J. Phys. Chem. 1981, 85, 1756-1762
1756
Multicomponent Diffusion of Electrolytes with Incomplete Dissociation. Diffusion in a Buffer Solution Derek G. Leaist and Philip A. Lyons* Department of Chemistry, Yale University, New Haven, Connecticut 06520 (Received: December 3 1, 1980; In Final Form: M r c h 2, 198 1)
Approximate expressions are developed to describe isothermal diffusion of multicomponent electrolyte solutions with incomplete dissociation. Provided the association constants and mobilities of the ions and ion aggregates are known, numerical estimates of the diffusion coefficients and L i b coefficients can be made. Ternary diffusion of a buffer solution consisting of a weak acid and its salt is examined; the flow of hydrogen ions along a gradient of pH increases the rate of diffusion of the salt and leads to strong coupling between diffusive flows. Ternary diffusion coefficients are measured at 25 "C for acetic acid/sodium acetate/water at low concentrations, using the conductimetric technique. The data are in satisfactory agreement with predictions.
1. Introduction
The main features of electrolyte diffusion in multicomponent solutions have been described.lI2 Diffusive flows of electrolytes in these systems are coupled by electrostatic forces that arise from differences in ionic mobilities. For mixed electrolyte solutions containing polyelectrolytes, and aqueous solutions containing acids (or hydroxides), such coupling is ~ t r i k i n g , ~ while - ~ the interaction between flows of electrolytes and nonelectrolytes is usually quite For strong electrolyte mixtures accurate transport coefficients (Lib and Dik)can be predicted from limiting ionic mobilities at low ionic strength.l~~@?~ Methods are available for estimating the diffusive properties of these systems at higher c o n ~ e n t r a t i o n s . ~ , ~ J ~ If ionic dissociation is incomplete, electrolyte diffusion is more complex. For binary diffusion of an associating electrolyte, equations have been developed14and used to describe the diffusion of weak acids,l5-I7bivalent salts in water,laJgand electrolytes in low dielectric media.12 How will ion association influence multicomponent transport? To answer this question a procedure is outlined for estimating the diffusive properties of mixed electrolytes with an arbitrary number of association reactions. For purposes of illustration, the ternary diffusion of a buffer solution consisting of a weak acid and its salt is examined in some detail. The rate of diffusion of the salt (1) W. Nernst, 2.Phys. Chem., 2, 613 (1888). (2) L. Onsager and R. M. Fuoss, J. Phys. Chem., 36, 2689 (1932). (3) V. Vitagliano, R. Laurentino, and L. Costantino, J. Phys. Chem., 73, 2456 (1969). (4) A. Revzin, J. Phys. Chem., 76, 3419 (1972). (5) H. Kim, G. Reinfelds, and L. J. Gosting, J.Phys. Chem., 77, 934 (1973). (6) D. G. Leaist and P. A. Lyons, Aust. J. Chem., 33, 1869 (1980). (7) D. G. Leaist, Dissertation, Yale University, 1980. (8) D. G. Leaist and P. A. Lyons, J.Phys. Chem., submitted for publication. (9) L. A. Woolf, D. G . Miller, and L. J. Gosting, J. Am. Chem. SOC., 84, 317 (1962). (10) P. J. Dunlop, J. Phys. Chem. 69, (1965). (11) M. V. Kulkarni and P. A. Lyons, J.Phys. Chem., 69,2336 (1965). (12)D. G. Leaist and P. A. Lyons, J.Solution Chem., in press. (13) D. G. Miller, J.Phys. Chem., 71, 616 (1967). (14) W. H. Stockmayer, J. Chem. Phys., 33, 1291 (1960). (15) G. T. A. Muller and R. H. Stokes, Trans. Faraday SOC.,63,642 (1957). (16) L. A. Dunn and R. H. Stokes, Aust. J. Chem., 18, 285 (1965). (17) (a) E. L. Holt and P. A. Lyons, J.Phys. Chem., 69,2341 (1965); (b) C. W. Garland, S. Tong, and W. H. Stockmayer ibid., 69,2469 (1965). (18) (a) H. S. Harned and R. M. Hudson, J.Am. Chem. SOC.,73,3781, 5880 (1951); (b) C. W. Garland, S. Tong, and W. H. Stockmayer, J.Phys. Chem., 69, 1718 (1965). (19) J. M. Creeth and B. E. Peter, J. Phys. Chem., 64, 1502 (1960). 0022-3654/81/2085-1756$01.25/0
is increased by the weak acid, and the cross-diffusion coefficients can be large. The key role played by the pH gradient is discussed. The diffusive properties of such simple buffers may be relevant to transport in biological systems. Experimental transport coefficients are reported for acetic acid/sodium acetatelwater and compared with values predicted by the procedure which was developed. 2. Calculated Transport Coefficients (a) Phenomenological Equations. The linear equations describing isothermal diffusion of electrolytes A.,B C,D, etc. dissolved in a neutral solvent may be written"1: s-1
Ji =
k=l
Lik(-d p k / d x )
(1)
Jiis the flux of electrolyte i relative to the solvent, Lik are electrolyte phenomenologicalcoefficients, and d &/ax are electrolyte chemical potential gradients. If there are s kinds of ions A z A , B", CzC,etc., then there are s - 1linearly independent electrolyte flows. In terms of electrolyte concentration gradients, the flow equations are s-1 Ji
= xDik(-dCk/dx) k=l
(2)
The diffusion coefficients, Dik, are related to the Lik coefficients by s-1
Dik
=
m=l
Lirn(dprn/dck)
(3)
Diffusion may also be described in terms of flows of arbitrary species NA
;i
=
c li&d
k=l
Pk/dX)
(4)
;iis the flux of species i, Iik are species phenomenological coefficients, and N is the number of species (s ions and N - s ion aggregates). If species i is electrically neutral, pi is its chemical potential. Otherwise pi is its electrochemical potential. For very dilute sol_utions2J4 the matrix 1 of lik coefficients is diagonal and lii = EiQ/'. Ei is the concentration of species i and Lip is its limiting mobility. Not all species flow independently. The local equilibrium assumption (that the rates of the ion association reactions are fast compared to the rate of diffusion14) provides N - s relations among the flows of species. An additional relation is provided by the requirement that, 0 1981 American Chemical Society
Multicomponent Diffusion of Electrolytes
The Journal of Physical Chemistry, Vol. 85,
for pure diffusion, the electric current must vanish.
z2;i = 0
= P 2 + P3 (18) (iv) the flux of the weak acid component equals the flux of molecular acid plus the flux of dissociated acid (19) J1 = 7o + ?1 (v) the flux of the salt component equals the flux of B+ J2 = j 2 (20) The result is (following tedious algebraic manipulations outlined in ref 6) given by eq 21. The limiting cases in Pz
N i=l
No. 72, 1987 1757
(5)
When these constraints are utilized, eq 4 may be converted to eq 1 and the Llk coefficients expressed in terms of Ilk coefficients. Approximations. For purposes of estimating the effects of ion association on diffusive transport, the dilute solution approximation lik 6.ck c^.&1 c (6) is used. Starting with eq 6, we develop approximate expressions for the Lik and Dik coefficients (an example is given later). The analysis includes changes in the diffusive properties with concentration due to changes in the proportions of the species and electrostatic coupling of ionic flows resulting from the electroneutrality requirement (eq 5 ) . Because I is assumed to be diagonal, and therefore symmetric, the resulting Lik coefficients are consistentz0 with the Onsager reciprocal relations Lik = The concentration dependence of the mobilities of the species is ignored, but, for diffusive transport, this assumption is acceptable. A t low concentrations the approximation made by eq 6 amounts to the neglect of small electrophoretic and time of relaxation effect^.^^^^^ ( b ) Diffusion of the Buffer HA(l)/BA(2)/Solvent.We examine here diffusion of a buffer consisting of the weak acid HA, its “strong” salt BA, and solvent. Anion A- is common to both electrolytes. This system is ternary with regard to both thermodynamic and diffusive properties. The electrolyte components are numbered HA(1) and BA(2). There are four species numbered as follows: 0, HA (molecular acid); 1, H+; 2, Bt;3, A-; and one association reaction A
Ht + A - = H A The degree of dissociation of the acid, a,is a = c^l/Cl
(7)
= i,,(-dfi,/dx)
(10)
32 = i 2 , ( - d P 2 / d x )
(11)
= i33(-dp3/dx)
(12)
33
To convert these equations to the electrolyte flow equations Ji = L i i ( - d ~ i / d x + ) ~512(-~3~2/dx) (13) J2
= L21(-dPl/dx)
+ Lzz(-dP2/dX)
eq 22 and 23 correspond to the Nernst limits for a none-
(ill=
-(g ) “L
o !L +l
= 0)
4, +
(14)
and thereby obtain expressions for the Ljk coefficients in terms of lii coefficients, we used the following relations: (i) zero electric current (15) j3 = j1 j2 (ii) equilibrium of the association reaction Po = P1 + P 3 (16) (iii) the chemical potential of each electrolyte equals the sum of the electrochemical potentials of its constituent ions P l = P 1 + P3 (17)
+
(20) D. D. Fitts, “The Thermodynamics of Irreversible Processes”, McGraw-Hill, New York, 1962, Chapter 8.
(22)
133
L a;tl (loo = c, = 0 ) h
\-;ll
+ i;, + i3,
lectrolyte/strong 1:l electrolyte and two strong 1:l electrolytes. In the former case, the coupling coefficients L12 and Lzl are zero. Diffusion Coefficients. The diffusion coefficients entering into the flow equations J1 = D11(-dcl/dx) + D12(-dc2/dx) (24)
J2 = D21(-d~1/dx)+ Dzz(-d~z/dx) (8)
Ljk Coefficients. The species flow equations are (approximately) j o = i,(-dp,/dx) (9) 31
L=
(25)
are obtained by use of eq 3 and 21: Dll = Lll(dPl/dcl) + Ll2(dP2/dCl)
(26)
D12 = Lll(dPl/dc,) + LlZ(dPZ/dCZ)
(27)
+ Lzz(dP2/dC1)
(28)
D21 = LZl(dPJdC1)
= J5zl(aPl/dc2) + Lzz(dP2/dC2)
(29) The expressions for the electrolyte chemical potentials in our example are 022
PI
= PI’ = PLZO
+ RT In [ac1(ac1 + C Z ) Y I Y ~ ~
(30)
+ RT In
(31)
[c2(ac1+ c2)y2y31 The degree of dissociation is obtained from ~2
KDcis the dissociation constant of the acid and y i are activity coefficients. We will assume that yo, the activity coefficient of the molecular acid, is unity and for each of the ions use the mean activity coefficient ya computed from the semiempirical expressionz1 In y + = - S f N Z / ( 1 N 2 ) (33)
+
at ionic strength
I = ac1
+ c2
(34)
(21) E. A. Guggenheim and J. C. Turgeon, Trans.Faraday Soc., 51, 747 (1955).
1758
The Journal of Physical Chemistry, Vol. 85,
No. 12, 198 1
Leaist and Lyons
lS8
t
D22
4
-i cn N
E
0
c
2
L
\
\
1 .o
0.1
10.01
-5
-4
log
(c,
+
10.0
I
'."b
l-
I
-2
-3 c2)
I
tD:o 1
0 :1
1 ..o
-5
-4
-3
-2
l 0 g ( c 1+ c 2 )
Flgure 1. Diffusion coefficients D,, and D,, for HOAc (1)lNaOAc (2)/H20 at 25 O C plotted against the log of the total electrolyte concentration for several values of the ratio f l = c , / ( c , c2).
Figure 2. Diffusion coefficients D12and D2, for HOAc (l)/NaOAc (2)/H20 at 25 O C plotted against the log of the total electrolyte conc2), centration for several values of the ratio f , = c , / ( c ,
The concentration derivatives of the electrolyte chemical potentials are
TABLE I: Parameters Used To Estimate Transport Coefficients of Acetic Acid (l)/Sodium Acetate (2 )IWater at 2 5 'Ca
+
2dc12 =
9 dc2 =
"[ c1
-[ + :( + $) c2
1
1
"1
I--- c1 1 - a dc,
+
+
(35) species 0, molecular acetic acid 1,H' 2, Na+ 3, OAc-
(37)
d In ya2 d In ya2 -- -yc
ac1
dc2
(39)
If we neglect activity coefficient terms and set c2 = 0, the expression for Dll becomes identical with the expression already derived for the binary diffusion coefficient of an associating 1:l electrolyte15J8 ffDionic+ 2(1 - cy)& D= 2-a Do is the diffusion coefficient of the neutral associated species and DiOicAisthe binary Nernst diffusion coefficient given by 2RTLi1u8/(Li1+ 1.2~). Unfortunately, for the general case the expressions for
349.64e 50.07e 40.88e
di x
6-i
x
105/cmZ 1016/mol s g-'d
c
s-l
1.20@
4.873
(9.331) (1.333) (1.089)
37.558 5.378 4.391
FQc= 1.753 X lo-'
mol L-' (ref 25), Sf = 1.175. Limitin ionic conductivity. Limiting diffusion coefficient. 'Mobilities (per mole) are calculated from diffusion coefficients and ionic equivalent condlctivities (in cmz ohm" equiv-') by using the relations Di = RT6-i and 6-i = 1.0742 X lO-''hi/kiI. e Reference 24. Reference 16 a
with
hi0/cm2 ohm-' equiv-I (45.37)
the ternary diffusion coefficients are not compact and, a t first glance, obscure. (c) Acetic Acid (1)lSodiumAcetate (2)/ Water, 25 O C . To establish the main features of diffusion of a simple buffer, the ternary diffusion coefficients calculated for acetic acid (l)/sodium acetate (2)/water at 25 " C are plotted against the total electrolyte concentration in Figures 1 and 2 for several values of the ratio cl/(cl + cz) = fl. The mobilities and other parameters used in the calculations are listed in Table I. The figures contain much information and merit further comment.22 ~~~
~
~~
(22) (a) The ionization of water is neglected. This is acceptable as long as the concentration of H+arising from the dissociation of the acetic acid is much greater than the amount contributed by the ionization of water. For extremely dilute solutions (near pH 7), where the two amounts are comparable, the diffusion is quaternary and corresponds to the system acetic acid/sodium acetate/sodium hydroxide/water. See ref 9. (b) The dimerization reaction 2HOAc F= HzOAc2is neglected, but, for dilute solutions, this assumption is valid. The association constants for this reaction is 0.047 L mol-]. at 25 "C.16 Thus if the concentration of acetic acid is about 0.01 mol L-l, the concentration of the dimer is less than 0.05% of the monomer concentration.
Multicomponent Diffusion of Electrolytes
The Journal of Physical Chemistry, Vol. 85, No. 12, 1981
The curve for Dll with f l = 1.00 represents the binary diffusion coefficient of acetic acid in water. As the concentration is reduced, a higher proportion of the acid ionizes to OAc- and highly mobile H+; Dll increases. If the equations provided in the preceding section are used it can be shown that for f l = 0, Dll is approximately equal to the average of the diffusion coefficient of molecular asetic acid, Do, and the tracer diffusion coefficient of H+, D1, given by Dll x (1- a)Bo ~yD1 (43) [For strong acid/strong salt mixtures, such as Hql (1)/ NaCl (2) and HC1 (1)/KCl (21, as f l 0, Dll 01.1 The cross-diffusion coefficient DZlmeasures how effectively a concentration gradient in acetic acid transports sodium acetate. At high concentrations DZlis small because the acid is almost completely associated and the nonelectrolyte/strong electrolyte limit applies. As the concentration is reduced, however, Dzl becomes large and negative. This behavior results from electrostatic coupling. Because of the large difference in mobility between H+ and OAc-, a concentration gradient in ionized acetic acid is accompanied by an electric potential gradient that acts to slow down the diffusion of H+ and increase the rate of diffusion of OAc-. The electric potential causes Na+ to diffuse in the direction opposite to the flow of acetic acid. Therefore, Dzl is negative. (These are the same arguments used to explain coupled diffusion of strong electrolytes.) As required by the flow equations, DZl 0 as f l 1.00. (It is impossible for a gradient in acetic acid to transport sodium acetate if the concentration of the salt is zero.) The cross-diffusion coefficient DI2measures how effectively a concentration gradient in sodium acetate transports acetic acid. At high concentrations where the acid is highly associated the coupling is weak. The mobility of Na+ is slightly larger than the mobility of OAc- (see Table I); therefore, on the basis of strong electrolyte behavior, we would expect D12to be small and negative as the concentration is reduced. D12is indeed negative, but is very large. To understand this remarkable behavior consider the situation where there is a uniform concentration of acetic acid and a concentration gradient of sodium acetate. Because of the common ion effect the concentration of H+ is lower in the region of higher sodium acetate concentration, and H+ diffuses “up” the sodium acetate gradient (“up” the pH gradient) toward the region of lower hydrogen ion activity; therefore, D12is negative. The high mobility of H+ is responsible for the large magnitude of D12. If the ratio of acid to salt is fixed and the total electrolyte concentration is varied the calculated values of D12pass through a minimum (Le., a maximum in the magnitude of D12,see Figure 2). This behavior is closely related to the rate of change of H+ concentration with salt concentration, cld a/dc2. At very high and very low total electrolyte concentrations, where the concentration of H+ is relatively insensitive to the salt concentration, D12 is small. A t intermediate concentrations, however, a salt concentration gradient is better able to cause a coupled flow of H+ and the magnitude of D12is larger. If D12 and clda/dc2 are plotted against total electrolyte concentration for different ratios of acid to salt, it may be verified that the extrema of both these plots occur at the same total electrolyte concentration. D22is the main diffusion coefficient of sodium acetate. The curves for D22with f l = 0.00 and 1.00 are, respectively, the binary diffusion coefficient of aqueous sodium acetate and the tracer diffusion coefficient of Na+ in aqueous acetic acid. Dz2 passes through maxima for other values of f l . (For a given value of f l the maximum in DZ2and the
+
I1
c
I
I
1
I
I
I
I
I
I
1759 I )
1.4
‘Cn &I
E
l . 1
\ u
1.0)
- -
- -
X
I
3
5
4
6
PH Figure 3. Diffusion coefficients D,, and D,, for HOAc (1)lNaOAc (2)/H20 at 25 OC plotted against the pH [= -log ( a c l y + ) ] for several values of f , = c , / ( c , c,).
+
minimum in D12occur at the same total electrolyte concentration.) As H+ diffuses “up” a sodium acetate gradient, Na+ is “pumped” in the opposite direction to satisfy the constraint of zero current flow. As a result, Dz2is larger than the binary diffusion coefficient of sodium acetate at the same concentration. DI2and DZ2are plotted against the pH in Figure 3. A change of a few tenths of a pH unit can produce large changes in these coefficients. Figure 3 illustrates that, although the H+ concentration of a buffer solution of acetic acid and sodium acetate is small and relatively constant, the diffusive properties are highly variable.
3. Experimental Transport Coefficients The system acetic acid (l)/sodium acetate (2)/water at 25 OC was chosen for an experimental test of the calculated transport coefficients of a simple buffer. Data for binary diffusion of aqueous acetic acid have been rep~rted.l~J’s~~ In addition, the ionic mobilitiesz4and the dissociation constant26and mobility of the molecular acid16are accurately known. ( a ) Experimental Section. Procedure. In order to make measurements on solutions which are sufficiently dilute (where an appreciable fraction of the acid is ionized and the diffusive properties are most interesting) we used the conductimetrictechnique. In that technique an electrolyte concentration gradient is created along a vertical column of solution by injecting a small amount of electrolyte into the base of the column. The electrical conductivity is measured at one-sixth of the distance from the top and bottom of the column by using pairs of small electrodes set in the column walls. From the rate of change of the conductivity, the ternary diffusion coefficients are obtained. The procedure and equipment have been de~~~~
(23) V. Vitagliano and P. A. Lyons, J.Am. Chem. SOC.,78,4538 (1956). (24) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions”, 2nd ed, Academic Press, New York, 1959, Appendix 6.1. (25) Robinson and Stokes, ref24, p 339.
1760
The Journal of Physical Chemistry, Vol. 85, No. 12, 1981
Materials. Mallinckrodt glacial acetic acid was used without further treatment. Mallinckrodt anhydrous sodium acetate was dried in a vacuum oven at 110 "C. Deionized distilled water was degassed prior to use. Solutions were prepared by weight and converted from molal (mi)to molar concentrations by using the density equation d/g cm-3 = 0.9970 + 0.0082ml + 0.040m2
(44)
Data Analysis. The injection of electrolyte into the base of the diffusion column creates the step-function initial conditions
Leaist and Lyons
m*(t) = KB - kKT
is determined, where k is the ratio of cell constants kT/kB for the top and bottom electrode pairs introduced to allow for inequivalence of the electrode dimensions. AK*(t)is directly proportional to AK(t). AKl*(t) is determined for the case where the initial concentration gradient is entirely in electrolyte 1 (AczO= b1(2)= 0), and, for the same final composition, AK2*(t)is determined for the case where the initial concentration gradient is entirely in electrolyte 2 (Aclo= bl(l) = 0). The apparent diffusion coefficients Dal(t) and D,&) defined by A2
D,i(t) = --a2 A is the height of the diffusion column, [ is the thickness of the injected layer, Acioare the step sizes, and Ei are the final concentrations. Solution of the time-dependent ternary diffusion equations with restricted diffusion boundary conditions provides the Fourier expressions7J'
+ bn(l)DzlIe-n2$D(2)t/A2 +
(55)
d In [ a Al'(C1B - C1T) A ~ / ( C ~-BC Z T ) ] / ~ ~ d In [ ~ ~ ( ~ B C-I ~BT C I T ) + h 2 / ( C 2 ~- C ~ ~ ) l / d t (56)
bn(1)D21]e-n2,2D(1)t/A2]] (48) where a is the degree of dissociation at the final compo-
for the concentration profiles along the column. The Fourier coefficients are b,(i) = (2Acio/nr) sin ( n r [ / A ) . and are the eigenvalues of D, the diffusion coefficient matrix, given by D(l) [Dll 0 2 2 (D11- D22) X
+
d In AKi*(t) dt
are obtained by numerical differentiation of the experimental data.6-8 For strong electrolyte mixtures the ternary diffusion coefficients can be obtained directly from the apparent diffusion coefficients.6-s For dilute solutions of acetic acid/sodium acetate, however, the strong variation with concentration of the degree of dissociation of the weak acid causes small departures from the linear expression for the conductivity difference between the bottom and top electrode pairs (eq 53). To correct the apparent diffusion coefficients, they are multiplied by Fi(t) Fi(t) =
[ bn(2)(D11 - D(l))-
(54)
+
(1 + 4D12D21/(D11 - D22)2)1/21/2(49)
D(2) = [Dll + D22 - (Dl1 - 0 2 2 ) x (1 + 4D12D21/(D11 - D22)2)1/21/2(50) The series expressions for the concentration differences A&) = ci(A/G,t) - ci(5A/6,t)converge rapidly; two to three days after preparation of the initial concentration gradients only the leading terms (with n = 1) need be considered:
b1(1)D21]e-~Dc2)t/A2 + [b1(2)(Dll- D(1))- bl(1)Dzl]e-~D(')t/A2 1 (52) Furthermore, for small concentration gradients, the conductivity difference AK(t) = KB - KTbetween the bottom and top electrode pairs is linear in Acl(t) and Ac2(t) M(t)= SlAcl(t) + SzAcz(t) (53)
Siare the conductivity increments aK/dci evaluated at the final composition cl, c2. In practice, the conductivity difference
sition and A{ and A; are the hypothetical equivalent conductivities of the completely dissociated electrolyes at ionic strength atl Cz. A long waiting period (-3 days) between the preparation of the initial concentration grais nearly dient and the collection of data ensures that Fi(t) unity (=0.99-1.01). As a result, the correction factors can be evaluated with sufficient accuracy for each run by using the approximation A[ = X+O + Aotogether with eq 47 and 48 and the diffusion coefficients calculated in section 2. Once the small conductivity corrections are applied to the apparent diffusion coefficients, the ternary diffusion coefficients can be obtained by the method of data analysis developed for strong electrolyte mixtures. By fitting the data to the equation^^,^
+
F1(t)Dal(t) =
D(l)+ E1D" exp[r2(D(l)- D2))t/A2] 1 + El expCr2(D(1)- D@))t/A2] (57)
F2(t)Da2(t)=
D(2)+ E2D(')exp[r2(D(2) - D(l))t/A2] 1 + E2 exp[$(D(2) - DU))t/A2] (58) and D2)and the parameters El and
the eigenvalues E2 given by
El = [D22 -
D2)- (S2/Sl)D211/ [Dll - D2)+ (S2/~1)D211 (59)
E2 =
[Dll -
-
(Sl/S2)D121/
ID22
- D1)+
(Sl/f32)Dl21 (60)
are determined. Finally, the ternary diffusion coefficients
The Journal of Physical Chemistry, Vol. 85, No. 12, 1981 1761
Multicomponent Diffuslon of Electrolytes
TABLE II:= Comparison of Experimental and Calculated Transport Coefficients of Acetic Acid (1 )/Sodium Acetate (2VWater at 25 "C
0.010 0.002 0.00920
0.006 0.006 0.00344
Cl c2
01
exptl
calcd
calcdb
exptl
calcd
calcdb
105~,, 1.2(+. 0.1) 1.235 1.234 1.24(+0.05) 1.269 1.267 -0.318 -0.310 -0.032 - 0.42(+0.1) - 0.029 1 0 5 ~ ~ ~ - 0.04(t 0.05) -0.03(*0.02) -0.034 - 0.033 - 0.015 - 0.015 105~,, - 0.01(t 0.02) 1.348 1.350 1.37(+(0.02) 105~,, 1.17(+0.02) 1.168 1.179 1.268 10l oRTL ,I 0.72(+0.05) 1.24(+0.05) 1.270 0.7412 0.7405 -0.013(+0.002) - 0.05(+0.01) -0.0394 -0.0386 10'" R TL , -0.0104 -0.0100 -0.0386 - 0.04(+0.02) -0.0394 10'"RTL,, -0.009(+0.01) -0.0104 - 0.0100 0.1442 0.146(+0.002) 0.1441 1O1"RTL,, 0.366(+0.005) 0.3659 0.3694 a Units: ci in mol L-';Dih in cm2 s - ' ; RTLih in mol cm: s-: Estimates of experimental precision are based on repeataWith lib', lib" terms. bility of the diffusion coefficient determinations.
N 'm
1.31
E
'i ^t 1.0 .-
am
-
./
c
0.8 k 3
i i
4
10-5
6
5
x t
7
/ s
Flgure 4. Corrected apparent diffusion coefficients plotted against the time at c , = 0.010 mol L-' and c2 = 0.002 mol L-'.
are obtained from D(l),D@),El, and E 2 by use of the relations
Starting with a trial value of D(2),we obtained least-squares estimates of D(l)and El by fitting the F,(t)D,,(t) data to eq 57. Then, holding D(l) constant, we obtained leastsquares estimates of D(2)and E2 by fitting the F2(t)Da(t) data to eq 58. The cycle was repeated until convergence was achieved. ( b ) Results. Experimental diffusion cofficients and the derived Lik phenomenologicalcoefficients for two compositions of the acetic acid/sodium acetate/water system are listed in Table I1 and compared with their predicted values. The agreement is good for DZl and D22,but poorer for Dll and D12. The trend toward large negative values of D12as f l increases is observed, and the increase in DZ2 with f l is accurately predicted. Although the solutions are dilute, less than 1%of the acid is ionized and the molar conductivity of the acid is small compared to that of the salt. As a result, it is more difficult to measure precisely flows of acid by the conductimetric technique. For this reason the salt transport coefficients are determined more precisely than those of the acid (see Table 11). Also listed in Table I1 are transport coefficients calcu[ated wi$h the electrophoretic and time of relaxation terms l i i and li{ included.61~These terms were evaluated at ionic strength acl + c2 by using the equations of Onsager, FUOSS, and Chen.% At the low concentrations of this investigation the electrophoretic and time of relaxation contributions to the transport coefficients are very small. Therefore, the approximation made by neglecting these effects [eq 61, and the resulting estimates of the transport coefficients, should be reliable. With the practical difficulties of the experiment in mind, the agreement between theory and experiment shown in Table I1 is satisfactory. 4. Discussion
S1/S2, k.e ratio of conductivity increments, was evLJated by using Onsager and Chen's mixed electrolyte conductivity equation at ionic strength aEl + E ~ . ~ ~ Typical data and fitted curves are shown in Figure 4. An iterative two-parameter fitting procedure27was used. (26) M. S. Chen and L. Onsager, J. Phys. Chem., 81, 2017 (1977). (27) H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry", 2nd ed, D. Van Nostrand, New York, 1956, p 517.
The procedure used in this paper to predict the diffusive properties of a simple buffer can be applied to systems containing larger numbers of ions and ion aggregates. The general procedure is as follows. First, the major species and association reactions are identified. Then, using lik = 8ik~iLi,, we developed approximate expressions for the L i k coefficients in terms of the concentrations of the species and their mobilities. If the mobilities and the association constants are known, numerical estimates of the L i k coefficients can be made. If activity data for the solutes are available, or can be estimated, diffusion coefficients can also be predicted. The results will reflect the electrostatic coupling between flows of charged species that is required to maintain zero flow of current, and the coupling resulting from association equilibria among the species. Interactions between flows of ions and neutral species (not connected by an association equilibrium) are neglected, as are other higher order effects such as the
J. Phys. Chem. 1981, 85, 1762-1767
1762
electrophoretic and time of relaxation corrections. Many methods are available for the determination of ion association constants.28 Among them are emf, conductimetric, spectroscopic (including NMR), solubility, and sound absorption techniques. The mobilities of simple ions may be obtained from conductivity and transference experiments on nonassociating or weakly associating systems. While it is more difficult to obtain the mobilities of ion aggregates, under favorable circumstances they can be estimated by fitting binary diffusion data of the associating electrolyte to an equation containing the mobility of the ion aggregate as an adjustable parameter. In this manner, (28) Robinson and Stokes, ref 24, Chapter 14.
mobilities of molecular organic acidd5J6@and metal sulfate ion pairsl8J9 have been determined. Thus with a few well-chosen binary experiments and some thought it should be possible to make a first approximation to the diffusive properties of complex multicomponent systemsB for which direct measurement may be difficult or not feasible. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. (29) The approximation = 6&Qi is a practical starting point for the description of diffusion of associating nonelectrolyte mixtures. For example, see (a) V. Vitagliano, J. Phys. Chem., 74,2949 (1970); (b) H. Kim., J. Solution Chem., 3, 271 (1974).
Electrostatic Potential of Bilayer Lipid Membranes with the Structural Surface Charge Smeared Perpendicular to the Membrane-Solution Interface. An Extension of the Gouy-Chapman Diffuse Double Layer Theory Gregor Cevc, * Institute for Biophysics, Faculty of Medicine, and J. Stefan Institute, E. Kardelj University Ljubljana, 6 1000 Yugoslavia, and Max-Planck Institut fur biophysikalische Chemie, Spektroskopie, D-3400, Gottlngen-Nikolausberg, Federal Republic of Germany
Saga Svetina, and BoZtjan Zek6 Institute for Blophyslcs, Faculty of Medicine. and J. Stefan Institute, E. Karde! University Ljubljana. 6 1000 Yugoslavia (Received: January 6, 198 I; In Final Form: February 27, 198 1)
An extension of the Gouy-Chapman diffuse double layer theory for bilayer (lipid) membranes is proposed, which takes into account the distribution of the structural charge perpendicular to the surface of the membrane. The linearized form of the Poisson-Boltzmann equation is solved for an arbitrary distribution of the structural charge normal to the surface plane, but numerical solutions of the corresponding nonlinearized form show that this linearization does not greatly affect the conclusions drawn from the model introduced. For the case of the exponential charge density profile perpendicular to the membrane surface, the explicit expressions for the electrostaticpotential and the electric field of the membrane are derived. It is demonstrated that the electrostatic properties of the membrane are little sensitive to the precise shape of the charge distribution function, but depend strongly on the average transverse displacement of the structural charge from the membrane surface plane, d. Because of the structural membrane charge smearing perpendicular to the membrane surface, the electrostatic potential profile becomes shallower as d increases, and a region of relatively small electric field is created near the membrane-solution interface. It is concluded that the distribution of the structural charge perpendicular to the membrane-solution interface markedly affects the electrostatic properties of the membrane either if the electrolyte solution is sufficiently concentrated or if the average transverse displacement of charge is sufficiently large. In the physiological ion concentrations range (c = 0.15 mol/L) the electrostatic surface potential decreases by a factor of 0.2 for d = 0.2 nm when compared to d = 0.
Introduction Some membranes carry structural surface charge which arises from the presence of ions on lipids, lipopolysaccharides, or proteins. When such membranes are in electrolyte solution, the surface charges attract oppositely charged counterions from the solution, while repelling co-ions of the same charge. Thus, a layer of unequal positive and negative ion concentrations in thermal equilibrium is formed at the membrane-solution interface. Gouy and Chapman, who first discussed the existence of such ion distribution, have termed this layer the diffuse double layer.lv2 The structural ions and the ions of the solution give rise to the electrostatic potential of the membrane which de(1) G. GOUY,J. PhyS., 9, 457-68 (1910). (2) D. L. Chapman, Philos. Mug., 25, 475-81 (1913). 0022-3654/8 1/2085-1762$01.25/0
pends on the number as well as on the spatial distribution of the charges. Calculations of mutual electrostatic interactions have usually assumed either that the structural ions form the surface of uniform charge density (uniform charge density model3) or that they are discrete and ordered in a perfect two-dimensional Recently, Tsien has further improved the discrete charge model by developing the virial expansion calculation.' Many experimental results have now been gathered which support the view that some properties of the bilayer membranes are influenced by the electrostatic potential at the membrane. Such is the case with regard to mem(3) S. McLaughlin, Curr. Top. Membr. Trunsp., 9, 71-144 (1977). (4) D. C. Grahame, 2.Elektrochem., 62, 264-74 (1958). (5) R. H. Brown, Prog. Biophys. Mol. Biol., 28, 341-70 (1974). (6) R. Sauv6 and S. Ohki, J . Theor. Biol., 81, 157-79 (1979). (7) R. Y. Tsien, Biophys. J., 24, 561-7 (1978).
0 1981 American Chemical Society
