2504
J. Phys. Chem. 1981, 85,2504-2511
Time-Dependent Cell Potential and Single-Ion Activity Coefficients for a Concentration Cell with Liquid Junction John H. Leckey and Frederick H. Horne” Depattment of Chemistry, Mlchlgan State University, East Lanslng, Mlchlgan 48824 (Recehred: August 1, 1980; In Flnal Form: Aprll 13, 198 1)
The time range of the dependence of the cell potential on single-ion activity coefficients has been delineated for several Ag/AgCl concentration cells containing a single chloride electrolyte. This delineation has been accomplished by using an analytical solution to the linearized system of partial differential equations derived from the equations of conservation of mass, the Onsager flux equations, and the Poisson equation, for ion transport in a concentration cell. A simple formula is obtained for the cell potential as a function of time, and in terms of Onsager coefficients, ionic valences, and the permittivity. Numerical solution of the nonlinear equations verifies the analytical solution and leads to an empirical formula for the coalescence time, which is the time in an experiment when single-ioneffects become undetectable. The coalescence time appears to be about twice the dielectric relaxation time TD. The results have two major implications: (1)they can be used as guidelines for choosing appropriate electrolytes, time domains, and boundary conditionsfor future experiments to determine single-ion activity coefficients; (2) they suggest complicated interdependences between electrical transport processes, on the one hand, and single-ion activity coefficients, on the other.
I. Introduction The significance of single-ion activity coefficients has not diminished despite their persistent experimental unattainability. Conwayl has listed several areas where information about single-ion activity coefficients and other individual solvated ion properties is required for further interpretation of observed phenomena. Among the more important areas are ion binding at proteins, specificity of ion-solvent interactions, and kinetics of electrode processes. Buck2 has emphasized the dependence of interfacial potential on single-ion activities. Frank3* and Goldberg and Franksb have reviewed the importance of single-ion activities and the apparent futility of their experimental determination. Following Goldberg and Frank,3b we have investigated concentration cells with liquid junction with the aim of determining the feasibility of measurement of single-ion activity coefficients. We report here both analytical and numerical results which delineate the time intervals during a concentration-cell experiment in which single-ion activity effects might be measured. A concentration cell can be made by bringing into contact two solutions of different electrolyte concentrations and containing identical electrodes. The cell emf (Ece“= C#JR- &, where r$ is the electric potential) that results will depend on the electrode potentials and the charge distribution near the region of contact between the two solutions. At t = 0, when the solutions make contact, there is no separation of charge in the liquid-liquid interface region. After the solutions make contact, the ions will begin to diffuse at varying rates. This creates a charge distribution, which produces the liquid junction potential. If the junction is formed from two solutions containing the same binary electrolyte, the junction potential, and hence the cell potential, obtains a constant value very rapidly. Eventually the cell potential will decay to zero as the entire cell reaches equilibrium. The plateau region begins when the liquid junction potential has risen to its maximum (1) B. E. Conway, J. Solution Chem., 7,721 (1978). (2) R. P. Buck, Cn‘t. Rev. Anal. Chem., 5, 323 (1976). (3)(a) H. S.Frank, J. Phys. Chem., 67,1554(1963);(b) R.N. Goldberg and H. S. Frank, ibid., 76,1758 (1972). 0022-3654/8 1/2085-2504$0 1.25/0
value and ends when diffusion begins to decrease significantly the electrolyte concentrations at the electrodes. During this time period the cell potential is constant. The relationship between plateau-region cell potentials and activity coefficients, for concentration cells with liquid junctions, has been described theoretically since Guggenheim.4 Bearmans derived the relationship from rigorous nonequilibrium thermodynamics. For the plateau region, most electrochemistry texts6 state that the cell potential for concentration cells containing a single electrolyte is independent of the activity coefficients of the individual ions and depends only on some mean activity coefficient. Although the time regions before and after the plateau region are not often discussed, MacGillivray’ has used singular perturbation techniques to obtain analytic solutions to the time-dependent Nernst-Planck equations for ion flux across a membrane under voltage-clamp conditions. Our results for the liquid junction potential do not rely on the (incorrect8) Nernst-Planck equations. Hickmang used singular perturbation techniques for investigating the junction potential created by diffusing electrolytes. His approach is not directly applicable to electrochemical cells because he used boundary conditions at infinity. Goldberg and Franksb also attacked the question of concentration cell potential dependence on single-ion activities. Their approach was entirely numerical, however, and they were mainly concerned with the plateau time region. They report early time behavior s) for a particular KC1 concentration cell which exhibits significant cell potential dependence on single-ion activities up to lo4 s, when significant effects vanish. Our principal goal here is to relate the liquid junction potential, as well as the full cell potential, to single-ion activity coefficients in a time-dependent analytical equa-
-
(4)E. A. Guggenheim,J.Phys. Chem., 33, 842 (1929). (5)R. J. Bearman, J. Chem. Phys., 22, 585 (1974). (6) (a) D.A. MacInnes, “The Principles of Electrochemistry”,Reinhold, New York, 1939; (b) J. O’M. Bockris and A. K. N. Reddy, “Modern Electrochemistry”,Plenum Press, New York, 1977. (7) A. D.MacGillivray, J. Chem. Phys., 52, 3126 (1970). (8)F. H.Horne in “Physics of Superionic Conductors and Electrode Materials”, J. W. Perram, S. deleeuw, and B. C. H. Steele, Eds., Plenum Press, New York, 1981. (9)H. J. Hickman, Chem. Eng. Sci., 25, 381 (1970).
@ 1981 Amerlcan Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 17, 1981 2505
Time-Dependent Liquid Junction and Cell Potentials
tion. This has the advantage that the final equation specifically reflects the effect that the input parameters have on the final result. In our case the input parameters are activity coefficients, Onsager coefficients, and ionic valences. Our starting equations are the equations of conservation of mass for the positive and negative ions, the Poisson equation, and the Onsager flux equations. The disadvantage of the analytical solution approach is that it is rarely possible to solve the equations exactly, which is the case here. We have therefore also solved the problem numerically, both to obtain accurate results and to check the validity of the approximations made in obtaining the analytical solution. Mean ionic activity coefficients, y a , have been experimentally determined and tabulated.1° For single-ion activity coefficients y + and y-, y + is defined by y+” = y+W+y-.”-
(1.1)
where u+ and u- are the positive and negative stoichiometric coefficients, and u = v+ + u-. Since y a is known, y + and y - may both be calculated if another independent relationship between them is known. The mean ionic activity deviation, defined by Frank,3a is (1.2) Thus, y + and y - can be expressed in terms of 6+ and y a by 6+V
= y+w+/y-y-
y + = (y+g,)~/(Z~+) y - = (y*/~*)~/(2”-) (1.3) Following Goldberg and Frank,3bwe assume that 6* depends on concentration according to In 6+ = BI (1.4) where B is a constant and I is the ionic strength
I = (1/2)(c+z+2+ cz-2) (1.5) where ci is the concentration of ion i and zi is its charge number. With the help of eq 1.4 we solve the transport equations for the cell potential as a function of time and the single adjustable parameter B. One of our goals is to determine whether there is a time range in which the cell potential depends strongly enough on B to determine y + and y experimentally. Our analytical expression for the liquid junction potential has the form EJ = tPs exp(-St) (1.6) for short times. Here EJ is the liquid junction potential, t is time, Ps is the preexponential factor, and S is the short-time time constant. Ps depends on single-ion activity coefficients, but S does not. During this time range the full cell potential depends on single-ion activity coefficients. A t s the expression for the liquid junction potential becomes Ej = PM (1.7) where PM depends on single-ion activity coefficients, but not on time. In this time interval the single-ion activity coefficients do not, however, appear in the expression for the full cell potential. At much longer times, the liquid junction potential is EJ = PL exp(-lt) (1.8) where PL depends on single-ion activity coefficients, but L does not. Single-ion activity coefficients do not appear in the expression for the full cell potential.
-
(10) G.N.Lewis and M. Randall, “Thermodynamics”, McGraw-Hill, New York, 1961.
We find analytically, as others have numerically,3bJ1that cell potential dependence on single-ion activity Coefficients vanishes very rapidly e) for concentration cells operating with a single binary electrolyte. We extend previous results by quantifying this effect for a series of chloride electrolytes. Our calculations for these electrolytes suggest interesting and complicated interdependence between electrical transport processes, on the one hand, and single-ion activity coefficients, on the other. 11. Transport Equations
Our fundamental equations are those of nonequilibrium thermodynamics.12 The systems which we are dealing with are isotropic, nonreacting, isothermal, binary electrolyte-water mixtures not subject to magnetic or gravitational fields. We assume that all concentration and electric field gradients are in one direction and that there is no flux of any kind through the walls of the cell. The mass conservation equations for the positive and negative ions are d c + / d t = -(d/dx)(J+H + c+uo) (2.1) d c _ / d t = -(d/dx)(J-H + c-u,) where t is time, x is the space coordinate, c+ and c- are the positive and negative ion concentrations in mol dm-3, J+H and J-H are the Hittorf diffusion fluxes, and uo is the velocity of the solvent (water). The Onsager equations which relate the fluxes to electrochemical potential gradients are -J+H = l + + ( d p + / d x ) + Z+-(dp-/dx) (2.2) -J-H = l - + ( d p + / d x ) + L ( d p - / d x ) where the 1, are the Onsager coefficients and p+ and p- are the electrochemical potentials of cation and anion P+ = P+p-
= p--
+ R T In (c+Y+) + z+F4
+ R T In ( c y - ) + z_F$
(2.3)
where pimdenotes the infinite dilution standard state, R is the gas constant, T is the temperature, F is Faraday’s constant, and 4 is the electric potential. The chemical potential derivatives of eq 2.2 in the variables electric field, concentration, and ionic strength are dp+/dx = ( R T / c + ) ( d c + / d x )+ R T [ u / ( ~ u + ) ] (+MB)(dI/dx) + z + F ( d 4 / d x ) dp-/dx
+
= (RT/c-)(dc-/dx) R T [ u / ( ~ u - ) ] (-MB ) ( d I / d x ) + zJ’(d@/dx) (2.4)
where
M
In Y + / ~ ~ ? T , P (2.5) An additional relation between electric potential and concentration is provided by Poisson’s equation (2.6) where t is the permittivity and p ( x ) is the charge density p ( x ) = F(z+c+ + Z-C-) (2.7) Thus d24/dX2 = -(47r/+(x)
d 2 4 / d X 2 = (-4?rF/t)(z+c+ 4-
(2.8)
Combining eq 2.1, 2.2, and 2.4, we find (11) D.R. Hafemann, J. Phys. Chem., 69,4226 (1965). (12) R. Haase, “Thermodynamics of Irreversible Processes”, Addition-Wesley, Reading, MA, 1969.
2506
Leckey and Horne
The Journal of Physical Chemistry, Vol. 85, No. 17, 1981
&+/at = @/ax)
x
+ a+-(dc-/ad + P + E ( d @ / d d - c+uol
[.++(ac+/dx) dc-/dt =
+ a - - ( d C - / a X ) + PL&#J/dx)
(a/ax)[a-+(ac+/ax)
(2.18)
~TFz+z-/E
(2.19)
with K
- c-uo] (2.9)
= -KD
a2b/aX2
E
a++/RT = (l++/c+)f/4(z+/z-)(z+- z-)[z+l++(M+ E ) - z-l+-(M - E)]
Thus, once D(x,t) is obtained, eq 2.18 can be integrated to determine the junction potential EJ = bR - &. The differential equations for S and D are, from eq 2.8 and 2.16 as/at = (elax) x [~ss(aS/ax)+ Y S D ( ~ D / ~+XYSJ&-'(&/~X) ) - SUO]
a+-/RT = ( l + - / ~ - )-
dD/dt = @ / a x ) X
where /3+E = F(z+l+++ ~-l+-)
0-E
F(z+l-+ + 2-l-J
(2.10)
+
'/,(z-/z+)(z+ - Z-) [z+l++(M E ) - z-l+-(M - E ) ]
[Y~s(as/ax)+ Y D D ( ~ D / ~+XYDEK-'(~~/~X) ) - Duo] (2.20)
a-+/RT = (L+/c+)-
+
&(z+/z-)(z+ - z-)[z+l-+(M E ) - z-l--(M - E ) ]
a--/RT =
where Yss = '/z(a++ + a--)- 1/2[(z+/z-)a+-
+
(L-c-) - Y~(Z-/Z+)(Z+
- ~-)[z+l-+(M B ) - z-l--(M - B ) ]
(2.11)
YSD
The stoichiometric coefficients and ionic valences are related by v- = z+[v/(z+- z-)I v+ = -z-[v/(z+ - 2-11 (2.12)
YDS
which are easily obtained from v+ + v- = v v+z+ + vz-= 0
= 0 = (d@/dX)a/P,t
(2.14)
where a is the length of the cell and the electrodes are at (a/2) and -(a/2). The gradients of composition vanish at the walls because the fluxes vanish there; the electric potential gradient vanishes at the walls because the walls are not charged. The initial conditions are c+(X,o)/v+ = c - ( X , o ) / v -
c+(x,O)/V+ = c-(x,O)/V-
< z < a/2 - ~ / 2< x < 0 (2.15)
= sRo
= SLo
0
where SRoand SLo are the electrolyte concentrations in the two cell compartments at the beginning of the experiment except for 2-2, 3-3, etc., electrolytes. If there were no charged regions in the electrolyte solution, the electric potential gradient would vanish everywhere, and c+ and c- would be related everywhere by z-c= -z+c+. In that case, eq 2.9 would be replaced by a single equation for the salt concentrations S, and the only transport process would be ordinary diffusion. When there is a charged region, ordinary diffusion will still be the dominant process, and its effects will dominate the solutions of eq 2.9. In order to separate the mass diffusion effects from the electromigration effects, consider the sum (S) and difference (D) functions (2.16)
Then C+
= z-(D - S) p
and eq 2.8 becomes
C- = z+(D = 2z+z_FD
+ S)
YSE
= -1/2(a++- a--)- '/Z[(z+/z-)a+- - (z-/z+)cu-+] -72(a++-
=
1/2(a++
a--)+ l/z[(z+/z-)a+- - (z-/z+)a-+]
+ a--)+ 1/2[(z+/z-)a+- + (z-/z+)cU-+] - Z+B+E) =
= (4TF/&B-E
-(~TP/E)(z+'~++ - zPL-) (2.13)
The quantity v/(z+ - z-) is unity except for 2-2,3-3, etc., electrolytes because in those cases v- # z+ and v+ # 12-1. The boundary conditions for eq 2.8-2.9 are i = +, ( d ~ ~ / d ~ ) - ,=/ ~0 , =~ (dci/dx),/2,t (a$/dx)-a/2,t
YDD
+ (z-/z+)a-+]
?'DE
= ( ~ T F / E ) ( ~ - ~ -z E +/3+~= ) 22+2-1+- + 2?1--) = (~T/E)X = TD-~ (2.21)
+
(~TF/E)(Z+~~++
where X is the specific conductance13and TD is the dielectric relaxation time.14 The rather long expressions for 'ym, ^/sD, YDS, and Y D D in terms of the lu, M, and E appear in Appendix A. In eq 2.21 and throughout this work, l+- = I-+. The boundary conditions for eq 2.20 are (dS/dx)-a/z,t = 0 = (dS/dx)a/z,t
(2.22)
(dD/ax)-a/,t = 0 = (dD/ax)a/,t
The initial conditions are S(X,O) SLo s(X,o) = SRO
D(x,O) 0 D(X,o) = 0
-Y~u < x < 0 0 6 a 2 / ~ s(Le., s for t 2 lo5 s for a l-cm cell), to better than 1% accuracy
&(t) = -(4/7dK(YDS/YDE)(SR0
- SL') exp(-R+lt)
(4.7)
with
R+i
( ~ / a ) ~ ( r s+s700) = ( * / ~ ) ~ ( a+ + a--) +
(4.8) (iii) For 5/yDE< t < 106a2/yss(i.e., for lo4 5 t 5 1s), to better than 1% accuracy, the liquid junction potential is independent of time
EJ = -K(YDs/YDE)(SR' - SL') (4.9) The expression for the lower time limit of eq 4.9, 5/yDE, is particularly important. 5/yDEis a measure of the onset of a constant liquid junction potential. Note from eq 2.21 that YDE is proportional to the conductance, A, and that 5/rDE = 54(4?rA) = 5q,. In the next section it will be shown that yDEis also directly related to the time interval in which the cell potential depends on single-ion activity coefficients. From Appendix A, eq 4.9 becomes, for 1-1 electrolytes and for 5/yDE < t < 105a2/yss. EJ = -(RT/IF)(SR' - SL')[(t+ - t-)(l + MI)+ (4.10) Note that (1+ MI) = [l + (a In y+/a In I)Tp].
V. Full Cell Equation Results The liquid junction potential is, of course, not measurable by itself. Any measurement will also include potential differences at the two electrode-solution interfaces. For definiteness and simplicity we calculate the full cell potential for the following single electrolyte concentration cell:
Ag(s)lAgCl(s)lMC1(SL,aq)lMC1(SR,aq)lAgCl(s)lAg(s) (A) The cell potential is
JL&)= EN+ EJ
(5.1)
Explicit equations for c+ and c- are easily found by combining eq 4.1 according to eq 2.17.
where
(15)(a) F. H. Home and T. G. Anderson, J. Chem. Phys., 63,2321 (1970); (b) S. E.Ingle and F. H. Horne, ibid., 59, 5882 (1973).
EN = (RT/F)[ln (SR/SL)+ In (Y-R/Y-L)I (5.2) and EJ is given by eq 4.5. S R and S L are the electrolyte
2508
The Journal of Physical Chemistry, Vol. 85, No. 17, 1981
Leckey and Home
111. It is more convenient to use the electric field, E , as a variable rather than the electric potential, 4. The relationship is sRo/ SLO/ E = -(&$/ax) (6.1) (mol (mol cell dm-’) dm-3) 1O3E,dC~/V lO3EeXp~/V For purposes of this computer simulation, we replace the Poisson equation, eq 2.18, with the displacement current KCl 0.10 0.040 21.16 21.17‘ equation at zero Faradaic current16J7 - 80.37 - 80.42’ 3.00 KCl 0.10 -4.13 -4.05b 0.080 NaCl 0.10 (dE/dt) = -(47~/e)(Z+J+~ Z-J-H) = -16.84b 0.040 -17.09 NaCl 0.10 9.94 9.95‘ 0.10 0.078 HCl K [ y D S ( a S / d x ) e r D D ( d D / d x ) - YDEE] (6.2) 36.21’ 0.040 35.39 HCl 0.10 where we have used eq 2.2, 2.4, 2.17, 2.18, 2.19, and 6.1. 103.4 118.aa HCl 0.10 0.005 Derivatives of the form (a/&) [a,(dU/dx)] were evalu’Reference 20. Reference 21. ated by using successive applications of a first derivative central difference analog. Values of the transport coefconcentrations at the right- and left-hand electrodes. ficients as functions of composition were calculated from Strictly speaking the chloride ion concentrations should the information in Miller’s arti~1e.l~Because the debe used, but SR/SL is equivalent to [C1-]R/[cl-]L. y - and ~ pendent variables S, D, and E change rapidly near the y-L are the chloride ion activity coefficients at the rightinterface ( x = 0) at short times, it was desirable to use a and left-hand electrodes. It is straightforward to put eq nonuniform space mesh similar to that used by Brumleve 5.2 in terms of y+ and B. By eq 1.3 and 1.4 and Buck.17 The grid points were 10% units apart near the interface and gradually expanded to 10-la units apart EN = -(RT/F)[ln (SR/SL)+ In (Y+R/Y+L) - B(SR- S L ) ~ at the electrodes. Ninety space grid points were used. Time steps of 10-lo s were used initially. The step size was (5.3) increased as the cell approached equilibrium. Since the where we have used the fact that I = S for cell A (1-1 partial differential equations were nonlinear, the resultant electrolyte). The results are not changed significantly for finite difference algebraic equations were nonlinear. Beother electrolytes. cause of this it was necessary to use a Newton-Raphsonla Combining eq 5.3, 5.1, and 4.5, and simplifying the iteration procedure to help solve them. The software preexponential factors using the expressions in Appendix programs used in solving the resultant band matrix A, we find equations were developed by the authors of “Linpack”.lg Numerical stability was checked by comparison of reEceu = -(RT/F)[ln (SR/SL)+ 1n (Y+R/YiL) - B(SRsults obtained by using different step sizes. The numbers of spatial and temporal grid divisions were varied by SI,)] + (4/r)KrDS(SR0 - s L o ) (-1Ik[(2k k=l factors of 3 and 10, respectively, without change in result. l)Qkl-l[exp(-R+d) - exp(-R+kt)l (5.4) For cell A operating at concentrations of S R = 0.11 mol dm-3 and SL= 0.09 mol dmT3,nonlinear computer simuNote that SR,SL,y+R, and yiL will depend on time only lation results and analytical solution results agreed to after diffusion has begun to change the electrolyte conwithin 5% for all time ranges and all electrolytes. Since centrations at the electrodes. the only approximation in the computer simulation is The implications of eq 5.4 are clearest during the time neglect of the solvent velocity, 5% is a good estimate of periods when eq 4.6,4.7, and 4.9 are applicable. We find the accuracy of the analytical solution for cell A operating that Eceudepends on B for t C (47DE)-’, while Eceuis inas indicated. dependent of B for t > (5/yDE). For (5/7DE) C t C Figure 1shows the numerically predicted charge density (106a2/yss), the cell potential equation can be expressed, surface near the liquid-liquid interface for a NaCl conafter considerable algebra, in the particularly simple form centration cell with an assumed value for B of zero (y+ = y-), Figure 2 shows the same cell at three different times. Ece” = (-2RTt+/F)[ln (SRO/SLO) -I-In (Y~R/?’+L)](5.5) As time increases, the maximum value of the charge denThis is the classical electrochemical cell potential derived sity decreases and moves away from the interface. by Guggenheim4assuming constant transference numbers Figure 3 is the same cell at lo-* s for three values of B. across the cell. As can be seen, changing B changes the magnitude of the Table I shows a comparison between eq 5.5 and some curve but does not change its shape. This corresponds in experimental results for concentration cells operating with our analytical solution to B appearing in the preexpoliquid junctions and moderate electrolyte concentration nential factor but not in the time constants. differences. The difference are quite small as long as the Figure 4 is a graph of cell potential vs. time for cell A cell electrolyte concentration differences do not become with M = Na, S R = 0.11 mol dm-3, and SL= 0.09 mol dm-3. too large. Although these cell potential comparisons are We choose the criterion that the cell potential no longer all in the constant cell potential time interval, they give depends on B when the E d vs. time curves for B = 1dm3 us some idea of the error induced by our linearization for mol-l and B = -1 dm3mol-’ differ from the plateau-region longer and shorter times. In the next section we describe the numerical technique (16) H.Cohen and J. W. Cooley, Biphys. J.,6, 145 (1966). used to check the validity of the analytical solution for (17) T.R. Brumleve and R. P. Buck, J. Electroanul. Chem., 90, 1 (1978). times less than 5/rDE = 5 7 ~ . (18) R.Camahan, H. A. Luther, and J. 0.Wilkes, “Applied Numerical Methods”, Wiley, New York, 1969. VI. Numerical Solution (19) J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart, “Linpack”, SIAM, New York, 1979. In an effort to validate the analytical results at short (20) T. Shedlovsky and D. A. MacInnes, J. Am. Chem. Soc., 69,503 times, we have solved the system of eq 2.20,2.22, and 2.23 (1937). numerically, with some slight modifications. The uo term (21) A. S. Brown and D. A. MacInnes, J. Am. Chem. SOC.,67, 1356 (1935). was omitted from eq 2.8 for the reason discussed in section TABLE I: Comparison of Experimental Cell Potentials (EexpU)for Cell A with Cell Potentials ( E c d d ) from Eq 5.5
5
The Journal of Physical Chemlstry, Vol. 85, No. 17, 198 1 2509
Time-Dependent Liquid Junction and Cell Potentials
Flgure 1. Predicted charge density surface near the Ilquld-liquid Interface at short time for a NaCl concentration cell operating between concentrations of 0.11 and 0.09 mol dm4. B Is zero for this simulation ( y + = y-). The time row at t = 0 has been omltted since it would obscure the lower portion of the surface.
-8
-4
-6
-2
2
0
4
6
8
-5,0Y
-5.5L
x / I0-8,
Figure 2. Constant time planes from Figure 1: (a) 5 X lO-'s; (b) 1 x 10-8 s; (c) 2 x 10-8 s.
-24
I
I
1
1
I
1
1
I
I
1
I 10 t / 10-9s
I
0.5
I
I .5
2.0
Figure 4. Cell potential vs. time for cell A with M = Na, S, = 0.11 mol dm-3, and SR = 0.09 mol dm-3: (a) B = 1.00 dm3 mol-'; (b) B = 0; (c) 8 = -1.00 dm3 mol-'.
241 I
I
1 1
.*
20
C
24
I
I
I
W I
-0 -6 -4 -2
I
I
I
I
I
0
2
4
6
8
x / I 0-8,
Figure 3. Charge density vs. cell position for dlfferent values of Bat t = 1 X lo-' s for a NaCl concentration cell operating between concentrations of 0.11 and 0.09 mol dm3: (a) 8 = -1.00 dm3 mol-'; (b) B = 0; (c) B = 1.00 dm3 mol-'.
cell potential by less than 5%. With this somewhat arbitrary criterion, Figure 4 indicates that, from a time of 1.6 X s on, this cell potential no longer depends on B. Figure 5 shows a graph of the coalescence time (t,) vs. YDE for several electrolytes. VII. Conclusions The cell potential dependence on single-ion activity coefficients vanishes very fast ( t < 10" s) for the systems which we have investigated. From Figure 5, it is clear that YDE is a good indicator of this effect for an arbitrary electrolyte. For the electrolytes studied, the relationship t, = ~ . S / Y D E= 1 . 9 ~ / ( 4 ~=X )1 . 9 ~ ~ (7.1)
4t I
I
4
I
8
I
I
12
16
I
20
1
1 1
24
TOE / 108s-I
Flgure 5. Coalescence time vs. yDEfor 0.1 mol dm-3 solutions, with 5 % criterlon of section VI: (a) HCI; (b) BaCI,; (c) KCI; (d) NaCI; (e) LiCI.
approximates the coalescence time to within 5 % . Our results can be interpreted as follows. Initially in the interfacial region the electrical potential gradient is zero and the activity gradient is infinite. Consequently the positive and negative ions begin to diffuse from the region of high activity to the region of low activity. Since the aij values are not equal, the diffusion begins at different rates for the two types of ions. This soon creates a separation of charge and an electrical force that pulls the positive and negative ions back together. Thus a steadystate balance of diffusive and electrical forces is estab-
25 10
The Journal of Physlcal Chemistry, Vol. 85,No. 17, 198 1
lished. Before this balance is established, the cell potential depends on the positive and negative ions separately. After this the forces and hence the ions themselves are coupled in such a way as to remove their individual identities. Buck2 has similarly described the behavior of the junction potential in terms of differential ion motion. This model is consistent with the expression for yDE (the last of eq 2.21)
+
YDE= (47rP/t)(2+21++ 22+2-1+-
Leckey and Horne
way to determine the feasibility of this experiment.
Acknowledgment. Support of J.H.L. in 1979 through a General Electric Foundation-Michigan State University Summer Fellowship is gratefully acknowledged. Appendix A. The yij in eq 2.20 and 3.3 are
+ z-~Z--)
= (4*/t)A =
YSS TD-~
(7.2)
Large values of I++ or l-- (HC1for instance) indicate rapidly diffusing ions, large values of yDEand A, small TD, and consequently a rapid vanishing of single-ion effects. The same is true for large values of z+ and z-. For BaC12,the z + factor ~ offsets the relatively small values of Z++ and I--, which leads to relatively large values of yDEand A, small TD, and a small coalescence time. In terms of our model, the valence number of +2 for Ba2+creates a larger electrical force sooner. This apparently overwhelms the tendency of Ba2+to diffuse slowly, and thus the coalescence time is relatively fast. For large values of t, yDEis small, TD is large, and the single-ion effect time is prolonged. This is consistent with the idea that a large value of E indicates that the positive and negative ions are relatively independent of one another. Because we are interested in such short times (ls). The equations to be solved to analyze this experiment are much more complex than those described in this paper, and a useful approximate analytical solution may not be possible. Preliminary calculations, using a numerical approach, are now under(22) H. S. Harned and B. B. Owen, ‘‘Physical Chemistry of Electrolyte Solutions”,Reinhold, New York, 1950.
= [RT/(2~+2-)1(ae- bg) = [RT/(2z+z-)l(-af
- bh)
yDS = [RT/(2z+z-)](-ce
+ dg)
YSD
YDD = [RT/(2z+z-)l(cf
+ dh)
(A.1)
with
-ZL+) b = (z+/c-)(z+~+-- 2 J - J
a = (Z-/C+)(Z+l++
+ 2L+)
c = (z-/c+)(z+l++
+ zL-) e = 1- $$(Z+/Z-)(Z+ - zJ2c+(M+ B ) f = 1- ~ / ~ ( Z + / Z _ ) ( Z +-~z?)c+(M + B ) d = (z+/c-)(z+~+-
g = 1 - 1/4(z-/z+)(z+ - 2-)2C-(M - B )
h = 1 + 1 / 4 ( 2 - / 2 + ) ( ~ +-~z-~)c-(M - B)
(A.2)
For 1-1 electrolytes a1
bl c1
= -c+-’(Z++ +) ,2 = c--’(l+-
+ 1.J
= --c+-l(l++ - L+)
dl = c--l(l+- - 1--)
el = 1
+ c+(M + B )
fl = 1 g , = 1 + c-(M - B)
hi = 1
(A.3)
and in that case
Yss = 1/2R7I[U+++ l-+)/C+l
+ [U+-+ Z--)/C-l + (Z++ +
21+- + L ) M + (I++ - I-JB)
J. Phy~.Chem. 1981, 85, 2511-2519
YDS/YDE
= [eRT/(8rWl[(t+ - t-)(l+ MI)+ BT] (A.8)
Appendix B In order to solve eq 3.3 for S and D, we write them in matrix form (B.1) PU = 0 where P is the linear operator matrix and U is the solution vector. If W is defined such that DET(P) W = 0 (B.2) then it can be shown that Vi = COFji(P) W 03.3) where j can be any i (the cofactor can be expanded along any row). For our system
(a/at) - rss(az/ax2) -rsD(azlax2) + ~
S E
( a / a t ) - r D D ( a z l a x z )+ Y D E
(B.4)
The equation for W is
2511
{A(a4/dx4) + B[d3/(dt ax2)] + C(d2/dx2) + ( d 2 / d t 2 ) + G(d/dt))W= 0 (B.5) where
A = YSSYDD - YSDYDS B = -(YSS + Y D D )
c = YSflDS - YDflSS G = YDE 03.6) Since W, in this case, must be an odd function about x = 0, we expand the solution in a Fourier sine series m
W(x,t)= C bk(t) sin ( k r x / a ) k=1
03.7)
Substitution of eq B.7 into eq B.5 produces the ordinary differential equation bk”(t) + [G - (kr/a)2B]b’k(t) + [ ( k r / ~ )-~(k~/a)~C]bk(t) A = 0 (B.8) The solutions of eq B.8 are bk(t) = L+k eXp(-R+kt) + L-k eXp(-&t) (B.9) where R+k and R-k are given by eq 4.2, and L+k E d L-k are determined by using eq B.3 and the initial conditions. The result is eq 4.1.
Growth and Size Distributions of Cetylpyridinium Bromide Micelles In High Ionic Strength Aqueous Solutions GrQgolre Porte Centre de Dynamlque des Pheses Condens&s,t U.S.T.L. Montpllier, France
and Jacqueline Appell
*
laboratolm de Spectrom6trieRayb&h-Brlllouln, U.S.T.L. Montpelller, France (Received: February 13, 198 I; In Flnal Form: May 13, 1981)
The growth of cetylpyridinium micelles in high NaBr concentration aqueous solutions is observed by the use of quasi-elastic light scattering. The variations of the mean hydrodynamic radius, R H , are measured over wide M), and added NaBr ranges of temperature (27 < T < 75 “C),detergent concentration (1.5 X 10-9-40X concentration (0.2,0.4,0.6, and 0.8 M). These results are interpreted in the frame of a multiple equilibrium description of the micellar elongation. This now classical description is here rewritten in order to incorporate the counterion influence through the use of the crude ion binding approximation. We then developed a quantitative geometrical model for the elongated micelles which, from our previous studies, are known to have the shape of long flexible spherocylinders. The obtained relation between RH and the mean aggregation number ( N ) allows us to compare quantitatively the experimentalresults to the predictions of the theory. The agreement is found to be very good in the range of concentration of the detergent where the ideality condition stands. Furthermore, the ion binding approximation appears to be unexpectedly good in predicting the variations of the micellar size with the temperature and the NaBr concentration. In the range explored in this work, the true N-micelle apparently behaves like a chemical species the enthalpy and entropy of formation and the chemical composition of which are independent of the temperature and of the added salt concentration. It is, however, concluded that this convenient equivalency is not necessarily significant and rather indicates that the functional dependence of the elongation is probably not discriminative between detailed a priori descriptions of the ionic environment of the micelles.
Introduction In recent years, Mukerjee‘ initiated a simple theoretical description of the evolution of size and shape and partic‘Laboratoire Associ6 au Centre National de la Recherche Scientifique (LA 233). *Equipe de Recherche Associee au Centre National de la Recherche Scientifique (ERA 460). 0022-3654/81/2085-2511$01.25/0
ularly of the sphere-to-rod transition of micelles appearing in solutions of amphiphilic molecules. This description is given on the basis of a thermodynamic theory of multiple equilibrium (or equivalently of mass action law). Israelachvili et al.2 have &own that the sphere-to-rod transition (1)P. Mukerjee, J. Phys. Chem., 76, 565 (1972).
0 1981 American Chemical Society
