J. Pkys. Ckem. 1994,98, 6003-6007
6003
Observable Electric Potential in Nonequilibrium Electrolyte Solutions with a Common Ion Javier Garrido' Departamento de Termodinhmica, Universitat de Valtncia, 461 00 Burjassot, Valencia, Spain
Vicente Compafi and Marla Lid6n L6pez Departamento de Ciencias Experimentales, Universitat Jaume I, 12071 Castellbn, Spain Received: December 7, 1993; In Final Form: March 29, 1994'
The electric potential in a nonequilibrium electrolyte solution with a common ion is studied. We have used an observable electric potential (that measured in the metallic phase of a n electrode in equilibrium with the solution) instead of the unmeasurable electric potential usually invoked. The phenomenological equations deduced from the dissipation function of irreversible thermodynamics describe the system in a precise, clear, and simple manner when we consider this observable electric potential. An interesting property of this formalism is the distinction between distribution equilibrium and electric equilibrium. The phenomenological coefficients are related to well-known variables such as the specific electric conductance, transference numbers, and diffusion coefficients. The utility of this new formalism is shown by evaluating the profile of the electric potential in a liquid junction.
Introduction The study of the electric potential in nonequilibrium systems is a topic which appears in many fields of science. This is not a simple question, but on the contrary, we find important difficulties to establish the concept of electric potential and to evaluate it. The first obstacle arises from the fact that we are not able to measure an electric potential difference between two parts of a system thermodynamically different, Le., which differ in temperature, pressure, or concentration.l.2 To overcome this difficulty, some postulates have been assumed. In some cases the activity coefficients of ionic species have been given.3~~ In others, specific equations have been proposed, as is the case of the Nernst-Planck-Poisson f~rmalism.s-~The results obtained have had important consequences. We find among them: (i) the salt bridge concept; (ii) normal potential of electrodes; (iii) the methods of measuring the pH; (iv) measurements of concentrations with ion-selective electrodes; etc. We also find efforts accomplished on the definition and concept of the absolute single-electrode potential.8-10 From the latest progress toward the measurement of the work function for the electrolyte solution, it can be deduced that, under certain conditions, the absolute potential is the negative of the electrochemical potential, referenced to vacuum, in the solution.* In this work we define the electric potential at a point as that measured by electrodes in equilibrium with the solution. This electric potential is related to the true, but unmeasureable, electric potential at this point. This substitution, in the frame of thermodynamics of irreversible processes, produces a profitable formalism which can describe the system in a complete manner. Then, the true electric potential, not accessible by experiment, is replaced by something we can measure. Besides, to present the new formalism, we will check its utility. We show that this formalism not only is sound from a theoretical point of view but also has practical value. To prove this point, we evaluate the observable electric potential (hereafter denoted by OEP) in a liquid junction when the electric current is zero. The selected system is the ternary solution NaCl-MgC12-H20. We assume thermal and mechanical equilibrium. The experimental data are taken from the literature. This study follows the line developed in previous papers.11-15 One of these describes a formalism for discontinuous systems Abstract published in Advance ACS Absrracts, May 1, 1994.
when electric processes are present." Another deals with the question of defining the volume flux through membranes when there is an electric current.12J6 A discussion on the use of salt bridges and electrodes with a gaseous phase when the electric potential is measured in membranes has also been given.13 Significant advances have been given in the explanation of the asymmetry potential in membranes.14 Finally, the presence of superfluous terms in some expressions of the electric magnitudes in gravitational and centrifugal systems has been detected.l5Now, in this work, the study of continuous systems is intended. We can find in the literature other papers which analyze the flows and forces in electrolyte solutions. One of these follows an alternative procedure to that employed here and describes separately the diffusion and electric conduction processes.I7Others evaluate directly the dissipation function of the system and reach in a more complex way results similar to those shown here.'*-20 Electrolytes with a Common Ion The phenomenological equations can be deduced from the dissipation function of irreversible thermodynamics. If thermal equilibrium and uniform pressure are assumed, the dissipation function of a system with k ion constituents in a neutral solvent is given by2J k
j i vjii
=i= 1
where ji is the flay density of matter in a solvent-fixed frame of reference, and pi the electrochemical potential of the ion constituent i. Thesolution is constituted by k - 1 electrolytes with a common ion. The chemical potentials of ion constituents ( i = 1, 2, ...,k ) and electrolytes ( i = k + 1, k + 2 , ..., 2k - 1 ) are related by
where uj and Uik are the stoichiometric coefficients. The dissipation function of this solution in a nonequilibrium state, and the phenomenological equations, can be expressed in a useful form. In order to obtain this result, electrodes reversible to the common ion k are assumed. The chemical species X of
0022-3654/94/2098-6003$04.50~00 1994 American Chemical Society
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Garrido et al.
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994
the electrodes is in equilibrium with the ionic species k of the solution kx = f i k
+ z&e
(3)
where Zk is the charge number of ion k and the subscript e denotes the electron. The new driving force is defined by
where I)is the OEP in potentiometric measurements with X/Xzk electrodes. We have assumed here that (i) the concentration of X cannot vary and (ii) the pressure is uniform. It is convenient to point out that under these assumptions a change in the nature of the electrodes X does not modify the profile of the OEP in the solution: OEP is an unambiguous quantity. For the other thermodynamic forces we obtain
If matrix notation is used, the new and older forces are
and the relationship between them is given by
X=MTX'
k-1
where 1, are the phenomenological coefficients. Thedriving forces V&+j depend exclusively on electrolyte concentration gradients. When vpk+j = 0 0' = 1,2, ...,k - l), the system is in distribution equilibrium. When VJ, = 0, the electric equilibrium is reached. The distinction between distribution equilibrium and electric equilibrium is an interesting property of this formalism. When vpk+j = 0 0' = 1, 2, ..., k - 1) and VI) # 0, a process of pure electric conduction takes place in the system. When i = 0 everywhere, the electrolytes diffuse as a whole and then -ji 1. i'
= jk+i
(8) 7iK
=uiF
(9)
we deduce
i = 1, 2,
..., k - 1
where K is the specific electric conductance and Ti are the Hittorf s reduced transference numbers. We have written above that OEP is an unambiguous quantity and does not vary when we change the electrodes. The same conclusion applies to the phenomenological coefficients lik, the specific electric conductance K,and the Hittorf s reduced transference numbers q. The diffusion coefficients of the Fick's laws,
We have employed the superscript T for the transposed matrix. From the relationship
where Y denotes the older flow matrix
(16)
There are differences between our formalism and that presented by Farland et a1.18 They both use the same thermodynamic force, but the fluxes involved are different: we work with ion fluxes and Flarland et al. with electrolyte fluxes. This is a consequence of their particular deduction of the dissipation function which involves some misunderstandings. We can also assert that the new electric potential I)generalizes the emf concept of a cell, constituted here by two electrodes and a nonequilibrium solution. This emf depends on the solution state, i.e. on concentration profiles and on electric current density, as we deduce from eq 15. The phenomenological coefficients are related to well-known experimental coefficients as follows:
where the matrix M is
Y'=MY
i = 1, 2, ..., k - 1
k- 1
(10) are related by
where
zi ji F is the electric current density. where i = The new dissipation function will be klj
= "'+I
Ij = 1, 2,
..., k - 1
ack+j
and the phenomenological equations
The electric driving force VI) is measured through electrodes in equilibrium with the solution. Undoubtedly we still do not know the true electric driving force. This fact was already pointed out in the beginning, and now we can assert that the true force is superfluous for the description of the system. We will be able to evaluate all the forces, Le. v p k + j (j= 1, 2, ..., k - 1) and AI), if we measure the concentration and OEP profiles, and then we
Observable Electric Potential
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994 6005
can calculate all the fluxes with the aid of the phenomenological coefficients and the phenomenological equations. For a fixed concentration profile, the value of the electricdriving force can be modified by two methods: (a) through working electrodes and (b) by changes in the electric charge density. The way in which the working electrodes act is manifested when we refer to the pure electric conduction process. Here all the forces v&+j0’ = 1, 2, ..., k - 1) are zero. We assume the system is initially at equilibrium. The process begins when we apply the electric driving force V$ through the working electrodes. When the electric charge density p in the solution is different from zero, the phenomenological equations, eqs 14 and 15, hold since eq 2 does not change. Then the formalism remains unaltered. Changes in p can produce variations in the electric driving force V$, but not in vp&+j0’ = 1, 2, ..., k - 1). The variations of p everywhere in the system are given by the electric current density through dp = -Vi dt.
0.2 0.3
0 .o
t
c 1
-1.2
-0.6
0.0
1 04y
Figure 1. Concentration profiles in the liquid junction (ref 22, Table I, series AM-2, experiment 4); y = x / 2 d , where x is the distance and t is the time. TABLE 1: Initial Conditions of the Liquid Junction in the Series AM-2, Experiment 4, by Mathew et a1.22
Nonobservable Electric Potential There are other formulations which study nonequilibrium electrolyte solutions. The electric potential they work with is usually a nonobservable variable. We are going to consider now two of them: (i) the formalism based on the electrochemical potential splitting and (ii) the Nernst-Planck-Poisson formalism. We assume again mechanical and thermal equilibrium. The phenomenological equations for the first formulation can be deduced from eq 1, i.e. k
ji = -
EAijViij
i = 1, 2,
..., k
j= 1
The electric potential terms1
++is defined by the splitting of iiinto the
1.2
0.6
(mol dm-’) cs (moldm-3) c4
xo
0.102 45
0.147 57 0.408 19
0.332 99
The liquid junction is formed from a sharp boundary established at t = 0 between two solutions which are above and below the position x = 0, respectively. The more concentrated solution is below. We refer here to the experimental series AM-2 worked by Mathew et a1.22 The initial conditions (r = 0) for the two solutes are given in Table 1. In this section the subscripts 1-5 denote Na+, Mg2+, C1-, NaCI, and MgC12, respectively. The system is at thermal equilibrium at 25 “C. At t # 0 the concentration profile will be (ref 23, eqs 38-39) c4 = 0.125 01 - 0.012 989(3.2822 X 104y)
+
0.035 549(2.8055
Zi = p i + zi F$’
(24)
c5 = 0.375 09 - 0.023 099(3.2822 X 104y)
where pi is the chemical potential. The two electric potentials are related by
X
104y) (28)
+
0.014 Sl(P(2.8055 X 104y) (29) where
2
9(q) = -s,’eqz dq
As v& cannot be measured, numerical results of $+ can only be obtained if the activity coefficient Yk is postulated. The Nernst-Pianck-Poisson formalism describes the evolution of the system through the phenomenological equations5-’
(
ji = -Di Vc,
+ z.c.F M_V$* RT
i = 1,2,
F k
V2$*
= -e
..., k
(26)
VG
and y = x/2& Figure 1 shows this profile. We assume that the diffusion coefficients are constant, since the concentration differences are small. The OEP in this liquid junction depends on the electrolyte gradients
F V+
z,ci
j=1
where Di are the diffusion coefficients of the ion constituents, $* is the electric potential, and e is the dielectric constant of the medium where transport occurs. As $* and Di are nonmeasurable variables, we can obtain numerical results for $* if the Di = Di (CI,c2, ...,ck) are postulated. A relationship between $* and $, similar to eq 25, is not possible here. Observable Electric Potential When i = 0 In this section we will study the OEP in a nonequilibrium state of the solution NaCl + MgCl2 + H2O when i = 0. An international collaboration22 is collecting sufficient experimental data to calculate the generalized ionic transport coefficients of irreversible thermodynamics. From these data we can also obtain the phenomenological coefficients Iij of the formulation described before.
-7,
Vp4 - r2 Vps
(31)
In the integration of this equation, we will take into account the chemical potential of the electrolytes:
+ RT In m4(m4+ 2 m , ) ~ , ~ p5 = pS* + RT In m5(m4 + 2mS)’y; p4 = p4*
(32) (33)
where miare the molalities, and y i the mean activity coefficients. These coefficients have been evaluated from the following isopiestic data: (a) the osmoticcoefficients and the mean activity coefficients of NaCl and MgClz aqueous solutions, from Hamer et al. (ref 24, eqs [2-81-[3-1 I ] and Table 16) and Rard et al. (ref 25, eqs 2-3 and Table VIII), respectively; (b) the osmotic coefficient and the mean activity coefficients of NaCl-MgC12H2O solutions, from Scatchard equations;26-28(c) the Scatchard parameters for NaC1-MgClrHzO solutions from Rard (ref 29, Table VII). Figures 2 and 3 show the values for the activity
Garrido et al.
6006 The Journal of Physical Chemistry, Vol. 98, No. 23, 1994 0.683
r4
a
-
0.07
~
0.681
b
'
0.677 0.10
0.12
0.14
0 10
0.16
m 4 (mol kg") 7 4 versus molalities from Scatchard equations, Hamer et al. data, and Rard et al. data: (a) m5 = 0.41; (b) m5 = 0.38; (c) m5 = 0.35; (d) m5 = 0.32 mol k g ' .
0.473
0.14
0 16
m 4 (mol kg")
Figure 2. Activity coefficient
30.472
0 12
I
1
I
I
r1 versus molalities from the Van Rysselbergheapproximation and transferencenumbers for binary systems from Miller: (a) m5 = 0.41; (b) m5 = 0.38; (c) m5 = 0.35; (d) m5 = 0.32 mol kg-I.
Figure 4. Transference number
0.29
I
I
0.471 0.470
0.469 0.488 0.32
0.35
0.38
0.41
m 5 (mol kg")
Figure 3. Activity coefficient 75 versus molalities from Scatchard equations, Hamer et al. data, and Rard et al. data: (a) m4 = 0.10; (b) m4 = 0.12; (c) m4 = 0.14 mol kg-I.
coefficients. It has been necessary to express the concentration profile in molalities. We have used (a) the Dunlop et al. expression (ref 30, eq [A-5]); (b) a modified form of Young's rule given by Albright et al. (ref 31, eq 3) for the density of NaCl-MgC12E -1.0 H20 solutions (the parameter E has been fitted todata of Mathew et al.,32 and the value E = 5.5 X 1 0 4 kg mol-1 has been obtained); 3 (c) the densities of NaCl and MgCl2 aqueous solutions obtained -2.0 t from Gon~alves33and Miller,34 respectively. The transference numbers for the ternary system have been evaluated from the Van Rysselberghe a p p r ~ x i m a t i o n ,and ~~,~~ the transference numbers for binary systems, from Miller.21J4 -3.0 -1.2 -0.6 0.0 0.6 1.2 Figures 4 and 5 show thesevalues. As we can see, the transference lo4y numbers vary with concentration. We will introduce these results Figure 6. Observable electric potential through the liquid junction. into eq 3 1 before integration. The OEP between two electrodes, one placed at x = --m and ment of the OEP profile requires a spatial distribution of electrodes the other at any position x of the liquid junction, will be in local equilibrium with the solution. The distinction between the distribution and electric equilibria is one of the properties of this formalism. The electrolyte chemical potential gradients express that the system is not in distribution where equilibrium, and the OEP gradient shows the deviation from the electric equilibrium. An interesting result of this formalism is that the OEP spatial (35) distribution appears as a consequence of thermodynamic properties of the system. Assuming electroneutrality is probably a solid In Figure 6 we show the OEP profile evaluated with the data hypothesis for many systems. We expect that only in very few given before. cases do we need to recourse to a spatial distribution of charges.5-7 In contrast to some formalisms where the electric potential Conclusions shows discontinuities as we move through an interface (Le., at the The formalism proposed has relevant properties from a electrode surfaces and at membrane interfaces: the Donnan potential), the OEP has the advantage of varying continuously theoretical point of view. The state of the system is determined by the spatial distribution of the intensive variables ck+j (j = 1, through the system. 2, ..., k - 1). From these we can evaluate the forces Vp(k+j (j = In conclusion, studies of this type can be of interest to explain 1, 2, ..., k - 1 ) . The OEP spatial distribution is given by eq 15 those phenomena where electric potential is a basic variable, when the values of the electric charge density are known. These because of the small number of simplifying hypotheses involved can be controlled through the working electrodes. The measureand the use of well-defined measurable quantities.
-
F
1
Observable Electric Potential Acknowledgment. This work is part of Project No. PB920516 of the DGICYT (Ministry of Education and Science of Spain). References and Notes (1) Guggenheim, E. A. J . Phys. Chem. 1929, 33, 842. (2) Haase, R. Thermodynamics of Irreversible Processes; Dover: New York, 1990 p 293. (3) Smyrl, W. H.; Newman, J. J. Phys. Chem. 1968, 72, 4660. (4) MacInnes, D. A. The Principles ofElectrochemisrry;Reinhold: New York. ) 133. ....., 1939: - - - - ,I r --(5) Manzanares, J. A,; Murphy, W. D.; Mafb, S.;Reiss, H. J. Phys. Chem. 1993, 97, 8524. (6) Horno, J.; Castilla, J.; GonzBlez-Fernindez, C. F. J . Phvs. Chem. 1992, 96, 854. (7) Riveros, 0. J. J. Phys. Chem. 1992, 96, 6001. (8) Reiss, H. J. Electrochem. SOC.1988, 135, 247C. (9) Trasatti, S.Mater. Chem. Phys. 1986, 15, 427. (1 0) Parsons, R. In Modern AspectsofElectrochemistry; Bockris, J. OM., Conway, B. E., Eds.; Academic Press: New York, 1954; Vol. 1. (1 1) Garrido, J.; Mafb, S.;Aguilella, V. M. Electrochim. Acta 1988,33, 1151. (12) Garrido, J.; Compail, V. Bull. Chem. SOC.Jpn. 1990, 63, 2454. (13) Garrido, J.; Compaii, V.; Aguilella, V. M.; Maf6, S.Electrochim. Acra 1990, 35, 705. (14) Garrido, J.; Compail, V. J . Phys. Chem. 1992, 96, 2721. (15) Garrido, J.; Compail, V.; L6pe2, M. L. Electrochim. Acta 1993,38, 877.
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