2330
ARTHURF. BUTLER AND ERNEST GRUNWALD
K values a=0.001 d = l b=O.OI e =lo
3.
-e
2.
0
a
Y
c
c
E L
J I.
-
Q,
0
\
0.
0
,
0
'.
U I
I
200
,
400
I
1
600
TIME (sec.). Fig. 5.-Case 111: Log i us. 1 for different values of K = k f / k b , calculated for (C,) (Yi)= M , p = 0.01 sec.-l, and kf = 10 sec.-l
+
can be obtained from log i-t curves. For case 111, generation of C by a preceding chemical reaction, on the other hand, rate constant data can only be obtained from log i-t curves, since in all cases noapp= n. Reactions of this type are indicated when integral, constant, noappvalues are obtained under a variety of experimental conditions, and log i-t curves show appre-
Vol. 67
ciable curvature, or have slopes which are smaller than usual. Both Icf and k b determine the shape of the log i-t curves. When Icf > 100 kb, 99% of the total species is initially present in the electroactive forms, aad the i-t behavior differs only very slightly from the uncomplicated behavior. When kf/lcb is small, then the current may take a long time to decay to the background level, even for kf values which are large compared to p (Fig. 5 ) . I n general, a preceding reaction with a K in the range of 100 to 0.01 can be investigated. Reaction schemes corresponding to the preceding cases are probably fairly cominon in controlled potential coulometry. Cases I and I1 occur whenever the electroactive species can react with the solvent, supporting electrolyte, oxygen, etc. Analytical coulometric data can be corrected for the effect of side reactions by techniques suggested in these studies. The reaction of a,adjphenylpicrylhydrazyl (DPPH) with bromide ion during the electroreduction of DPPH in an acetonitrile solution of tetrabutylammonium bromide was studied coulometrically by this technique.* Reactions of the type described in case I11 might have been responsible for the curvature in log i-t curves obtained during the coulometric determination of tin.6 A recent communication on the coulometi-y of the uranium(VI)uranium(1V) system in a tripolyphosphate medium indicated that the electrolysis time for the reoxidation of U(IV) was much longer than that for the reduction of C(VI).6 This difference in electrolysis times under eseeutially identical cell conditions may be due to a slow chemical reaction step during the anodic oxidation. Acknowledgment.-The authors are grateful to the Kational Science Foundation for support of this work (Grant No. G 14478). (4) E Solon and A. J. Bard, t o be submitted for publication.
(5) A. J. Bald, Anal. Chzm. Acta, 22, 577 (1960). (6) H. E. Zittel and L. B. Dunlap, Anal. Chem., 36, 12.5 (1963).
SALT EFFECTS ON THE ACTIVITY COEFFICIENTS OF HYDROCHLORIC ACID ASD IONIZED a-NAPHTHOIC ACID AND ON THE DISSOCIATION CONSTANT OF a-NAPHTHOIC ACID IN 50 WEIGHT PER CENT DIOXANE-WATER1" BYARTHURF. BUTLERAKD ERNEST GRUIU'WALD'~ Chemistry Department, Florida State University, Tallahassee, Florida Received April 16,196s Activity coefficients are reported for millimolal hydrochloric and ionized a-naphthoic acid (H,Cl and €&A) in 50 wt. yo dioxane-water in the presence of 0.04-0.6 m SaC1, xaClO4, (CH8)&Cl, and sodium p-naphthalenesulfonate (SaiYs) a t 25". The observed salt effects indicate that salt-induced medium effects make only a relatively small contribution to YH-/C~ and T H Y 4 and are masked by other interactions. There are significant specific differences between the salt effects of NaC1 and NaC104 on the one hand, and those of (CH3)4NC1and N a y s on the other, in such a direction as t o suggest stabilizing interactions with the large organic ions. I n particular, there is evidence for specific attractive interactions between the two naphthyl anions 8-and Ns-. The values of YHYA are based on measurements of the solubility2 and acid dissociation constant of a-naphthoic acid in these salt solutions.
?Ve wish to report B study of salt effects 011 the activity coefficients of hydrochloric acid ( Y H Y C ~ ) and ( 1 ) (a) Work supported by Office of Naval Research: reproduction in whole or in part is permitted for any purpose of the United States government: (b) Bell Telephone Laboratories, Ino.. M u r r y Hill, N. J. (2) E. Grunwald and A. F. Butler, J . A m . Chem. Soc., 82, 5647 (1960).
ionized a-naplithoic acid (YHYA) and on the acid dissociation constant of a-naphthoic acid, in a solvent consisting of 50.00 wt. yodioxane-50.00 wt. yowater a t 25'. The dielectric constantof this solvent is 35.35.2 (3) F. E. Critchfield, J. A. Gibson, and J. L. Hall, ibid., 76, 1991 (1953).
SALTEFFECTS ON ACTIVITY COEFFICIEXTS OF ELECTROLYTES
Nov., 1963
-
-k
L
Hf
A-
HA
The salts were SaC1, KaClO4, (CH3)4SCl, and sodium P-naphthalenesulfonate (NaSs) . The concentratioiis of the substrates were small compared to the salt concentrations, which ranged from 0.04 to 0.6 m. The fact that the solvent consists of two components introduces the possibility of special interactions, such as salt-induced medium effects, that are typical of mixed solvents. A simplified model of the salt-induced medium effect is that the selective solvation of the added salt changes the effective solvent composition for the ~ u b s t r a t e . ~It has already been shown that salt-induced medium effects make relatively large contributions to the observed salt effects on activity coefficients and first-order reaction-rate constants of nonelec? ~ we now wish to trolytes in 50% d i o x a n e - ~ a t e r , ~and evaluate the relative importance of these effects in the interaction of electrolytes. One qualitative consequence of a large salt-induced medium effect would be that specific interactions are important even between ions of like charge, so that Bronsted’s principle of the specific interaction of ions6 would fail. We are also interested in finding out whether salts with large organic ions, such as (CH8)45+or Ns-, interact with electrolytes by specific additional mechanisms slnalogous to those involved in their interaction with nonelectrolytes.*J Our results indicate that the salt-induced medium effects make only a relatively small contribution to the observed salt effects and are masked by other iiiteractions. There are, however, significant specific differences between the salt effects of KaC1 and NaC104 on the one hand, and those of (CH,)&CI and XaYs on the other. In particular, there is evidence for a specific attractive interaction between a-naphthoate ion and pnaphthalenesulfonate ion. Experimental Method of Meiisuremeiit.-Activity coefficients of HC1 were measured using a cell without liquid junction, as shown in eq. 1.
glass electrode/HCl(ma), iUX(m4); in 50 wt. % ’ dioxane-water/AgCl-Ag
(1)
AgC1-Ag was a silver mirror-electrolytic AgCl electrode.8” The glass electrode was of the type (Beckman No. 1190-42) that has been shown to give accurate results in measurements involving the hydrogen ion activity in aqueous-organic solventssb and was stored in 50 wt. % dioxane-water between experiments. I n the experments without added MX(m4 := 0), m3 varied from 0.001 to 0.45 m. Otherwise ma was about 0.002 m and m4 varied from 0.04 to 0.6, 0.3, 0.2, and 0.2 m, respectively, for NaC1, NaC104, (CHa),NCl, and h’ah’s. K Avalues for a-naphthoic acid were measured by a modification (4) J. O’M. Bockris and H. Egan, Trans. Faraday Soc., 44, 151 (1948)’ Ezperientaa. 3, 453 (1947); E. Grunwald, in “Electrolytes-International Symposium,” B. Pesce, Ed., Pergarnon Press Ltd., London, 1962. ( 5 ) E. F. J. Duyniitee, E. Grunwald. and M. L. Kaplan, J . Am. Chem. Soc., 82, 5654 (1960). (6) J. N. Brdnsted, ibzd., 44, 877 (1922); 4 6 , 2898 (1923). (7) R. L. Bergen, Jr., and F. A. Long, J . Phvs. Chem., 60, 1131 (1956). (8) (a) E. L. Purlee and E. Grunwald, abzd., 69, 1112 (1955): (b) E. L. Purlee and E. Grunwald, J . Am. Chem. Soc., 79, 1366 (1957); A. L. Bacarella, E. Grunwald, H. P. Marshall, and E. L. Purlee, J . Phys. Chem., 62,866 (1958).
2331
of cell 1 in which the solutes: HA(m’3), NaA(m’’3), and WaCI(ma”’)were substituted for HCl(m3). Values of w’a ranged from 0.001 to 0.005, values of m”3 were less than 0.003, and values of m3’” were about 0.002 except in those experiments in which MX was NaC1. Solutions were prepared from purified reagents2 by standard gravimetric and volumetric methods. The effect of each salt was measured in a t least two independent series. All concentrations are expressed in moles/kg. of dioxane-water mixture. Calculations.-Molal activity coefficients, 7, for HCl were calculated from the measured e.m.f., E, according to eq. 2.
In
YHYCl =
[(E’ - E)F/RT] - In
mHm1
(2)
The quasi-standard e.m.f., E*, was deduced from experiments on cell 1 without added M X and with m3 < 0.005 and was checked frequently throughout the course of these experiments. Values of Y H Y C l a t these dilutions were close to the limiting-law values and were actually calculated from eq. 3, in which the coefficient, 7.732, is the appropriate Debye-Huckel limiting slope. Equa-
In
YHYCl =* - -
-7.732m3’Ia m3 1 2.474m3””
+
< 0.005, m4
=
0 (3)
tion 3 actually fits the e.m.f. data for HC1 within experimental error up to concentrations well over 0.01 m. Since in the experiments with added salt, m3 is small in a n absolute sense and very small relative to m4, we may neglect the contribution of HCl(m3) relative to that of MX(m4)in the interpretation of YHYCI. For the same reason we may neglect, in the K A determinations, the effects of Hh(mfa),NaA(m03), and NaC1( m t t f arelative ) to that of i%X(ma). Since the value of YHYCl for a given salt solution is therefore the same, whether the substrate be dilute HC1 or a dilute buffer consisting of HA, NaA, and h’aC1, we may combine activity coefficients obtained in the former case with e.m.f. data for the latter case to calculate mH according to eq. 2 . The value of K A for the given salt solution is then given by eq. 4, in which ma/mnAis equal to m“3/m 3.
KA =
mNmA
-
(4)
mHA
It will be noted that K A is a quotient involving molal concentrations. K A is related to the thermodynamic constant, KA’, by eq. 5, in which the 7’s are molal activity coefficients. K.4’ was obtained by a differential potentiometric m e t h ~ dusing ,~ dilute solutions of the substrates without added salt, in 45, 50, and 55 wt. 70 dioxane-water. Activity coefficients could be estimated with sufficient accuracy under these conditions of low ionic strength from oDebye-Huckel expressions analogous to 3, assuming a value of 5 A. for the “distance-of-closest-approach” parameter.
Results I n most of our salt effect studies, we are measuring the activity coefficient of the substrate (component 3) in the preseiice of salt (component 4) a t essentially zero substrate concentration. We find it convenient to represent our results as power-series functions of the salt molality,1° as illustrated in eq. 6 and 7. In these equations, and in the rest of this section, we shall use symbols with commas (such as Ka, C1 or H, C1) to denote electrolytes, and symbols without commas (such as HA) to denote nonelectrolytes.
+
~ H , C I : X ~ , C I ? ? % ~ ~D ’ ~H
+
(log Y H A ) ~ ~== Jo C W A . N ~ , C ~~ ~H ~A Na,C1
+
, c ~ . N ~ , c. .I .~ ~ (6) ~
+ ...
: N ~ , c I ~ ~ ~
(7) (9) ,4.L. Bacarella, E. Grunwald, H. P. Marshall, and E. L. Purlee, J . O w .Chern., 20, 747 (1966). (10) G. Scatchard and S. 9. Prentiss, J. Am. Chem. Soe., 66, 1486 (1934); 56,2320 (1934).
ARTHURF. BUTLERAND ERXEST GRUNWTALD
2332
Vol. 67
the coefficients of m4 and mq2. That is, we let the coefficients of m40/2,as well as those of m45/', 77$, . . . , be zero. This decision was partly based on an analogy to a previous study of the activity coefficients of sodium chloride in 50 wt. % d i o x a i i e - ~ a t e r ,in ~ ~which it was found that the complete series, involving all powers of m'I2up to m6/', did not fit any better than an abbreviated series involving only rn1I2,m, and m2. As a matter of fact, the fit of the present data to the abbreviated series is satisfactory, the worst of the fits being shown inFig. 1. The standard deviation of 49 values of yf for HC1 in our salt effect experiments from the respective power series function was 2.4%, that of 35 K Avalues was 1.2%. Values of the parameters that give best fit to our data are listed in Table I. The table also lists activity coefficients calculated from these parameters for m4 = 0.2. Table I1 summarizes our measurements of KA' for a-naphthoic acid, and certain derived functions that are of interest in connection with the salt-induced medium effect. TABLE I SALT-EFFECT P A R A M E T E R S A S D -4CTIVITY C O E B F I C I E Z T S FOR T H E SUBSTRATES H,Cl, HA, AND H,A IN 50 WT. 70 DIOXANE-WATER AT 25'
01
I
I
0
I
I
02
1
I
0.4
0.6
m4. Fig. l.--SaJt effect of sodium chloride on the activity coefficient of hydrochloric acid (top) and on the acid dissociation constant of or-naphthoic acid in 50 wt. yodioxane-water a t 25". The solid lines me calculated from the parameters in Table I. The substrate concentrations are very small compared with that of sodium chloride.
The power series for electrolyte substrates involves successive powers of m*'/', that for nonelectrolyte substrates iiivolves successive powers of m4.lo The coefficient of the leading term in eq. G can be predicted from the Debye-Huckel limiting law and has the value X = 1.679 for 50 wt. % dioxane-water at 25'. All other coefficients in eq. 6 and 7 are obtained by adjustment to the data. It followsfromeq. 5,6, and 7 that thesalteffectsonlog KA a t zero concentrd,tion of substrates are represented similarly by power series of the form 8, in which the coefficients have a significance as shown in eq. 9. (log KA)sa,cl = log K A O
+ 2~?n4'/'-
cNa,Clm4a'2 bSa,CI =
C N ~ , C I= dNa,Cl
=
-
bNa,Clm4
dNa,clm2.
k H A Na,CI BH,AN a , C 1 CH,ANa,C1 D H ,N~a , C l - HA Sac1
-
..
(8) (9)
Coefficients k and 1 for the salt effects on the activity coefficient of a-naphthoic acid in 50 wt. yo dioxanewater have been reported previously. * In applying eq. 6 and 8 to represent the salt effects on Y H Y C ~and on KA, we felt that our data were sufficiently extensive and accurate to warrant the use of two adjustable parameters, but not of three or more. We decided, somcwhat arbitra13y, to let these parameters be
Salt
HC1
BH.CI:sslt
5.17 -4.81
DH9C1:sde halt &It
kEA:dta BH,.k:salt DH,A:salt -,EyCI ( 0 . 2 ??Z) YHY.4 (0.2 m! ?Cl/YA a
(0.2 Vl)
From ref. 2 ;
...
...
0.12
... ...
0.218
... ...
NaCl
NaClOa
(CHa)&CI
NaNs
4.45 -2.90 -5.19 5.24 -0.10 5.10 -5.24 0.167 ,203 .92
4.76 -4.01 -4.60 4.46 0.31 4.91 -4.46 0,195 I200 .97
4.45 -4.61
4.43 -4.05 -4.38 4.05 -0.25 4.13 -4.05
7.33 -0.19 5.22 -7.33 0.157 ,177 .68
0.167 ,145 1.15
Z ~ . k : ~= ~ l0.00 ~
TABLE I1 ACID DISSOCIATION OF ~-XAPHTHOIC ACID MIXTURES AT 25" Dioxane (wt. 70)
a
-5,40
DIOXAEE-WATER
KAo 21
45 0,8567 50" ,8302 55 ,8000 In 50 wt. % dioxane-water -2.303RT d log K ~ " l d Z 1 dF,"/dZ1 for HA (ref. 2 ) dF,"/dZ1 for H +
IP;
+ A-
x lo6
(molal concn.)
1.10 1 0 . 0 4 ,532 f ,007 ,220 f ,001
- 16.7 ked. 10.9 kcal.
_____.
-
5 . 8 kcal.
Discussion As far as we know, this is the first investigation of salt effects on the activity coefficients of electrolytes for a variety of structural types in a binary solvent. Alt'hough the accuracy of the data is only moderate, a few conclusions concerning the relative importance of rarious interactions can be reached. It will be sufficient t o compare activity coefficients at a single salt concentrat.ion that is high enough for specific effects to be appreciable. We have therefore tabulated YHYCl, YHYS, and yC1/yAa t m4 = 0.2 (Table I). Although the accuracy of these values is not uniform, we estimate that differences in excess of 5yo are significant, with a statistical probability of a t least 0.67. Thus me obtain t'he salt-effect sequeiices (11) 1,;. G r u n w d d nn,l .I1,. . Hacairll;i, .I. ,lm. C h c r r r . Sei,., 80,3810 (l!lcj8).
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).
