556
R. H. ARANOTV
T‘ol. 67
THE STATISTICAL MECHANICS O F NICELLES1 BY R. H. ARANOW RIAS, Baltimore 12, Maryland Received July 23, 1962 A theory of micelle statistics is developed using the extended theory of dilute solutions, the dielectric continuum model of the solvent, and the statistical mechanical treatment of physical clusters a t constant pressure, and n connection is mhde between the mass action approach and the two-phase approach currently in use for examining micelle behavior. Both ionic and non-ionic micellev are treated. A discussion is given for non-ionic micelles of the meaning of averages, fluctuations in size, ideality, and variation of c.m.c. with temperature. A more general theory of micelle statistics is formulated by the elimination of the continuum model of the solvent and the oxtended theory of dilute solutions. Reich’s model for non-ionic micelles is used t3 illustrate the theory.
Introduction Micelles, large clusters of molecules in equilibrium, are sometimes treated theoretically as a product of the reaction of single molecules to form clusters with an associated equilibrium ~onstant.2,~Sometimes the micelles are considered to be a different phase and the thermodynamics of phase equilibria is i n ~ o k e d . ~Hoeve J and Benson,6in a treatment mhich can be considered as a simplification of Hill’s theory of molecular clusters of imperfect gases,’ studied the statistical mechanics of micellar systems using the canonical ensemble. The purpose of this report is to develop a more general theoretical treatment of micelle statistics using Hill’s theory of physical clusters and the ensemble natural for constant temgerature, pressure and quantity of solvent. The more general theory provides a common foundation for previous different approaches and reveals the relationship between them as well as the source of some of the uncertainty underlying the interpretation of micelle behavior. Some of the problems which arise in the study of micelles are the meaning of the temperature variation of the critical micelle concentration, the effect of interaction of micelles upon the distribution of micelle sizes, the meaning of “ideality,” the nature of the spread of micelle sizes about the average value, and whether or not consideration of a single micellar size can be used as a basis for a theoretical treatment of micelles. We shall endeavor to answer some of these questions or a t least to state the question in a quantitative manner. I n section I of this paper we shall present a more general ensemble for non-ionic systems using the physical cluster theory of Hill, the extended theory of dilute solutions of Fowler and Guggenhe ,g and the assumption of incompressibility. Thi nsemble yields information about osmotic pressurehandthe effect of interaction of clusters. The meaning of the law of mass action and tionships for non-ionic (1) This remarch was supported b y the United States Air Force through Directorate of Chemical Sciences, Air Force Office of Scientific Research, under Codtract Number A F 49(638)-735. Reproduction in whole or in part is perlnittsd for any purpose of t h e United btates Government. (2) (a) I. Reich, J . Phys. Chem., 60, 257 (1956); (b) D. Stigter, Rec. trav. chzm., 73, 593 (1954). (3) M. J. Vold, J . Colloid Sei., 6, 506 (1950). (4) K. Shinoda, Bull. Chem. Soc. Japan, 26, 101 (1953). (5) (a,) E. Matijevic and B. A. Pethica, Trans. Faradag Soc., 64, 587 (1958); (b) E. Hutchinson. A. Inaba, and L. G. Baily, Z. phgsik. Chem., 5, 344 (1955); (e) G. btainsby and A. E. Alexander, Trana. Faraday Soc., 46, 587 (19501; (d) K. Shinoda and E. Rutchinson, J. Phys. Chem., 66, 577 (1962). (6) C. A. J. Hoeve and G. C. Benson, ibid., 61, 1149 (1957). (7) T. L. Hill, J . Chem. Phys., 23, 617 (1955). (8) R. Fowler a n d E. A. Guggrnheim, “Statistical Thermodynamics,” Cambridge, 1956, Chapter VIII, p. 372.
micelles will be derived concerning fluctuations in micelle size and the effect of dimerization on quantities related to the critical micelle concentration. I n section 111 ionic micelles will be treated using a modified constant pressure ensemble, the physical cluster theory of Hill, the continuum model of the solvent, and the extended theory of dilute solutions. Suggestions for a more general treatment of micellar statistics will be discussed in section IT’. I n section V calculations for an analytically tractable model proposed by Reich will be presented in order to demonstrate the relationships developed in preceding parts of this report. I. Constant Pressure Solution Theory.-In order to develop a more general formal structure for studying micelle systems, we use the ensemble suitable for constant pressure, constant temperature, and constant amount of solvent which Hill considered for a constant pressure solution t h e ~ r y . ~ The partition function for this ensemble is
r(Na, p , T,p,)
= , - - ~ a ~ a /= k ~ N p 10
AS, ( N u , p ,
T)eNpPo/kT
(I)
where AN@
V
&(Ne, N,, V , T ) e - p V l k T
and &(Ne, Np, V, T ) is the canonical ensemble partition function. N is the number of molecules, p is the chemical potential, V is the volume, p is the pressure, and T is the absolute temperature. Subscript a refers to the solvent and subscript p to the solute. We restrict our discussion to a binary non-ionic system. The leading term, A,, of the sum in (1) refers to the pure solvent. The extended theory of dilute solutions8 requires that the total free energy be the sum of the free energy of the component parts and the partition function for a binary solution must be the product of two partition functions of the form &solution
=
&solvent
(T,V
- Npqpr Ne) X &solute
( N a jNp, V j T ) (2)
where Qsolute contains all solute-solvent interaction terms and ZIP is the partial molar volume of a molecule of solute. If both the solvent and solution are regarded as incompressible and the extended theory of dilute solutions applies we have
QiN,, N P == 0, V , T ) = Qo(N,, VCI,T)G(V - J‘o) (9) T. L. Hill, J . Am. Chem. Xoc., 79, 4885 (1957); J . Chem. Phys.. 80, 93 (1959).
STATISTICAL MECHANICS OF MICELLES
March, 1963
Nn =
and
Q(N,, Npt V , T ) = Qo(Na, - N, up, 7’) X 6(V - [Vo 4-NB v g l ) Q ~ p ( N , , Ng, V,T> (3) where 6 is the Dirac &function. Substituting these relations into eq. 1we have
r
Qo QN, e N @ P p / k T =
q-p(Vo+N@8B)/kT
= Nolo
QNa AbN, NRLO
Qo
e-PvJkT
(4)
where AB is a constartt quantity for the ensemble and is, by definition, exp (pa pvp)/kT. Let us introduce the following notation
-
Ps = s(I*,) V8
= SbP)
As
= ( A 2
(5)
(6)
QN
where QN refers t o the partition function for a set of clusters N = N1, AT2, . . .Ns,, , . and the summation is taken over all possible sets subject t o the restriction
2 sN,
=
8-1
N,
(7)
The product Q N XoNP ~ can then be rewritten as & ~ )iNP p =
N
(8)
&N (n[ Ashrs) 8
where the N , is from the set N. Then
C
=
Q N ~
NP20
QN
(nA s N a )
NL0
(9)
=
Q2
=
&3
... &zoo. . . Q~oo...
Qiio..
A,n+l(Qloo. .i)
. . .1 . . (12)
+.
When the clusters are non-interacting as in the theory of Hoeve and Benson, only the leading terms appear. The Q’s of the leading terms are the partition functions for single clusters in the system. The interaction of the cluster with the solvent but not with other clusters is contained in the leading term. When the first inter action term is included, the effect of interaction of monomer with cluster is taken into account. The set of eq. 12 leads to ‘L1auTof mass action” equilibrium quotients
RIn Rn
Q100. . . . &OW.*.
-
.-
Qooa
(13)
. . -1
where the first term in Xp accounts for interaction of monomer with monomer and with the cluster of size n. Extension of the development to ideal mixed micelles can be made by considering the clusters themselves to be ideal (perfect) solutions’l internally a t constant temperature and pressure. Then the summation in r(N,,p,T,PcLa,Pr) would be taken over all values of N B and N , and then over all cluster distributions N where each distribution corresponds to a particular combination, N p N,. Since no further concepts are introduced, the methods are essentially the same as we have discussed and the assumptions are equivalent to those of the two phase treatment by Shinoda12and the mass action treatment of Mysels and Otter,13 we shall not consider this topic further. 11. Applications.-One further result of constant pressure solution theory which is useful in micelle studies is the relationship of the ensemble to osmotic pres~ure.~ Xoting that
+
N,p,(p,T,O) = -kT In AO
.
(14)
where p,(p,T,O) is the chemical potential of pure solvent and defining I*a’(ptT, f l P )
QlOO.
+ Qoio. +
+
8
where the summation is over all cluster sets corresponding t o all possible values of N , and the relationship between QN and QN is displayed, for example, by &I
. . .1
Qioo.. .Qooo..
(n- 1 ) Q l O O . .
Using the methods (leveloped by Hill in the theory of molecular clusters in imperfect gases the partition func~ be written in a way in which the clusters tion Q N can appear explicitly QNP =
XpnQooo.
557
=
~ * a ( p , Tf ,l p ) - ~.ca(p,T,O) (15)
we have
..
+ Qooi.. .
(10)
The subscript notation 04the right reveals the nature of the distribution. The position in the sequence reveals the number of particles in the cluster, ie., s, while the number in that position reveals N , . The equilibrium or average number of s clusters is obtained by applying
But for osmotic equilibrium between a solution a t p,T,N@,and the pure solvent at p - II, T , we have
Cla(p,T,q9) = Pa(P - II,T,O)
(17)
to eq. 4 using‘eq. 9. After the differentiation we set Ape. Thisgives
As =
E ’ 1 = -!2
=
A,
&loo.
A,2Qolo.
+ .. . + Ap3(&110.. . ..
(2Q200..
Qmo2.
Ql0c.
. .)
. . QOlO.
+ .. . . .) + . ,
(10) B. D. Flockhart, J . Colloid Sci., 16, 484 (1961). (11) R.Fowler and E. A. Guggenheim, ref. 8, Chap. VIII, p. 353. (12) K. Shinoda, J . Phys. Chen., 86, 541 (1954). (13) K.J. Mysels and R. J. Otter, 2. Colloid Sci., 16, 474 (1961).
R. H. ARANOW
558
Usiiig t,he same assumption of incompressibility we used to develop the form of r we have the relationships
Therefore, the ensemble sum has a clear physical interpretation. When the maximum term method can be used for N p and for N and micelles do not interact, the expected result for an ideal dilute solution can be deduced that
2-
N ,*
1cT where the asterisk denotes values in the most probable (maximum) distribution. Let us return now to the distribution of average micelle sizes in order to compare the statistical treatment to a tmo-phase equilibrium problem. For the non-ionic detergent, ignoring cluster interactions we have, letting Qooo. . .1 = Q s
Vol. 67
The interpretation of the changes of c.m.c. with temperature is clear for the case in eq. 26.
For eq. 27 a similar computation may be made. Deviations of heats of micellization computed using eq. 28 from those found calorimetrically are sometimes attributed to non-ideality or to dimerization.1° Equations 12 indicate the nature of the terms which must be considered to include monomer and cluster interactions if these are the source of non-ideality. The fluctuation of micelle size about the average of the distribution for non-interacting clusters of non-ionic detergents can be related to experimental obserrahles. Using the definitions
N
=
C nS=
For the monomer we compute pLp= pvp - kT In Q1
Rp
spLB=
S
+ kT hi ivl
For the cluster s j v p - kT In Q8
(22)
+ k T In ns
At the maximum RSin the distribution, asterisk denotes the maximum value
sIVs
=
(23)
XQ:s S
S
=
C XX,"Q,
(29)
S
we compute b(/XKVp)/bXB. For experiments where X p is varied only by the introduction of more solute, Q s is not an implicit function of X p as long as the system is still dilute. The result is
where the
b In Rs* ~-0 bS
Thus pp = pup -
kT
b In Qs*
___
bS Equating 22 to 25 we have the relationship
- LT-b IndSQs* =
- LT In Q1
(25)
= PLia - P U P
IC T
A t ordinary pressures p v p / k T is constant and negligible. Thus
+ kT In iTl
At the critical micelle concentration when dimers have not formed appreciably is the critical micelle concentration. This relationship is comparable to the one used by Shinoda4in his treatment of the effect of chain length on critical micelle concentration. (See section 111of this paper.) Shinoda's energy calculations for the micelle "phase" clearly are related to the energy changes involved in adding or removing a molecule from the most probable micelle. This relationship displays the reason for the success in some calculations of theories which postulate only a single micelle size: only the most probable micelle need be considered in any ~ $ 1 culation involving chemical potential. (Vold3 has discussed other aspects of this problem for ionic micelles in terms of the mass action law.) When dimers are appreciably formed at the critical micelle concentration, we need only substitute the relationship N / V = c.m.c. = 2172)/V to obtain
+
N _ -- 1- and In A, 17 s
I n the ordinary micellar system the activity and the average micelle size change very little above the critical micelle concentration; hence, very sensitive measuremerits mould be required to determine the fluctuation. 111. Derivation of the Constant Pressure Ensemble for Ionic Detergents.-Development of a comparable cluster theory for ionic detergents is attended by certain difficulties which are part of the general problem of electrolyte theory. Among these difficulties is the necessity of assuring weak interaction among the members of the statistical ensemble while allowing the number of molecules of a charged species to vary from zero to infinity. Clearly, the condition exists that some members of the ensemble will have infinite charge and in this case will interact strongly with other members of the ensemble. I n order to avoid these problems of interaction we shall assume the restriction of the members of the ensemble to electrically neutral systems or to systems of some arbitrary closeness to electrical neutrality. By the use of this restriction we assure the condition that the members of a statistical ensemble be weakly inter-
STATISTICAL MECHAXICS OF MICELLES
March, 1963
acting. The resultant restricted ensemble is no longer the same as that for the non-ionic case but is a hybrid constructed by the sum over canonical, incompressible, almost neutral, and neutral ensembles. When the fluctuation of charge of the real system about the neutral point is very small, the resultant hybrid ensemble should give a correct representation of properties. The extended theory of dilute solutions applies only to non-electrolytes. We find an analog of this theory, however, in the continuum model of the solvent in an electrolyte system described by Fowler and Guggenheim.14 The statistical mechanics of the system is studied by considering the solvent to be a dielectric continuum of dielectric constant E. The ions which interact are immersed thus in a continuum and the effect of the solvent on the interaction of the ions is contained in the term E. The effect of the solvent on the ions themselves is assumed constant for all configurations of the ions If we generalize these definitions and require that the effect of the solvent on the potential energy due to charge and the effect of the solvent on the potential energy of the remaining part of the ion be separable, then the effect of the solvent on the potential energy due to charge can be treated as constant for all configurations of the ions while the remaining potential energy can be treated just as in the case of noli-ionic molecules. The Helmholtz free energy of a system then can be expressed as
F
= Fsoivent
(N,, P'
in ideal soln.
-
C Nstls, T ) + 8
C Fs uncharged -I- 8'"' solute s
(32)
where s refers to all solute species. The additional free energy of the electrolyte, Fel will be a function of the volume. The assumption of incompressibility will again be applied although the assumption of additivity of volume is a poorer assumption for electrolyte solutions. The use of the continuum model for electrolyte solutions as well as the use of the extended theory of dilute solutions for non-ionic solutioiis obviously is incorrect for members of the ensemble where the amount of solute is of the same order or much greater than the amount of solvent. However, for systems where iT8 corresponds to a highly dilute system, the contribution of such concentrated members of the ensemble to the ensemble is negligible. Hence, the use of these models for concentrated ensemble members has a negligible effect on the averages but is a great convenience mathematically. I n the derivation that follows, n-e shall consider only one type of solution for the sake of simplicity: namely, an ionic detergent and uni-univalent salt with a COMmon ion in aqueous solution. The application to other types of systems is obvious. Ionic detergents shall be treated as systems where each type of ion is regarded as a different species. The following notation shall be used DL = solvent species B = detergent ion y = detergent counterion and added counterion from a simple salt 9 = ion from simple salt of same charge as detergent ion
The ions all are tyeated as solutes at constant pressure. The ensemble sum as in eq. 4 is (14) R. Fowler and E. A. Guggenheim, ref. 8,Chap. IX, p, 383.
559 - Pv,
~ ( N , , P , T , P ~ ) P , ,= C LQoe ~)
c
IcT
ATbNy3Nq
QW,,Np,Ny,N,, v , T ) X e
Np(/@
- PV@)
+ Nybr
-
PVY)
+
Nqbq
+ PV,)
kT
(33)
where Qo no longer represents the complete partition function of the solvent but rather the partition function of the solvent in a solution of discharged ions. Q(N,,Np,AT,,N,, V , T ) represents the product of the partition function of the discharged solute and the additiorial term, Qe1, due to presence of charge. Thus
F, (discharged solute)
+ F"' =
S
-kT In Q ( N ~ , N B ~ N , ~ N , , F ,(34) T) We use the following definitions X p = ~ X P ( P-~ ~up)/JcT
A, = exP(Py
- PV,)/kT
A, = exP(P,
- PV,)/LT
(35)
As in eq. 6
where QN is the partition function for the grouping, N, of Np, N,, N , molecules into a particular arrangement (set) of various size clusters. The summation over N is the summation over all possible sets consistent with the restriction that the total amount of molecules of 0, y and 7 be Np, N,, and N,, respectively. Let us decompose the product XpNp X,NY, X,Ns into a product of terms of the type Xpsp X,sr X,sv which correspond to clusters containing sp pions and s, y-ions and s, q-ions. Then the summation in eq. 33 is taken over all cluster sets, where the cluster sets range over all values of No, N , , N , but restricted to set values close to electrical neutrality. If we then define the variable Xspsys,
= Xp8pXy%,87
(37)
we have the relationship as in eq. 11
Thus from the ensemble sum using eq. 36 we can express the properties of the system taking all possible clusters into account explicitly. I n general, the average number of systems Rsp,sr,sq as derived using eq. 38 will be infinite series involving terms of the type XpnQ,",X,"~
where np 2 $6, n, term of the type
2
s,, and n, XpS@XySY
2 s,. In particular a
+IX,%
accounts for the interaction of a cluster (sa s, s,) with a single monomer ion of y. We shall now focus attention on terms of the type L V ~ ~where , ~ , sp is equal in number to s,. The distribution of terms of this type can be represented as lying in a plane of the distribution space where the axes cor-
R. H. ARANOW
560
respond to s = sp = s, and LTsgsy.Terms of this type correspond to uncharged clusters and hence interaction terms with other particles and clusters should he relatively small, particularly in dilute solution. Thiis we have approximately
-
X p S p X y s ~Qm.
AVapq,
. . .I ( s ~ ~ Y y~p) ,
=
S,
(39)
m-here the partition function Quo”.. .,(sg,sr) is the partiIf s,. tion function for a single cluster of size s, we now differentiate the logarithm of eq. 39 with respect to s = sp = s,, we obtain
+
b In
-
.ii;ispsr
-
dS
+ b In Qooo. . .
In (XX, ),
Vol. 67
refers to the positive species and y to the negativc species, KC have
This relationship may be compared with eq. 8 from the paper of Shinoda and Hutchinson which differs only i n the use of mole fraction in place of L7T/V. Obviously, this difference depends upon the definition (15). Ilefinition 4.5 is used here since it is analogous to the relationship used for ideal gases
l(sa,sr)
dS
sg =
,
s, = s
(40)
T e also have the relationship
I n the plane defined by +yaBsy and s, the existence of a leads to the relationship maximum, iT*spsy,
The average number of monomer particles is given by equations of the type
-
1V1p =
X~[QIOO(”)
+
1 m.n
-4 I ~ ~ ~ A ~ X ~ ~ 8X0 ,0 0, ~ ] ,0 (42)
When the interaction terms are assumed negligible, which would be the case for ideally dilute solution, we have
-
XI,
g A,
Q1m.
. .( 1 0 )
A,
Q1oo.
. , (IT)
iYly
(43)
Combining eq. 43 and 41 we have the relationship
IU
XpX,
=
1x1 iTIpT1, - 111 Qloc. . . (16) Qloc. . .(1,) =
This result can be compared with the two-phase uncharged micelle model of Shinoda and HutchinsonEd which assumes monomer ideality. Define the standard state by the equations (pa (p, -
- pv,)O = -1iT In
pv,)O
Qloo.
= -1;T 1n QILiO. ..
,
. . .(la) (17)
+ k T In V
+ kT In TT
(33)
Thus
(pus -
pzip)(pY -
pv,)
=
lcT I n
(%)(%) +
At ordinary pressures the pv terms are negligible. Thus
Since -kT ln QOOO. . . l ( 3 @ s ~ ) * , s 8 = S r may be interpreted as the Helmholtz free energy of the maximum neutral micellar cluster, the derivative with respect to s may be interpreted as the chemical potential of the “mean” particle which is removed from the micelle leaving the micelle in an uncharged state. Calculation of temperature changes of all these quantities follows according to the derivation given by Shinoda and Hutchinson. Ilowever, it must be observed that the maximum in the distribution may be a function of T . Thus, in the interpretation of the quantity bp* bl’, consideration of the shift of the maximum of the distribution cannot be ignored. We shall now comment on the comparison of the ensemble to the mass action model. For uncharged micelle clusters the mass actmionquotients derived as for eq. 13 should have negligible interaction terms. These quotients therefore have a constant value and the mass action approach is a good one. For charged micelle dusters, the interaction terms most certainly will not be negligible and deviations from ideality may be considerable. As a first approximation to a correct distribution, interaction with gegenion only might be considered. Then we would have for the distribution an expression of the form
where the term in brackets represents an effective single cluster partition function as a function of gegenion concentration. One might also consider a fully charged micelle
-
-Yap = X p s a
(Qooo.
..
+ C L4in,nA$XpmX~]
When terms due to interaction of micelle cluster with particles and clusters of the same charge as the micelle may be considered negligible, the term in the brackets may be considered independent of 0 and 7 and the distribution becomes n s p = Xp”
The equation W J Y
=
(M)
defines a mean chemical potential,
k). Thus if p
(52)
Inn
Qetf(X,)
(53)
Equatians of the type of (23)-(26) may also be derived for this case. Estimation of Qeg(Xr) by a particular model of electrical interaction leads to the charged phase separation model of Shinoda and Hutchinson.
STATISTICAL MECHANICS OF MICELLES
March, 1963
IV. Improvement of Statistical Treatment.-Two methods for improving the statistical treatment of micelles are suggested. Either more and mo% interaction terms may be included as in eq. 12 or the separation of the solvent and solute partition functions may be abandoned. In the latter case the ensemble SUM in (4)would take the form
As we have seen from eq. 13, this ratio in the ideal case corresponds to
and then s(SoT
-
Ne = XBQ,
N s 20
where Q’ is the canonical ensembbe partition function for the solute and the solvent. An equation of the type of ( 9 ) can still be miritten
Q’(N,,Nb,
vo + .N,v~,T)X,~P= CN Q N ’ ( ~A s N s )
(55)
8
but the QN’ is to be interpreted $6 the partition function for a particular arrangement of solute molecules unci?the solvent. We still have the relationship
Then since
r may be written as
r
Q’(Na,o,Vo,Tl( 1
= e
=
X,S
Q18
e
+ €(I - 8-’18)) kT
(62)
Let us now consider this problem from the “two-phase equilibrium’’ viewpoint.
b In R, Q1 ~- 1nX-
v
dS
+ SOT + e(1k T - s-”~) +-3 k T e~-’”
(63) Then
where s*
=
smm.
But
+
561
m - _ XQI la-
Then
if the sum of the X p power series terms in the brackets, { 1, is less than unity, then the inverse of may be expanded in a power series in { ]. Thus
At the critical micelle concentration sidered the total solute. We also have
R1can be con-
(58) Thus after differeintiation as in eq. 56 using eq. 55, changing X, to ABs and after collection of terms in Xp” we have for example
(Q‘mI.
’)’)
+
...
(59)
According to Reich, phase separation (P.S.) occurs when
where Xo is the value of X when phase separation occurs. Thus, just before phase separation
where Q‘(N,,O,Vo,T) = Qo Formally the new series may be generated from the series in equation 12 with the substitution, for example, Q’OOO. . . I/QO for Qoo(,. .I. This formulation enables explicit account to be taken of specific interactions of the solvent with the micelle (such as hydration). A similar derivation may be given for the ionic system. V. Model Calculations.-Reich’s model for explaining stability of micelles is based upon a law of mass action treatment.1 The equilibrium ratio of concentrations for formation of a micelle of size s is given by
.
where So is the entropy change per molecule when a cluster of size s is formed from monomer, SE is the internal energy change and es2/8is the surface energy change.
where er = ?,/Ao and Then
r
is negative and close to zero.
(69) and
When r approaches zero the exponential, esr, may be expanded in a power series in r and higher terms ignored. Then
fl, ~ ( +1S r ) e - ( e / k T ) eZ/a (71) We can also compute the maximum value of N , in terms of T .
KANGYASG
562
But, as we have seen, r is equivalent to In (X/XO) or
Since the pua term is negligible and for all practical purposes constant, we have (74) where pLp0 is the chemical potential when phase separation occurs. We also observe
R,
In S
-
(75)
= s
b7
It can be deduced using eq. 71 and replacing the summation by an integration that Z -V, is a linear function S Br where A and B are non-zero of r of the form A coefficients. Thus
+
s=-
B
A
+ Bi-
From eq. 7 2 me have s*
=
(!3 2 IcT
9
3
r
(77)
TTol. 67
As expected, the average value and the maximum value are quite different. Reich has discussed his model extensively from a mass action viewpoint. We shall not restate his argument. It has been our purpose to illustrate, using Reich’s model, the results which are to be obtained when Reich’s model is analyzed from the two-phase viewpoint and from the constant temperature and pressure ensemble viewpoint. Conclusions.-The general statistical approach contains within it the various models used for the study of micelles and the relationship between these models. The nature of the approximations involved is laid bare. The appropriate interaction terms necessary for improving existing theory are displayed in the general distribution equations for iv,. The basic assumptions of incompressibility and additivity of volumes and the assumption of separability of partition functions used to separate solvent and solute contributions to r are of course inexact. However, they enable us to simplify the ensemble sum and do lead to results in accord with existing theoretical treatments. The separability of solvent and solute contributions can be eliminated easily in the formulation, leading to equations which represent an improvement over existing treatments. Acknowledgment.-The author wishes to express sincere appreciation and gratitude to Dr. Louis M7’itten and Dr. John Gryder for many helpful and stimulating discussions and to the Chemical Directorate of the Air Force ORce of Scientific Research for their support of this work.
FREE RADICAL REACTION INITIATED BY IONIZING RADIATIONS. 111. PARAFFIN REACTIVITIES I N HYDROGEN ATOM ABSTRACTION REACTIONS BY RANG YANG Radiation Laboratory, Continental Oil Company, Ponca City, Oklahoma Received J u l y $7, 196.9
+
+
Rate constants, k = BT1I2exp( - e / R T ) , for these reactions H 31 + Hz R have been determined by investigating the temperature dependence of hydrogen yields in the yradiolysis of paraffin-propylene systems. Assuming E = 2.2 kcal./mole for the hydrogen atom addition reaction with propylene, the following E values are estimated: ethane, 8.6; propane, 7.0; n-butane, 6.3; isobutane, 4.7. The LCBO treatment of an assumption that the energy required for isolating two electrons in a bond to be broken from the rest of the c-electron system SCH?. Here N C H plays the decisive role in determining paraffin reactivities leads to the expreseion: E = E is the number of additional CH bonds formed by the carbon atom that forms the CH bond to be broken. TWO constants, 4 and 7,have these meanings: E is the activation energy for the hydrogen atom abstraction reaction involving diatomic CH molecules; and 7 represents the major structural contribution to the reactivity and comes from the fact that the bond to be broken undergoes etabilizing interactions with neighboring CH bonds. Examination of experimental data in the light of the above equation indicates that each such interaction contributes 2.0 kcal./mole to the activation energy, thus reducing the paraffin reactivity about 30-fold a t room temperature.
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Introduction Structurally similar compounds often exhibit markedly different reactivities in a series of similar radical reactions.1 I n the case of radical addition reactions with carbon-carbon double bonds, this difference is attributable to a difference in the energy, Eloc, required for localizing a n-electron a t the reaction center2-6; thus it is demonstrated that’ (1) See, for example, M. Srwaro, J . Phys. Chem., 61, 40 (1957). (2) G . W. Wheland, J . A m . Chem. Soc., 64, 900 (1942). (3) C. A. Coulson, J . Chem. Soc., 1435 (1955).
aEloc - b (E-1) where E is the activation energy for the hydrogen atom addition reactions in gas phase, and two constants, a and b, are expressible in terms of parameters characterizing potential changes involved in these reactions. Attempts halve been made to extend this successful localization energy concept to the case of abstraction E
=
(4) J. H. Rinks and M. Saw-arc, J . Chem. Phys., 30, 1494 (1959). ( 5 ) S. Sato and R. J. Cvetanovic, J . Am. Chem. Snc., 81, 3223 (1969). (6) K. R. Jennings and R. J. Cvetanovic, J . Chem. Phys., 36, 1233 (1961). (7) K. Y a w , J . Am. Chem Soc., 34, 3795 (1962).
