DISCUSSION AND INTERPRETATION O F T H E THERMODYNAMIC PROPERTIES, CONDUCTIVITY, AND DIFFUSION OF A TYPICAL COLLOIDAL ELECTROLYTE: LAURYLSULFONIC ACID I N AQUEOUS SOLUTIOW PIERRE VAN RYSSELBERGHE Department of Chemistry, Stanford University, California Received A u g u s t 1 , 1999 I. INTRODUCTION
The exact quantitative interpretation of the various properties of colloidal electrolytes meets with a number of difficulties, and progress is possible only through the use of drastic simplifications and assumptions. Any theory that tries to take account of more factors than can be handled simultaneously by rigorous methods will never reach a stage where significant comparison with experimental data can be carried out. On the other hand, simplifying assumptions should be systematically applied to Ihe interpretation of the data before any opinion as to the respective merits and defects of these assumptions is expressed. The method should be the following: Having a t one’s disposal reliable data corresponding to one or several properties of the systems under consideration, one sets up the exact equations describing these properties according to the simplifying assumptions previously adopted. These equations are written for a number of different concentrations, and solved for the unknowns corresponding to the assumptions chosen; for instance, as in the work discussed in the present paper, the numbers of positive and negative simple ions associated in the average micelle. If the equations are compatible, and if their solutions are a t all plausible, something tangible with some element of truth has probably been obtained. One a t least knows exactly “where one stands” regarding the assumptions a t the basis of the treatment. The present confusion of the subject and the conflicting theories so far proposed should be clarified to some extent by the use of the method here suggested. In a previous paper (16) the author has tested in this manner the application of the Debye-Huckel theory to the interpretation of osmotic data of two colloidal electrolytes, Congo red and sodium thymonucleate, in aqueous solution, the assumption being that a negative micelle with a large charge is present. The equations were solved for 1 Presented a t the Sixteenth Colloid Symposium, held at Stanford University, California, July 8-8, 1939. 1049
1050
PIERRE V A N RYSSELBERGHE
the ionic diameter a of the Debye-Huckel theory, the charge of the negative micelle being assumed constant but large enough to make the function u ( K ~of ) the theory smaller then unity, as required to make a positive. When a turned out to be fairly constant over the whole range of concentrations, a reasonable model for the electrolyte was assumed to have been obtained. The method, in spite of several imperfections, works in the right direction and explains, a t least qualitatively, the very low osmotic pressures exhibited by these electrolytes. The assumptions involved in such a treatment have been discussed by Hartley (2). Very recently, the main ideas of the Debye-Huckel theory have been applied by Levine
x,z
300
200
IO0
01 C,
WLES PER LITER,
02 C,
03
04
05
06
07
(
3
MOLES PER LITER
FIG.2 FIQ.1 FIG.1. Variation with concentration of the osmotic coefficient and of the conductivity ratio of aqueous solutions of laurylsulfonic acid. FIG.2. Number ( 2 ) of laurylsulfonate ions and number (z) of hydrogen ions in the average micelle of lsurylsulfonic acid solutions. Apparent degree of ionization ‘((2
- z)/2).
(6) to the problem of stability in hydrophobic colloidal solutions. The Debye-Hiickel theory is also used in the theoretical considerations recently published by Langmuir (5). At the suggestion of Professor McBain we have attempted to interpret the properties of aqueous solutions of lauryl(or dodecy1)sulfonic acid, for which conductivities (lo), freezing points (ll),and electromotive forces ’ (12)have been measured by McBain and Beta, and diffusion coefficients by Mrs. McBain (7). Further conductivity and freezing point data are reported in a recent paper by Mrs. McBain, Dye, and Johnston (8). Gn figure 1 are plotted the osmotic coefficients, i, and the molar conductivities, A. The i values correspond t o van’t Hoff’s definition and are deduced
PROPERTIES OF A COLLOIDAL ELECTROLYTE
1051
for round concentrations from Johnston’s data by means of linear interpolation between consecutive experimental points. At concentrations larger than 0.1 we have assumed i constant and equal to its value 0.280 a t 0.1, while McBain and Bets’ values for the range 0.1 to 0.8 vary from 0.326 to 0.318, with a minimum of 0.290 a t a concentration of 0.36. Differences of this order have a negligible influence on our calculations. The h values are those of Mrs. McBain (8) and of Beta (9), obtained a t 25’C. These are used in conjunction with i values corresponding to the freezing point. The presence of colloidal micelles in these solutions is obvious and, without adopting any a priori assumption as to the possible presence of several types of micelles, we shall attempt t o account for these two properties (osmotic coefficient and conductivity) on the basis of an average micelle whose composition changes with concentration. The results will then be used in a discussion of diffusion coefficients. Although interionic forces are undoubtedly a t play, their effect is certainly small compared with the ideal properties of the associative model, just as in the case of dilute strong electrolytes ideality in terms of complete dissociation is responsible for the main portion of the thermodynamic properties. With many colloidal electrolytes the model is still to be found, and refinements or corrections taking electrostatic and other forces into account have but little significance as long as the problem of finding the correct model is not solved. Ideality implies constancy of mobilities for the simple ions. The mobility of the average micelle, however, changes with size and charge. We shall calculate this mobility according to Stokes’ law, along the lines suggested long ago by McBain (9). A posteriori interpretation of diffusion data seems to confirm the validity of this application of Stokes’ law. 11. OSMOTIC COEFFICIENTS AND CONDUCTIVITIES OF LAURYLSULFONIC ACID IN AQUEOUS SOLUTION EXPLAINED ON THE BASIS OF AN AVERAGE NEGATIVELY CHARGED MICELLE
At each concentration we assume that we have exclusively an “average micelle,” H,L,, consisting of z hydrogen ions (H+) and z laurylsulfonate ions (L-), together with the compensating amount of free hydrogen ion. The charge 5 - z of the micelle is balanced by that of z - z free hydrogen ions. Calling C the total concentration of acid in moles per liter of HL (exact formula: HCI~H&O~), and designating by (H+) and (H,L,) the respective concentrations of free ions and micelles, we have the following two stoichiometric conditions: (H+)
+ z(H,L.) z(H.Lz)
=
C
(1)
C
(2)
1052
PIERRE VAN RYSSELBERGHE
The osmotic coefficient, i, is such that (H+)
+ (H,L,)
=
iC
(3)
We shall assume that this law, strictly valid only a t low concentrations, applies to the whole range of concentrations considered here, i. e., up to 0.8 molar. Calling A+ and A- the molar conductivities of H+ and H,L,, the molar conductivity, A, of the electrolyte is such that (H+)A+
+ (HzL*)A- = CA
(4)
For each particular concentration C we have four unknowns: x, z, (H+), and (HzL2),but we also have four independent equations. Let us note that any incomplete dissociation is automatically taken into account in the average micelle. This likewise includes any free simple sulfonate ions. A treatment of this type could actually be developed for an electrolyte like acetic acid, for which one would find an average negative ion intermediate between the acetate ion and the neutral acetic acid molecule, the charge of this “average ion” being a function of the total concentration. In formula 4 we shall take A+ = 350, the limiting value of the molar conductivity of hydrogen ion a t infinite dilution a t 25’C. For the simple laurylsulfonate ion, i. e., when z = 1, z = 0, we have A- = 22, according to the latest experimental data from this laboratory (8). On the basis of Stokes’ law the mobility u(z, z ) of the micelle in centimeters per second per unit field is
in which e is the elementary charge of positive electricity, q is the viscosity of the medium, and r ( z , z ) is the radius of the micelle which is assumed to be spherical, The mobility u(1, 0) of the simple ion is
From equations 5 and 6, neglecting the variation of viscosity, we deduce
The volume of the micelle is proportional to the number (z) of laurylsulfonate ions and is practically independent of the number ( z ) of hydrogen ions. Hence
PROPERTIES OF A COLLOIDAL ELECTROLYTE
1053
and, according to equation 7, u(z, z, =
2 - 2
51/3 u(lJ O)
(9)
The molar conductivity of the micelle is then related to that of the simple ion by the formula
Equation 4 becomes
Combining equations 1 and 3 we get
z = (1 - i ) x
+1
One easily finds that
Taking equations 12, 13, and 14 into account in equation 11, we get 350
(i - 2'>+22- (ix - 1)' d'3
= A
or, also, 1 p3 (O.063i2z2- O.126iz - z'"
A + 0.063) = 350 -- i
(16)
This equation is solved by successive approximations for a number of round concentrations. The particular concentration for which the difA
ference - - i is equal to zero (0.058 molar) is also included. From 350 x and i one calculates z by means of formula 12 and the concentrations (H+) and (H,L.) by means of formulas 13 and 14. The results are reported in table 1, in which are recorded, in successive columns, the stoichiometric molar concentration, the osmotic coefficient, i, the molar conductivity, A, the values of z, those of z, and the concentrations (H,L,) and (H+). The values of z and 2, as well as those of (z - z)/z, are plotted on figure 2. The latter values are a measure of the relative charge of the
1054
PIERRE VAN RYSSELBERGHE
micelle; they also represent the ratio of the hydrogen-ion concentration to the total concentration:
The concentrations (H,L,) and (H+) are plotted (at different scales) on figure 3. The results are entirely acceptable. The maximum in the (H,L,)-curve is due to the combined effect of two factors: increase of concentration of the simple ions and increasing size of the micelles, our H,L, being an average of all micelles, molecules, and simple sulfonate ion. TABLE 1 Composition of the “average inclusive micelle” and concentrations of that micelle and of the hydrogen ion i n aqueous solutions of laurylsulfanic acid, assuming no other constituents to be present C
z
0 0.005 0.01 0.02 0.03 0.04 0.05 0.058 0.06 0.07 0.08 0.09 0.1 0.2 0.4 0.6 0.8
1.00 1.02 1.16 2.91 5.09 7.80 13.8 18.9 20.2 30.1 48.6 92.3 166 219 270 321 321
0.990 0.640 0.512 0.432 0.396 0.388 0.350 0.322 0.300 0.280 0.280 0.280 0.280 0.280
246 168 147 142 139 137 132 132 139 146 156 167 176 176
(H+)
2
0.00 0.11 0.12 1.03 2.83 4.81 8.81 12.4 13.4 20.6 33.9 65.5 120 158 195 232 232
0 0.00490 0.00862 0.00687 0. 00589 0.00513 0,00363 0.00307 0.00297 0.00232 0.00165 0.000975 0.000604 0 ,000915 0.00148 0.00187 0.00249
0 0.00446 0.00897 0.01292 0.01332 0.01532 0.01800 0.01995 0.02022 0.02119 0.02408 0.02610 0.0275 0.0552 0.1112 0.1662 0.2216
Above the maximum, the latter effect is by far the predominant one. The application of Stokes’ law in the manner explained above implies rather large mobilities and conductivities for the micelles. For instance, a t C = 0.07, the concentration at which the conductivity goes through a minimum, the velocity of the average micelle is 3.1 times that of the simple anion and 0.19 times that of the hydrogen ion; its conductivity is 29.4 times that of the simple anion and 1.9 times that of the hydrogen ion. has a velocity equal to 13 times that The largest average micelle, Hea~Lsel, of the simple anion and 0.82 times that of the hydrogen ion; its conductivity is 1156 times that of the simple anion and 73 times that of the hydrogen ion.
1055
PROPBRTIES OF A COLLOIDAL ELECTROLYTE
It is intewsting to note that the electric field at the surface of this largest micelle is only 1.9 times that at the surface of the simple anion, when both are assumed Spherical. Actually, the field at the active end of the simple anion is much larger than for a hypothetical spherical ion and the true ratio of the fields is certainly much smaller than 1.9. The above results would of course be modified to a certain extent if viscosity changes could be taken into account. Viscosities of laurylsulfonic acid solutions have been measured by Mrs. McBain (8) and are used in the nextzsection of this paper in connection with diEugion data.
C, MXES
RR
LITER
C,
hlOCES PER LITER
FlQ.3 FIQ.4 FIG.3. Variation with total concentration, C, of the concentrations of the average micelle and of the hydrogen ion in aqueous solutions of laurylsulfonic acid. FIQ.4. Curve 1, experimental values of I)/Doratios (integral values of diffusion coefficient); curve 2, calculated values of DIDOratios (differential values of diffusion coefficient); curve 3, calculated values of b / D o ratios; curve 4, calculated values of D/Do ratios tentatively corrected for viscosity.
Structural viscosity is probably predominant in the more concentrated solutions, and it is impossible to carry out viscosity corrections with any degree of certainty. 111. DIFFTJSION COEFFICIENTS
OF
LAURYLSULFONIC ACID
IN
AQUEOUS
SOLUTION CALCULATED ON THE BASIS OF THE AVERAGE MICELLE. PREDICTION OF TRANSPORT NUMBERS
Integral diffusion coefficients of laurylsulfonic acid have been measured by Mrs. McBain (7). The ratios of the average diffusion coefficient b to the limiting value D Ofor infinite dilution are given in table 2 (column 2)
1056
PIERRE VAN RYBSELBERGHE
and are plotted on figure 4 (curve 1). The theoretical value of D Ois given by the Nernst formula:
in which u$ and u ! are the mobilities of the simple ions at infinite dilution. With u! equal to 350 and u! to 22, we find, at 25OC., D o= 0.952 TABLE 2 Differential and integral diffusion coeficiente of laurylsulfonic acid i n aqueous solution calculated on the basis of the average micelle C
0 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.4 0.6 0.8
and
1.000 0.919 0.872 0.735 0.625 0.473 0.402 0.383 0.373 0.499 0.567 0.586 0.720 0.762 0.762 0.762
D/Do
51Do
(CAICULATBD)
(CALCULATED)
1.OOO 0.855 0.866 0.639 0.414 0.376 0.423 0.445 0.481 0.547 0.691 0.813 0.915 1.001 1.064 1.064
3/00 (CORRECTEDFOR VIWOOITY)
1 .Ooo 0.927 0.893 0.823 0.724 0.642 0.593 0.566 0.552 0.547 0.555 0.575 0.719 0.839 0.903 0.943
per day. For an unsymmetrical salt dissociating into negative ions, we have
1.000 0.907 0.865 0.760 0.664 0.581 0.531 0.502 0.484 0.475 0.477 0.485 0.556 0.554 0.504 0.446 v+
positive
v-
In our case we have, for each molecule of acid, (z - z)/z hydrogen ions and 1/z micelles. The solution being assumed ideal in terms of the average micelle, we calculate the differential diffusion coefficient D a t each concentration by means of the Nernst formula or the ratio D / D o by means of the following formula deduced from formulas 19, 18, and 9:
D - (z-z++~)(z-z~). 22"' 1
5 0 -
1 .063
+ 0.063s*z
(20)
PROPERTIES OF A COLLOIDAL ELECTROLYTE
1057
The results are reported in table 2 (column 3) and plotted on figure 4 (curve 2). In column 4 we give the corresponding integral values, which are directly comparable with the experimental results given in column 2 (see also curve 3 of figure 4). In the last column of table 2 (and also in figure 4, curve 4) we give B / D , ratios tentatively corrected for viscosity, according to the formula proposed by Gordon (1) and Van Rysselberghe (17). The correction is applied to the differential values of column 3 and the new values of the ratios DIDO are then transformed into integral values D/Do. We have
in which 7 is the viscosity of the solution referred to pure water. The viscosities used in our calculations have been measured by Mrs. McBain (8). I t is indeed remarkable that, as shown in figure 4, our calculated values, both without and with viscosity correction, exhibit a minimum a t practically the same concentration &s the experimental curve. At concentrations higher than 0.1 the viscosity increases very rapidly and may be of a structural type, which renders the correction applied to diffusion coefficients uncertain. In fact we do not attach much weight to the viscosity correction as a whole in the present case, since the model was established by means of calculations on conductivities in which the change of viscosity with concentration was neglected. An interesting application of diffusion data to the determination of the size of micelles was recently published by Hartley and Runnicles (4).2 It is also possible to predict, on the basis of our model, the transport numbers of hydrogen and of the simple laurylsulfonate radical. The formulas are (Ht Lz)
x - 2
7.x.22
T- = (H+).350 - (H.Lz) ’ s 2 * 2 . 2 2
-
+ (H,L,)
5 - 2
F . x . 2 2 (22)
X
+
15.9~”~x - z T+ = 1 - T - =
15.9s”’ - z 15.9~”’ x - z
+
* N o t e added i n proof: While this paper was in the process of publication, G. 5. Hartley (Trans. Faraday Soc. 86, 1109 (1939)) has given a qualitative interpretation of the minimum in Mrs. McBain’s diffusion curves on the basis of an ionic micelle of constant size and charge in equilibrium with the aimple ions.
1058
PIERRE VAN RYBBELBERQHE
With the values of z and z reported in table 1 we h d T- = 0.060 and T+ = 0.940at C = 0.005, and T- = 1.620and T+ = -0.620at C = 0.8, the transport number of hydrogen becoming negative between the concentrations 0.09 and 0.1. Further refinements in our calculations, such as the viscosity corrections mentioned above, might alter these predictions considerably. Ty. AVERAQE, IONIC, AND NEUTRAL MICELLES
It would be highly interesting and useful to have some precise information as to the actual micelles and ions of which our H.L, micelle is the average. McBain (13) has been of the opinion that an ionic micelle of high mobility and high conductivity, and a neutral micelle, or a t least a practically neutral one of low conductivity3, are necessary in order to explain all the properties of soap solutions, and hence those of an electrolyte like laurylsulfonic acid. This point of view has been criticized by Hartley (3) and discussed by Tartar and his collaborators (15, 18, 19). The distribution of our average micelle among two or more types must, of necessity, be baaed on some rather arbitrary assumptions. In order to clear the ground we shall first show that the decomposition of H,L, into an ionic micelle, L,, and an exactly neutral one, HmLm,or into &Lb and HmL,, is incompatible with Stokes’ law as applied above. 1. Decomposition o j H,L, into L,
+ H,L,:
The electroneutrality condition gives n(L,) = z
- z(H.L=)
Since (L,) $: (HmLm) = ( H L ) = iC - (E+) we have, obviously, from equations 24 and 25:
The conductivity would require
Dividing equation 27 by equation 24 we get
a Estimates of this conductivity can be deduced from Mrs. McBain’s work on the migration and electrokinetic propertiee of soap solutions and soap curds (J. Phys. Chem. 28, 678 (1924); Trans. Faraday Soc. Si, 163 (1935)).
PROPERTIES OF A COLLOIDAL ELECTROLYTE
1059
The second inequality (formula 26) gives, since both n and x - z are larger than unity,
> (x - 2)2/a
n*/8
But, since 1
2 - 2-is
or
n218
> x*/a(1 - >,a,
(29)
smaller than unity, (1
> 1 - -2
-
(30)
2
which shows that formula 29 is incompatible with formula 28. 2. Decomposition of H,L, into H,Lb
+ HmL,:
The electroneutrality condition gives (b
- a)(&Lb)
= (z
- z)(H&=)
Since
(=La) 4- (HmLm)
= (HsLz) =
IC - (H')
(31)
(32)
we have, obviously, from equations 31 and 32: (&La) < (H.Lz) and b
-a>2 -z
(33)
- z)
(34)
- z)'(H.L,)
(35)
The conductivity would require
b-i/a(b - a) =
x-1/y2
Dividing equation 34 by equation 31 we get
- a)*(&La) and, hence, since b - a > x - z, b-'la(b
