GIBBS O S ADSORPTIOS BY W I L D E R D. B A X C R O F T
There is a footnote on adsorption by Gibbs,‘ which seems not to have received the attention which it deserves. “If liquid mercury meets the mixed vapors of water and mercury in a plane surface, and we use kl and p 2 to denote the potentials of mercury and water respectively, and place the dividing surface so that ri = 0, Le., so that the total quantity of mercury is the same as if the liquid mercury reached this surface on one side and the mercury vapor on the other without change of density on either side, then rl(l) will repre@entthe amount of water in the vicinity of this surface, per unit surface, above that which there would be, if the water-vapor just reached this surface without change of density, and this quantity which we may call the quantity of water condensed upon the surface of the mercury) will be determined by the equation 1’2(1)= -da’dp?. (In this differential coefficient as well as the following, the temperature is supposed to remain constant and the surface of discontinuity plane. Practically, the latter condition may be regarded as fulfilled in the case of any ordinary curvatures.) “If the pressures in the mixed vapors conform to the laws of Dalton, we shall have for constant temperature dp2 = y2dp2, where p2 denotes the part of the pressure in the vapor due to the water-vapor, and y?,the density of the water-vapor. Hence we obtain r2C1) = -yda dp?. For temperatures below 100’ centigrade, this will certainly be accurate, since the pressure due to the vapor of mercury may be neglected.” If we postulate that water vapor is described by the simple gas law, and if we write z for the amount of water vapor adsorbed per unit surface of mercury, the first equation becomes z = - d a / R T dlog p?. If the variation of the surface tension at constant temperature is proportional to the change in the amount adsorbed, we can write d a d s = -nRT. Substituting in the preceding equation, we have ndz ’x = R T dlog p which integrates to Freundlich’s equation xn = kp. In so far as Gibbs is right this is the condition for the absolute accuracy of the Freundlich equation. If du, d s is not a constant for constant temperature, the Freundlich equation will not hold. Furthermore, for the first time, we learn the physical significance of the esponent n. IVith a liquid adsorbent, is a measure of the change of the surface tension with the amount adsorbed. :cientiF.c ’ Papera,” 1, 235 (1906)
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The data of Patrick’ on the adsorption of mercurous sulphate from solution by mercury can be represented moderately well by Freundlich‘s equation, Table I.
TABLE I Formula x5 6 = k c where x = amount adsorbed; c = equilibrium concentration, both in niillimols per liter; k = 0.0046 (459). X
c found
c calc
0.10;
0.000
0.190
0.020
0.020
0.261
o.oj4
0 .I 1 8
0.31j
0,315 0.692
0.321
0.358
0.0OI
0.692
The third figure is not as much out as it appears to be. Changing the amount adsorbed from 0.261to 0.240-1,a change of less than ten percent would bring the concentration up to 0.75-0.74 and would make it fit exactly with the formula. This point lies off the smooth curve as Patrick has drawn it and is therefore in error to some extent. Patrick gives a curve and not the data for surface tension against equilibrium concentration. Guessing at the surface tension values from the curve and plotting them against the amounts adsorbed gives pretty close to a straight line. The real curve cannot be a straight line because the straight line, when prolonged, does not come anywhere near the surface tension of pure mercury for zero adsorption. This means that the Freundlich equation with these constants only holds over a portion of the adsorption curve, which is all that anybody expects of the Freundlich equation. Patrick’s data for salicylic acid, picric acid, and new fuchsine cannot be represented with any satisfactory degree of accuracy by the Freundlich equation, though they give a smooth curve, indicating adsorption. The data of Iredale* for the adsorption of methyl acetate vapor by mercury cannot b? used because he did not determine the amounts adsorbed. Years ago Freundlich3 started from exactly the same equation, but substituting concenc 6u trations for pressures. He wrote u = - - - where u is what I have R T 6c’ called x. He wished to get the adsorption in the surface of a solution and postulated an empirical formula (ubi - q,)= sc’ where uA1is the surface tension of the pure liquid, uL the surface tension of the solution, and s is a s ‘-1 and u = CI = constant. From this he deduced du’dc = - - c nRT n (uc””, where a is a new constant. This deduction rests of course on the validity of the empirical equation connecting the surface tension and the conZ.physik.Chem.,86,545(1914). 2
Phil. Mag. (6)45, 1088 (19~3). “ICapillarchemie,” 51,65, jj (1909)
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centration. This is admittedly not accurate. Since u J ~is a constant at constant temperature, the empirical equation can of course be written’ 41f
- UL - S I C ’
us1
On p. 61 of the English translation, Freundlich says: “A theoretically well-founded formula which gives the connexion between u and c does not at present exist. Its deduction, for instance, from the van der Waals theory of liquids is no simple matter. We only need to consider that it has hardly been found possible to represent a single property of a solution-say the compressibility or density-by means of this theory. Too little is indeed known about the extent to which the attractive force acting between unlike molecules (the quantity a12 of van der Waals) must be taken into account, and the attractive forces a1 and a2 between like molecules; and, further, we know too little about the degree and nature of hydration. In other words, the lyotropic properties of solutions cannot yet be expressed quantitatively, and referred to simple quantities. An empirical formula, which gives the relation between u and c very satisfactorily, is due to von Szyszkowski.2 I t runs thus:
when uIl and uL have the significance previously given and b and c are constants.” While this is true, it resembles the flowers that bloom in the spring because one can eliminate practically all the difficulties to which Freundlich refers, by sticking to the case, cited by Gibbs, of the adsorption of a gas by mercury. The connection between Freundlich’s relation and the one I have used is quite simple. If we assume that Freundlich’s empirical equation and his adsorption equation hold simultaneously for small changes of surface tension, then uxf - uL = sc’ = kix, this is equivalent to du/dx = constant, which is the condition, according to Gibbs, for Freundlich’s equation being accurate. The relation between surface tension and adsorption is for a liquid adsorbent and of course we are usually more interested in adsorption of a gas or a solution by a solid. Eo far, I have not been able to find any definite statement in Gibbs that a similar relation holds for solids; but he certainly implies that there is a great similarity between the two cases.3 “We have hitherto treated of surfaces of discontinuity on the supposition that the contiguous masses are fluid. This is by far the most simple case for any rigorous treatment, since the masses are necessarily isotropic both in nature and in their state of strain. In this case, moreover, the mobility of the masses allows a satisfactory experimental verification of the mechanical conFreundlich: “Colloid a n d Capillary Chemistry,” 66 (1927).
* Z. physik. Chem., 64,385(1908). 3
“Scientific Papers,” 1 , 3 1 4 (1906).
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WILDER D. BANCROFT
ditions of equilibrium. On the other hand, the rigidity of solids is in general so great, that any tendency of the surfaces of discontinuity to variation in area or form may be neglected in comparison with the forces which are produced in the interior of the solids by any sensible strains, so that it is not generally necessary to take account of the surfaces of discontinuity in determining the state of strain of solid masses. But we must take account of the nature of the surfaces of discontinuity between solids and fluids with reference to the tendency toward solidification or dissolution at such surfaces, and also with reference to the tendencies of different fluids to spread over the surfaces of solids. “Let us therefore consider a surface of discontinuity between fluid and a solid, the latter being either isotropic or of a continuous crystalline structure, and subject to any kind of stress compatible with a state of mechanical equilibrium with the fluid. We shall not exclude the case in which substances foreign to the contiguous masses are present in small quantities at the surface of discontinuity, but we shall suppose that the nature of this surface (Le. of the non-homogeneous film between approximately homogeneous masses) is entirely determined by the nature and state of the masses which it separates, and the quantities of the foreign substances which may be present. The notions of the davidzng surjace, and of the superficzal densities of energy, entropy, and several components, which we have used with respect to surfaces of discontinuity between fluids (see pages 219 and 224), will evidently apply without modification to the present case. We shall use the suffix with reference to the substance of the solid, and shall suppose the dividing surface to the determined so as to make the superficial density of this substance vanish. The superficial densities of energy of entropy, and of the other component substances may then be denoted by our usual symbols (see page ‘35)J % ( I ) , %(I). r w . F3(1). etc. Let the quantity u be defined by the equation u = cB(I)- t4s(I)- p 2 r Z ( * -) p ~ ~ ( ~ ~ - e t c . , (659) in which t denotes the temperature, and p 2 , p 3 , etc., the potentials for the substances specified a t the surface of discontinuity,” p. 314 Equation ( 6 j 9 ) is nearly the same as equation (514). d o = - ?s(m) dt - Fz(r)dpz - r3(,)&a - etc., (514) from which Gibbs started in deducing the relation between surface tension and adsorption for liquid absorbents. Of course, there is a difference, as pointed out by Gibbs. “As in the case of two fluid masses (see page 2 j 7 ) , we may regard u as expressing the work spent in forming a unit of the surface of discontinuityunder certain conditions, which we need not here specify-but it cannot properly be regarded as expressing the tension of the surface. The latter quantity depends upon the work spent in stretchzng the surface, while the quantity u depends upon the work spent in forming the surface. With respect to perfectly fluid masses, these processes are not distinguishable, unless
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the surface of discontinuity has components which are not found in the contiguous masses, and even in this case (since the surface must be supposed to be formed out of matter supplied at the same potentials which belong to the matter in the surface) the work spent in increasing the surface infinitesimally by stretching is identical with that which must be spent in forming a n equal infinitesimal amount of new surface. But when one of the masses is solid, and its states of strain are to be distinguished, there is no equivalence between the stretching of the surface and the forming of new surface. “This will appear more distinctly if we consider a particular case. Let us consider a thin plane sheet of a crystal in a vacuum (which may be regarded as a limiting case of a very attenuated fluid), and let us suppose that the two surfaces of the sheet are alike. By applying the proper forces to the edges of the sheet, we can make all stress vanish in its interior. The tensions of the two surfaces are in equilibrium with these forces, and are measured by them. But the tensions of the surfaces, thus determined, may evidently have different values in different directions, and are entirely different from the quantity which we denote by u, which represents the work required to form a unit of surface by any reversible process, and is not connected with any idea of direction. “In certain cases, however, it appears probable that the value of u and of the superficial tensions will not differ greatly. This is especially true of the numerous bodies which, although generally (and for many purposes properly) regarded as solids, are really very viscous fluids. Even when a body exhibits no fluid properties at its actual temperature, if its surface has been formed at a higher temperature, at which the body was fluid, and the change from the fluid t o the solid state has been by insensible gradations, we may suppose that the value of u coincided with the superficial tension until the body was decidedly solid, and that they will only differ so far as they may be differently affected by subsequent variations of temperature and of the stresses applied to the solid. Moreover, when an amorphous solid is in a state of equilibrium with a solvent, although it may have no fluid properties in its interior, it seems not improbable that the particles at its surface, which have a greater degree of mobility, may so arrange themselves that the value of u will coincide with the superficial tension, as in the case of fluids,” p. 3 1 5 . Freundlichl considers that we are forced to assume a surface tension for a solid. “ I t is the mobility of liquids which enables us to recognize and measure their surface tension, but when we come to consider the interfaces of solid substances we encounter quite different conditions. Here the particles suffer mutual displacement with great difficulty, so that we cannot recognize the surface tension directly or measure it. h’evertheless it is found expedient to assume a surface tension of solids against a gaseous space. . . , Just as the increased vapour pressure of minute drops is related to the surface tension of liquids, so we may also connect the phenomena depending on the increased
’ “The Elements of Colloidal Chemistry,” 37.
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vapour pressure of minute crystals with a surface tension for crystalline solids. We would therefore, on the basis of these observations, postulate such a surface tension for solids. “A few phenomena may still be mentioned which can be explained on this hypothesis. Since tension tends to reduce the surface to a minimum, it should also tend to round off the sharp corners and edges of a crystal. Such crystals with rounded edges and corners are actually known. If, for instance, we heat a piece of metal to a high temperature which is still appreciably below the melting-point, surface tension may begin to overcome the rigidity of the individual crystals constituting the metal, the so-called crystallites, which then become rounded at the edges and corners. Surface tension is also the cause of recrystallization, in which the small crystals unite to larger ones a t such temperatures below the melting-point.” Liquids spread on solids apparently just as they do on liquids and a slight excess may draw up into single drops in the two cases. A thin film of gold leaf becomes granular when heated, unquestionably as a result of surface tension. The melting-point could be considered as the temperature at which the surface tension overwhelms the crystalline forces. So far as we now know, the adsorption by liquids, amorphous solids, and crystalline solids is similar in nature. The theory of peptization postulates a change of surface tension with adsorption. All the phenomena point to the fact that we have a change of surface tension or of some equivalent property when adsorption occurs, regardless whether the adsorbing agent is a crystalline solid or a liquid. We are therefore justified, at least for the present, in applying the conclusions from the Gibbs equation to the case of adsorption by solids. The general conclusions of this paper are : From the work of Gibbs it follows that the necessary and sufficient I. criterion for the applicability of the Freundlich equation to the case of the adsorption of a gas by a liquid, in which the gas is practically insoluble, is that the change of the surface tension is proportional to the change of adsorption. 2. The exponent n in the Freundlich equation is a measure of the change of the surface tension with the adsorption in the case of a liquid adsorbent. 3 . It seems probable that the necessary and sufficient criterion for the applicability of the Freundlich equation to the case of the adsorption of a gas by a solid is that the change of the surface tension or of some equivalent property is proportional to the change of adsorption. 4. I n so far as conclusion 3 holds, conclusion z can be extended to cover the case of a solid adsorbent. Cornel2 University.
