CONVENTIONS APiD ASSUMPTIONS I N THE INTERPRETATION OF EXPERIMENTAL DATA BY MEANS OF THE GIBBS ADSORPTION THEORENP P I E R R E VAN RYSSELBERGHE
Department of Chemistry, Stanford Cniversity, California Received July 1 , 1038 I. INTRODUCTION
Yarious conventions used in the interpretation of surface tension data by means of the Gibbs adsorption theorem (4) have been discussed in a number of recent contributions, such as those of Guggenheim and Adam (5), Butler and Wightman (2), and Rice (8). We shall briefly recall the essential points of these methods. Let us consider a two-component, two-phase system, for instance, a binary solution and its mixed vapor, separated from each other by a plane interface. The Gibbs adsorption theorem (4) states that, when the inhomogeneous layer a t the interface is in complete equilibrium with the liquid and the vapor phase at a given temperature,
in which u is the surface tension of the solution, p1 and ~2 the molar chemical potentials of the two components, and rl and F2 the surface excesses in moles per unit area. I t has been demonstrated by Gibbs (4) that, when the mathematical dividing surface is plane, equation 1 is imariant in form with respect t o a change in the position of this surface and du has the same value for all positions of the surface. The surface excesses rl and Fz, on the contrary, depend on the position of the surface, but the expression I'I
NZ - J
-
F2J
in which iV1 and Nz are the mole fractions of the bulk
N1
solution, is invariant with respect t o a change in the position of the dividing surface, sincc, on account of the Gibbs-Duhem formula Ari dpi
+ Nz dpz = 0
(2)
Presented a t the Fifteenth Colloid Symposium, held a t Cambridge, Massachusetts, June 9-11, 1938. 1021
1022
PIERRE VAN RPSSELBERGHE
(in which the concentrations of both components in the vapor phase are neglected), formula 1 may be rewritten
The two differentials du and dpz being invariant, the expression
is also invariant. A special position of the mathematical dlvidirlg surface corresponds to what Gibbs has called the “surface of tension,” for which certain equations have “the same form as if a membrane without rigidity and having a tension u , uniform in all directions, existed a t the dividing surface” (4, page 229). Defay (3) has established the conditions of mechanical equivalence between the real system with an inhomogeneous layer of finite thickness and the idealized system in which the two phases are separated by the surface of tension. These conditions are only of theoretical interest and cannot yield explicit information concerning the exact position of the surface of tension until the complete structure of the layer and the distribution of tensions are known. Let us note, however, that even in the case of plane interfaces, in spite of the invariance of U, there is a definite surface of tension, determined, a8 shown by Defay ( 3 ) , by conditions of equivalence of total torques. It is necessary, however, to have recourse to other dividing surfaces if one wishes to calculate values of rl and from which 3ome information as to the structure of the layer can be derived. It is customary t o locate the dividing surface in such a manner that the surface excess rl of the solvent is equal to zero and to calculate the surface excess I’* of the solute from the formula
which gives us a. convenient and suggestive method of recording the results of surface tension measurements. If, however, one wishes more precise information concerning the inhomogeneous layer, it is preferable t o put the dkiding surface immediately underneath the layer. Since the concentrations in the vapor phase are neglected, the values of I’l and represent the total amounts of components 1 and 2 present in a column of 1 cm.* croqs section on top of the homogeneous liquid. So far, we have only one equation t o determine the unknowns rland I’2, namely, the Cribbs theorem and I’z. 3. We thus ha3;e t o set up a second relationehip between
THE GIBBS ADSORPTION THEOREM
1023
For instance, we may assume that the layer is one molecule thick and write
rlAl + rzAz= 1
(5)
in which AI and Az are the areas per mole of each component. Guggenheim and Adam (5) and also Butler and Wightman (2) have shown, in the case of water-alcohol mixtures, that the simultaneous equations 3 and 5 yield reasonable values of rl and rzwhen plausible constant values of AI and A2 are adopted and all other disturbing effects are neglected. The surface mole fraction rz/T1 I'z obtained by these authors is not steady and exhibits a slight maximum before increasing to the value 1 in pure alcohol. Guggenheim and Adam (5) ascribe this effect to a possible lack of constancy of the molar areas A1 and Az, while Butler and Wightman (2) suggest that the layer may not be truly monomolecular. Various mixture rules for the surface tension of binary solutions have been used in the older work on the subject without any reference to the Gibbs adsorption theorem. For instance, Morgan and Griggs (6) and Morgan and Scarlett (7) use the mixture rule
+
u = xu1
+ (1 -
2) u2
(6)
in which UI and uz are the surface tensions of the pure liquids, and z and 1 - z the percentages by weight of the components in the bulk solution. No discussion of adsorption a t the interface is presented, and the Gibbs theorem is not mentioned. Whatmough (9) used mixture rules of the same type, but took for z and 1 - z mole fractions, weight fractions, or volume fractions of the bulk solution, again without reference to the phenomenon of adsorption. I n this paper we wish to present and discuss the simplest possible mixture rule in which adsorption is taken into account, namely
This formula is derived from the logical assumption that the rl and rz values actually responsible for the surface tension will contribute, as a first approximation, the portions
to the total surface tension u . The Gibbs theorem 3 and our formula 7 are simultaneous equations, the unknowns being rland r2.
11. DISCUSSIOX O F ‘THE MIXTURE 12CI.E A X D 01” THE GIBBS ADSORPTIOS
THEORLIU
The terms “positive adsorption” and “iieg:itive :idsor.ption” of the solute at tlie surface of tlic solution are Iiawd upon tilt Fign of pi1) of formula 4, which is obviously opposite to that of d u j d p z or also opposite to that of du/dNz, since diL,/dSz is always positive. From formula 3, howevcr. nothing can be said about the sign of r2 until r, is known, or vice vrrsa. Our mixture rule (equation 7 ) requires both r1and rz t o be positive. We shall associate the ternis “positive adsorption” and “negative adsorption,” respectively, with the following two inequalities
Formula 7 gives
Combining equations 3 and 7 we get du
?;ow,
rz being poqitive,
we have
or also
Our mixture rnlc is, therefore. in agreement with the familiar interpretation of tlic Gibbs throrem based upon the convention rl = 0. Figurc 1 corresponds to positive adsorption of alcohol (component 2) at the surface of water-alcohol mixtures. ‘The curve represents the surface the straight line rrpretension plotted against the bulk mole fraction SZ; sents the surface tension plotted against the mrface mole fraction I’z/ r1 rz, The analogy with phase rule diagrams is obvious. If the surface tension curve exhibits a minimum or a maximum at some particular composition (K:.Xi) our mixture rule cannot be applied without modification. At the extremum we have
+
1025
N& FIG.1. Surface tension of water-alcohol mixtures plotted against bulk and surface mo1e:fraction of alcohol
FIG.2. Surface tension curve exhibiting a minimum
1026
PIERRE V A S ETSSELBERGHE
and, according to equation 3,
In other words, the surface mole fraction and the bulk mole fraction are identical a t the extremum. We should have, according t o our mixture rule,
which is impossible, since it would require the segments A B and AC of figure 2 t o be equal. We can, however, overcome this difficulty by applying the mixture rule separately between u1 and go and between u0 and Q. We then have positive adsorption of the mixture N : between ul and u0 and negative adsorption of this mixture between uo and 6 2 . The analogy with the boiling point diagram of a n azeotropic mixture is obvious. 111. INTERPRETdTION OF SURFACE TENSION DATA FOR TSATER-ALCOHOL MIXTURES AND DESCRIPTION OB THE SURFACE LAYER
In order to test the plausibility and usefulness of our mixturerule we now present a new interpretation of the surface tension data of wateralcohol mixtures. The values of da/dpr are obtained from the paper of Guggenheim and Adam (5), where -- da/dpz is called r;'), being the value of rZ corresponding t o the usual convention r1 = 0. The values of u are those of Bircumshaw (1). Similar values for both u and du/dpz mere obtained by Butler and Wightman (3). In table I \?*e give, for a keries of mole fractions iVz of alcohol, the surface tension u in dynes per centimeter, the derivative da/d,uz in moles per cm.*>the surface excebses rl and r3as calculated by means of our formulas 7 and 10, in moles per cm.2. the surface mole fraction I?2/r1 TZ,and the area Az in em.' occupied by one mole of alcohol and calculated as indicated below. The temperature is 25°C. Our alms of ri and are of the right order of magnitude for a monomolecular layer. A comparison with those obtained by Guggenheim and Adam ( 5 ) from equation 5 shows that, while their values of Tzincrease with S z , our values decrease, indicating that the area occupied per mole of alcohol increases with Nz. This area is constant in Guggenheim and .\darn's calculations. Their values of rl decrease when Xz increases and 50 do ours, but our T1's are smaller, in agreement with the idea of a n increasing area for the alcohol molecules, since the area of mater molecules can probably be assumed constant and independent of possible changes of orientation, while the area occupied by alcohol molecules will vary ronsiderably, the two extreme cases being molecules standing up on the
+
1027
THE GIBBS ADSORPTION THEOREM
surface and molecules lying flat on it. We have calculated the area occupied by one mole of alcohol at the various mole fractions iVZ by assuming the area A1 for water t o be 0.06 X loin cm.? per mole, i.e., the second of the three values considered by Guggenheim and Adam (5). The results are reported in the last column of table 1. Extrapolating to N Z = 0 we see that the limhing value of Az is in perfect agreement with the value 0.12 X loLo per mole adopted by Guggenheim and Adam ( 5 ) for molecules of alcohol oriented perpendicularly to the surface. Extrapolating to Nz = 1 we find AZ = 0.45 x 1Olo cm.? per mole, R result which would indicate that in pure alcohol most of the molecules are lying flat on the surface. The intermediate values of Az can be considered as giving a measure of the average orientation of alcohol molecules at the surface. It is of course true that, if orientation changes with concentration, the TABLE 1 Surface tension and surface composition of water-alcohol mixtures Na 0
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
U
72.2 45.0 36.4 29.7 27.6 26.35 25.4 24.6 23.85 23.2 22.6 22.0
- dgt 2 x 1010
As X
0
0.0
5.85 6.3 6.45 5.9 5.1 4.25 3.4 2.9 2.5 2.2 2.1
5.2 2.65 1.2 0.78
6.1 6 6 6 75 6.25
0.54
~
0'95 0.965 0.975 0.99 1 00
(0.12) 0.11 0.135 0.155
0.845 0.89 0.935
10-10
'
, ~
1
0.215 0.265 0.31 0.36 0.405 (0.45)
values of u2 in the mixture rule (7) also depend on concentration, but in an unknown manner. It is, however, probable that such a variation is small and that our interpretation of surface tension data is just as plausible as that based upon the initial assumption of a monomolecular layer and upon the corresponding formula (formula 5 ) in which the area A%i s considered as constant. The mole fractions rz/Tl FZ reported in table 1 increase with N Zin a continuous and steady manner, as is shown on figure 3 by the heavy curve I, while the dotted curve I1 represents the mole fractions obtained by Guggenheim and Adam (6) for A I = 0.06 X loLo. Similar results were obtained by these authors with A I = 0.04 X 1 O l n and 0.08 X loLo,A2 being 0.12 X 1O"J in the three cases. It is interesting to note that our interpretation of the composition of the surface layer agrees with a suggestion made by Rice (8, page 573) : "NOWit might happen
+
1028
PIERRE VAN RYSSELBERGHE
that with increasing concentration of alcohol, the more polar water molecules being replaced by weaker alcohol molecules, there would be a decrease in orientation with an increase in area occupied, caused by each alcohol molecule lying flat in the burface.” It iq interesting t o note that the distance between :ur dividing surface = 0 varies from 1.09 A. for N z = 0.05 to and that corrrsponding to 0.16 A. for ,LIZ = 0.9. A displacement of the dividing surface smaller than the thickncis of a monomolecular layer has thus a profound influence on 1.0
0.3, 0.3.
01.
ai 0
0.1
0.1
0.3
0.4
05
0.6
0.7
0.6
0.9
N, FIG. 3. Surface mole fraction of alcohol plotted against bulk mole fraction in water-alcohol mixtures
the values of rl and F2. When the surface corresponding to I’l = 0 is used me are unable t o obtain any information regarding the amount of component 1 which, together with the amount FZ of component 2, is responsil~lefor the surface tension, while our mixture rule gives a first approximation for both rl and FZ. IV. SUMMARY
1. Conventions and aasumptioiis in the inkrpretation of surface tension data of binary mixtures a,re briefly discussed. The concept of surface of tension is recalled. 2 . h mixture rule for the surface tension of binary mixtures involvirlg
T H E CIBBS ADSORPTION THEOREM
1029
thc hurfacc excesses of both compoiierits is proposed. This rule and the Gibtis adsorption theorem form a system of two simultarleous tquations in n-hich these surface excesses arc the unknowns. 3 . It, is shown that the mixture rule agrees with the familiar concepts of positive and negative adsorption usually defined by means of the convention = 0. The case of surface tension curves with ai1 extremum is discussed separately. 4. The method is applied t o water- alcohol mixtures. The results show t’hat the surface mole fraction of alcohol increases rapidly and steadily when the bulk mole fraction increases, that the surface excess of water decreases rapidly, that the surface excess of alcohol decreases after passing through a maximum near N z = 0.2. If the inhomogeneous layer is monomolecular, the alcohol molecules, oriented perpendicularly to the surface in very dilute solutions, lie flat on the surface in pure alcohol. REFERESCES (1) BIRCUMSHAW, L. L.: .J. Chem. SOC.121, 887 (1922).
(2) BUTLER,J . A . V., A X D WIGHTMAN, A , : J. Chem. S O C . 1932,2089. (3) DEFAT,R . : fitude Thermodynamique tie la Tension Superficielle, pp. 97-105, 150-151. Gauthier-Villars, Paris (1934). (4) GIRHS,J. W.: Collected Works, Tol. I, pp. 219--331. Longmans, Green and Co., Xew York (1928). ( 5 ) GUGGEKHEIM, E. A . , A K D ADAM,K. K . : Proc. Roy. soc. (London) A139, 218 (1932). (6) ~ I O R G A JX. ,1,. R., A K D GRIGGS,11.A . : J . A m . Chem. SOC. 39,2261 (1917). (7) MORGAN, J. L. R., AND SCARLETT, A. J . : J. Am. Chem. Soc. 39, 2275 (1917). (8) RICE, J . : A Commentary on the Scientific IVritings of J . W.Gibbs, Vol. I, pp. 505-708. Yale University Press, Kew Haven (1936). (9) \l”.%mocnH, W.H . : Z. physik. Chem. 39, 129 (1902).
