Solubility of Gases in Liquids
(21) (22) (23) (24) (25) (26)
The Journal of Physical Chemistry, Vo/. 82, No. 26, 1978
of AH(fusion) were determined at the melting points, T,. The difference between T , and T, is small enough that the variation of &/(fusion) with temperature can be neglected. W. T. Lee and Z. A. Munir, J . Nectrochem. Soc., 114 1236 (1967). R. A. Reynolds, D. G. Stroud, and D. A. Stevenson, J . Electrochem. Soc., 114 1281 (1967). A. G. Sigai and H. Wiedemeier, J. Electrochem. Soc., 119, 910 (1972). L. Torpor, J . Chem. Thermodyn., 4, 739 (1972). See, for example, (a) L. V. Woodcock, Chem. Phys. Lett., 10, 257 (1971); (b) J. W. E. Lewis, K. Singer, and L. V. Woodcock, J . Chem. Soc., Faraday Trans. 2, 71, 301 (1975). It is probably reasonable to assume here that each ion is on average surrounded by ions of the opposite charge. See section 3 and ref
2801
25. (27) For most of the alkali halides, only -60-70% of the total mass in the vapor is monomeric (e.g., KCI), the rest being largely dimeric (e.g., [KCI],); for the sodium salts the proportion of monomer is around 35-50% (see ref 24). The bH(per gram formula weight) for boiling to the dimer is probably just about ' / 2 of that for boiling to the monomer (ref 28); this means that the slopes of the In p vs. T-' line will be similar for either process. Thus literature (e.g., ref 3) values of AH, which were presumably deduced from vapor pressure data assuming a monomeric vapor, will still be a good estimate of the true A H f o r boiling to the monomer. (28) See, for example, I. G. Murgulescu and L. Torpor, Rev. Roum. Chim., 13, 1109 (1968); 15, 997 (1970).
Solubility of Gases in Liquids from the First-Order Perturbation Theory of Convex Molecules Tomas Boublikt and Benjamin C.-Y. Lu" Department of Chemical Engineering, University of Ottawa, Ontario, Canada K I N 6N5 (Received June 19, 1978; Revised Manuscript Received October 2, 1978) Publication costs assisted by the University of Ottawa
The perturbation theory of convex-molecule systems was applied to determine the Henry's law constant, KH, partial molar volume, V2,and heat of solution, m2, at infinite dilution of Ar, CH4,N2,02,and C 0 2 in benzene and in CCl, at 298.15 K. The Kihara acentric pair potential was used to characterize the intermolecular interactions of the given molecules, hard cores of which were assumed to be points in the case of Ar and CH4, rods in the case of N P ,02,and C02,a hexagon for benzene, and a tetrahedron for CC14. The method of separation of the predominantly repulsive-forcesregion as that proposed by Barker and Henderson was followed to define the reference system. A simple approximation of the dependence of the average correlation function on the shortest surface-to-surface distance, based on Monte Carlo data, enabled the determination of the first-order perturbation term. The calculated values of KH and V 2agree well with the experimental values available in the literature, while the AH, values agree only qualitatively.
Introduction In recent years, we have witnessed an increasing interest in the solubilities of gases in liquids. As a result, several reviews and original papers have appeared dealing with the solubility both from the standpoint of the experimental determination and the theoretical calculation. The first attempt to estimate the solubility of simple fluids within the framework of modern theories, which employ distribution functions to the characterization of the system structure, was connected with the derivation of a hard sphere equation of state from the scaled particle theory (SPT)by Reiss et a1.l However, in their application of the derived equation to gas solubility, the effect of attractive forces was neglected; Pierotti2 improved this approach by adding an attractive contribution term in a physically intuitive way. Wilhelm and Battino3 employed this intuitive method too (where a solution of a solute molecule in a solvent is considered as a two-step process, the creation of a cavity for the solute molecule in the solvent and the introduction of the solute molecule into the cavity plus the full coupling of intermolecular forces) and in addition to dispersion forces, they also considered interactions of dipole moments etc. Several oversimplifying approximations were introduced within this approach; thus, the success in its application to a variety of systems, including that with water (see Wilhelm et aL4),obviously results from I n s t i t u t e of Chemical Process Fundamentals, Czechoslovak Academy of Science, Prague, Czechoslovakia.
0022-3654/78/2082-2801$0 1.OO/O
the fortuitious cancellation of errors. The use of experimental values of solvent volumes, compressibilities, and thermal expansion coefficients seem to be inevitable with this method. Snider and Herringtons employed a modified van der Waals equation of state, in which the repulsive SPT term was combined with the vdW attractive term, for both the determination of the excess functions and the Henry's law constant. An exact treatment of the solubility of simple fluids (interacting via the Lennard-Jones pair potential) was given by Neff and McQuarrie6 who derived expressions for the Henry's law constant, KH,the partial molar volume of solute, V2,and the heat of solution, AHz, from the Lennard-Henderson-Barker (LBH) theory of simple fluid mixtures,' with use of the Percus-Yevick equation of state (P-Y(c)) and radial distribution functions. Derived relations were successfully applied to describe the behavior of the system neon in argon. Goldmans used the same expressions as Neff and McQuarrie for the contribution of attractive forces but he also used an improved hard-sphere term as obtained from the extended Carnahan-Starling equation of state. In addition, he considered quantum corrections for the systems Hz in N2,Ar, and CH4,He in N2, Ar, and CH4, and Ne in N2 and Ar. A similar approach was followed in his application of the variational method to the interpretation of gas solubilities for the same binary r n i x t ~ r e s .In ~ addition, Goldmanlo employed the Weeks-Chandler-Andersen perturbation theory of simple fluidsll as extended to mixtures by Lee and LevesquelZand applied it to gas
0 1978 American
Chemical Society
2802
T. Boublik and E. C.-Y. Lu
The Journal of Physical Chemistry, Vol. 82, No. 26, 1978
expansion of the Helmholtz energy in the reference system (characterized by energy UNO = Cue) is given by the following expression:
similarly for a mixture: A, - A," -- A,O - As* NAkT NAkT
+
Figure 1. The geometry of two interacting particles. (Shaded bodies are cores; parallel bodies are the representative hard convex bodies.)
solubilities for the above-mentioned systems. Although this approach is believed to characterize the effect of repulsive forces more appropriately than the LBH theory, he found that the calculated results were only slightly better. More recently, Goldman,13 in an effort to extend statistical mechanical methods to the description of dense fluid mixtures comprised of nonspherical molecules, employed the hard convex-body equation of state14 (which resembles the P-Y (c) equation) and an expressionlj for the Helmholtz energy of a mixture of convex molecules in the determination of P, of Ar, CH4, and N2 in benzene and in CC14. In the paper, it was assumed that repulsive forces were interpreted by means of the hard core interaction. The contribution of attractive forces was not evaluated from the theory but adjusted to experimental values of KH. Experimental volumes of solvents as well as values of compressibilities were used. In the present study, an attempt was made to evaluate all the characteristics from the molecular data. The validity of the Kihara core pair potential was assumed and the first-order perturbation expansion for the compressibility factor was used to determine the required properties of pure solvents. An improved expression which relates to the best available equation of state of hard convex-body mixture16 was employed for the contribution of repulsion forces to the chemical potential of component i in the mixture. An approximation for the average correlation function based on the Monte Carlo data was used to determine the first-order contribution of attractive forces.
In eq 4, the subscript s denotes mixture properties, and A, AO, and A* represent the Helmholtz energy of the studied, reference, and ideal gas systems, respectively. N A is Avogadro's number, k is the Boltzmann constant, p (= NA/W is a number density, T refers to temperature, xi represents the mole fraction of the ith component, gi? is the average correlation function of an i-j pair of molecules, and S,,+l+,, represents the mean surface area of a body formed by a center of a molecule j when it moves around a molecule i with the constant surface-to-surface distance 1; the quantity gz,avSci+l+,~ dl represents the average number of molecules in a volume element with a surface-to-surface distance between 1 and 1 dl. The behavior of the reference soft convex body system can be determined through the properties of a suitable chosen system of the representative hard convex bodies; these convex bodies are parallel to the given molecular cores with some yet unknown thickness F , This thickness can be determined by performing another perturbation expansion of the Helmholtz energy Ao in a system of hard convex bodies. The Barker-Henderson method (see ref 7), in which a suitably chosen coupling parameter enables the softness of the repulsive part of the interaction potential to change gradually, is used. The following expression can then be obtainedl8JQfor a mixture: A,O - A,* A$b - A * - NAk T s + NAk T
Theory In the proposed approach, it is assumed that the total potential energy is pairwise additive and the Kihara core interaction potential characterizes pair intractions:
in which the quantity Ashcbrepresents the Helmholtz energy of the representative hard convex body system, d ,
u,,(l) = 4 ~ , , [ ( g , , / l ) '-~ (~,,/Vl
(1)
Here cZ, and g,]are characteristic interaction parameters and 1 is the shortest surface-to-surface distance between two hard convex cores ascribed to the given molecules of types i and j as depicted in Figure 1. The geometry of each convex core is characterized by three fundamental rneasures:l7 volume of the core, V,,, surface area, S,,, and the ( 1 / 4 T ) multiple of the mean curvature integral, Rc,. Similarly as in the case of simple fluids,6 the BarkerHenderson's division of the range of intermolecular distances into two intervals with predominant action of repulsive or attractive forces is used; thus the reference and perturbation potentials (uoand up, respectively) are defined by U,,O(1)
= UJl)
U,O(l)
=0
u,P(1) = 0 U,P(l)
= uz,(l)
1 < u,, 1>
+
=
E, + E, G,, = gZlhcb(l = 4,)
(6)
is the contact value of the average correlation function of an i-j pair in the hard convex body mixture, and 6ij is given by the following expression: 6, = ~ " ( -1exp[-u,,0(l)/kT]) dl
(7)
By nullifying the perturbation terms for the like pairs (i-i, j-j) in eq 5 , the thickness of the representative hard convex body of the given type is determined by Fi
= l/zd,i =
Ydzi
(8)
Because the equation of state used for the evaluation of the thermodynamic functions
(2)
UZ]
As was shown by one of us,15 the first-order perturbation
was developed16 under the assumption that d, = + [ j , the contribution due to (dij - 6 i j ) must be taken into ac-
Solubility of Gases in Liquids
The Journal of Physical Chemistry, Vol 82, No. 26,
count in the perturbation expansion. In eq 9, u = pCxiVi, s = pCxiSi, q = pCxiRi2,r = pCxiRi,and p = C N ; / V . From the known equation of state (eq 9), an expression for the chemical potential (p;hcb- pi*) of the component i in the hard convex body mixture (which gives the contribution of repulsive forces to KH) can be derived as follows: &hcb -
= In (1 - u )
+
[
=
S ~ + x d L , + ] / d ~ ] 2=
x
9u3
-
+ vjp + + Ris +(1Sir - u) R;2s2+ 2S,qs + 9V,rsu Vlqs2 (2 5u + u2) In (1- u )
-
In KH = In (NAkT/V,')
(10)
The quantities pibcband p,* denote the hard convex body and ideal gas chemical potentials of component i, respectively. The quantities R,, s,,and v, are the geometric fundamentals of the representative hard convex body of the given type i (i.e., a parallel convex body to the core of the given molecule) and are related to R,,, S,,, and V,, through R, = R,, + tl S I = Sc, + 8rR,,[, + 4rEI2 (11)
v~= VCI+ S C I+~ 4I " R ~ ~ +t ~"/3rt1~ 2 It should be mentioned that for the special case of a hard sphere mixture, eq 10 reduces to thatz0corresponding to the extended Carnahan-Starling equation of state for a mixture. An expression for Henry's law constant in the framework of the perturbation theory of fluids is based on the thermodynamic definition of KH: kT In KH = (p20(')- pZo(g)) (12) where pLz0(l) and p$(g) are the chemical potentials of solute (related to one molecule) in the respective standard states of infinite dilute solution and gas a t a pressure of 0.101 MPa (1 atm), respectively. In the statistical thermodynamic treatment the vapor phase is assumed to be ideal and the chemical potential pz0(')is determined by partial differentiation of the expression for A, with respect to the number of solute molecules, N2. The expression for A, may be obtained by combining eq 4 and 5:
+ 4ax2 (16)
+ (pZhcb p2*)'/kT + -
-
(~2''")
'/ kT
(17)
where the degree denotes the limit for x 2 0 and Pl. is the partial molar volume of the solvent. The quantity (p2hcb- ~ , * ) " / k Tis given by eq 10, while
(p2corr)o/kT = ~~(112+ ' 111')
-
9u2(1 - u)3
S ~ + ] / d ~+] 2 8nx(R, + R,)/d,
Then, by substituting the derivative of eq 14 into eq 12, we have for the Henry's law constant (cf. ref 13)
-4
R?s iuLSlqs 2Viqs2
kT
9u(l - u)2
S~+xd+l*
7978 2803
- plI12nd0
(18)
with 112'
=
d 1 ~ ~ l ~ c 1~ 122-P l( ~l ) g 1 2 h c b ( X ) o S 1 + x d ~ ~ dx +2*
(l9)
and (21) G 12hcbS 1+2[d12 - 8121 The contact value GI,of the average correlation function, corresponding to the above-mentioned hard convex body equation of state, possesses the following form: (SITl+ SIT,)s + S,S,(r - t ) I12nd0 =
GI1 =
l
+
+
(1 - u)2s,+l
2s,s,qs
(22) 9 0 - u)3s,+, where t = pCx,T, and T I = 4aRI3/S,. At present the theory does not provide an analytical expression for the dependence of the average correlation function, gllhcb,on the reduced distance. However, Monte Carlo simulations in the hard prolate spherocylinders21v22 and hard spheres23(a special case of the convex body systems) offer the basic information for formulating an approximation for this dependence. Hence, the approximation
G,
g,,hCb(x)
-
1
= -exp[-G,x] cos
l+x
( P X / ~ B +, ~1)
(23)
with
B,, = (1- u)(R,+ Rl)/2d,, As gLIhcb equals zero for 1 < d and can be suitably expressed in terms of the reduce2 distance x for 1 > d,, i.e. x = ( 1 - dil)/dll it is advisable to first express A, in the following way:
The mean surface area SI+]as well as Sl+nd+J* can be easily expressed in terms of the geometric measures of individual representative hard convex bodies: SI+,= S I + S, + 8rR,R1 (15) and
(24)
is employed. Equation 23 fits the MC data reasonably well at high densities for asphericities, which may be expressed by a parameter a ( a , = R,S,/3V,) with its value equal to or smaller than 1.2.
Calculations and Results In the process of evaluating the Henry's law constant, the heat of solution and the partial molar volume of the solute in the system under consideration, we first calculated the molar volume, the coefficient of thermal expansion, and the isothermal compressibility of a pure solvent a t the given temperature (298.15 K) from the equation of state that followed from eq 3 by taking the derivative with respect to density. As usual in statistical 0 thermodynamic treatments, liquid volumes as P (instead of a t P = 0.1 MPa) were determined. The Henry's law constant was then calculated from eq 17 where the calculated molar volume of the solvent was substituted for Vl; the same value was also used for the
-
T. Boublik and B. C.-Y. Lu
The Journal of Physical Chemistry, Vol. 82, No. 26, 1978
2804
TABLE I : Geometric Characteristics and Interaction Parameters of Pure Compounds S&m2 Vc/nm3 u /nm ( e l k )/K 0 0 0.3430 122 0 0 0.3678 167 N* 0 0 0.3207 117 0 2 0 0 0.2851 151 co, 0 0 0.2940 316 C6H6C 0.100394 0 0.3250 586 cc1,c 0.144050 0.002827 0.2950 653 a Data were taken from ref 20. Data were taken from ref 24. Functionals of cores evaluated from data of Kihara" e l k and u determined from heats of vaporization and liquid densities. compd Ara CHi a
R ,Inm 0 0 0.02325 0.02575 0.05750 0.10425 0.13150
TABLE 11: Solubilities of Ar, CH,, N,, 0, and CO, in Benzene and in Carbon Tetrachloride a t 298.15 K -
KHexp,t/ MPa-
cm mol-'
cm mol-'
95.5 55.1 199.9 76.7 16.3
115.0a 48.0 227.2 124.1 10.4
40.0 43.8 47.1 31.4 43.0
43b 52 53
77.8 44.7 158.7 64.3 14.3
75.0 35.3 156.4 84.4 9.6
40.6 44.1 48.4 38.0 42.9
44 52.4 52.5 45.3 48
KHcalcp/
svstem Ar-C,H, CH,-C6H, N 2 6'-
H6
' 2
H6
6'-
C0,-C,H, Ar-CCl, CH, -CCl, N, -CC1, 0, -cc1,
co, -cc1,
MPa-
Evaluated from data of Wihelm and Battino." From Pierotti.* a
evaluation of the geometric functionals r , s, q , u , and t , while R,, Si,and V, were determined by employing eq 8 and 11. The geometric rule was assumed for the cross parameter t12, i.e. cc14
All the necessary parameters, used in calculations, are summarized in Table I. In this treatment, pointwise cores were ascribed to molecules of Ar and CH4, rods to N2,02, and COz, a hexagon to benzene, and a tetrahedron to CC4. The cores and the representative hard convex bodies are shown in Figure 2. Calculated values of the Henry's law constants, KH,and the partial molar volumes, V2,are listed in Table IT for Ar, CH4, N2,02,and C 0 2 in benzene and in carbon tetrachloride at 298.15 K. Heats of solutions are given in Table 111. For the purpose of comparison, experimental data available in the literature are also listed in Tables I1 and 111. It is seen that the proposed method yields very good predictions of the partial molar volumes of the solute (practically within the accuracy of experimental determination) and fair estimations of the Henry's law constant. The calculated heats of solution did not agree as well with the experimental data, but showed (small) positive values for sparingly soluble gases and negative values for very soluble ones in agreement with the experimental evidenceaZ6On the other hand, it is noted that discrepancies of available experimental data may amount to 1-1.5 X lo3 J/mol. Deviations are mainly due to the rough approximation used for the average correlation function, and insufficient knowledge of molecular interaction parameters as well as the shapes and sizes of the cores. For example, in the case of benzene two sets of interaction parameters (determined from the second virial coefficient) are available which differ considerably and lead to either too great or too small representative convex bodies. In this particular case, properties of liquids were employed for obtaining the suitable set of parameters; a similar approach was also used
' 6 H6
Figure 2. The cores and the representative hard convex bodies of the compounds considered in this investigation.
TABLE 111: Heats of Solution (AE,) of Ar, CH,, N,, 0,, and CO, in Benzene and in Carbon Tetrachloride at 298.15 K
-
-
calcd/
system Ar-C,H, CH,-C,H, O,-C,H, CO,-C,H,
- 2.42
10.33
Ar-CCl, CH,-CCl, N,-CCl, O,-CCl, co2 - CCl,
- 1.41
- 0.43
-4.10 0.90 - 2.45 - 10.22
2.99 2.36 0.03
-
1.27
Jmol-' x
1.24a 1.28 4.25 1.71 - 9.31
N2-C6H6
a
Jmol" x
A l l , expi/
-- 3.90
1.09
--
Evaluated from data of Wilhelm and Battino."
for C C 4 but no effort was made to find optimum values of these parameters nor to improve on the geometric rule for t12. Thus, introducing a correction factor of f = t12/(t11t22)1/2 which varies from 1 to 0.96, some 10-20% increase of the values of KH was obtained; a t present the use of this correction factor cannot be justified because of the inaccuracy of other molecular parameters. In addition, we would like to note that, when taking data from Koide and Kihara,24the effect of permanent multipoles was neglected. Interactions of permanent multipole moments cannot be included into this approach without the lost of simplicity. However, the quadrupole moment, the higher multipole moment, and triple interactions can be taken
Dynamics and Structure of Micelles
into account by employing the "effective Kihara potential parameters". The presented results show that the perturbation theory of convex molecule systems represents a powerful tool for the prediction of equilibrium behavior of nonpolar mixtures, yielding values that could be used in chemical engineering design. I t must be stressed that the method is completely consistent in the sense that all of the properties, including the molar volumes, compressibilities, and expansion coefficients of the solvents, are calcukted from the perturbation theory employing the given sets of the Kihara parameters. In comparison with the previously known methods, a comparatively broader class of systems can be considered. Wider practical applications of the described approach depend mainly on the increase of our knowledge of the parameters of the Kihara core pair potential.
Acknowledgment. The authors are indebted to the National Research Council of Canada for financial support. References and Notes (1) H. Reiss, H. L. Frisch, and J. L. Lebowitz, J . Chem. Phys., 31, 369 (1959). (2) R. A. Pierotti, J . Phys. Chem., 67, 1840 (1963).
The Journal of Physical Chemistry, Vol. 82, No. 26, 1978 2805
E. Wilhelm and R. Battino, J . Chem. Thermodyn., 3, 379 (1971). E. Wilhelm, R. Battino, and R. J. Wilcock, Chem. Rev., 17, 219 (1977). N. S.Snider and T. M. Herrington, J. Chem. Phys., 47, 2248 (1967). R. 0. Neff and D. A. McQuarrie, J. Phys. Chem., 77, 413 (1973). P. J. Leonard, D. Henderson, and J. A. Barker, Trans. Faraday Soc., 66, 2439 (1970). S. Goidman, J . Phys. Chem., 81, 608 (1977). S. Goldman, J . Solution Chem., 6, 461 (1977). S. Goldman, J . Chem. Phys., 67, 727 (1970). J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys., 54, 5237 (1971). L. L. Lee and D. Levesque, Mol. Phys., 26, 1351 (1973). S. Goldman, J. Phys. Chem., 81, 1428, (1977). T. Boublik, Mol. Phys., 27, 1415 (1974). T. Boublik, Collect. Czech. Chem. Commun., 39, 2333 (1974). T. Boubiik, J . Chem. Phys., 63, 4084 (1975). T. Kihara, Adv. Chem. Phys., 5, 147, (1963). T. Boublik, Mol. Phys., 32, 1737 (1976). T. Boubiik, presented to CHISA Conference, Prague, 1978. T. M. Reed and K. E. Gubbins, "Applied Statistical Mechanics", McGraw-Hill, New York, N.Y., 1973, p 250. T. Boublik, I.Nezbeda, and 0. Trnka, Czech. J . Phys., 826, 1081 (1976). I.Nezbeda and T. Boublik, Czech. J . Phys., 828, 353 (1978). J. A. Barker and D. Henderson, Ann. Rev. Phys. Chem., 23, 439 (1972). A. Koide and T. Kihara, Chem. Phys., 5, 34 (1974). E. Wilhelm and R. Battino, Chem. Rev., 73, 1 (1973). H. L. Clever and R. Battino in "Solutions and Solubilities", Part I, "Techniques of Chemistry", A. Weisberger and M. R. Dack, Ed., Wiley, New York, N.Y., 1975.
Dynamics and Structure of Micelles and Other Amphiphile Structures Gunnar E. A. Aniansson Department of Physical Chemistry, University of Goteborg and Chalmers University of Technology, Fack, S-402 20 Goteborg, Sweden (Received June 21, 1978)
A previous calculation of the rates of exit and entrance of tenside molecules and ions out from and into micelles is extended to the dynamics and extent of partial exits. The results indicate a very pronounced and varying head group protusion from the hydrophobic core. Several implications for the structure and properties of micelles are discussed and extended to amphiphile mono- and bilayers.
Introduction Within the context of the kinetics of stepwise micelle formation and dissolution1 the elementary process whereby one tenside molecule or ion (hereafter called "monomer") enters or leaves the micelle was considered in a quantitative way.2 A precursor to this latter treatment developed by the author and based on a treatment3 of reaction rates when the mean free path along the reaction coordinate is smaller than the width of the activation battier kT under its top was included in a later, joint paper of three groupsS4 The simple picture emerging was that of a largely uncurved monomer moving diffusively essentially a t a right angle to the micellar surface into and out of the micelle in a free energy field composed of the hydrophobic bonding energy of about one kT per exposed CH2groupI3 and the potential energy of the hydrophile end group, when charged, in the electric double layer. The average time required for a monomer with a 12 carbon atom chain to leave the micelle will be of the order of s when the electric field is absent and otherwise shorter in essential agreement with experimental result^.^ In this paper the dynamics and extent of partial motions of the monomers out into the aqueous environment and back is explored. Some important consequences are discussed but confined mainly to qualitative statements in anticipation of a fuller treatment taking into account, 0022-3654/78/2082-2805$0 1.OO/O
inter alia, the finer details of the electric double layer.
Dynamics Since the motion is a diffusive one there will be partial movements out from and back into the micelle without the monomer leaving the micelle. The time scale of these motions is easily calculated using the methods of Smol u c h ~ w s k i .For ~ example, the average time ( t ) required for the protrusion x to occur can be obtained by solving the following diffusion problem. J monomers are added per unit time at x = 0 and removed when they reach x = p . At steady state there will hold, simply ( t )= n / J (1) where n is the number of monomers between x = 0 and x = p. The flow equation to be used is dc D d V J = -D- - _dx kT dx where D is the diffusion constant, c(x) is the number of monomers per unit length at distance x from the core, V(n) is the free energy potential, and kT has the usual meaning. An analytical solution in terms of definite integrals is obtained in a straightforward way. It is particularly simple when, for example, the charge effects are absent or neutralized by a high ionic strength. (l/kT)(dV/dx) will then 0 1978 American
Chemical Society
