HYDROLYSIS IN AQUEOUS SOAP SOLUTIONS
383
M E C H m I S M OF HYDROLYSIS IN AQUEOUS SOAP SOLUTIONS MELVIN A. COOK Department of Metallurgy, University of Utah, Salt Lake City, Utah Received March 17, 1960
The degree of hydrolysis (j3) in soap solutions of fatty acid salts as a function of soap concentration (C) was measured for a number of soaps by Powney and Jordan (5). Their j3-C curves exhibited very interesting, but extremely perplexing, minima and maxima. Recently Stainsby and Alexander (6) published a theoretical discussion of parts of these curves, emphasizing conditions associated with equilibria in bulk solution. We shall consider in this paper the entire curves on the basis of a model involving in addition surface equilibria and their effects on the bulk solution equilibria. The j3-C curves for the fatty acid salt solutions have the shape indicative of nonequilibrium. They resemble, for example, supercooling curves. However, several more fundamental arguments strongly oppose the nonequilibrium hypothesis: (I) The curves are remarkably reproducible and consistent at all temperatures. (2) Ample time was allowed for nonequilibrium effects either to have been observed or to have disappeared. (3)By comparison with surface tension (u)-concentration curves, one finds that the C,h and C,, of the j3-C curves correspond to high surface tension (low surface pressure) and low surface tension (high surface pressure), respectively. In fact, concentrations of the maxima (C,-) on the j3-C curves correspond closely to the concentrations of the minima on the corresponding u-C curves. While nonequilibrium conditions are observed at the minimum of the u-C curves, they do not appear (significantly) at lower concentrations, Le., in the region from Cmin to C,, of the j3-C curves. Surface tension, as a matter of fact, is extremely sensitive to small variations in pH in this range, and any nonequilibrium hydrolysis conditions would thus manifest themselves critically in the measurements of surface tension. The absence of such critical effects along the 4 curves from Cminto C, of the j3-C curves is therefore strong evidence that the 8-C curves represent equilibrium conditions. If one adopts the equilibrium hypothesis on the basis of this evidence, he is immediately impressed by the supposition that an analysis of the equilibrium conditions aasociated with the anomalously shaped 8-C curves will surely uncover a great deal of information concerning the nature of aqueous soap solutions. The present analysis seems to provide ample justification for this supposition. We consider evidence first from experimental observations, and next from deductions based on the equilibrium postulate, that the bulk equilibrium conditions are affected in an important way by those existing in the solution-air and solution-hydrocarbon or solution-glass interface. Actually, one fmds that surface hydrolysis, expressed in terms of bulk solution concentrations, is unimportant except at the lowest soap concentrations. However, an important condensati.on product, so far as hydrolysis is concerned, evidently cannot form
3 84
MELVIN A. COOK
in the bulk solution but only via the surface or interface. As a result, when surface saturation is reached, the lowest free-energy state otherwise available to the system then becomes unavailable. Still the hydrolysis reaction may be treated BS an equilibrium one, because the energy barrier to the particular reaction in question is insurmountable in the bulk solution (in the sense that reaction taking place over it is nil), but is small or absent at the surface. The u versus log C data of Lottermoser and Lottermoser and Giese (4) show evidence of discontinuities at the C,,n of the p-C curves. This is clearly visible in the 40°C. curves of sodium laurate and sodium myristate. Moreover, the log C us. u curves are substantially straight lines between C,,, and the maxima (CmJ of the 8-C curves. This means a substantially constant surface excess (rf))and surface concentration in the range from C,,, to CmaXof the p-C curves. Also, from the slopes, one finds F;') values amounting to around 1.5 X lo-'' to 3 X lo-'' moles/cm.2 for the soaps, or about 100 A.* to 50 .&.* per molecule. The Fh') values for sodium palmitate correspond roughly in concentration to the liquid-expanded films described by Langmuir, Harkins, and Adam. The slope of the sodium palmitate curve, however, is somewhat larger than for sodium oleate. This may be significant, if it is realized, as shown by Adam ( l ) , that the sodium palmitate monolayer begins to undergo a transition from the condensed to the expanded state at 37.5"C. This transition is apparent in the 0-C curves. The coincidence of the concentration of surface saturation and the C,,, of the p-C curves thus appears very significant. It is enlightening to consider mathematically the conditions under which the p-C curves can, under equilibrium, shorn minima and maxima. In the first place, clearly this would not be possible unless some type of soap condensation were to take place in the region of the minima. The recognized micelle region, however, commences just below C,,,; hence one is dealing in the C,,, region with submicelle particles. By submicelle is meant particles smaller than the well-recognized micelles but containing at least two molecules or ions per particle. Particles referred to here as submicelles were recognized by Pouney and Jordan ( 5 ) in their qualitative explanation of the 0-C curves, and later by Stainsby and Alexander ( 6 ) , who mentioned also earlier suggestions of Jones and Bury. The hydrolysis constant may be expressed as follows: X/Y = Kh/Z
(1)
where z is the free acid (HX) concentration, y the soap anion concentration, Z refers to (OH-), and Kh is the hydrolysis constant. Let M,, represent generally both micelles and submicelles of composition (HX),(NaX), in which NaX may be either a neutral molecule or the corresponding ions. Here i and j may take on all values from zero to infinity, but their sum is 2 or more. In addition, one may take into consideration adsorbed soap, designating its constituents by x. and y I . Micelles and submicelles in the surface may be included in the general micelle and submicelle terms. The hydroxyl-ion concentration is given by:
385
HYDROLYSIS IN AQUEOUS SOAP SOLUTIONS
where Zoaccounts for any (OH-) not generated by hydrolysis, e.g., any deliberately added. Along the 6-C curves of Powney and Jordan 20 is effectively zero. The total soap concentration is given by the equation:
c = z + 5.
+ Y + Y. + C i ~ i+j C j M i i
(3)
f
I
and, by definition (for ZO = 0):
p
=
z/c
(4)
From equations 1, 2, and 3, one obtains:
P-'
= (K'Z
+ 1) + (Y. + C j M i j ) / ( z+ + C iMii)
(5)
2.
I
J
If M , j were zero, no minima or maxima could occur in the 6-C curves, since the term (K'z 1) will always increase monatonically with C , or, upon approaching a free acid saturation value, become substantially constant. For the p-C curve to go through a minimum the second term on the right of equation 5 must be finite at Cminand the denominator, moreover, must increase with increasing C enough more rapidly than the numerator to make the whole righthand side of equation 5 decrease with C. Clearly this imposes rather stringent 2,) conditions. Incidentally, the minima cannot be accounted for by y,/(z (with M i j = 0); it is not possible to account for both the absolute value of p and its minimum by surface effects alone, since generally y. < < Cmin.(Notice that the surface concentrations should here be expressed in bulk concentration units.) One thus obtains, under the equilibrium hypothesis, the surprising result that C j M , j is finite at Cmi,,.As a matter of fact, in some cases this term ac-
+
+
i
counts for more than 75 per cent of the total soap in solution a t Cmh! Let us consider, for example, the 60°C. sodium palmitate curve of Powney and Jordan. 1) will not decrease with Here (3 is 0.04 at Cminand 0.17 at Cmu. Since (K'z C in the region Cminto C,, the second term on the right of equation 5 is at least four times greater at Cminthan the first and decreases to less than onefourth of its Cminvalue a t Cm., This means that at least 75 per cent of the soap in solution should exist in the form of submicelles (i j 2 2) a t Cmin.Moreover, at least some free acid constituent is present in the submicelle particles at Cmin.Otherwise, one could not account for the observed magnitude of 6, because the last term in equation 5 would then be excessively large. These conditions may be seen more readily by expressing 6-l in the form:
+
+
8-l
= (fi'z
+ 1)
/( -
Y.
1
+ CjMij) h
A simple general consideration indicates the possible source and nature of the submicelles most prominent at Cmin. It is important to realize that the term C j M i j in the region of Cminis independent of the ion y. To show this one i
may express this term by a function of y, thus
j M i j = f(y), in which f(y) 1
386
MELVIN A. COOK
would necessarily involve powers of y not less than unity. For example, f(y) vanes as yz for j = 2, y' for j = 3, etc. Consider the case 2y ~1 in which the equilibrium constant a' is given by:
+
a' = y2/y2 and U'(&'XZ)~ = y l In this case, the second term in equation 5 would be:
(6)
(y, is negligible.) Since 2 always increases with C,a minimum in the 8-C curve would be impassible in this case. For j = 1, the last term in equation 5 would be a constant and would thus not describe a minimum. For j > 2, the last term in equation 5 would increase in proportion to powers of 2 of 2 or more. This analysis shows therefore that the minima and maxima could not exist along the 8-C curves if the bulk solution equilibrium relations
X - + HzO
* HX +.OH-
\\ /
MQ were to apply without exception! This presents a very perplexing situation in the light of the equilibrium hypothesis. However, this dilemma may be removed, and the equilibrium hypothesis retained, by taking into account conditions associated with surface saturation. SUBMICELLE EQUILIBRIA
The formation of y2 (2y -+ y2) is at a disadvantage in the bulk solution at low concentrations because of the ionic repulsion of the ionized head group. At Cmin the average radius of the ionic atmosphere of the y ion is about 300 A. The barrie- for y2 formation in solution would begin to increase at about this distance and would be too large at the distance where condensation of two y ions could set in, to allow the reaction 2y -+ yz to take place at all in the temperature range of the 8 - 4 curves. On the other hand, the free energy of formation of y2 directly from y ions would be favorable in the configuration in which the head groups are at opposite ends of the di-ion, and the hydrocarbon chains overlap appreciably in sufficiently long-chain acids. The potential of hydrocarbon chain interaction may perhaps be something like half that of Gibbs' adsorption a t the air-solution interface, which, according to Traube's rule, is about 625 n (n is the number of carbons in the chain). The free energy of formation of y2 from y particles would then be something like 5-8 kcal. for n 2 12. The potential barrier for the reaction 2y + y~ would be Ze2/Dr, plus the activa$ion energy for diffusion. While this would not ordinarily be excessive, there would also be an unfavorable transmission coefficient K in the rate equation associated with the fact that for condensation the ions would be required to move a distance of about 300 A. relative to each other under a repulsive force.
HYDROLYSIS IK AQUEOUS SOAP SOLUTIONS
387
In the surface the soap concentration is large (-10‘ times the solution concentration at Cmin),and x,/y,>> x/y, &s shown, for example, by 4 data of Long, Nutting, and Harkins (3). The avtrage ionic atmosphere radius under surface conditions would be only about 3 A., and the reaction 2y. + yb would thus not be limited by an ionic repulsion. The yz particles would result by migra: tion of the ylr di-ions into the solution, or by migration of xzr(with head groups at the opposite ends of the particle) into the solution, followed immediately by ionization. The reactions 22 ---i XZ, and 2 2 --+ y2 2Hf are probably not an important source of y2 submicelles, because the ionization of the x2 particle with head group interaction should result in y instead of yz particles, owing to electrostatic repulsion. The 22 particle without head group interaction, on the other hand, will probably not form at all in solution because of the low z concentration and the low reaction potential in the absence of head group interaction. I n fact, if it did, it would be a source of yz particles. The experimental observations of minima along the 6-C curves are thus sufficient evidence that this reaction is unimportant. The significance of the coincidence of surface saturation with Cmin of the 0-C curves now becomes evident. Below surface saturation one would have y. s y and y2. y2, giving the impression that 2y e y2. In other words, owing to the equilibrium that would exist between the solution and the surface at concentrations below C,i,, the fact that the direct reaction 2y --+ y~ does not occur would be of no consequence, because as long aa slow diffusion processes were eliminated, e.g., by stirring and/or by allowing ample time for equilibrium, the reaction could still take place by way of the surface. Above surface saturation, however, a new situation presents itself. Mechanically, the equilibrium between the solution and the surface would be destroyed, i.e., C f C, above surface saturation. Even so, this would not prevent the equilibria
+
+
22. 22. e yz, yz from being maintained. But, since y can now condense to y2 only by passing into the surface, and since y. f y above surface saturation, the equilibrium between y and yt would no longer exist above the concentration of surface saturation. Despite this condition the hydrolysis reaction along the 8-C curve may still be regarded and treated as an equilibrium reaction. One may summarize these equilibrium conditions as follows:
2Y
e 2Y. $ yzr e yz 2y
4 YZ
(nil)
+
1
x, y. = C, e C J 2Y f 2Y, e YZ. e yz 2y z,
-P
yz
+ y8 = C.
(nil)
c < c,,,,
\
1
(sat’n) f CJ ay: = Yz
c > Cmm
(6s)
388
MELVIN A. COOK
It is, of course, not at all unusual to have equilibrium under conditions in which lower free-energy states are not accessible to the system, but the present type of nonaccessibility may be rare. FORMATION OF MICELLES A N D BUBMICELLES
For the purpose of providing an adequate model for consideration of the @-Ccurves, let us consider in more detail possible mechanisms of submicelle formation. As indicated above, the di-ion yt might have either of two structures:
'-
and that represented by ( b ) ', HOW0 ever, structure (a) would be unstable at low concentrations because of the repulsion of the head groups. Structure (b) would be stable when the overlapping hydrocarbon chain produces enough interaction to offset the head group repulsion and thermal effects. A third y particle cannot condense on the y2 particle, however, until the radius of the ionic atmosphere becomes small, i.e., until the ionic head groups can be stabilized by counter-ion ( 0 ) interaction. At C,, the average radius of the ionic atmosphere will be only about 1/10 of that a t Cmin, and counter-ion stabilization is therefore much more probable at Cma.Moreover, the free-energy barrier for y condensation would be lowered and the transmission coefficient K increased considerably at C,,,. Hydrocarbon interactions could therefore come into play at ambient temperatures. The nonequilibrium effects observed in surface tension measurements at the minimum of the u-C curves (or to Cm& may perhaps be due largely to a state of transition in the magnitude (particularly the thickness) of the free-energy barrier for y polymerization in solution, from the high value at which no condensation can occur to a low value where equilibrium is readily established. In other words, this transition in the energy barrier for y condensation would be caused by the closing in of the ionic atmosphere around the ionic head groups as the soap concentration increases. These arguments would limit the neutral soap constituent of the submicelles to the particles Miz. Also, since X i M i j is (a) that represented by
@
only a small fraction of c j M i 2 at Cmin,and probably i i
>> 2,
I
one may, as an
approximation, consider that the only appreciable neutral soap condensation in the submicelle region is represented by the y2 particles. One may carry the discussion a little farther and consider a possible mechanism for lamellar micelle formation. At high soap concentrations (relatively 0 high ionic strength) the particles M a and Mia would serve as nuclei for y condensation. In fact, a pure neutral soap micelle M o could ~ then form as represented by the structure A below. This lamellar submicelle unit perhaps has considerable free acid constituent, as mentioned. The strength of the ionic head group and counter-ion interaction for the hypothetical pure neutral soap submicelle A would increase (attractively) as the size of the particle increases, in accord with the theory of the Madulung constant for ionic structures. As a result, after the A particle has grown to a particular critical size, there would be a favorable potential for polymerization of A type particles by interaction at
+
HYDROLYSIS
389
IN AQUEOUS SOAP SOLUTIONS
the hydrophilic ends (possibly, but perhaps not necessarily, with the removal of thesolvation sheath from the ions @ -( and O)), to form the B lamellar micelles. The conditions both for growth of A particles and for polymerization to B particles would be even more favorable if A were either a pure free acid or a mixed free acid-neutral soap micelle. However, pure (or nearly pure) free acid micelles of type C could polymerize in such a way aa to tuck away the active head groups and produce a particle more or leas stable toward further lamellar growth aa illustrated by D. C submicelles could also add, head end to head end, to the A and B micelles, and thereby tend to stabilize these particles toward further growth.
I . I
D
Next, consider free acid reactions with Mo,particles. Here z would tend to add to yz and A type particles with a much more negative free energy of reaction than y particles, because the head group interactions would always be favorable in this case. That is, the z particle (0 ) would react with the negative head group both by dipole-ion interaction and by hydrogen bonding. One may therefore account for an appreciable concentration of free acid in the micelles A and B. In fact, at low soap concentrations this reaction will predominate over the y Moj-1 + Moj reaction despite a low x/y ratio. The lamellar micelles approach the condition j >> i only at high soap concentrations, but at low soap concentration, Le., in the region of Cmin,the conditions i >> j would apply. This is in accord with the discussion of Stainsby and Alexander. The reaction 22 -+ xz with only hydrocarbon interaction would be extremely unfavorable because of the very low 2 concentration in solution. As mentioned previously, the type with both hydrocarbon and head group interaction
+
(-
:)
would be much more favorable, and the xz particles may be rela-
tively stable toward ionization, owing to self solvation (e.g., by hydrogen bonding). Moreover, still further condensation in the hypothetical pure free acid submicelle would improve the stability by increased coordination in both hydrocarbon and head group (self-solvation) interaction. HYDROLYSIS CONSTANT8
The hydrolysis constants of the homologous fatty acid salts increase slightly (at constant temperature) from CIto about Cr, but are substantially constant
390
MELVIN A. COOK
above C6.The high degree of hydrolysis in C14to CU members of the series implies that perhaps this constant suddenly begins to increase above about C14. This was suggested by the work of Stainsby and Alexander. It is difficult to reconcile this supposed increase in K A for chain lengths of Cl4 and higher with the physical factors involved. The mere addition of CH2 groups f a r removed from the head group is surely not responsible for an increase in KA.That the hydrocarbon group, by folding back over the head group, might tend to stabilize the head group toward ionization suggests itself, but this is not a very well supported hypothesis. It is not in evidence, for example, in the formation of insoluble fatty acid monolayers, which usually spread spontaneously and rapidly, and the observed hydrophobic nature of the head groups. In fact, it is doubtful that such an effect, if it did occur, would really tend to exclude solvent from the head group and influence ionization. The evidence for a changing K A in the higher homologs is merely circumstantial, and is aasociated with computations needed to compensate the previously unrecognized conditions connected with minima in the p-C curves and with submicelle formation. While we shall use the asymptotic value of Kh, determined from the lower members of the homologous fatty acid series, the actual value of Kh is not important in the present study, because it always occurs aa a product (KAc)with the free acid micelle equilibrium constant. The value of K,,, of course, atrects the determination of x (but not y above Cmi,,). On the other hand, x will still be negligible for any reasonable value of Kh. It is clear, therefore, that K,+cannot be found from an analysis of the p-C curves. One is concerned also with the hydrolysis constant pertaining to the surface phase. Experimentally (1, 2, 3), the ratio x,/y, is about unity at a solution pH of 8 to 9 in the higher homologs, e.g., Cl4 to C17, where x/y would be unity around pH 5. The surface hydrolysis may be expressed by the equation: d y . = K:/Z,
-
where K : is the true surface hydrolysis constant and 2,is the (OH-) concentra10aKA,that Z, is corretion in the surface. One must aasume either that K: spondingly smaller than Z , or that a situation between these extremes exists. The second condition seems most unlikely. While inorganic ions generally show negative adsorption, this is small relative to the large positive adsorption found for soap solutions. Evidently the true surface hydrolysis constant is largely responsible, therefore, for the large x./y, ratio in comparison with x/y at a given pH. I t is possible to justify a large increase in K: over 'KAby considerations of relative adsorption potentials of the ions and free acid. K: for the solution-air interface should be related to K Aby the following equation: where A,, A,, and A, are the contributions to & of free acid, anion, and hydroxyl groups, and XO appean in the equation for the total Gibbs adsorption potential (A = & 625n) of the soap, obtained from Traube's rule (Adam ( 5 ) , p. 122). Here n is the number of carbon atoms in the chain and XO is the contribution
+
HYDROLYSIS I N AQUEOUS SOAP SOLUTIONS
39 1
from the head group alone. In an equilibrium mixture Xo is the sum of the separate terms, A., A,, A,, weighted according to their relative concentrations per mole of total adsorbed soap. Experimentally, different head groups influence A,, considerably. Adam listed A,, values differing by 1.5 kcal. Tacitly assuming A, N 0, one obtains A. A, 4-5 kcal. from the observed x,/y, ratio. This is an upper limit to A. A,, because A. will be negative, but perhaps small. For computational purposes, we shall define K. by the equation K , = K : Z / Z , , from which one obtains:
-
-
-
%8/y# =
K,/Z
(7)
Here K , is an apparent surface hydrolysis constant which differs from the true constant by the factor K, may be treated on an equal basis with K:. This applies to computations of the temperature coefficient of the ratio x,/y,, as well aa to ita variation with 2. In ofher words, the temperature coefficient of K , haa the same form aa tha;t of K. because of the factor e-'''RT relating them. The w e of K8 instead of K , haa the advantage of eliminating the necemity of evaluating 2..This makes use of an apparent surface pH equal to the bulk pH. Besides surface tension studies, one finds further evidence for K. values 10' (to aa much as IO') times greater than Kh in the experimental contact bubble data of Wark and others obtained in flotation research (2). THEORETICAL EVALUATION O F
8 4
CURVES
A quantitative treatment of hydrolysis in aqueous soap solution requires the evaluation of the two terms X i M i j and E j M , j in equation 5 . According t o i
i
the above considerations, the neutral soap micelle and submicelle term should be divided into two parts aa follows:
C> jMij
+
2 ~ 2 C'jMij
(8)
where the second summation now excludes the particle y2. The second term represents normal neutral soap micelle formation occurring to an appreciable extent only in the region from slightly less than Cmsx to all higher concentrations. The equilibrium relations for free acid condensation, for a given j (i.e., j') would be: AF = -RT log, ci Mi-l,j, HX e Mij,; (9)
+
where the ci's are the equilibrium constants. One is interested, according to the proposed model, in two types of free acid condensation: (1) that producing a pure free acid micelle M j o , and (9)that initiated by possible condensation of free acid on particles M,?. Case 1
By combining all possible equation 9 reactions, one obtains:
392
MELVIN A. COOK
and
The ci’s are all interrelated with respect to coordination effects. They will increase sharply from c2 to about cg because there will be an increased coordination of both hydrocarbon chain and head group interactions in this range. However, they will also quickly level off to a constant value when the Ma particle becomes large enough so that each additional condensing x particle may experience optimum coordination. The cl’s actually are rather complicated functions, in the sense that there are at least three types of interactions affecting AF. These are head group interactions alone, hydrocarbon chain interactions alone, and head group and hydrocarbon chain interactions occurring together. Moreover, hydrocarbon chain interactions are expected t o be variable, depending on the amount of chain overlap. The ci’s used in this treatment may thus be considered to be statistical equilibrium constants. Coordination effects will vary in the smallest type submicelles, possibly approximately in proportion to i/i’ (i < i’), where i’ is a particular i above which complete coordination will be achieved, Le., for i 2 i’, the c,’s will all be equal. Hence, one may write the product of c,’s as follows:
where f is a small positive constant which is greater than unity, because of the lower limit of the product. Using equation 11, equation 10 now becomes: x i ci Mi0
= CJ
The total free acid in the Mio submicelles and micelles is then m
m
CiMio= 2
ixici
-= 2
CJ
-
x2c(z’)(2 zc) (1 - xc)Z
(12)
Case d Adding all equation 9 equilibria for Mi2 particles, one obtains: y2
+ i x s yzxi;
A F = -RT log, c:
Treating the product of c: in the same manner a~ above, and realizing that c: = ci = c ( i 2 i“), where i” is the number of x particles present in the Ma submicelles required for optimum coordination, one obtains the relations: Mi2
y2 xici
C”
393
HYDROLYSIS IN .\QUEOUS SOAP SOLUTIONS
and 02
i M i 2 = y2c“-”’z/(1
- zcy
1 02
02
The relative importance of the terms z i M i o and z i M i 2depends on the M e r 2
1
ence f - f’, which is probably around zero or less. That is, according to the lower limits of these sums, it should be unity. But, since the potential of interaction of free acid head groups should be considerably greater than for that of a free acid and a neutral soap, the potential effect will about offset the greater coordination for condensation of z on the two-particle submicelle y2, than on itself, to give 2 2 . I n the region of Cminone may usually neglect yI and z in the second term of equation 5, where they occur alone. Moreover, z > M , l = 0 and
-
I
c 104/y2 in this region. Hence, at C,,, written :
the last term in equation 5 may be
+ C’j~ii)/(z+ + 7ihfti)
(2~2
Under condition l5a one eliminates by cancellation the troublesome y2 term. The difference between equations 15a and 15b however is only minor, because the important part of the expression is the factor 1/(1 - zc)’, which becomes critical as zc approaches unity. Consider now the term z ’ j M i j in equation 8. In the region where neutral I
soap condensation becomes important (aside from the y2 particle formation), Le., as the concentration approaches C,., the free acid condensation will have become large enough so that the i subscript in Mij, on the average, will be relatively large, at least along those curves where om= is much greater than &in and C, >> Cmin.One may therefore treat neutral soap condensation also as made up of two types: ( I ) pure neutral soap Maj and (6)condensation onto free acid submicelle nuclei Mi and Mi2. The direct condensation of neutral soap involves the counter ion 0 as well as the anion y. These come into the micelle in pairs. The pure neutral soap reaction, comparable with equation 9 for pure free acid micelles, is: 0
+ X- + Mi,,j-i e
Mij;
AF =
--RT log. b j
Treating the M o micelle ~ in the same way as the M a ones, and regarding 0 as a single particle, equation 17 is obtained as follows:
(16)
+y
394
MELVIN A. COOK
where g here has the same significance as f in equation 11. Likewise, condensation of neutral soap onto Migo and Mit2leads to equations of the same form aa equation 14, with Mi.0 or Mil2 replacing y2, i.e.,
and 0
jMi,,j+* = Mirzb(l-u')y/(l
Finally, for the total neutral soap in all
Mij
- yb)'
(18)
micelles, one obtains:
Neutral soap micelle formation corresponds to the condition where yb approaches unity, just aa zc approaches unity during free acid condensation. Equation 19 may be written:
In this case, g - g' < 1, because the interaction of neutral soap with itself would be less than with a free acid particle. In general, C ( M i t o Mi,2)
