J. Phys. Chem. 1988, 92, 1696-1700
1696
Diffusion Coupling In Electrolyte Solutions. 1. Transient Effects on a Tracer Ion: Sulfate Jean-Pierre Simonin,*f Jean-Francois Gaillard,: Pierre Turq: and Embarek Soualhiat Laboratoire d'Electrochimie, UA430, Bat. F, Tour 74, Universite Pierre et Marie Curie, 8, Rue Cuvier, 75005 Paris, France, and Laboratoire de GPochimie des Eaux and I.P.G.P., UniversitZ Paris 7, 2, Place Jussieu. 75251 Paris Cedex 05, France (Received: June 17, 1987; In Final Form: September 27, 1987) Experiments performed with the closed-capillary technique are used to investigatediffusion coupling processes in multicomponent aqueous electrolytes. The transient effects observed on the sulfate ion during the migration of LiCl and NaCl are compared with theoretical predictions based on solutions of the diffusion equations. These equations are solved by using normal-mode analysis and finite-difference calculations and the influence of ionic association is considered explicitly.
1. Introduction Sulfate is the second-largest anion in seawater and a major constituent of natural brines. During early diagenesis of marine coastal sediments, the reduction of sulfate, through bacteriamediated reactions, produces large concentration gradients and enhances sulfate migration from seawater.' Moreover, the presence of sulfate ions seems to inhibit the formation of dolomite CaMg(C03)2,an important mineral entering in the basic composition of oil reservoirs.2 In interstitial aqueous solutions, ion migration is controlled by diffusion processes. Hence, diffusion data on multicomponent aqueous electrolyte are essential for a complete understanding of transport in aqueous natural solutions. Through these practical applications this paper addresses the more fundamental problem of the importance of ion association and its influence upon diffusion proce~ses.~Sulfate is supposed to be relatively associated with magnesium ions in seawater, as confirmed by sound-speed mea~urements.~ The extent to which such association can modify the basic ionic interactions during diffusion processes is still poorly d ~ c u m e n t e d . ~ This work reports on the experimental techniques utilized for continuous monitoring of transient coupling effects and the corresponding theoretical treatment. The results of the experiments are compared to the theory and discussed in the framework of further experimental and theoretical approaches. 2. Experimental Section Coupling effects during ionic diffusion processes69 are greatest for systems where anions and cations have large differences in mobility. Hence, after a first experiment performed with seawaterlike solutions, we chose more appropriate solutions in order to enhance transient effects. Consequently we prepared aqueous solutions with NaCl and Na2S04,respecting the Cl/S04 seawater ratio, and, afterward, solutions containing LiCl, Na2S04,and MgCI2. Solutions of these salts were prepared gravimetrically by using Merck Analytical reagent grade and deionized water from a Millipore system (Milli-Q). The diffusion technique used in this study is the closed-capillary Briefly, the closed capillary presented in Figure 1 consists of two cylindrical sections of plastic glued together. A bore of 1-mm diameter was drilled through the center of the cylinder. The bottom section, of length L = 1.5 cm, is made from scintillating plastic (Altustipe), whereas the top section is made of ordinary plastic of the same length L. The total length of the capillary is then 2L = 3 cm, corresponding to characteristic diffusion times in the system of about 2 or 3 days. The aqueous solutions are spiked with tracer amounts of (NH4)235S04 radioactive aqueous solutions from the Radiochemical Centre, Amersham (U.K.). The progress of an experiment consists in filling up the bottom compartment of the closed capillary, using a Pasteur pipet, with
the more dense aqueous solution containing the diffusing electrolyte (LiCl), possibly a defined amount of MgC12, and a tracer concentration of Na2S04spiked with 3sS042-. The second solution is then added on top as carefully as possible in order to avoid mixing. This solution is identical with that contained in the bottom compartment except for the major salt LiC1, which is not present in the upper compartment. The capillary is then positioned vertically in a scintillation counter (Packard) and the activity of the bottom compartment is recorded as a function of time. The temperature in the counting chamber is 28 f 1 "C.
3. Theoretical Section Three experiments will be presented in this section. The variations of activity and the transient effects resulting from these experiments are presented in Figures 2, 3, and 4. Before examining these results, we give the theoretical basis of the diffusion processes and their mathematical expressions. We distinguish two different cases: the first one involves the treatment of the diffusion equations without ion-pairing reactions, whereas the second involves the treatment of the diffusion equations with ion-pairing reactions. In each case we perform a normal-mode analysis8 and use a finite-difference computing technique to solve the diffusion equations. Lastly, in this paper we suppose that the solution behaves ideally. In the conclusion we give the result of a first simple attempt to treat activity coefficient influence. The careful1 treatment of deviations from ideality and its consequences on diffusion coupling are the subject of our present research and will be addressed next. (a) Normal-Mode Analysis. Without Ion Pairing. The starting point is a set of continuity equations describing the transport processes occurring in the closed capillary: (1) i_nvolvingthe local concentrations C, of the ith species and its flux J, given by 5, = ClZ, (la) (1) Berner, R. A. Early Diagenesis. A Theoretical Approach; Princeton University Press: Princeton, NJ, 1980. (2) Kastner, R. Nature (London) 1984, 311, 410. (3) Turq, P.; Orcil, L.; Chemla, M.; Barthel, J. Ber. Bunsen-Ges. Phys. Chem. 1981,85, 535. (4) Fisher, F. H. Science 1967, 157, 823. (5) Applin, K. R.; Lasaga, A. C. Geochim. Cosmochim. Acta 1984, 48, 2151. (6) Turq, P.; Chemla, M.; Latrous, H.; M'halla, J. J. Phys. Chem. 1977, 81, 485. ( 7 ) Turq, P.; M'Halla, J.; Chemla, M. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 982. (8) Turq, P.; Orcil, L.; Chemla, M.; Mills, R. J . Phys. Chem. 1982, 86, 4062.
(9) Mills, R.; Perera, A.; Simonin, J. P.; Orcil, L.; Turq, P. J. Phys. Chem. 1985, 89, 2722.
* Author to whom correspondence should be addressed.
'Universite Pierre et Marie Curie. 'UniversitC Paris 7 .
0022-3654/88/2092-1696$01.50/0
(10) Liukkonen, S.; Passiniemi, P.; Noszticzius, Z.; Rastas, J. J . Chem. Soc., Faraday Trans. I 1976, 72, 2836. (1 1) Simonin, J. P.; Perera, A.; Turq, P.; Mills, R. J . Solution Chem. 1986, 15. 12.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1697
Diffusion Coupling in Electrolyte Solutions
m
E E
0
5
0
Figure 1. Closed-capillary setup. The bottom part is plastic, the upper part of ordinary plastic.
ma^
0
of scintillating
40
20
60
80
100
Time (hrs)
Figure 4. y as a function of time. Conditions: LiCl see Figure 2; MgCI2, 0.1 M; Na2S04,lo4 M. Results: curve e, experimental points; curve 1, normal mode without ion pairing; curve 2, finite difference without ion pairing; curve 3, normal mode with MgSOt pairs (K = 200); curve 4, finite difference with MgSO: pairs (K = 200); curve 5 , finite difference result with the Fuoss value for K,K = 40.
I
-E h
M
15
E 0 10
5
I 0
20
40
60
80
temperature, pi the chemical potential, zie the charge, and E the electrical field classically used to describe the transport of ions. Its expression can be obtained by a local electroneutrality assumption as we see below. In a solution containing a tracer T plus an electrolyte with ions 1 and 2 we have three equations of type 1. We notice that in the present situation the tracer flow is governed by the electrical field created by the salt, which is in much larger amount, and is given by 3, = -DT6CT CTwTZTeE (IC)
+
100
Time (hrs)
Figure 2. y as a function of time. Conditions for the first experiment: LEI, 1 M in the bottom compartment;no magnesium; Na2S04, M in the whole capillary, at t = 0. Results: curve e, experimental points; curve 1, normal-mode result without association; curve 2, finite-difference
result without association; curve 3, normal-mode points with LiSOL ion pair; curve 4, finite-difference points with LiS04- ion pair.
where_wT= &/kT is the mobility of the tracer. In this particular case E is the internal field created by the diffusion of the binary electrolyte 1-2. If we take, like in our case, a binary symmetrical electrolyte (z2= -2, = -z), the field is readily evaluated from the equality
5, = j 2
(2)
expressing the local dynamical electroneutrality condition
25
CZiJ = a i
20
K
applied to our system. As 3, = CICl = C l w l ~ itl , follows that
Clwl[ - k TW, F
15
+ zlez
v
EE
Hence we express the electrical field 2 using this equality and obtain the Henderson formula
:
10
E = -kT- - Di 5
ze D ,
- D2 ~ C S
+ D2
C,
(4)
with C, the concentration of the major electrolyte: C, = C,= 0
20
40
60
10
100
Time (hrs)
Figure 3. y as a function of time. Conditions: LiCl, see Figure 2; MgCI2,0.01 M; Na2S04,10-4 M. Results: curve e, experimental curve; curve 1, normal mode without ion pairing; curve 2, finite difference without ion pairing; curve 3, normal mode with MgSO: pairs (K = 200);
curve 4, finite difference with MgS0: pairs. where iJi,mean velocity of particles i at position x and time t, can be derived according to the linear transport theory to yield CiDj 3, = -[-$Mi + zgZ] kT Di is the diffusion coefficient, k is the Boltzmann constant, T the '
c2.
Then we get for the tracer flow the following equation
5T = DTtCT
+ aCTDT-acsc,
(5)
with the dimensionless parameter a a = zT - - D1 - D2
z D1
+ Dz
If we introduce this expression into the tracer flow equation, after keeping only the first-order terms in the concentration heterogeneities with respect to equilibrium (for which quantities are denoted with the superscript m)
Simonin et al.
1698 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
SC, =
&x,
= CT - c T m
6cT
it remains then
c, - e,(6)
< 2L.
cT(x, t )
Thus a t T can be expanded in a Fourier series m
G(x, t ) e X A n ( t ) COS ( q d ) n=O
*
asc,
--
at
ale,
a2wT
D +~ a - a=xO 2 2 ax
(7)
for our system in one-dimension x, with the definition of a, a = &(cTm/c$"), having the dimension of a diffusion coefficient. Thii equation is solved by using both Laplace transform on time (denoted by a tilde) and Fourier transform on space8 (denoted by an asterisk). We obtain 6cT*(q, s)[s
for 0 < x
t)
+ q 2 D ~=] 6cT*(q, t = 0) + aq2c,*(q,s)
(8)
(13)
where q, = ( ~ / 2 L ) nand , the functions A , are given by
It can be shown that A2k = 0 and
The electrolyte 1-2 has a diffusion motion independent from the tracer, expressed by
ac, -- D,-a2c, _ at
(9)
ax2
Therefore we can deduce the function y ( t ) defined by eq 1 1
8
with
y(t) =
which can be obtained with eq 2, 3, and 4 and represents the Nernst-Hartley diffusion coefficient of the salt. The solution of eq 9 in the Fourier-Laplace space is C,*(q, s) = C,*(q, t = O)/(s
+ q2DJ
(9a)
Hence one obtains simply the solution in the (q, t ) Fourier space ScT* = 6CT*(')
+ 6CT*(m)
(10)
In this expression the first term is the self-diffusion propagator of the tracer 6CT*(')(q,t ) = 6cT*(q, f =
O)f?-q24r
(loa)
while the second describes the migration of the tracer in the diffusion field of the salt
-ZT-
DT
D1 - D2
D, - DT D I + D2k-0
e-q2k+12&I - e-q2k+1'DJ (2k
+ 1)'
(16)
As we will see below, we obtain here an approximate solution, which gives a good estimate for that observed in the experiment. We can use it in particular to evaluate the order of magnitude of the maximum of y ( t ) and its location in time, which are the two main points to look at in an experiment. With Zon Pairing. We consider first the case where the (monovalent) cation of the salt is implied in an ion-pairing reaction with the tracer but is in high enough concentration with respect to the latter for its diffusion process not to be affected; this will be the case of the experiment shown in Figure 2 in which we suppose the eventuality of an association between lithium and sulfate ions. Writing the continuity equations for the three species, the cation of the supporting electrolyte, the free tracer ion, and the ion pair, assuming a local equilibrium between them with a constant K , and proceeding to the same linearization as before, we obtain, it can be shown, instead of eq 7
with the following definitions From eq 10 we deduce any diffusional behavior for the tracer, from pure self-diffusion to purely coupled motion. From now on let us consider more explicitly the following situation: if we take a tracer whose concentration is initially homogeneous in the whole capillary, it will undergo a transient concentration gradient arising from the diffusing supporting electrolyte. The observable quantity is in this case the relative variation of the amount of the tracer in compartment A
Note that y(0) = y(m) = 0. Then, using eq lob, in the situation where initially the electrolyte (LiC1) is contained in half, say part A, of the capillary and diffuses in the other half, part B, we get sin ( q L )e-q2LW - e-q2Dsf
6cT*(q, t ) = 4aCSm-
4
Ds
- DT
(12)
after having symmetrized the concentration profiles with respect to the origin x = 0, which allows us to take the cosine of the Fourier transform of C,at t = 0. To go back to the real space, the system is then repeated periodically, with a period 4L, so that &x, t ) = ~ T ( x+ P ~ Lt ,) = &-X, t) (with p = integer) and
and
the index P designates the ion pair. It suffices then to replace these parameters in eq 16 to obtain the function y ( t ) in the present case. Second, it is also possible to perform a normal-mode analysis for the case where the tracer ion forms an ion pair with an ion which does not belong to the major salt (for example Mgz+ with SO:-, as for the experiments of Figures 3 and 4 where magnesium is added to the solutions). In this case we neglect the presence of association between lithium and sulfate ions with respect to the ion pairing of magnesium and sulfate, which has proved, according to the literature (seediscussion in following paragraphs), to be a much more evident event. We then have here to handle the whole of eq 1 in the spirit of normal-mode treatment. This is a harder task than in the previous situation, since it requires the resolution of 5 coupled linearized transport equations. The fundamental features of this study have been given extensively
Diffusion Coupling in Electrolyte Solutions in previous paper^.^^*^^ Doing so, we can obtain the expression of the observable quantity y ( t ) in the case where magnesium is added and possibly forms an ion pair with sulfate. ( b ) Finite-Difference Treatment. A more rigorous treatment of coupled diffusion eq 1 consists in a finite-difference algorithm. The basic ingredients for the application of this procedure to the coupling of diffusion processes of ions has been given previously.11 We took a number of cells N = 200 and a time step shorter than the mean diffusion time between contiguous cells for the faster ion, in such a way that if Ax and At are the space and time steps
The Journal of Physical Chemistry, Vol. 92, No. 6,1988 1699 where apand afstand for the proportions of, respectively, the ion pair and the free ions. So, eq 19 gives a nontrivial result for D that can be obtained by a normal-mode analysis. Moreover, it is interesting to point out that this single mode appears to be very simple for a particular initial profile of concentration: any species obeys the Fick macroscopic diffusion equation
and therefore we chose M = 5 or 10; for the diffusion of a single species, M = 2 is the lower limit for the convergence of the algorithm. Furthermore, close attention must be paid to the conservation in time of the total amount of matter in the system. For the ith species of charge Z i the diffusion equation reads
So, following the same treatment as given by eq 13 and 14, we see that, for the initial profile K ( X ,
t = 0) =
sc, cos (q1x)
(21)
with q, defined by eq 13, and K O
