DENNISR. HAFEMANN
4226
Charge Separation in Liquid Junctions
by Dennis R. Hafernann Department of Chemistry, University of California, San Diego, L a Jolla, California 08058 (Received June 85,1966)
Liquid junction potentials are calculated by a computer simulation method. The charge distribution generated by the interdiffusion of two electrolyte solutions is computed explicitly and used to evaluate the liquid junction potential as a function of time. A rise time of about lW9see. is calculated for the potentials of junctions between ordinary salt solutions. The steady-state results agree with previous calculations which were made by assuming electroneutrality throughout the junction and performing a thermodynamic integration of transport numbers. The application of the simulation method to more complex systems, including biological membrane systems, is discussed.
Introduction Steady-state values for the electrostatic potential difference across variously constrained liquid junctions are calculable by several methods.l-1° I n some methods reversible thermodynamics is applied to an irreversible process, Such treatments can be expected to be valid only for the analysis of diffuse junctions on a long-time scale. Other methods treat the irreversible phenomena more or less correctly, but they depend on the assumption of electroneutrality throughout the junction. This assumption is self-contradictory because there can be no potential difference if there is no charge separation. To rationalize the electroneutrality condition, it has been assumed that the deviations from electroneutrality are so small that they have no effect on the result. No electrostatic charge distributions have previously been calculated to support this rationalization, The generation of an electrostatic potential between two initially electrically neutral solutions is the subject of the present work. The electroneutrality condition is eliminated, and transient charge distributions are calculated using a digital computer. The results show that the electroneutrality assumption is approximately valid for the time scales on which experiments are usually performed. A liquid junction which is formed from a sharp initial boundary and not subsequently constrained with respect to width is shown to give rise to a charge distribution that changes with time in such a way that a steady liquid junction potential results. The Journal of Phgsical Chemistry
Method Consider an experiment in which two uncharged solutions are initially brought together to form a planar junction of large area. I n the limiting case of infinite area, neglecting gravity and assuming no bulk flow or convection, the one-dimensional nature of the initial condition will persist throughout the experiment. The x direction is defined to be perpendicular to the plane of the junction. To construct a digital model, one subdivides the re1 “cells,” separated gion around the junction into rrz by planes parallel to the junction a t fixed intervals along the 5 axis. The cells are numbered in spatial sequence from 0 to m. The model is represented by an n(m 1) concentration matrix C
+
+
(1) H. von Helmholtz, “Wissenschaftliche Abhandlungen,” J. A. Earth, Leipzig, Vol. 1, 1882, p. 840; Vol. 2, 1895, p. 979. (2) W. Nernst, 2. physik. Chem. (Leipaig), 2, 613 (1888). (3) M. Planok, Ann. Phys. Chem., 39, 161 (1890); 40, 561 (1890). (4) P. Henderson, 2. phgsik. Chem. (Leipzig), 59, 118 (1907); 63, 325 (1908). (5) H. Pleijel, ibid., 72, 1 (1910). (6) P. B. Taylor, J . Phys. Chem., 31, 1478 (1927). (7) R. Schlogl, 2.physik. Chem. (Frankfurt), 1, 305 (1954). (8) H. Cohen and J. W. Cooley, Biophys. J., 5, 145 (1965). (9) S. R. de Groot, “Thermodynamics of Irreversible Processes,” North-Holland Publishing Co., Amsterdam, 1952, pp. 133-140. (10) D. T. J. Hurtle, J. B. Mullin, and E. R. Pike, J. Chem. Phys., 42, 1651 (1965).
4227
CHARGESEPARATION IN LIQUIDJUNCTIONS
where n is the number of distinct species (ions and neutral molecules) considered. It is assumed that each of the cells is homogeneous, the entire difference between cij and ci,j+l being displayed as a discontinuity 1. in ct(x) at the boundary between cell j and cell j The charge density p j in cellj is
+
Nonlinear force-flux relationships and coupling of fluxes can be treated by the method described here. Assuming constant temperature and pressure, the equation12for the flux J i of species i is
n
0)
=
Cm, i=l
(2)
where z( is the charge a’fspecies i. Let us consider all {of the charge in cellj to lie on a plane at the midpoint of the cell. The surface charge density adon this plane is aj =
Pld,
where d j is the thickness of cell j . The planes of charge are located xo, 21, . . ., xm, respectively. The surface charge densities are subject to the condition Ea, = 0
(4)
because the system as a whole has no net charge. Consider a point with coordinate x lying between . application of Gauss’ theoremll xk and z ~ + ~By one can show that the electric field intensity E at x is (5)
where E(X)is the permittivity of the medium, which is in general a function of x. The upper limit on the summation of (5) is chosen so that the sum includes only the charges lying to the left of point x. Formally k is determined by the inequality xk
< x < xk+l
where yi is the activity coefficient of species i, $J is the electrostatic potential, and 5 is the Faraday. For uncharged species
D1:
gi = -
..I
j=O
where ci is the concentration, 9%the mobility (si Lid/ci), and pi* the electrochemical potential of species i. A more detailed expression, neglecting the polariaability13of the component species, is
(6)
The potential difference V across the system is given by (see Appendix)
where d is the constaat cell thickness and E is assumed to be independent of x. For detailed computations, (A3) is a more general equation. To simulate the effect of elapsed time, the numbers in the C matrix must be changed according to an algorithm using the methods of irreversible thermodynamics. I n the present work, it is assumed that the flux of species i depends on the electric field intensity and the gradient of the activity of i at the point in question but that it does not depend on the gradient of the activity of any other species. This oversimplification is made to provide a simple test for the method.
RT
where Di is the solvent-fixed diffusion coefficient. For ionic species
where ti is the solvent-fixed transference number of species i in a solution of conductance A. In the present calculation, solvent flow is neglected. Both g, and yc are assumed for simplicity to be functions of total ionic strength only. Single-ion activity coefficients are estimated by use of the MacInnes assumption14 yCI-MC’ = yc,-KC’
= YfKC1
(12)
Since these activity coefficients are not thermodynamically measurable, 16-18 there use introduces an error of unknown magnitude. However, it is better to include an estimate in the calculation than to ignore (11) L.Page and N. I. Adams, “Principles of Electricity,” D. Van Nostrand Go., Inc., New York, N. Y., 1931,p. 26. (12) H. S. Harned and B. B. Owen. “The Phvsical Chemistrv of Electrolytic Solutions,” Reinhold Publishing “Corp., New York, N. Y., 1968,p. 119. (13) I. Pregogine, P. Mazur, and R. Defay, J . chim. phys., 50, 146 (1963). (14) D.A. MacInnes, “The Principles of Electrochemistry,” Reinhold Publishing Gorp., New York, N. Y., 1939,p. 242. (16) E. A. Guggenhiem, J . Phys. Chem., 33, 842 (1929); 34, 1640 (1930). (16) J. G. Kirkwood and I. Oppenheim, “Chemical Thermodynamics,” McGraw-HiU Book Go., Inc., New York, N. Y., 1961,p. 211. (17) I. Oppenheim, J. Phus. Chem., 68,2969 (1964). (18) H. S. Frank, ibid., 67, 1664 (1963).
Volume 69, Number 12 December 1965
DENNISR. HAFEMANN
4228
activity coefficients altogether. It is possible that calculations of the kind described here may eventually make possible more accurate estimates of single-ion activity coefficients. At time 0 matrix C is set to the initial conditions of the experiment being simulated, and values of bci/bx and b In y a / b x for every species i and b#/bx are determined at the intersections of adjacent cells. Then the fluxes J a are calculated from (9). These fluxes are assumed to remain constant over an appropriately short time interval At, and a new set of values for C is determined. From these values bcJbx, b In yt/ dx, and b#/bx are again calculated, and the process is repeated as often as desired. A typical calculation, 1 = 100, and the At iteration is for which n. = 2, m repeated 500 times, requires about 1 min. of computation on a CDC 3600.
I
I
1
2
I
I
I
I
5
6
+
Results The junction XaCl(O.099570 M)-NaC1(0.049833 M) ha,s been studied, using the data1g-21in Table I. The values in this table have been adjusted to provide the most accurate interpolation possible in the concentration range of interest. Figure 1 shows the calculated variation of V with time. The three curves are results obtained with three sets of values for At and d. The shortest value of At probably gives the most reliable curve. Because V reaches its final value very rapidly sec.), no comparison of the rise time with published experiments can be made. It is possible, however, that methods could be devised to measure the radiofrequency energy which is generated by a flowing junction. After the initial fast rise, the liquid junction potential remains constant (except for some overshoot in cases where At is too long), in agreement with experiment. Table I: Data Used in the Computations” Conon., M
Ln = In ycl-
0 0.05
-0,079 -0,202 -0,264 -0.302
0.10 0.15 a
See ref. 19-21.
LnrNa+
-0,077 -0,188 -0,240 -0.264
gKtb
7.54 7.02 6.78 6.57
UCI-
gNa+
7.83 7.31 7.07 6.85
5.08 4.62 4.41 4.20
The units of g are 10-la mole sec. g.-l.
0
The Journal of Phgsical Chemistry
30
Figure 1. Liquid junction potential vs. time after formation of a sharp junction between NaCl(O.099570 M) and NaCl (0.049833 M ) : top curve, At = 5 X sec.; middle curve, At = 10-’0 eec.; bottom curve, At = sec.; At/# = lo4 sec. cm.-*; T = 25” for dl curves.
time after formation from an initially sharp boundary. At 1.2 X lo-* sec. after formation of the junction, there is a 0.36% excess of CI- ions at the trough of the see., the excess charge distribution. After 6.2 X is reduced to 0.075%. It is only for a very sharp junction that the electroneutrality assumption is incorrect. The junction NaCl (0.1 M)-KC1 (0.1 M ) has also been studied because this is a case in which the Planck and Henderson methods fail to predict the correct result.22 I n Figure 3 the concentrations of Naf, K+, and C1- are shown 6.2 X loF8sec. after the formation of a NaC1-KC1 junction. Note that the chloride concentration profile is distorted by the inequality of the mobilities of the two counterions, in qualitative agreement with the calculation of Taylo9 on the junction
HCI-KCI. Figure 4 shows the dependence of liquid junction potential rise time on the assumed dielectric constant ~~
Figure 2 shows the broadening of the junction that is continually taking place. Although a junction of specified width will reach a steady-state charge distribution, the charge distribution in a junction unconstrained with respect to width is a strong function of
3 4 Time, 10-S aeo.
~~~~
(19) See ref. 12,Appendix A. (20) R. A. Robinson and R. H. Stokea, “Eleottrolyte Solutions,” Butterworth and Co. Ltd., London, 1969. (21) A. 8.Brown and D. A. MacInnes, J. Am. Chsm. floc., 57, 1358
(1935). (22) See ref. 14,p. 236.
4229
CHARGESEPARATION IN LIQUID JUNCTIONS
I
I
I
I
0.10
0.08
%
.# 0.06 + 3
8
uB
0.04
I
-3
-2
I'
-1 0 +1 Distance, 10-8 am.
I
+2
-
Figure 2. Concentration of excess charge vs. distance in the junction NaCl(O.099570 M)-NaCl(O.049833 M ) a t 10" sec. (heavy curve) and a t 6.2 X 10-8 see. (light curve) after formation. For both curves At = 10-lO see., A t / @ = lo4see.
of the solution. A low dielectric constant decreases the time for attainment of a steady potential difference.
0.02
0
Distanae, 10-6 om.
Figure 3. Concentration of Na+, K+, and C1- us. distance in the junction NaCl(O.1 M)-KC1 (0.1 M) a t 6.2 X 10-8 see. after formation. At = sec., A t / @ = lo4 see. cm.-a.
Discussion The following two conditions have been found to be important to the model's stability and to the fidelity with which it simulates a physical process. The ratio At/d2 must be such that d is approximately equal to the root-mean-square distance a molecule travels in time At. For diffusion of uncharged species At/d2 = O(0-l). For charged species the ratio is also affected by the strengths of the electric fields present and by the dielectric constant of the medium. If A t / @ is too small, the result will be only slightly in error, but, if At/d2 is too large, the calculation will be unstable and lead to meaningless results. The interval At must be short compared with the natural rise time of the liquid junction potential. The method may give an accurate long-time value of the liquid junction potential if this condition is not met, but it cannot yield the correct time dependence. As At +0, keeping A t / @ constant, the fidelity of the model increases monotonically, but less rapidly than the computation time increases. The value of At used in a calculation must be determined by a compromise between high fidelity and short computation time. The potential calculated for the NaCl (0.1 M)KCl (0.1 M ) junction, 4.83 mv., differs significantly
0
1
2 3 Time, 10-8 SBO.
4
62
Figure 4. Liquid junction potential vs. time for the junction NaCl (0.1 M)-KC1 (0.1 M ) , At = lo-'* see., At/# = lo4 see. em.-*; top curve, dielectric constant K = 19.63; second curve, K = 39.27; third curve, K = 78.54; bottom curve, K = 157.08. Top curve overshoots because the value of A t / @ is too high.
from the measuredzz GI- ion potential difference of 6.42 mv. Essentially the same result, 4.86 mv., is obVolume 69, Number 18 December 1966
DENNISR. HAEEMANN
4230
tained by the Planck and Henderson methods of calculation. 22 Therefore, one of the following assumptions is probably incorrect. (1) The activity coefficient of C1- is the same in a given concentration of NaCl as it is in the same concentration of KC1. (2) The flux of a given component is independent of the flux of any other component, except that the fluxes of charged species will influence each other electrostatically. (3) The mobility of a given component depends only on the total ionic strength of the solution. (4) The dielectric constant of the solution is constant and equal to that of water. This includes the assumption that the effect of the force tending to pull the most polarizable species to the point of highest field strength is negligible for the solutions treated here. The use of the bulk dielectric constant, or of any dielectric coonstant, on the distance scale encountered here (10 A.) is questionable, but, since the value of the dielectric constant affects only the rate of change of the potential and not its steady-state value, this is probably not the cause of the discrepancy. (5) The junction is isothermal. If the heat of mixing of the two solutions were large, this simplifying assumption could not be made. (6) Solvent flow is negligible. The error probably results from the first and second assumptions. The NaCl (0.099570 M)-NaC1 (0.049833 M) results agree within 1% with experiment.21 Table I1 shows that the result obtained using the Guggenheim assumption (yx+ = ycl- = ykMC1)is not significantly different from that obtained using the MacInnes assumption. However, complete neglect of activity coefficients leads to an 8% error in the calculated cell potential.
Calculation of this cell potential by means of the thermodynamically exact Helmholtz method, with experimentally determined transport numbers and activity coefficients, leads to a much more accurate result than any quoted in Table I1 but gives no information about time dependence or activity coefficient asymmetry. Although the present investigation has been limited to the study of one-dimensional isothermal liquid junctions between simple salt solutions, the method developed also is applicable to: (1) more complex geometries, providing the electrostatic part of the problem can be solved; (2) nonisothermal systems, if the irreversible thermodynamic treatment is extended to include the effect of temperature gradients; (3) solutions which are reacting chemically, provided the rates of reaction, equilibrium constants, and heats of reaction are known; (4)diffusion in ionic solids, in which experimentally observable rise times may be predicted ; (5) membrane phenomena, including analysis of the generation and time variations of electrostatic potentials across biological membranes. The temporal and spatial variations of the ionic mobilities which are important in the description of biological membrane phenomena could easily be included in the treatment described above.
Acknowledgments. The author wishes to thank Dr. Stanley L. n!tiller, Dr. Donald G. Miller, and Dr. Alexander Mauro for their helpful criticisms and suggestions. The work was supported by U. S. Public Health Service Grant No. 11906.
Appendix Potential Di$erence across a System of InJinite Planes of Uniform Surface Charge. From elementary electrostatics
Table 11: Effect of Activity Coefficient Assumption on the Calculated Potential of the Concent,ration Cell Ag; AgCl, NaCl(O.099570 M)-NaC1(0.049833 M ) , AgC1; Ag
A.
Concentration potential" B. Liquid junction potential C. Calculated cell potential (A B) D. Observed cell potentialb E. Deviation (C - D)
+
a
Ideal aoln. assumption
MacInnea aaaumption
Guggenheim assumption
17.783 - 4.062 13.721
16.191 -3.611 12.580
16.345 - 3.745 12.600
12.692 +1.029
12.692 -0.112
12.692 -0.092
MI potentials are in millivolts. 'See ref. 21.
The Journal of Physical Chemistry
If eq. 5 is substituted for E($),(Al) becomes
Making explicit the discrete nature of the charge distribution, we write
if e has the value ea a t all points between xa and xa+1. The upper limit on the first summation in (A3) can be changed to m since by (4)the added term will equal
423 1
HEATSAND ENTROPIES OF DILUTION OF PERCHLORATES OF Mg AND Sr
zero. For the special case where E is independent of x and the planes are equally spaced d,
=:
Evaluating the final sum in (A5) and using (4) t o simplify the expression, we arrive at the result
d
for allj, (A3) can be reduced to which has been used in all the calculations reported above.
The Heats and Entropies of Dilution of the Perchlorates of Magnesium and Strontium'
by H. S. Jongenburger and R. H. Wood Department of Chemistry, University of Delaware, Newark, Delaware
(Received June 28, 1966)
The heats of dilution of magnesium perchlorate and strontium perchlorate have been measured at 25" from 0.002 up to 4.4 m. The results were extrapolated to infinite dilution using an extended Debye-Huckel equation. The values of the relative apparent molal heat content are intermediate between the corresponding chlorides and nitrates. The corresponding entropies of dilution have been calculated. The results indicate that the perchlorates are not strongly ion paired.
This paper is one of a series2 on the heats and entropies of dilution of electrolyte solutions. The regularities found in the entropies of dilution offer some hope of achieving a deeper understanding of the structural effects in concentrated electrolytes. The perchlorates of the alkali metals show an increasing degree of ion pairing as the alkali metal gets heavier. The same sequence holds for the nitrates of the alkali metals and the alkaline earths. This study was undertaken to see if the alkaline eartlh perchlorates showed a similar behavior.
Experimental Section Materials. All chemicals used were reagent grade. The magnesium perchlorate was prepared from MgO and HC104. The oxide was digested several times with large amounts of water to reduce further the alkali salts content. After three recrystallizations the stock
solution was prepared by dissolving the wet salt in the required amount of water and adjusting the pH t o about 4. Strontium perchlorate stock solution I was prepared from SrCOa and HC10,. The procedure was similar to the above. For stock solution 11 the SrCOa was prepared from Sr(NO& and (NH&COa. The SrCOa was dried at 600" to remove all ammonium salts. Stock solutions were analyzed for Na, K, and Ca by flame photometry using a Beckman DU spectrophotometer, Model 2400, with an oxygen-acetylene flame attachment. The procedure was to compa.re a solution of the salt with one to which known amounts of (1) The authors thank the National Science Foundation for iinancial aid under Grant NSF-G14304. (2) (a) R.H. Wood, J. Phys. C h m . , 63,1347 (1959); (b) F.R.Jones and R. H. Wood, ibid., 67, 1576 (1963).
Volume 69, Number 1I December 1966
