J. Phys. Chem. 1082,86,2463-2468
2463
Solublllratlon of Phenothiazine in Aqueous Surfactant Micelles Yoshlklyo Morol,’ Kelko Sato, and Ryohel Matuura Department of Chemise, Faculty of Science, Kyushu University, Higashi-ku, Fukuoka 8 12, Japan (Received: March 20, 198 1; In Final Form: December 16, 198 1)
The maximum additive concentration (MAC) of phenothiazine (PTH) in aqueous micellar solutions of sodium dodecyl sulfate, manganese(II)dodecyl sulfate, and zinc(I1) dodecyl sulfate was determined spectrophotometrically at 20, 25, and 30 “C. The plots of MAC against surfactant concentrations could be divided into two straight lines. The surfactant concentration at their intersection was in good agreement with the critical micelle concentration (cmc) determined by conventional methods. The thermodynamic parameters of the interaction between monomeric phenothiazinemolecule and micelle without any solubilizate were evaluated from the stepwise association equilibria between solubilizates and micelles. Through the analysis of the data from the viewpoint of aggregation numbers and kinds of gegenions of micelles, it turned out that the aggregation numbers played a more important role than the gegenions for an incorporation of phenothiazine into the micelles. The Poisson distribution is highly suggested for the distribution of phenothiazine among the present micelles.
Introduction Surfactant molecules aggregate to form micelles in aqueous solution above a narrow concentration range which is called the critical micelle c0ncentration.l Hydrophobic groups of surfactants form a micelle core which is liquid hydrocarbon like in character, while their hydrated polar groups constitute a micelle outer shell in contact with water. Because of these characteristics, physicochemical properties of micelles, particularly micelle formation, have been studied extensively.2-5 Solubilization also provides the basis for surfactant action of amphiphilic molecules. Solubility increase of nonpolar organic materials in a surfactant solution a t concentrations above the cmc is ascribable to an incorporation of hydrophobic molecules into micelles in surfactant solutions. Indeed, many studies have been reported on solubilization into surfactant micelles,- but their interpretations have been rather qualitative, while the papers involving theoretical discussions are very ~ ~ W . ~ J O This work was intended to make clear the solubilization in micellar solutions from a thermodynamic point of view. A special attempt has been made to highlight the theoretical background to a mechanism of solubilization and a determination of association constants of solubilizates with micelles.
Equilibrium Distribution of Solubilizates among Micelles What has been derived concerning solubilization in the preceding paperlo can be applied to the present system. The monodispersity of micelles in the absence of solubilizate is assumed to remove the difficulties arising from their polydispersity, which will be discussed in Appendix I. As is clear from this appendix, the following discussion remains essentially the same. The ideality of chemical species in a solution is also assumed because of their low concentrations. The association equilibrium between surfactant monomers (S) and micelles (M) is presented by
where K,,, is the equilibrium constant of micelle formation and m is an aggregation number of micelles. The stepwise association equilibria between micelles and solubilizates (R) can be represented schematically as follows: K
M + R ~ M R ,
MR,
... MR,-l
(1)P. Mukerjee and K. J. Mysels, Natl. Stand. Ref. Data Ser. (U.S., Natl. Bur. Stand.), No. 36 (1971). (2)J. N. Phillips, Trans. Faraday SOC.,51, 561 (1955). (3)K. Shinoda, ‘Colloidal Surfactants”,Academic Press, New York, 1963,Chapter 1. (4)P. Mukerjee, J.Phys. Chem., 76,565 (1972). (5)Y. Moroi, N. Nishikido, H. Uehara, and R. Matuura, J . Colloid Interface Sci., 50,254 (1975). (6)P. H. Elworthy, A. T. Florence, and C. B. Macfarlane, ‘Solubilization by Surface-ActiveAgents and its Application in Chemistry and the Biological Sciences”, Chapman and Hall, London, 1968. (7)(a) P. Mukerjee, ‘Micellization, Solubilization, and Microemulsions”, Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977,p 153; (b) P. Mukerjee and J. R. Cardinal, J.Pharm. Sci., 65,882 (1976); (c) P. Mukerjee and J. R. Cardinal, J. Phys. Chem., 82,1620 (1978). (8) N. Nishikido, J. Colloid Interface Sci., 60,242(1977);K.S. Birdi, H. M. Singh, and S. U. Dalsager, J. Phys. Chem., 83,2733(1979);A.Goto, M. Nihei, and F. Endo, ibid., 84, 2268 (1980). (9)E. Ruckenstein and R. Krishnan, J.Colloid Interface Sci., 71,321 (1979). (IO) Y. Moroi, J. Phys. Chem., 84,2186 (1980). 0022-365418212086-2463$01.25/0
+ R &M R ~ + R & MR,
(2)
where MRi is the micelles associated with i molecules of solubilizate, Kiis the stepwise association constant between MRj-, and a monomer molecule of solubilizate, and n is an arbitrary number. The total number of components of this system is n + 4 including water molecules, and the number of phases is two, micellar solution phase and solid or liquid solubilizate phase. A solubilizate phase is inevitable in the case where the maximum additive concentration (MAC) is a point of discussion. The n + 1 equilibrium equations for the micellar system reduce the number of degrees of freedom of the phase rule by n + 1, resulting in 3 degrees of freedoms. Hence, at constant temperature and pressure, only one other intensive variable can be selected to specify the thermodynamic system. We selected here the total surfactant concentration for the variable. From eq 1and 2, we have the following equations for the total micellar concentration ([M,]), the total 0 1982 American Chemical Society
2464
Moroi et al.
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 T h e rmos t a t
1 Ji
Magnetic
0,8
Cell
~
stirrer
I
aE
8
rl
0,6
\
z
0 IMembrane f i l t e r
4, t-
5 0,4
Figure 1. Apparatus for solubility determination.
equivalent concentration of solubilizate ([R,]), and the average number of solubilizate molecules per micelle (R): [Mtl =
5 W3,I
i=O
z
0 0
+ a S
0,2 (3)
[R,] = [R] + =
[R] + K,[SIm
5 i[MR;]
i=l
0
2 i(fi K;)[R]'
i=l
j=1
(4)
(5) R = W t 1 - [RI)/[M,l The succeeding three equations can be derived with infinite n (see Appendix 11) from three ways of distributing solubilizates among micelles: lo random (or geometrical) distribution (6) [RJ = [Rl{1+ Ki(1 + B)[Mt11 Poisson distribution (7) [%I = [R1(1 + Kl[M,I) Gaussian distribution (8) [R,] = [R][1 + K,R exp(h2(1- 2R)j[M,]]
Experimental Section Materials. Phenothiazine (PTH) of extra pure reagent grade as a solubilizate was purchased from Nakarai Chemicals, Ltd. The reagent was purified by recrystallization 3 times from pure benzene and stored in the dark. The melting point was 184-185 "C, and the elemental analysis agreed with the calculated values within 0.1 % . Sodium dodecyl sulfate (SDS), manganese(I1) dodecyl sulfate (Mn(DS)2),and zinc(I1) dodecyl sulfate (Zn(DSI2) were prepared and purified by the standard procedure.'l Their purity was checked by the cmc determination and the differential thermal analysis technique."J2 The water used was distilled twice from alkaline permanganate. Solubilization. A suspension of phenothiazine powder in surfactant solution was stirred initially at room temperature by disk rotors in 10-mL injector tubes. The injectors were then dipped into a thermostat until equilibrium was reached, during which any light was shut out from the injectors, and the temperature was kept constant (11) (a) Y . Moroi, K. Motomura, and R. Matuura, Bull. Chem. SOC. Jpn., 44, 2078 (1971); (b) Y. Moroi, T. Oyama, and R. Matuura, J . Colloid Interface Sci., 60, 103 (1977). (12) Y . Moroi. K. Motomura, and R. Matuura, Bull. Chem. SOC.Jpn., 45, 2697 (1972).
0
20
10
SDS
30
40
CONCENTRATION / m o l dm-3
Figure 2. Solubility change of PTH with SDS concentrations.
at 20,25, and 30 "C and controlled within 0.01 "C (Figure 1). The run was timed from this point and proceeded for more than 2 h. A t hourly intervals, 2.5 mL of filtrate was withdrawn by applying pressure upon the injector through the filters of 0.2-pm pore size (Millipore; FGLP 01300). Phenothiazine has two absorption bands in the ultraviolet region whose maximum wavelengths are 248 and 307 nm in the intermicellar bulk phase and 251 and 315 nm in the micellar phase.13 The bands at 248 and 315 nm were used for MAC determination for the surfactant concentrations below and above the cmc, respectively. The molecular extinction coefficients were e248nm = 3.81 X lo3 and t315m = 4.12 X lo2 m2 mol-'.
Results and Discussion The variations of MAC of PTH with SDS concentrations are illustrated in Figure 2. As is evident from the figure, the plots of MAC against the surfactant concentrations can be divided into two straight lines; one for below the cmc and the other above it. The former has zero slope irrespective of surfactant concentrations, whereas above the cmc MAC increases linearly with increasing surfactant concentrations. The intercept and the slope of the two lines giving the best fit with the experimental points were obtained by linear regression analysis. From this result and from an intelligent inspection of expressions of eq 6-8, we can see that (1)where [M,] is zero below the cmc, [&I remains constant as the saturated concentration ([R]) of PTH at the specified temperature and (2) where [M,] increases with surfactant concentra-tions above the cmc, the solubilization increases keeping R and [R] constant up to 40 mmol dm-3 of SDS, since otherwise a linear relationship between [RJ and SDS concentrations would not (13) Y. Moroi, M. Saito, and R. Matuura, Nippon Kagaku Kaishi, 482 (1980).
Solubilization of Phenothiazine in Micelles
The Journal of Physical Chemistry, Vol. 86,
TABLE I: Cmc Values Determined by Solubilization and Other Methods cmci(mmo1~ i m - ~ ) temp, solubili- conduc- calcusurfactants K zation tivitv lationa SDS
293.15 289.15 303.15 293.15 298.15 303.15 293.15 298.15 303.15 293.15 298.15 303.15
SDS/0.15 M
NaCl
Mn(DS)* WDS),
8.21 8.34 8.46 1.38 1.29 1.45 1.18 1.05 1.16 1.29 1.14 1.14
No. 13, 1982 2465
i 1; 293.15 K
2 ; 298.15 K
so
3; 303.15 K
8.24 8.20 8.25 1.22 1.28
-
1.34
60
(r
1.18 1.16
I
\
-
h
1.18
E
1.21 1.20 1.22
I
.- 40 LT I
a
v
By the method given in the ref 5.
result. In addition, we can say that the present surfactant concentration range is in a state where the micelles are too far apart to influence one another. The surfactant concentration and the MAC at an intersection of the two lines are considered to be the cmc of surfactant' and the monomer concentration of the solubilizate at the cmc, respectively. The cmc values thus obtained are in good agreement with cmc's determined by other conventional methods (Table I). Rearranging eq 6 and 7, we obtain the following equations for determination of K1. Here, the Gaussian distribution is omitted, because it is clear that an application of the distribution to the present case where R is less than 1.5 is not appropriate: random distribution (9) ([&I - [RI)/[Rl = Ki(1 + R)[MtI Poisson distribution (10) ([Rtl - [RI)/[Rl = K'[MtI where [M,] = (C - cmc)/N. Thus, the slope of the line obtained by plots of ([&I - [R])/[R] against C - cmc gives K J N and K1(l R)/N for the Poisson and random distributions, respectively. Figure 3 shows good linear relationships at 20, 25, and 30 "C,which confirm that the analysis is valid within the experimental accuracy. The point is that K1/N values decrease with increasing temperature, notwithstanding that the solubilized PTH amount increases as temperature increases. This is due to the fact that higher solubility at higher temperature is largely counterbalanced by the increased concentration of R at higher temperature as shown in Figure 2. Now attempts will be made to probe further into the physicochemical meaning of the association constant K1. As is shown in eq 2, K, is a parameter which relates to the free energy change caused by an incorporation of one monomeric PTH molecule into micelles free from any PTH molecule. In terms of K, the free energy change can be expressed in the following form: AGO = - R T In K1 (11) Furthermore, AGO is made up of three contributions as follows:
+
AGO = peMR, - peM - peR
(12)
where peml is the standard chemical potential of MR1 at infinite dilution, and peM and p e R are the corresponding potentials. The difference pam1 - peM should depend on the surfactant micelles into which a PTH molecule incorporates, in particular on the environment around the
20
0 0
20
10 SDS
CONC, (C
30
-
40
C M C ) / m o l dm-3
Flgure 3. Plots of ([R,] - [R])/[R] against micellar concentration (C - cmc) of SDS for determination of K , value.
site where a PTH molecule sits. It was found in the previous paper13 that PTH molecules locate in a palisade layer of micelles and that a complex formation takes place between PTH molecules and divalent metal ions. Considering these findings, we can suggest that the aggregation number and the kind of gegenion of micelles must have a strong influence on the K1 value. The aggregation number of SDS micelles can be varied by adding excess NaCl to the solution, the gegenion being kept the same. On the other hand, the effect of the kind of gegenion must be examined by using micelles of the same aggregation number and different gegenions. The aggregation number of SDS micelle at 25 "C is 64, which is deduced from many data, and its temperature coefficient, d In N/dT, is near -0.01.14 Thus, these lead to 62.6 and 65.4 at 20 and 30 OC, respectively. This is reflected on slight cmc change with temperature (Table I). On the other hand, cmc values and the aggregation number of divalent metal dodecyl sulfates are almost the same independent of the kinds of gegenions. The cmc's are 1.22 f 0.04 mmol dm-3 and the aggregation numbers are 95 f 7 at 30 OC.15J6 Here, the N values are for dodecyl sulfate ion. If a molar unit is used, N becomes 47.5 for the divalent metal dodecyl sulfates. The NaCl concentration which makes the aggregation number of SDS micelles 95 was around 0.15 mol dm-3, found by referring to many literature va1~es.I~In Figure 4 are shown the plots of solubility of PTH against surfactant concentration for four surfactant systems, SDS, SDS/0.15 M NaC1, Mn(DS)2,and (14) N. Muller, "Micellization, Solubilization, and Microemulsions", Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977, p 229. (15) I. Satake, I. Iwamatsu, S. Hosokawa, and R. Matuura, Bull. Chem. SOC.Jpn., 36, 205 (1963). (16) Y. Moroi, K. Motomura, and R. Matuura, J. Colloid Interfaace Sci., 46, 111 (1974). (17) K. J. Mysels and L. H. Princen, J. Phys. Chem., 63, 1696 (1959); H. F. Huisman, Proc. K. N e d . A k a d . Wet. Ser. B: Phys. Sci., 67, 367 (1964); M. Emerson and A. Holtzer, J.Phys. Chem., 71, 1898 (1967); N. A. Mazar and G. B. Benedek, ibid., SO, 1075 (1976); S. Hayashi and S. Ikeda, ibid., 84, 744 (1980).
Moroi et al.
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982
2488
TABLE 11: Association Constant ( K , ) and Thermodynamic Parameters of Solubilizationa 10-5~,, A G " , kJ AH", surfactants temp, K mol-' dm3 mol-' kJ mol-' SDS (N= 64)
293.15 298.15 303.15 293.15 298.15 303.15 293.15 298.15 303.15 293.15 298.15 303.15
SDS/0.15 M NaCl ( N = 95)
Mn(DS), ( N = 95)
Zn(DS), ( N = 95)
a
1.43 1.18 1.06 2.40 1.94 1.64 2.02 1.73 1.45 2.09 1.78 1.51
AS",J
K-'mol-' 24 24 24 7 7 7 25 25 25 19 19 19
- 28.9
-21.9
- 29.0 -
29.2
- 30.2 - 28.2
- 30.2 - 30.3 - 29.8 - 29.9 - 30.0 - 29.9 -30.0 -30.1
- 22.4 - 24.2
Based on the Poisson distribution.
t
dd
-a: 100 -. - 75 I
h
a:
U
a:
I
v
50
25
0
0
0
10
20
30
40
SURFACTANT CONC /mol dm-'
10
20
SURFACTANT CONC, ( C
Flgwe 4. Solubility change of PTH with the surfactant concentratabns at 298.15 K: (a) SDS, (b) SDS/O.15 M NaCI, (c) Mn(DS),, (d) Zn(DS),.
Zn(DS)2. That the inflection point of plots of the SDS/ 0.15 M NaCl system is almost the same as those of Mn(DS)2and Z ~ I ( D Ssuggests )~ an approximately equal aggregation number for the three surfactant systems. The surfactant concentrations at the intersections are given in Table I. Now it is observed that the cmc determination by the solubility method is good when a monomer concentration of solubilizate is lower than mol dm-3. The plots for K1determination are shown in Figure 5. The K1 values thus obtained lie in the order SDS/0.15 M NaCl > Z ~ I ( D S> ) ~Mn(DW2> SDS (Table I). The slope for SDS/0.15 M NaCl in Figure 5 is steeper than that for SDS, whereas the slope for SDS/0.15 M NaCl above the cmc in Figure 4 is more gentle than that of SDS. This result comes from the fact that a steeper slope for SDS in Figure 4 is counterbalanced to a large extent by a higher concentration of monomeric PTH molecules in the aqueous bulk phase; namely, aqueous salt solution has lower solvent power for PTH than salt-free water. This fact gave rise to larger K1values as can be derived from eq 9 and 10. From the variation of AGO with temperature, the enthalpy and entropy changes, AHo and ASo, of the association can be determined. AHo = - R[d In Kl/d(l/T)] (13) ASo = (AHo - A G o ) / T
0
(14)
30
-
CMC)/"ol
dm-3
Figure 5. Plots of ([R,] - [R])/[R] against micellar concentration (C - cmc) for determination of K , values at 298.15 K: (a) SDS, (b) SDS0.15 M NaCI, (c) Mn(DS),, (d) Zn(DS),.
The thermodynamic parameters derived from eg 13 and 14 are summarized in Table 11, where the Poisson distribution is employed. The reason for its employment will be mentioned later. Of course, AHo and ASo values change with temperature because of effects of temperature on both micellar structure and chemical potentials of other components. However, present attention is paid to the values at 25 "C which can be considered as the mean values across the temperature range 20-30 OC. We may here make a few remarks about the behavior of the association constant K1 with changing the gegenions and the aggregation number of surfactant molecules. The following can be deduced from the K1values in Table 11; (1) the association of PTH with the micelles is an exothermic reaction; (2) an increase in the aggregation number is favorable for the present association; and (3) differnece in the gegenions does not play an important role in the incorporation of PTH into the present micelles. The third conclusion is contrary to our expectation, because PTH forms a complex with divalent ions when solubilized. However, the result comes mainly from such weak complex formation that the halflife of the reaction is more than 1 day. Thus, the increase of K1 values for SDS/O.15 M NaCl solution is mainly due to an increase of hydrophobicity of the palisade layer with an increase of the aggregation number.
Solubilization of Phenothiazine in Micelles
The Journal of Physical Chemistty, Vol. 86, No. 13, 1982 2467
TABLE 111: Average Number of PTH Molecules per Micelle and the Difference in Standard Free Energy Change between Poisson and Random Distributions
surfactants
temp, K
PTH/ micelle
SDS
293.15 298.15 303.15 293.15 298.15 303.15 293.15 298.15 303.15 293.15 298.15 303.15
0.93 1.03 1.14 1.28 1.38 1.50 1.21 1.30 1.42 1.27 1.35 1.48
SDW0.15 M NaCl MNDS 1 2
ZnPS),
AGr."AGP-",
kJ mol-' 1.60 1.75 1.92 2.01 2.15 2.31 1.93 2.06 2.22 2.00 2.12 2.29
Consider now the distribution patterns of PTH among micelles. From eq 9 and 10, the free energy difference between the Poisson (AGp7") and random (AGrp") distributions can be expressed as AGr," - AGpr" = R T In (1 + R )
(15)
From the aggregation number (N),64 for SDS and 95 for the other surfactant systems, we can determine the R values from eq 5, using the relation [M,] = (C - cmc)/N. The problem is, however, whether the temperature change of N goes beyond the experimental error. As mentioned above concerning SDS micelles, the N value changes about 2% across the temperature change of 5 "C. In addition, as is seen from Table I, the cmc change is very little for each present system over the temperature range 20-30 "C, which strongly suggests almost constant aggregation number over the temperature range. Consequently we used the above-mentioned numbers for N. The AGO differences from this term are given in Table 111. The reasons that the Poisson distribution is more suitable than the other one are the following: (1)If we adopt the random distribution, AGs" values become almost equal irrespective of the surfactant systems, in spite of the fact that the palisade layer where the PTH molecule locates should change in its physicochemical properties with the aggregation number. (2) The analysis of many data for the micelle-catalyzed reactions strongly suggests that the Poisson distribution is most preferable.18 Besides, Nmethylphenothiazine molecules take the Poisson distribution among CU(DS)~ mi~e1les.l~(3) The condition for the association constant Kj = K,/j which was used to derive the Poisson distributionlo is really acceptable when solubilized molecules are much less in number than the micellar aggregation number, just as in the present case. In view of these results, it is highly likely that the Poisson distribution is much more suitable for the present systems. However, the free energy difference attributable t o the distribution patterns is in any event less than 10% of total free energy change. Finally, we may now make a few remarks about distribution patterns from a mathematical point of view. The phenothiazine distribution for the present micellar systems turned out to obey the Poisson distribution to a good approximation, which results from the fact that both solubilizate molecules and micelles are not only indistin(18)T.Harada, N. Nishikido, Y. Moroi, and R. Matuura, Bull. Chem. SOC.Jpn., 54, 2592 (1981).
(19)M. Maestri, P. P. Infelta, and M. Gr&tzel,J. Chem. Phys., 69,1522 (1978). (20)P.Mukerjee, 'Micellization, Solubilization, and Microemulsions", Vol. 1, K.L. Mittal, Ed., Plenum Press, New York, 1977,p 171.
guishable but also indpendent (Appendix 111). This result can be expected from small R values less than 2, just as in the present case. Thus, it can generally be said that a discussion of solubilization in terms of a Poisson or binomind distribution is reasonable for small R values and for micelles and microemulsions of large volume, because both micelles and solubilizate molecules can apparently act independently and indistinguishably. On the other hand, the random or geometrical distribution can be one choice for consideration when the R value increases, and as a result some restrictions become relevant. At this point, it should be emphasized that the distribution of solubilizate molecules among micelles is determined only by their association constants with micelles, not by mathematics, as mentioned in the preceding paper.1° As a matter of fact, each association constant is too difficult to determine. Therefore, we have analyzed the solubilization with the help of mathematics and examined which distribution pattern can be approximated by the present solubilization model. In conclusion, it can be said from Table I1 that the solubilization of PTH in the present micelles takes place as a compromise between the effects of the hydrophobic interaction between PTH and the micelles on the one hand and of randomness on the other. However, the former effect, AH",has a much greater contribution than the latter, -TAS", to the total free energy change AGO, where the TAS" term comes mainly from the melting of structured waters around PTH molecules in the bulk phase on moving from the bulk phase to the micellar phase. In addition, an advantage of the present approach over the two-phase model for solubilization is the determination of Kl the standardized index for interaction between micelles and solubilizate molecules, since the latter approach gives an overall value of Ki.
Appendix I Having discussed the solubilization only from the monodispersity of a micellar aggregation number, we now attempt to gain insight into the effect of polydispersity on the thermodynamic expressions of solubilization. A considerable effect of polydispersity is a variation of the association constant Kj. We define a new association constant Kim,which is K, of the micelles of aggregation number m. Suppose the polydispersity of the aggregation number of micelles ranges from a to p. The concentration of micelles of aggregation number m and with no solubilizate association becomes [M"] = Km[SIm
(16)
Summing up [MmR,]over the aggregation number from a to /3, we obtain for the concentration of micelles associated with i solubilizate molecules 8
[MR,I = C [MmR,I m=a
B
= C (K,m[R])i[Mm]/i! m=a
(17)
where we used the same relation as eq 24 in the preceding paper,l0 Kjm = Klm/j. Hence, the total solubilizate concentration becomes [Rtl = [R] + i = l i[MRi]
Moroi et al.
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982
2468
Hence, in the same manner as eq 10, the following expression results: B
([%I
-
[RI)/[Rl = C K,"[Mt"I
(19)
m=a
However, the difference in the mathematical expression between eq 10 and 19 will disappear if the following operations for averaging are made:
although the mathematics is not the main object of this appendix. Consider a random distribution of r balls in q cells, where both of them are independent and indistinguishable. The probability P(i) that a specified cell contains exactly i balls is given in the form21a
B
C Kl"[Mt"I
---
where r and q are on the order of Avogadro's number and i is very small compared with them, less than 10 for the present case. This equation is a special case of the so-called binominal distribution. P(i) can be rearranged as
m=a
B
[M,] = C [M,"] = (C- cmc)/N
(21)
m=a
where N is the mean aggregation number. Thus, the thermodynamic parameters given in Table I1 are mean values over the aggregation number from a to P. On the other hand, the distribution of the micellar aggregation number has already been discussed,18and the standard deviation is less than 10% against the mean aggregation number. Appendix I1 As mentioned before eq 6 and 7 have been derived by summing n to infinity. As n goes to a large number, the physical state of the micelles becomes unsuitable for the present solubilization model. In fact, however, the R values are less than 2, and the summations of probabilities over the n range from 0 to 6 for N = 64 and from 0 to 10 for N = 95 are the following:
Using the following approximation equation which is reasonable for the present condition 1 - l / q = exp(-l/q)
(25)
we obtain
where R is the average number of balls per cell, r/q. Because of the conditions as to r, q, and i values, P(i) becomes
For our present case, as is clear from the above table, the probability that the number of solubilized PTH molecules per micelle is within 10% of the micellar aggregation number is more than 99% even for the random distribution. For the solubilization to the above extent, the present stepwise association model can be used without problem. Thus, the foregoing expressions can be applied with more than 99% probability confidence irrespective of the distribution patterns, while there is no problem at all from the Poisson distribution.
which is the Poisson distribution. The Gaussian distribution can also be obtained from the binominal distribution.21b In this sense, both distributions are complete random distributions free from any restriction. However, the difference between Gaussian and Poisson distributions is that the former is a function of three variables, R , i, and a standard deviation from which the latter is free, as is clear from eq 26. It is also very enlightening to consider the distribution where some restriction is present. In such a case, the balls are not independent of each other even though interball interactions are absent. The distribution where balls are indistinguishable and cells are distinguishable is one of the above examples (Bose-Einstein distribution). As for solubilization, if the number of solubilized molecules in a specified micelle has an influence on the succeeding distribution of solubilizate molecules remaining among the other micelles, they cannot be solubilized independently. The random distribution based on the above concept is the random (or geometrical distribution) of the Appendix in our preceding paper,l0 which is an answer to problem 14 on page 61 of ref 21. However, the difference between the cases with and without restrictions becomes small with decreasing R value, as is clear from eq 6 and 7.
Appendix I11 Special mention should be made of probability theory in order to make the present report understood by readers,
(21) (a) W. Feller, 'An Introduction to Probability Theory and Its Applications", 3rd ed, Vol. 1, Wiley, New York, 1967, Chapter 11, p 35; (b) ibid., Chapter VII.
Poisson -
R
1 1.5
i p(n)
n=n
? p(n)
11: n
random
i R(n)
n:o
1.00
0.99
1.00
I?= n
~ ( n )
1.oo
where for the Poisson distribution P(i) = R' exp(-R)/i!
(22)
and for the random distribution
