forms in water supersaturated solutions with a marked capacity for maintaining the condition of supersaturation. It was found that the melt of this substance also shows a strong tendency for delayed crystallization, and can be undercooled over a considerable temperature range (AT = up, to ca. 90°C.) before crystallization becomes unavoidable. Such possibilities would be unexplainable if all molecules of the melt, independent of some kind of activation and orientation, could contribute to the formation of nuclei of crystallization. For small aggregates will always be formed frequently, thermal motion and the law of chance bringing about innumerable local density fluctuations. One may infer that of the molecules (ions) of the melt-and mutatis mutandi of the solute-maybe one in 100,000, perhaps even still a smaller fraction, is sufficiently activated, located, and oriented as to be capable of contributing to the formation of aggregates having those particular properties as to structure and activation which allow them to act as appropriate and effective, lasting, and growth-active nuclei of crystallization. VII. APPLICATION O F FORMULA

2 TO UNDERCOOLED MELTS O F 1)-TOLUIDINE

Hinshelwood and Hartley (6) investigated the change in particle size of some organic substances, the particles of which could be considered as being in equilibrium with their melts for various degrees of undercooling. They derived a relation which, for p-toluidine, takes the form:

‘ = 2a’d

1

((19.5

) om.

- 0.000195T2)E

(3)

For an explanation see Haber ( 5 ) . Evidently one would be able t o increase preferential adsorption, e.g., by using an excess of either cations or anions, but i t is assumed that such complications are avoided.

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LAWRENCE HARBURT

in which E stands for the mechanical equivalent of heat. They find the following values for r as a function of A T : A T (in “C.).. , . . . .. . . . . . . . , . , . . . . . , 8.3 r ( i n ~ n i t s o f l O - ~ X 2 u / d c m, ., .). . 1 0 . 2 4

13.3 10.15

18.3 10.11

~

23.3 28.3 0 . 0 8 6 1 0.072

These values for r are practically inversely proportional to AT, but this result could also be expected according to formula 2. VIII. SIGNIFICANCE O F ORIENTATION AND SIZE FOR GERMINATIVE ACTION AND 6’

In a properly prepared, sufficiently undercooled melt (or supersaturated solution, respectively) aggregates of activated and oriented molecules (ions) will be formed spontaneo~sly.~As soon as these aggregates, with more or less loosely packed building stones (molecules, atoms, ions), attain a certain size and sufficient life expectancy, there mill be time available for an internal transformation, under the influence of forces of reorientation,’O Le., for a transition into small crystals with characteristic, though not yet quite normal, lattice. A marked degree of reorientation will be required indeed, before such nuclei can actually serve as potent sources of crystallization. For it is known from experience in the field of recrystallization phenomena, in materials which have been subjected t o mechanical deformation, that even a relatively small lattice deformation markedly decreases the rate of recrystallization. It, has already been mentioned (Section II1,B) that germs may readily pass through ordinary filter papers. This is so even when the material for filtration is iised in thick layers and the process of filtration is repeated many times. Later it WA~Sfound that effective crystallization nuclei may even be able t o pass socalled germ-free filters. Eventually it became clear that an extremely small size of germs must generally be considered effective in bringing to crystallization highly supersaturated solutions or strongly undercooled melts. The inference is inevitable that particles exerting a germ action may be of much smaller size then the “critical” particle size which Wi. Ostwald initially inferred. The greater the degree of supersaturation or of undercooling, respectively, the smaller will be-ceteris paribus-the minimum size of a nucleus which will still be capable of acting as a center of crystallization. At the extreme limits of oversaturation and undercooling, when the size of the smallest effective germ might be termed a “critical” one, the density of such a tiny particle and its specific energy content may markedly differ from normal. Should such a critically small particle comprise only a small number of molecules (ions), the contrast between specific free surface energy and lattice energy may even almost The value of u’ will then be onZy a small fraction of u. vanish (Sect,ion“A). IX. “WALL ACTION”

It may further make a difference whether these small germs are formed inside the carefully prepared solution or melt, or are generated on a solid surface such 9 10

This process may be acconipanied by a certain degree of desolvation. This process too may be accompanied to some extent by desolvation.

PHYSICAL PROPERTIES AS FUNCTIONS OF PARTICLE SIZE

197

as the wall of a test tube or the surface of some dust particle. A tendencx for a preferential formation of ‘‘mall germs” (as compared with the spontaneous generation of “intra liquid germs”) has to be accounted for. A distinction is desirable, accordingly, between a; and u;,l1 and especially so for tiny nuclei in equilibrium with supersaturated solutions. In the case of chemically pure substances, the crystal lattice of which consists of atoms or atom-ions,lZ it is not possible to obtain a permanent state of substantial undercooling,‘a unless one works with extremely thin layers obtained by condensation of a very rarefied (metal) vapor on a deeply cooled wall. For this class of substances, the direction in which the molecules hit each other is almost indifferent for the formation of growth-active germs. I n the case of more complicated molecules (ions), however, it is definitely required for the formation of an active germ with specific structure that the molecules (which have to be taken from the liquid and attached to the aggregate that has eventually to become an active germ) obtain a favorable location, a direction, and activation. A “wall action” may be quite helpful in meeting these requirements. Also, since the stability of a small centre of aggregation and crystallization, adsorbed on a wall, tends to benefit by the adsorption energy, when located on a suitable wall, it may be expected-ceteris paribus-that the smallest size of effective “intra liquid germs” will be larger than that of “wall germs.” X. SOME EXPERIMENTAL EVIDENCE O F A WALL EFFECT ON THE

MINIMUM GERM SIZE

The inference to which we came in the preceding paragraph finds support in some experimental results (as t o minimum germ sizes) obtained by methods for which no knowledge at all either of u or of a’ was required. Thiessen found that the critical size of gold germs14capable of growing in rather strongly supersaturated gold solutions corresponded to a diameter of around 1.1 mp. Such particles, one may imagine, can consist of three lattice layers of 3 X 4 gold atoms each. On the other hand, i t was found by Langmuirls for cadmium and by Reinders and coworkers for silver, that particles with only about three metal atoms adsorbed on a glass wall could act as germs, in one case (cadmium) for supersaturated cadmium vapor, in the other case (silver) for strongly supersaturated silver solutions. The suffixes w and I refer t o “wall germ” and (intra) “liquid germ.” Such is the case with most pure metals or with solids like sodium chloride. As explained by Haber. Supposed t o have the shape of small spheres. l6 This author pointed out that aggregates of two cadmium atoms, adsorbed on a glass wall, will remain adsorbed for a longer time than can be expected of single cadmium atoms, and inferred t h a t a pair of cadmium atoms could function as a nucleus of condensation for cadmium vapor. The published data, however, do not exclude the possibility that actually aggregates of three cadmium atoms, adsorbed on the glass wall, are required for effective germ activity. A corresponding amendment is also desirable in view of the fact that, under the experimental conditions, some mobility has still t o be attributed t o single cadmium atoms adsorbed on the glass wall. l1

12

*

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LAWRENCE HARBURY XI. FURTHER EVIDENCE FOR LOW a’ VALUES

Reinders has described an electrochemical method for the determination of In his laboratory Beukers (1) found values for c/co of over 50 when using an iron malonate developer, acting on germs present in the form of a latent image produced by a yery weak exposure of a film. The developer hence acted on tiny silver germsl6attached to the surface of silver bromide grains which were protected by gelatin. With stronger types of developers, however, one is able t o get a marked development-corresponding to still higher values of c/co-of a so weakly exposed film that the iron malonate developer fails to produce any development thereon. It looks safer, therefore, t o assume a higher limit for C/CO, and to estimate crnax./c~as being of the order of 100.’’ The critical germ here is an aggregate of about three silver atoms of which it may be assumed, on the basis of energy considerations, that they are lined up in a row. This means that r is about 1.3 X cm.18 Substituting M / d = 10.6 cm.3 and i = 1, one finds for a; around 70 ergs/cm2. This value is about one-tenth of the value mentioned in the literature for the specific free surface energy of molten silver versus air. When comparing, more generally, a; values t o be attributed to “critical wall germs” of various substances, it appears that a; actually always amounts to only a small fraction of the u values of inorganic substances in contact with their solutions, as estimated, e.g., by Dundon. Although the said fraction varies more or less from substance to substance, some parallelism between the values of u and a; (in their dependence on the chemical nature of the substances used) does not seem to be entirely lacking. The same applies to a:, as might be anticipated, and to which we shall revert on another occasion. C/CO.

VALUES OF c/co ONLY FOR r VALUES 10-100 1.OR LESS The substitution of low values of u‘ in the afore-mentioned formulas implies that c/co (or A T , respectively) can assume large equilibrium values only in tke presence of particles that have a representative size r of the order of 10-100 A. or less. This is so in spite of statements to the contrary in the literature, which are based on faulty experimental procedures or unacceptable assumptions. This position, further, is in full agreement with an experimentally verified absence of large c / c , values for pure, finely divided and dispersed substances, as ascertained by Cohen and Blekkingh, Jr. Before determining c, these authors took care to remove from the filtrate small particles passing ordinary filters. However, in the case of damaged lattice surfaces which may be obtained, e.g.,

XII. MARKED INCREASE OF EQUILIBRIUM BELOW

18 Within the scope of this section i t is immaterial that part of the germs may have contained silver sulfide instead of silver, i t being known that both silver sulfide and the metal are able to act as centres of crystallization for silver atoms. 1 7 If one prefers to substitute C / C O = 50, a value of around 60 is found for u’ instead of 70. 1s One might consider the needle of three silver atoms as a cylinder with a radius equal t o half the diameter of a silver atom.

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199

by grinding processes of a sufficiently sweeping nature, it is possible-within a somewhat larger range of particle size-to obtain values for c/co that substantially, though only temporarily, surpass unity. But then C/CO does not represent an equilibrium value. XIII. SUMMARY

The formula (formula 1) of Ostwald-Freundlich-Dundon is unsuitable, even as a first approximation, for indicating relative solubility (C/CO) as a function of particle size r , unless u is replaced by u’. This quantity u’ (which at lower values of r is no longer independent of the particle size) should be considered as a “catchall” for a set of important corrections which are needed in order to compensate for various sources of error. Especially for extremely small values of r, u’ is only a small fraction of u . Though having the same dimensions as u (ergs/cm.”, u’ has no longer the physical meaning of a specific free surface energy. Working with solutions of pure, easily crystallizing substances in contact with small crystals having an undamaged surface, one can only expect high equilibrium values of:/co (say ’> 2) when the representative particle size T is of the order of 10-100 A. or less. For a group of substances such as potassium nitrate, potassium chlorat,e, potassium dichromate, and sodium sulfate decahydrate and using water as solvent, it is found that In cm.Jc0 is proportional to the molar surface. This is in accordance with what could be expected theoretically, if both the number of molecules present in the smallest crystallization nuclei of such substances (in equilibrium with their highly supersaturated solutions), and their. u’ values, are approximately equal and small. Similar remarks apply to undercooled melts. REFERENCES

(1) BEUKERS, M. C. F.: Thesis, Delft, 1934. (2) BLEKKINGH, J. J. A., JR.: Thesis, Utrecht, 1938. J. J. A., JR.:Proc. Acad. Sci. Amsterdam 30, 158 (1936). (3) COREN,E., AND BLEKKINGH, (4) DEBYE,P.: Physik. Z. 36, 100 (1935). (5) HABER,F.: Ber. 66, 1731 (1922). (6) HINSHELWOOD, C. N., AND HARTLEY, H.: Phil. Mag. [SI 43, 78 (1922). (7) J A F FG.: ~ , Z. physik. Chem. 43, 565 (1903). (8) JONES, W. J.: Ann. Physik. [4] 41, 441 (1913). (9) KOSSEL,W. : Die molekularen Vorgange beim Kristallwachslum. Leipsig (1928). (10) LANGMUIR, I.: Proc. Natl. Acad. Sci. U. S. 3, 141 (1927). (11) OSTWALD, WI.: Z. physik. Chem. 34, 503 (1900); Lehrbuch der allgemeinen Chemie, Leipsig (1896-1902). (12) REINDERS, W.: J. Phys. Chem. 38, 783 (1934). (13) REINDERS, W., AND HAMBURGER, L.: Z. wiss. Phot. 31, 32 (1932). . (14) REINDERS, W., AND D E VRIES, R . W. P . : Rec. trav. chim. 66, 985 (1937). (15) SAAL,R. N. J., AND BLOT,J. T. F.: Physica 3, 1099 (1936). (16) THIESSEN, P. A.: Z. anorg. allgem. Chem. 180, 10 (1929). W.: Phil. Mag. [41 42, 448 (1881). (17) THOMSON,

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