Dennis J. Miller lnstitut fur Physikalische Chemie der Universitit Stungari Pfaffenwaldring55 D-7000 Stungan 80 (Vaihingen) W-Germany
The Distribution of Solubilized Molecules among Micelles
The distribution of particles among "boxes" can serve as a useful model for statistical mechanical problems. A wellknown example ( 1 ) is the derivation of the Boltzmann distribution, where the "boxes" are energy levels. A case where molecules are almost literally confined to boxes is the s o h hilization of a substance by a micellar solution (Fig. 1).The distrihution is of interest in connection with the reactions of electronically excited states (e.g., excimer formation), because the micelle acts as cage within the lifetime of the excited state. In ~articular.s i n a h occunied micelles cannot ~ a r t i c i ~ ain te thi~xximer form&n)n reaction, A r A' ~ ~ ' . ' G e n e r &the furm of the distrihution must bt:assumed in order tu inter11rt:t the experimrntal results quantitativrly. Confltctiny view.; have heen put forward on thederivati,m of the distribution of solul~ilrzedmoleculei among micelltts (2-5). 'I'his stems from failure to consider the arrangement ofthesoluhilized molecules within the micelles. In the treatment presented here enthalpy effects are ignored as they are not amenable to a simule dls. general theorv. Thev.orobahlv . favor crowding of the molecules and so make the distribution narrower. Sometimes it is not oossihle tosolubilize more than a small number of molecules in the same micelle. Pyrene in aqueous Na-dodecyl-sulfate appears to have a maximum occupation number of two (2). Let there be N molecules and Z micelles with Z, micelles containing n solubilrzed molerulri. Kememht:rlng that thr solubilized molecules are t'undnm~ntallyindtitincuishal~le. the number of ways of arranging them among the micelles ("boxes") is w=- Z !
-.
-
Figure 2. Geometrical (-I
and Poisson (------)distributions.
nz,!
According to basic statistical-mechanical principles, the Z,s will he such that W is a maximum. The fixed numbers of micelles and solubilized molecules provide constraints to this problem, which can be solved by the standard method of Lagrangian multipliers. If W is maximized under the conditions
ZZ" = z ZnZ,
=N
(2)
we obtain the geometrical distribution eiven hv Dorrance and Hunter (2):
n+
Figure 3. Examples of binomial dishibution for maximum occupation numbel r=4.
ii"
geometrical (3) where p, is the probability that a micelle contains n soluhilized molecules and ii is the average occupation numher. The geometrical distrihution is a strange result, for, according to eqn. (3), however large the average occupation numher, there are more nnoccunied micelles t,han those containing Ti solubilized molecules. This is illustrated by Figure 2 which comDares the Poisson and eeometrical distributions. T h e anomaly is a co&equeuce of the fundamental indistinguishability of molecules. An analogous treatment of the distrihution of distinguishable (macroscopic) particles among hoxes leads via eqn. (4) to the Poisson distribution, eqn. P" = (1 +$"+I
Figure 1. Salubilizstion by svfactam solutions. Pyrene, fw example, is soluble in aq. Nadadecyl-sulfate solutions but not in water. 776 / Journal of Chemical Education
(5).
ii" n!
Pn =-=-"
-
Poisson
(5)
The Poisson distribution can. however, be derived even assuming indistinguishable particles. quat ti on ( 1 ) does not consider the arrangrmmt of the soIubili7ed molecules within the micelles. A simple treatment assumes each molecule requires a volume v.This volume will be larger than the molecular volume, because most of the occupied micelle consists of surfactant. If the micelle volume is V,the maximum occupation number is T = Vlv.Assuming there are m configurations of the molecule in the volume u , we can derive eqns. (6) and (7).
In this case the binomial distribution is also obtained for distinguishable particles. The binomial distribution, which is illustrated in Figure 3, can he considered a generalized Poisson distribution, for in the limit T eqns. (5)and (7) become identical. Though the geometrical distribution does not apply to solubilized molecules, it could be relevant to the distribution of aggregation numbers in some systems. Chung and Heilweil (6) . . have sueeested that some non-aaueous surfactant solutions may behave as ideal associating solutions and gave a statistical mechanical treatment leading- to a -geometrical distribution of aggregation numbers. I am grateful to Doz. Dr. M. Hauser and Dr. U. Klein for helpful discussions.
-
"-
Literature Cited T
VZ"! pn =
(F)
a);(
(I
- ;)T-"
binomial
(7)
(11 See, for example, Mmre, W.J., "Phydesl Chemistq': Prentim-Hall. Englewaal Cliffa, 1963. p. 619. (2) Hauser, M.,and Klein,U.K.A., AclaPhya. Chem Univ S z q r d , 19.363 (19731. (3) i3orranco.R. C..and Hunler,T. F..J. Chsm. Soe FaradqvI.68, 1312(1972l. (4) Dorrance,R. C.,andHunler,T.F., J. Chem Soc.. Faraday 1.70.1572 (19741. ( 5 ) Khuanga,U., Sel1nger.B. K.,and McDonsld,R.,Aurt. J. Chem., 29, 1!19761. (61 Chung,H.S..andHeilueil, I. J . J . Phys. Chem.,74.4S8!19701.
Volume 55. Number 12, DBcember 1978 1 777
