ACTIVITYCOEFFICIENT OF
Nov., 1963 For
YHYCI,
HC1
> NaC1 w
NaC104 > (CHs),NCI
A
SALT11'; A CHARGED MICROCAPILLARY
H
NaNs For
YHYA,
the system dioxane-water2$bboth indicate that given by an equation of the form
2333 sa4
is
NaCl s SaC104 > (CH8)4NCI > KaNs
For YCI/YA, NaCl M NaC104 = (CH3)4KC1< XaNs
It is seen in the first two series that the organic ion salts interact so as to reduce the activity coefficient relative to the inorganic salts, in spite of their somewhat larger size. This stabilizing interaction is especially strong between H,A and Na,Ss, and here the high value obtained for ycl/yB shows that the additional interaction involves specifically the or-naphthoate ion and the p-naphthalenesulfonate ion. The effect is probably the result of an attractioii between the naphthalene rings that is strong enough to overcome the electrical repulsion between these ions. Our chief aim in undertaking this study was to look for salt-induced medium effects. In first approximation, the salt-induced medium effect on log 73 can be represented as an additive term proportional to m4, as in eq. 10. log
log
+
(10) Theory and experimental results for nonelectrolytes in y3
=
73'
s34m4
where F,," is the standard partial molal free energy (on the m-scale) of the ith component, 21 is the mole fraction of water in the binary solvent, and the subscripts 3 and 4 denote the substrate and salt, respectively. A and B are parameters, the value of B being about 0.004 according to the previous result^^,^ for nonelectrolyte substrates, if F,' is expressed in kcal. In the present case, values of dFm"/dZ1in 50 wt. % dioxane-water are -10.7, -5.8, -16.6, -6.8, -16.3, and -7.3 kcal., respectively, for H,C1, H,A, Xa,C1, Na,C104, (CHa)J,Cl, and Ka,Ns (ref. 2 and Table 11). Hence, a t m4 = 0.2, this mechanism acting along should increase YHYCl in the presence of sodium chloride by about 21% relative to the value in the presence of sodium perchlorate, and there should be a similar increase of 11% in YHYA, assuming that B is again 0.004. Nothing of the sort has been observed. Acknowledgment.-It is a pleasure to thank Dr. E. L. Purlee for helpful advice about the e.m.f. measurements.
A VARIATIONAL PRINCIPLE FOR THE POISSON-BOLTZMANN EQUATION. ACTIVITY COEFFICIENT OF A SALT IN A CHARGED IMICROCAPILLARYl BY LAWRENCE DRESNER Seutron Physics Division, Oak Ridge ,Yational Laboratory, Oak Ridge, Tennessee Received April $9,1983
A functional which is a minimum for solutions of the Poisson-Boltzmann equation is given. The minimum of this functional is shown to be related to the electrostatic contribution to the free energy of the system. Using a simple trial function, an illustrative problem of interest in water desalination is worked.
Introduction The Poisson-Boltzmann equation has been used to describe many phenomena in solution chemistry, colloid c h e m i ~ t r yand , ~ the chemistry of polyelectrolytes"Ib and ion-exchange materiah6"Vb Because of the complexity created by its nonlinearity, only a few exact solutions to the Poisson-Boltzmann equation are k n ~ w n . ~ ~ , I~n, general, ~ ~ , ~ the J powerful methods of solution developed for linear partial differential equations, such as separation of variables, ex(1) Work performed for the Office of Saline Water, U . S.Department of the Interior, Oak Ridge National Laboratory, Oak Ridge, Tennessee, operated by Union Carbide Corporation for the U . S. Atomic Energy Commission. ( 2 ) (a) G. Gouy, J . Phys., [41 9, 357 (1910); Ann. Physik, [9] 7 , 129 (1917); (b) D. L. Chapman. Phil. Mag., [SIM, 476 (1913); (c) 0. Stern, Z. Elektrochem., 30, 608 (1924). (3) P. Debye and E . Hiickel, Phgsik Z., %4, 186, 305 (1923). (4) E . J. W. Verwey and J. T. G. Overbeek, "Theory of Stability of Lyophobic Colloids," Elsevier Publishing Co., Inc., New York, N. Y., 1948. ( 5 ) (a) R. N. Fuoss, A. Katohalsky, and S. Lifson, Proc. Xat2. Acad. Sci. U . S., 87, 579 (1951); (h) T. Alfrey, P. W. Berg, and H. Morawets, J. Polymer Sci., 7 , 543 (1951). (6) (a) L. Lazare, B . T . Sundheim, and H. P. Gregor, J. Phys. Chem., 60, 641 (1956); (b) L. Dresner and K. A. Kraus, ibid., 61,990 (1963). (7) The solution independently found b y Fuoss, et a1.Y and by Alfrey, et Q Z . , ~ has ~ also been given b y H. Lemke [ J . Math., 142, 118 (191311 and b y G. W. Walker [Proc. Roy. SOC.(London), 8 4 1 , 410 (1915)l. I t is a special case of a very general two-dimensional solution to the equation b2u = eu stated by J. Liouville [ J . Math., (1) 18, 71 (185R)].
pansion in orthogonal functions, use of Fourier and other transformations, etc., cannot be applied to the Poisson-Boltzmann equation.8a However, one of the well known approximate methods of solution developed for linear problems, the variational method, can be applied to the nonlinear Poisson-Boltzmann equation.8b I n this paper, a futictional will be exhibited that is a minimum for solutions of the Poisson-Boltzmann equation and whose minimum value is related to the electrostatic part of the free energy of the system. With the help of this functional and a simple trial function, the following illustrative example will be worked : the electrostatic contribution to the mean activity coefficient of 1: 1 electrolyte invading a small, surfacecharged microcapillary mill be estimated. The Poisson-Boltzmann Equation.-The PoissonBoltzmann equation describing the equilibrium of a mixture of N types of ions and a solvent in a volume V depends on two assumptions : (i) The concentration c t [ions ~ m . - ~of ]the ith type (8) (a) Interestingly enough, in a certain special case conformal mapping ie applicable. See R'I. v. Laue, "Jahrbuoh der Radioaktivitat und Elektronik," Band 15, Heft 3, 206, 1918; (b) the possible use of the variational method in the Debye-IIiickel case was mentioned in a footnote b y S.Levine, . I Ciirm. . Phiis., 7,836 (1937).
LAWRENCE DRESNER
2334
of ion depends on the electrical potential 4 through the Boltamann factor, i.e. (1) where zl is the valence of these ions, e is the charge of the proton, IC is Boltzmann's constant, and T is the absolute temperature. (ii) The discrete nature of the ionic charge can be neglected and the electrical potential 4 related to the ionic concentrations by Poisson's law
Secondly
in
clae-se+/kT
N
V Z =~ (-I/€)
i=l
zieci
,-z,e+/kT
sT'
e--z,e+/kT
d3r
(3)
dar
e--z&*/kT
in
=
e--zm/kT
d3r (74
111 Jf.(y) evdy
=
(7b)
where y = -x,ex/kT and f(y) dy is defined as the integral of e-z*e+*/kT over that part of the volume V iii which y lies in the interval dy. According to a theorem of Hardy, et al.10
(2)
Here t: is the dielectric constant.g The Poisson-Boltzmann equation is obtained by substituting a suitably normalized form of (1) into ( 2 ) . Writing (1)as ct = rile -z,e+/kT
Vol. 67
$f(d
$SWdY/ exp
dy 2
{ $Y W Y /
JmdY
1
(8)
Substitution of (8) into (7b) allows the latter t o be written
In
J7-1
ST,
d3r 2 ln
e-z@#/kT
V-1
Sv
,-Ste+*/kT
d3r -
where nedenotes the total number of ions of type i in V , we find the result
Equation 4 is the Poisson-Boltzmann equation. If the volume V is all space, eq. 4 is valid everywhere. If the volume Ti is finite by virtue of the ions being confined in a container, eq. 4 is valid inside V, while Laplace's equation v2+ = 0 is valid outside. As is usual in electrostatics, 4 and the normal component of the electric displacement are continuous across X, the surface of V , except when there are surface charges on X, etc. The Variational Principle.--Let us now consider the functional 1
"
If we now multiply (9) by kTni and sum over i and then add the result to (Gb), we get
H[41 2 H[4*1
+ 2A
sm
t:(Vd2d3r-
ST'
d3r]d3~ (10)
e-sie+*/lcT
The bracketed expression in the last integral vanishes because +* satisfies (4). Since the second term on the right-hand side is never negative, H [+I is always 2 H-
b*l. where a subscript a3 denotes an integration over all space. If 4 is allowed to vary over all functions which vanish a t least as rapidly as l / r a t a ,then H will vary over a semiinfinite range of values bounded from below and attain its minimum when 4 = qi*, a solution of the Poisson-Boltzmann eq. 4. This statement is proved as follows. Let 4 = 4* x. Then, first
Relation to the Free Energy.-The minimum value ET[$*] of the functional H is related to the electrostatic part of the free energy of the system. Starting from (a), one can easily show by an application of Green's theorem that
+
2
$a
+
$-
1 ~ ( V 4 ) ~= d ~-r ~(V$*)~d~r 2 1 t : ( V ~ ) ~ d ~ r t:V+*.Vx d3r (sa) 2
Using (3) and (4) we can rewrite the first term on the right-hand side as
lrn + lrn
=
2
$m
1 ~ ( ~ ~ $ * ) ~ d-3 r ~ ( V x ) ~ d~r 2s-
+
n
J-
niin
v-i
JV
d3r = ne In (nl/V)-
e-zie+*/ky
+ In ci]
exV24*d3r (Gb)
Here the second equality has been obtained from the first by use of Green's theorem and the fact that xV4* vanishes a t least as rapidly as l/r3 at 0 0 . (9) The units are rationalized m.k.s. units. constant of free apace is 8.854 X 10-12 f./m.
The summands in the second term on the right-hand side (11) can be rewritten as follows.
I n these units, the dielectric
nl[xeec$*/kT
sv
=
(13a)
ni In ( n e / V )-
c,[xle+*/kT
+ In ci]d3r
(13b)
(10) G. H Hardy, J. E. Littlewood, afid G. Pblya, "Inequalities," Cambridge University Press, London and New York, N. Y., 1934,Theorem 184.
ACTIVITYCOEFFICIESTOF
Nov., 1963
A
SALTIN
The first equality follows from use of eq. 3; the second follows from the observation that according to eq. 3 the bracketed ter:m in eq. 13a is actually independent of position. Combining (12) and (13b) with (ll),we find that
.”I:
A
2335
CHARGED MICROCAPILLARY
1000-
1
I
I
I
I
l
5001-3--u
+
- H[4*1 = iId [-kTniIn (ndV> =. 1
200
RiIarcusll has shown that the right-hand side of (14) is the electrostatic contribution to the free energy of the system. With -H [d)*] now identified as the electrostatic free energy, it is possible to find expressions for the activity coefficients of the various ions in the mixture. Let us begin by finding the derivative of H with respect to ni. A short calculation shows that
100
50 1 -.
y+’ 20
10
hT In V-l
sv
d3r
5
(154
Here &*/ani is given by -
bnt
-
lim AneO
$*(nl, n2,
[
..
2
nl
+ Ant, . . . n
~ --) 4*(nl, n ~.,.
.)
I
nt, . . ., n ~ >
An,
(l5b)
where q5*(nl, rh2, . . . , n,) is the solution of (4) corresponding to the ionic population nl, n2,. . . n,. Since +* satisfies Laplace’s equation outside of V and eq. 4 inside of V , the first two terms in (15a) drop out, and the derivative bH/dni is given by the last term in eq. 15a. From this fact, it follows that the electrostatic contribution to the chemical potential of the ith ionic type is )
pL.lez =
-kT In V-’
sv
e-zied*/kT d3r (16a)
This means that the quantity
is the electrostatic contribution to the activity coefficient of the ith type of ions. This result has also been obtained by Marcus.’l Illustrative Example.-The electrostatic contributions to the io.nic activity coefficients and/or the free energy can be numerically evaluated with the help of the functional 6 in the following way. An n-parameter family of trial functions is chosen, and the functional H is evaluated in terms of these as yet undetermined parameters. The parameters are then chosen so that the value of the functional. is a minimum. This specifies a (11) R. A. Marcus, J . Chem. Phys., 2.3, 1357 (1055).
I
1
I
0
0 2
06
04
08
10
P2‘
Fig. 1.-The reciprocal mean squared activity coefficient l / ~ * ~ of a 1 : 1 electrolyte in n surface-charged capiIlary plotted against the dimensionless salt concentration pz for various dimensionless radii [ R .
“best” member of the n-parameter family. With this “best” trial function approximate values of the various physical quantities of interest can be evaluated. It is important to note that the physical quantity represented by the functional itself is more accurately approximated than any other; if the error in the trial function is of order 7, then the error in the approximate value of the functional is of order q2. In general, the better the “best” trial function is able to approximate the behavior of the true solution, the more accurate will be the approximate values of the various physical quantities. Let us consider a cylindrical capillary of radius R and length L >> R on whose inner surface a fixed electric charge is distributed with surface density (T. Let the interior of the capillary be filled with a solutioii in thermodynamic equilibrium with an infinite external solution of a 1: 1 electrolyte. We wish t o estimate y&, the electrostatic contribution to the mean activity coefficient of the electrolyte in the microcapillary. When the electrolyte concentration is vanishingly small compared t o the average fixed charge conceiitration 2u/R in the capillary, ylt can be calculated ex-
A. TJ. TOBOLSKY, R. H. GOBRAN, R. BOHME,AXD R. SCHAFFHAUSER
2336
Vol. 67
actly using Liouville’s solution to the equation V2u = eU.’ This has been done by Dresner and Kraus,6bwho found the results 1/yrt2 = 1
+ tR4/(192+ 2452)
(17a)
where ER
=
KR
(17b)
and K~ = 2e/a]/kTeR (17c) The value of the electrical potential corresponding to these results is
4*
=
(kT/xie) In [l
+ (tR2- 52)/8]2
5 R (17d) = 0 r 2 R (17e) where [ = KT, r is the radius vector (0 5 r 5 R), and XI r
I$*
is the valence of the ions of opposite charge to the fixed charge cr (counter-ions). When the electrolyte concentration is not vanishingly small, y& can be estimated variationally by use of the trial function
4
=
(kT/xle) In [l
+ b(5R2 -
$2)]e
4=0
r 5R T
(18a)
2 R (18b)
where b and c are as yet undetermined constants. If we introduce the notation PI for the ratio of nl, the number of counter-ions in V , to 2lrRLfal/e, the number of monovalent “fixed” ions in V , and the notation pz for the ratio of n2, the number of co-ions in V , to 27rRLI al/e, then a straightforward calculation yields the result
Here use has been made of the coiiditioii of charge neutrality in the form 1 pz = PI. The coefficient of p2 in eq. 19 is the value of In (l/yi2) = In (l/ylyz) defined by eq. 16b that corresponds to the trial function (18). Shown in Fig. 1 are values of the reciprocal mean ~ with the trial activity coefficient l / y ~ tcalculated function 18, the constants b and c having been chosen (by trial and error) to minimize the value of H given in (19). The 02 = 0 values are the same as those given by (17a), since for Pz = 0, the values b = ’/*and c = 2, which make (17) and (18) identical, will minimize H . The ease with which these estimates have been obtained demonstrates the power of the variational method ; on the other hand, a disadvantage of the present calculation is that it gives no clue to the magnitude of the error involved. Nevertheless, it is the author’s feeling that the precipitous drop in l/yi2 with increasing Pz near p2 = 0 is realistically described because the trial function 18 is exact a t pz = 0 and because it can accommodate itself through two independent constants to changes in p2. The capillary system discussed in this section has been used by Dresner and KrausGbas a model of a suggested salt filter composed of a porous bed of ion-exchange active particles. Their studies of the thermodynamic equilibria of such salt filters in contact with electrolyte solutions mere based on the 0 2 = 0 mean activity coefficient yi given by (17). Since in their work pz was always 50.01, the curves of Fig. 1 indicate that their results do not stand in need of correction.
+
HETEROGENEITY INDEX DURING DEAD-END POLYMERIZATION BY A. V. TOBOLSKY, R. H. GOBRAK, R. BOHME,ASD R. SCHAFFHAUSER Frick Chemical Laboratory, Princeton University, Princeton, N e w Jersey Received April 89, 1963 Expressions are developed for determining the cumulative number and weight average degrees of polymerization throughout a free radical polymerization.
Introduction
It can be shown1 that in the absence of retardation of the termination reaction, radical-initiated polymerization may cease short of complete conversion even when the initiator does not undergo wasteful side reactions. This phenomenon, termed (‘dead-end polymerization,” is due to the fact that the initiator may be depleted before the polymerization has gone to completion. The theory of this type of polymerization has been checked2 quantitatively for the polymerization of styrene using 2,2’-azobisisobutyronitrile (azo-1) as the initiator. It was found that in the absence of the (1) A. V. Tobolsky, J. Am. Chem. Soc., 80, 5927 (1958). (2) A. V. Tobolsky, C. E. Rogers, and R. D. Briokman, ibid., 82, 1277 (1960).
Tromsdorff-Korrish effe~t,~.4 the actual amount of conversion agreed quite well with the predicted value. The effect of chain transfer to monomer was neglected in the above study since only the amount of conversion was considered. In the present treatment, the effect of chain transfer to monomer is considered and expressions are_developed for calculating the cumulative values of Pn and P, (the number and weight average degrees of polymerization) a t any time during the polymerization. Theoretical I n a free radical polymerization where the growing radicals do not undergo transfer reactions with solvent (3) E. Tromsdorff, H. Kohle, and P. Lagolly, Illakromol. Chem., 1, 169 (1948). (4) R. G. W. Sorrish and R. R. Smith, Nature, 150, 336 (1942).
