SURFACE CONCENTRATION AND
THE
GIBBSADSORPTION LAW
133
Surface Concentration and the Gibbs Adsorption Law. The Effect of the Alkali Metal Cations on Surface Behavior’
by Ira Weil Lever Brothers Company, Research Center, Edgewater, h‘ew Jersey
(Received J u n e Z8,1966)
To investigate the thermodynamic behavior of the surface phase, direct measurements of the surface excess concentrations of lithium, sodium, and potassium dodecyl sulfates were obtained by a foaming technique. The surface concentrations at constant bulk concentration are in the order Li+ < Na+ < K+. These data, in conjunction with surface tension measurements, show that the Gibbs adsorption law with the “factor 2” applies for all three dodecyl sulfate salts. Further comparison of the data indicates that at the same surface concentration the relative order of surface tensions is Li+ < S a + < K+. This finding is consistent with the surface behavior of the insoluble alkyl sulfates. However, the behavior of the alkali metal salts of long-chain fatty acids is opposite to that of the alkyl sulfates. These observations suggest and, indeed, it is found by comparison with simple salt solutions that the thermodynamic nature of surface behavior is similar to that of concentrated solutions. Thus, surface behavior could be described by an activity coefficient of the surfactant salt in the surface phase. A thermodynamic equation of state is obtained which indicates that the observed effects of the alkali metal cations on surface behavior are due to the nonideality of the surface phase.
Introduction Investigations of the surface behavior of adsorbed monolayers of surface active salts have been primarily concerned with the application of the Gibbs adsorption law to ionic materials.2-I’ Because of the importance in establishing the exact form of the relationship and resolving the experimental and theoretical discrepancies, other considerations of adsorbed monolayers have been largely ignored. One of these is the thermodynamic behavior of the surface phase. Generally, this facet of surface behavior is interpreted in terms of the properties of the solution phase which gives rise to the adsorbed monolayer. However, the assumption of a surface phase implies that its thermodynamic behavior, at constant temperature, will be dependent upon the composition and the nature of the components in the same phase. Since the parameter most uniquely associated with the surface is surface tension, a study of its dependence upon these variables should lead to a more complete description of adsorbed monolayers.
I n the absence of an extant thermodynamic relationship between the surface parameters, it is difficult to obtain information of a thermodynamic nature from only the concentration dependence of surface tension. However, if, in addition, the components are varied, a comparison of the data can be used in interpreting the results from a thermodynamic viewpoint. It will, (1) Presented at the 148th National Meeting of the American Chemical Society, Chicago, Ill., Sept. 1964. (2) A. P. Brady, J . Phys. Chem., 53, 56 (1949). (3) D. J. Salley, A. J. Weith, Jr., A. A. Argyle, and J. K. Dixon, Proc. Roy. SOC.(London), A203, 42 (1950). (4) J. T.Davies, Trans. Faraday SOC.,48, 1052 (1952). (5) M.A. Cook and E. L. Talbot, J . Phys. Chem., 56, 412 (1952). (6) B. A. Pethica, Trans. Faraday SOC.,50, 413 (1954). (7) E.G. Cockbain, ibid., 50, 875 (1954). (8) G.Nilsson, J . Phys. Chem., 61, 1135 (1957). (9) E. Matijevic and B. A. Pethica, Trans. Faraday SOC.,54, 1382 (1958). (10) J. W. James and B. 8.Pethica, Proc. Intern. Congr. Surface Activity, Srd, London, 2, 227 (1961). (11) J. E.Bujake and E. D. Goddard, Trans. Faraday SOC.,61, 190 (1965).
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134
therefore, be necessary to alter the components in such a manner that the comparison will most easily lend itself to logical inference. The simplest component variation consistent with the objective is that of the cation of a single surface-active anion. The most convenient series of surface-active salts for this study, from both ease in preparation and information available, are the alkali metal dodecyl sulfates. The concentration of the surfactant salt in the surface phase i; generally obtained from surface tension data by the application of the Gibbs adsorption law. However, as indicated above, uncertainties may be introduced by this procedure. I n order to avoid this possibility, the surface excess concentrations have been determined by direct measurement, employing a foaming technique. These determinations, obtained independently of surface tension measurements, not only permit conclusions about the thermodynamic behavior of the surface phase but also allow for an examination of the application of the Gibbs adsorption law to ionic surfactants.
Experimental Section MuteriaEs. Lithium, sodium, and potassium dodecyl sulfates (JIDS, 11 = Li+, Na+, K+) were prepared by neutralizing dodecyl hydrogen sulfate with the highest grade commercial alkali metal hydroxides. The organic acid was obtained by sulfation with SO3 vapor of l-dodecanol of purity greater than 99.6%. LiDS was purified by two recrystallizations from 2-propanol in the presence of charcoal, followed by three recrystdlisations from 2-propanol alone and extraction with ethyl ether. S a D S and KDS were recrystallized twice from redistilled water and twice from absolute ethanol in the presence of charcoal. Both salts were subjected to three additional recrystallizations from ethanol, followed by ether extractions. The surfave terision-concentration curves of all three salts were free of minima, indicating the absence of free l-dodevanol. Prior to measurement of surface tension or surface concentration, the test solutions were foamed for about 2 hr. to remove any divalent cations which might have been introduced with either the alkali metal hydroxides of the redistilled water. Xethod. Sgrface excess concentrations were measured by the foaming technique described by Walling.lz For the present work his apparatus was modified as shown schematically in Figure 1. Commercially purified nitrogen is bubbled through water a t approxiniately 35 ml./min. and passed through a glass-wool trap before entering the surfactant sohtion through a T h e Journal of Physical Chemistry
N ~ +
Figure 1. Schematic diagram of the foaming apparatus.
glass nozzle having a diameter of 0.5 mm. This opening produces bubbles of a diameter of approximately 3 inm. Throughout the determination, the surfactant solution is stirred vigorously with a Teflon-covered magnetic stirrer. The foam thus generated rises into a tube, 18 mm. in diameter and 120 em. high, at a rate of about 15 cm./min., which allows for adequate drainage of the lamellar fluid without film rupture. The foam then passes into a collection flask where it is collapsed by cooling. Kitrogen continues into a wettest meter, which measures the total gas volume. The surfactant content of the collapsed foam and also that of the hulk solution were determined by a Hyamine titration using methylene blue as the indicator. l 3 The Hyaniine 1622 titrant was standardized against sodium alkylbenzenesulfonate. The precision of the method is better than =tO.5%. Surface tension measurements of the surfactant solutions were taken with a Wilhelmy balance using a sandblasted platinum plate. The temperature of the surface excess concentra1"; tion determinations was maintained a t 25 that of the surface tension measurements was 25.0 f 0.1". (12) C . Walling, E. E. Ruff, and J. 56, 989 (1gj2).
(13)
s. R.
Epton, Trans. Faraday
L.Thornton, Jr., J . Phys. Chem.,
SOC.,
44, 226 (1948).
SURFACE CONCESTRATION AND
THE
GIBBSADSORPTION LAW
Discussion of Method. The difference in surfactant content of the collapsed foam and that of the bulk solution represents the surf ace excess provided the lamellar fluid has the same composition as the bulk solution. That this condition prevails was ascertained by analysis of the drainage liquid whose composition proved to be within =t2% of that of the bulk solution. This finding also confirms the gross observation that film rupture was negligible and, further, indicates that the bubbles are in equilibrium with the bulk solution before they become part of the foam network. Further assurance of equilibrium conditions was obtained when the residence time of the bubble in the bulk solution was varied. For flow rates of 20 to 60 ml./min., the values of the calculal ed surface excess concentrations agreed to within =t2%, which is equivalent to the over-all reproducibility of the measurements. The total surface, S, generated in a foaming experiment depends upon the total volume of foam, Vi, and the bubble diameter, db. For a sphere, these variables are related by the expression S = 6Vr/d~,. Since the volume of the collapsed foam is negligible in comparison to the total volume of nitrogen, the latter, as measured by the wet-test meter, can be equated to vi. Xeasurement of db is less direct. Although in the bulk solution. the bubbles are spherical, the foam network appears to be predominantly made up of rhombic and regular dodecahedra. The diameter, d~,,was obtained from the average of 10 to 20 measurements of the cross section of the individual dodecahedra as they appeared on the column wall after stopping the flow of nitrogen. To make certain that db thus measured is valid for an expression based on a spherical shape, the magnitude of the bubble volume was determined by two additional methods. In the first, high-speed photographs were taken of the spherical bubbles in the bulk solution. In the second, the foam network was passed into a calibrated capillary tube, and the number of bubbles per unit length was counted. All three methods gave bubble volumes which agreed to within *3%. The agreement is an indication of the reliability of the absolute magnitude of the surface excess concentration
Calculations The surface excess concentration, S,, can be defined as the total surface excess of surfactant, E,, divided by the total surface generated during the determination, 8, = E,/X, where E, is expressed in moles and S is in square centimeters. The total surface excess, in turn, is given by E, = Wf(Cf - Cb)10--3, where Wf is the
136
weight of the collapsed foam in grams, and Cf and cb are the concentrations of surfactant in moles per liter, in the collapsed foam and the bulk solution, respectively. (The density of the collapsed foam is assumed to be unity.) With this expression and the equation for the total surface given earlier, the surface excess concentration can be written as
This expression assumes that the bulk concentration remains constant during the determination. However, since the foam is enriched in surfactant, Cb should be an average of the initial and final bulk concentrations, This introduces a minor correction which involves the final bulk concentration, Cz, and the volume in milliliters of the initial solution, V,. Thus, the expressions for calculating the average bulk concentration and the surface excess concentration are
The experimental data for NaDS and the calculated results are shown in Table I. Table I: Surface Excess Concentration of Sodium Dodecyl Sulfate (Temperature 25 i 1'; V , = 500 ml.) 1 0 8 ~ ~ 10*Cf, ~
Wy,
lO-aVr,
db.
lo%,
lO' moles/ "sc,
M
llf
g.
ml.
cm.
*If
cm.2
3.00 3.47 4.15 5.08 6.27 7.30 8.91 13.45
2.27 1.55 2.11 2.3i 2.14 2.11 2.14 3.62
0.965 1.860 1.440 1.310 1.790 1.930 2.270 1.320
3.68 3.95 3.85 3.40 3.55 3.59 3.46 3.58
0.320 0.310 0.305 0.295 0.300 0.320 0.295 0.285
3.02 3.49 4.17 5.10 6.30 7.33 8.94 13.48
2.76 2.92 3.21 3.51 3.80 3.95 4.03 3.95
Deiinition and Use of Surface Concentration Terms Since the data described are to be used with the Gibbs adsorption law, it is appropriate at this point to discuss the physical meaning of S, and its relationship to the quantities defined by Gibbs and other related quantities. The measured surface excess concentration can best be understood by first considering the hypothetical situation of zero adsorption. I n this case, the concentration of the components in the surface is the same Volume 70,Number 1 January 1966
136
IRAWEIL
as in any equivalent dimensional plane in the bulk solution. I t is apparent that in this instance, So will be equal to zero. If, as found in the experimental work described, S, is greater than zero, it must represent the excess of the solute in the surface over that which exists in the equivalent plane in the bulk solution. I t thus appears that So is very similar to, if not identical with, Gibbs’ superficial density, r2, which is the surface excess concentration of the solute a t the surface of discontinuity or the surface of tension.14 The commonly used Gibbs adsorption equation, derived on the basis of a hypothetical dividing surface in which the excess solvent concentration is zero, calls for the use of I’2c1).1b This abstract term, the surface excess concentration at a dividing surface where rl, the surface excess of the solvent is equal to zero, can be avoided by using Gibbs’ treatment of the normal case in which the dividing surface is the surface of tension.16 I n this case, the expression at an air-water interface becomes dr -dP2
43.5E
1
83.0x 2.5-
5
2’0
3.0
nl
s, = r2= r2(1).
It is also of importance to approximate the surface concentration at Gibbs’ surface of tension. The earlier statements of this section can be expressed by the equations
r2 = r2’- rzO’ rd/rlo’ = n2/n1 where rZ’is the surface concentration and I’t0’ and ria' are the surface concentrations of the solute and solvent, respectively, which would obtain at zero adsorption. When these equations are employed for highly surface-active solutes, r2”’will be negligible and the surface concentration, r2’,will be very nearly the same as the surface excess concentration.
Results Data showing the surface excess concentration as a function of the bulk concentration for LiDS, NaDS, and KDS are given in Tables I and I1 and are shown graphically in Figure 2.
4.0
5.0
60
7.0
c x io3 (moles/liter) Figure 2. Variation of surface excess concentration with bulk concentration.
Table 11: Surface Excess Concentration (Temperature 25 3= lo)
2 r2- n-r,
where n2 and nl are the bulk solution concentrations in moles per cubic centimeter of the solute and solvent, respectively. This equation indicates that measurement of rz alone is not sufficient to describe the variation of surface tension with 12, unless the second term on the right-hand side is negligibly small. With highly surface-active materials this is certainly true. Thus, for the purpose of calculating surface quantities,
The Journal of Physical Chemistry
\
LiDS
KDS loscb,
M
10’0S,, moles/cm.l
IM
1O’OS, , moles/cm.2
4.54 4.74 5.11 5.72 6.55 7.09
2.39 2.48 2.52 2.75 2.86 3.05
2.79 3.15 3.61 4.40 4.87
3.48 3.64 3.88 4.11 4.25
108cb,
The magnitudes of the surface excess concentration are quite different for the three salts. At equal bulk concentrations, So for KDS is almost twice as large as that for LiDS, the general order being Kf > Na+ > Li+. Above its c.m.c. (7.5 X lod3 M ) , the surface excess concentration of NaDS is sensibly constant a t 4.00 X 10-lo mole/cm.2. As So is substantially equal to the surface concentration, this value leads to an area of 41.5 A. 2/mo1ecule, which compares well with published values derived from surface tension, l7 radioactive tracer,s and other measurements.l2!1*,19 Below the c.m.c., the variation of surface excess con(14) “The Scientific Papers of J. Willard Gibbs,” Dover Publications, New York, N. Y., 1961, p. 224. (15) See ref. 14, p. 234. (16) See ref. 14, p. 236. (17) E. J. Clayfield and J. B. Matthews, Proc. Intern. Congr. Surface ActiBitg, Bnd, London, 1, 172 (1957). (18) H. Kishimoto, Kolloid-Z., 192, 66 (1963). (19) A. Wilson, M. B. Epstein, and J. Ross, J . Colloid Sci., 12, 345 (1957).
SURFACE CONCENTRATION AND
THE
GIBBSADSORPTION LAW
centration with bulk concentration appears to be consistent with the Langmuir adsorption isotherm. This empirical behavior has been noted by previous investigators.20Pz1 I n Figure 3, a plot of the reciprocal of S, os. the reciprocal of the bulk concentration yields good straight lines for the three salts. Although the slopes of the lines are different, they have a common intercept. This intercept, an area term a t infinite bulk concentration, should correspond to the limiting area of the alkyl sulfate salts. The intercept, 1.72 X 109 cm.Z/mole, corresponds to an area of 29 W.2/molecule, which is in good agreement with soluble and insoluble monolayer data.6r18 The overriding importance of this finding is that this value is independent of the alkali metal cation. The direct measurements of surface excess concentrations can be employed, together with surface tension data, to determine the form of the Gibbs adsorption law for ionic surfactants. The general expression can be written as - d r = xRTS, d In aZB
+B
,
r
I
I
I
I
1
I
(1)
where a2Bis the activity of the solute in the bulk phase and x is a numerical coefficient which is 1 for unionizable materials, as originally derived by Gibbs, or 2 for ionic compounds.22 The controversy as to the value of the coefficient for ionic compounds arises from the idea that the surface-active species may exist as the undissociated acid, in the case of surface-active anions, in the s ~ r f a c e . ’ ~This ~ ~ ~phenomenon is designated “surface hydrolysis.” Although recent studies tend to discount the importance of this hypothesis for the dodecyl sulfate type of surfactant^,'^^^^^^^ the evidence is not as direct as that available from this investigation. Surface tension data for the alkali metal dodecyl sulfates, which can be employed to determine the coefficient of the Gibbs expresssion, were obtained and are shown in Figure 4. Although these data are generally employed to calculate surface excess concentrations, large errors can be incurred unless the measurements are of a very high degree of precision. However, since tho surface excess concentrations here obtained can be expressed as a function of the bulk concentration, a procedure is available which is not critically dependent upon the precision of the surface tension measurements. As shown in Figure 3, S, is empirically related to the bulk concentration by the Langmuir expression
l/Xc = m/Cb
137
(2)
This equation can be employed to obtain an integrable form of the Gibbs expression. If it is assumed that the
K 30 4 .O 50 60 7.0
38
cx
io3 (moles/iiter)
Figure 4. Variation of surface tension with bulk concentration.
solutions are sufficiently dilute to be considered in the Debye-Hiickel limiting law range, and the activity is set equal to Cbfb, the activity coefficient is given Performing by the expression -1n f b = 1 . 1 6 f l b . the integration of eq. 1 after the necessary substitutions, one obtains
G
-y
=
(3( + 3-
x - In 1
-C
1,164:
1.16-
tan-’
+
5 1 ) (3)
(20) B. von Seyskowski, 2.physik. Chem., 64,385 (1908). (21) A. Frumkin, ibid., 116, 466 (1925). (22) E. A. Guggenheim, “Thermodynamics,” North-Holland Publishing Co., Amsterdam, 1949, p. 327. (23) C. Walling, E. E. Ruff, and J. L. Thornton, Jr., J . Phys. Chem., 61, 486 (1957).
Volume 70,hTumber1 January 1966
IRAWEIL
138
’‘1 \
i
NaDS
, 34
715
sion at comparable concentrations are surprisingly large. The differences cannot be ascribed to the activity of the salts in the bulk phase since these are generally accepted to be within the Debye-Huckel limiting law range. Furthermore, the three salts obey the Gibbs adsorption law. Thus, the observed behavior most probably results from a phenomenon in the surface phase. This phenomenon manifests itself even more immediately in the variation of the surface tension with surface excess concentration. Corresponding values of the two variables were interpolated from the curves in Figures 2 and 4, and plotted in Figure 6. hJ A comparison of the curves shows that at equal surface excess concentrations the surface tension increases in the order Li+ < Na+ < K+. This finding indicates the source of the observed surface behavior.
10.0
12.5
15.0
17.5
L
Figure 5. Plot of eq. 3. The experimental points shown are from the surface tension data. The values of B and m for the three salts obtained from surface excess concentration data.
where G is the constant of integration. Within the limits of applicability of the Langmuir expression, as indicated by the data, the coefficient of the Gibbs equation can be ascertained from eq. 3. Using the values of m and B given in Table I11 and RT at 25’ equal to 2.48 X 1O1O ergs/mole, the surface tension data are plotted in Figure 5 according to the expression in eq. 3. 25
Table 111: Constants from Eq. 2
S, x
Compd.
1o-sTn
10-OB
LiDS NaDS
11.13 5.80 3.19
1.74 1.72 1.71
KDS
The plot in Figure 5 exhibits reasonably good straight lines with slopes from which an average value of 2.0 f 0.05 is obtained for the coefficient of the Gibbs expression. The validity of the “factor 2” form of the Gibbs adsorption law is thus confirmed for the alkali metal salts of dodecyl sulfate in the concentration range studied. This result indicates that the differences in the behavior of the dodecyl sulfate salts are not explicable in the terms which are explicit in the derivation of the Gibbs adsorption law.
Discussion The surface behavior of the alkali metal salts of dodecyl sulfate is shown in Figures 2 and 4. The differences in surface excess concentration and surface tenThe Journal of Physical Chemistry
30
3.5
40
45
10’0 (moles/cm2)
Figure 6. Variation of surface tension with surface excess concentration. The solid lines are obtained from the interpolation of the data shown in Figures 2 and 4.
The two major factors which control the adsorption of ionic surfactants are the van der Waals or cohesive forces between the hydrocarbon chains and the repulsive forces between the ionized portions of the molecules. At equal surface concentrations, the cohesive forces will be the same, and the different surface tensions exhibited by the individual salts can only result from differences in the ionic forces in the surface phase. Since this phenomenon occurs in the surface phase, similar behavior should be observed with insoluble ionic monolayers. Rogers and S c h ~ l m a nin , ~their ~ study of octadecyl sulfates, found that the surface pressures of the alkali metal salts increased in the order K+ < Na+ < Li+ at constant molecular area. Since in(24) J. Rogers and J. H. Schulman, Proc. Intern: Congr. Surface Activity, 2nd. London, 3, 243 (1957).
SURFACE CONCENTRATION AND THE GIBBSADSORPTION LAW
creasing surface pressures correspond to decreasing surface tensions, this behavior is identical with that found for the adsorbed monolayers of the dodecyl sulfates. The observations presented thus far are associated with a single surface-active anionic species, vix., an alkyl sulfate. If the observed surface phenomenon is brought about by the specific nature of the cation, the same or similar effects should also be found with other types of anions. I n a recent investigation by Sears and Schulman,*5 the surface pressures of the alkali metal salts of stearic acid were observed to increase in the order Li+ < Na+ < K+. This is the opposite of that found for the alkyl sulfates! Thus, the effect of the cations on surface behavior is not consistent with different surface-active anions and therefore does not permit an interpretation solely in terms of properties inherent in the cation. The observations here presented suggest, rather, that the properties of the specific cationic-anionic system determine surface behavior. A phenomenon that is basically dependent upon the concept of cationic-anionic interactions in an aqueous medium is solution behavior. The implication is that there exists a similarity between surface behavior and solution behavior. Since the parameter that uniquely describes solution properties is the activity coefficient of the solute, it is necessary to consider if a similar parameter is ayailable to account for the behavior of the surface phase. To approach the problem from this viewpoint, it is necessary to wcertain the functional dependence of surface tension on surface parameters. Such a relationship is implicit in the basic adsorption law of GibbsZ6 -dy = ridpi
+ rzdpz
(4)
since the variables can all relate to the surface phase. As eq. 4 is generally employed, dp is defined in terms of the bulk phase, which leads to the familiar expression shown in eq. 1. However, if dp is defined in terms of surface phase parameters, an expression relating only the variables of the surface phase will result. The thermodynamic potential or partial molar free energy of a component in the surface phase has been discussed by a number of authors. 27-31 Fundamentally similar conclusions have been reached. I n essence, the partial molar free energy of a component in the surface phase is not only a function of concentration but also of the surface free energy. Using the formulation of Sawyer and F o w k e ~ ,one ~ l can write dpZ8 =
2RT d In a28
- &dy
(5)
where a28 is the activity of the salt, Le., the product of
139
the surface mole fraction and the activity coefficient of the salt in the surface, x~:28fia,and a12 is the partial molar area of the salt. From eq. 4 and 5 and the Gibbs-Duhem relation for the bulk phase, the expression
can be readily obtained. However, to employ this expression conveniently, it is necessary to convert the surface excess terms to surface concentrations. From the relationships given earlier, eq. 7 is obtained. Since
(7) the second members of the terms on the two sides of the equat,ion are negligible in the bulk concentration range normally encountered, eq. 6 can be written in the form
without any loss of its original significance. Equation 8 contains only surface phase parameters. Since the surface concentration, the activity, and the partial molar area are concentration-dependent terms, eq. 8 relates the surface tension to the concentration in the surface phase. Thus, the expression can be considered the thermodynamic equation of state for adsorbed monolayers. (The same equation can be obtained for insoluble monolayers, by equating the partial molar free energy of the water in the bulk and surface phases. The problem is treated as in a distribution of a component between two immiscible phases.) As might be expected, the form of eq. 8 is quite similar to that of eq. 1. Since an equilibrium between the surface and bulk phases is assumed in the Gibbs derivation, the dependence of the surface tension on the activity of the solute in either of the two phases is a necessary consequence. This point is emphasized, as the (25) D. F. Sears and J. H. Schulman, J. Phys. Chem., 68, 3529 (1964). (26) See ref. 14,p. 230. (27) J. A. V. Butler, Proc. Roy. SOC.(London), A135, 348 (1932). (28)A. Schuchowitzky, Acta Physicochem. U.R.S.S.,19, 176, 508 (1944). (29) J. W. Belton and M. G . Evans, Trans. Faraday SOC.,41, 1 (1945). (30) E. A. Guggenheim, ibid., 41, 150 (1945). (31)W. M. Sawyer and F. M. Fowkes, J . Phys. Chem., 62, 159 (1958).
Volume 70,Number 1 January 1966
140
application of eq. 8 will be primarily concerned with the dependence of the surface tension on the activity of the solute in the surface phase. The exact form of eq. 8 will not affect this aspect of the expression. The expression in eq. 8 can be obtained in the form
since both the surface tension and activity are functions of the surface mole fraction. This expression can readily be used to examine the behavior of surfactants having a common surface-active ion with respect to the thermodynamic nature of the surface. If the surface phase is ideal, then by definition, d In a;/d In xZsis unity, and the partial molar area, to2, will be independent of concentration. Since the limiting area of the surfactant molecules is independent of the alkali metal cation, vide supra, the surface mole fractions of the salts will be equal when the surface concentrations are the same. Therefore, the slopes, d r / d In ,z: for the different salts will be equal a t a given surface concentration. From the boundary conditions of the solutions ( i e . , at zero concentration the surface tensions are the same), it follows that the surface tensions will be equal at equal surface concentrations. I n addition, if the surface phase is not ideal but the different salts have the same activity coefficients at equal surface concentrations, as exemplified in the Debye-Huckel limiting law region, it is apparent that the result will be the same as for the ideal surface. It is quite clear, however, that this result is not in accord with the facts. If, on the other hand, the surface phase is nonideal and the salts have different activity coefficients a t equivalent concentrations, then both d In az"/d In xZsand the partial molar areas will be different a t the same surface concentrations. It, obviously, follows that the surface tensions will be different a t equal surface concentrations. The phenomenon in the surface phase, which has been considered in the foregoing discussion, proves to be consistent with a nonideality generally associated with solutions whose concentrations are greater than that of the Debye-Huckel limiting law region. The validity of this analysis depends upon independent determination of the activity coefficients. Unfortunately, no direct measurement is feasible, as the surface phase is inaccessible for the known
The Journal of Physical Chemistry
IRAWEIL
techniques of determining activity coefficients. Moreover, the high concentrations existing in the surface phase cannot be duplicated in a bulk phase because of micelle formation. However, according to the above analysis, the activity coefficients of the alkali metal salts of the alkyl sulfates and those of stearic acid should show the reversal observed in their surface behavior. Since the differences in surface behavior being considered involve only the ionic portion of the molecule, a comparison of the activity coefficients of simple salts with ionic groups similar to those of the surfactants should also show a reversal. Activity curves for a number of alkali metal salts can be obtained from the data compiled by Robinson and Stokes.32 In choosing a simple anion to represent the monovalent alkyl sulfate, it would be reasonable to select either a nitrate, a chlorate, or a perchlorate even though, for the purpose of the argument, the divalent sulfate can be chosen. The alkali metal salts of these anions all show the same qualitative behavior. The activity curves of these salts at the higher concentrations clearly indicate that the slopes, d In a/d In Z, are in the order Li+ > Naf > Kf. On the reasonable assumption that the alkyl sulfate salts behave similarly, the expression in eq. 9 predicts that at equal surface concentrations, the order of the surface tensions would be Li+ < Na+ < K+. This is the experimentally observed order. The activity curves of the acetate salts can be used as the counterparts for the long-chain fatty acid salts. The alkali metal acetate curves have slopes in the reversed order compared to the salts indicated above. This reversal is consistent with the observed surface behavior of the salts of stearic acid. Therefore, it can be concluded that the thermodynamic property describing the effect of the alkali metal cations on surface behavior is the activity coefficient of the corresponding surfactant salt in the surface phase.
Acknowledgment. The author wishes to thank Mr. E. R. Tulp for his assistance in performing the experimental work, Dr. E. D. Goddard and Dr. J. D. Justice for their helpful criticisms and suggestions, and Dr. A. Cahn for his help in preparing the manuscript. The author is grateful to Lever Brothers Co. for its permission to publish this paper. (32) R. A. Robinson and R. H. Stokes, Trans. Faraday SOC.,45, 612 (1949).
