Calculation of Liquid-Junction Potentials and Membrane Potentials on the Basis of the Planck Theory Werner E. Morf Depadment of Organic Chemistry, Swiss Federal Institute of Technology, Universitatsstrasse 16, CH-8092 Zurich, Switzerland
The Pianck relation for the dlffuslon potential has been derived In a new and simple way. The exact result Is available in a form comparable to the Henderson approach and allows ready computation of ilquld-junction potentlab using a pocket or desk calculator. Extension of the theory leads to a new Planck-type solution for the membrane potential. This rather universal formalism can be used to descrlbe the response behavior of any type of Ion-selective ilquid-membrane electrode.
are the rather voluminous derivation as well as the unwieldy implicit form of the result which requires numerical evaluation by trial-and-error, or graphical methods (see also MacInnes (13)). Here, a new and less circuitous derivation of the Planck relation is presented. The result is obtained in a more transparent form which allows ready calculation of liquidjunction potentials by an iterative method. In addition, it may be evolved into an expression for the membrane potential which offers a basis for the understanding of ion-selective liquid-membrane electrodes.
LIQUID-JUNCTION OR DIFFUSION POTENTIALS Theory. The liquid-junction potential is produced by the External reference electrode
Salt bridge (or reference electrode
Sample solution
diffusion of ions across the aqueous contacting layer between sample solution and salt bridge solution (see cell (1)). Assuming an idealized plane diffusion layer of thickness d (81, we may describe the flux of any component by the Nernst-Planck Equation 2 as a function of concentration gradient and potential gradient:
1
Internal Internal selective solution reference electrode 1:mbrane Ion-selective membrane electrode (indicator electrode) (11 The EMF of such a cell reflects the two potential contributions that are influenced by the sample solution. These are, first, the so-called membrane potential (electrical potential difference between sample solution and internal solution) and, second, the so-called liquid-junction potential (electrical potential difference between sample solution and salt bridge solution). The membrane potential is found to be the fundamental part since it clearly describes all the performance of the ion-selective membrane electrode. In an ideal system, the membrane potential, and herewith the EMF of the electrode assembly, follows Nernstian behavior. The theoretical treatment of the membrane potential was set forth mainly by the physiologist Eisenman and his school who proposed different models for glass membranes (3),liquid ion-exchanger membranes ( 4 ) , and neutral carrier membranes (5)(see also (1,6, 7)). Recently, these individual models, which offer some insight into the ion selectivity of membranes and membrane electrodes, were condensed to one theory (8). A thorough knowledge of the liquid-junction potential is of practical interest insofar as this contribution may lead to severe deviations from an ideal behavior of the ion sensor. Thus, inconstancy of the liquid-junction potential has the same effect as a change in the measured activity. Methods for the calculation of liquid-junction potentials are available because of the pioneering work of Planck (9) and Henderson (10) (see also (1,ll-15)). Although Planck's solution is generally more convincing from the theoretical standpoint, the Henderson approximation is much more frequently used in practice. In fact, the principal drawbacks of the classical Planck theory 810
ANALYTICAL CHEMISTRY, VOL. 49, NO. 6, MAY 1977
(0 d x G d ) where J , = mass flux, ci = concentration, ui = absolute mobility (diffusion mobility) and zi = charge of a species. 6 = electrical potential, R = gas constant, T = absolute temperature, and F = Faraday constant. For special cases where J , = const, this may be integrated to (7,12, 16, 17): c , (d ) . e z i W ( d ) IRT - C i ( O ) . e q F @ ( 0 ) / R T J. =
-u.RT--'
(3) j e z j F @( ~ ) / R T ~ X 0
The Nernst-Planck equation 2 represents an extended form of Fick's diffusion law; the second term considers the interaction of charged species with the electrical potential. Another interrelation between the fluxes of ions is given by the electrical current density, j , which approximates zero for potentiometric measurements:
j / F = E /z,l
J , - E Iz,/ J, = 0
(4)
Here, the sums include all cations M and all anions X present within the diffusion layer. Combination of Equations 2 and 4 leads to the following general description of the diffusion potential (8, 13):
C Z m Z U , c,
(x)+
cz,2u,c,(X)
Obviously, a thorough knowledge of the concentration profiles is usually required for the exact evaluation of Equation 5. An approximate integration method was set forth by Henderson (10) which arbitrarily assumes linear concentration profiles for all ions within the diffusion layer, Le.:
(7) This leads to the well-known Henderson formula:
CIZ,IU, Ac, - ZIZ,IU,AC, Zzm2u,Ac, + Zz,Zu,Ac, Zzm2u, c, (0) + Zz,2u,c, In Zzm2u, c , ( d ) + Zz,2u, c,
ED =
RT F (0) (d) -X
(9)
(10)
(c) restriction to one class of mobile cations and one class of mobile anions (i.e,, ions of the same charge z,, respectively, z,; the original paper restricts to monovalent ions). With Equation 9 it is possible to introduce so-called mean mobilities, ai,characteristic of each ion class:
-
u, =
(JX 1%)
1
= const(x)
= const(x)
With this pivotal substitution, Equation 4 can be rewritten in the form:
j / F = IzmIiimZ(Jm/um)- Iz,IE,Z(J,/u,)
0.05 0.18 0.45 0.82 1.21
10-4
0.00 (1)
0.00
lo2
(x)= Zlzxlcx( x )
ZJm
103
0.18. . . 0.45 (1) 0.82 (1) 1.21 (1)
10'
(b) assumption of electroneutrality:
Z ( J m /Urn
0.01
1
Ji = const(x) (for all ions)
- = u,
0.01 (1) 0.05 (1)
lo-'
Equation 8 allows an easy and rather close characterization of the diffusion potential in terms of boundary concentrations and mobilities of diffusing ions (see below). In the case of the liquid junction, the boundaries a t x = 0 and x = d correspond in their composition to the given adjoining solutions. An exact solution was offered first by Planck (9) and is rederived here in a new form. It is based on the following restrictions concerning the diffusion layer: (a) assumption of a steady-state, i.e.:
Clz,Ic,
Table I. Liquid-Junction Potential Values at 25 "C, Calculated from the Planck Theory and the Henderson Approachagb Relative concn of E D inmV E D in mV sample according to solution, according to Henderson Sample x c i ( d ) l Planck, Eq. 8 solution Z C i ( 0 ) Eq. 12-14' KCI 0.00 (1) 0.00 10-3 0.00 (1) 0.00
=
0
(4a)
After insertion of the fluxes according to Equation 2, one arrives at a relationship similar to Equation 6 where, however, all individual ionic mobilities uiare replaced by mean mobilities ai:
..
Integration is now easily accomplished by using the electroneutrality condition, Equation 10. Finally one gets the following solution, as might be gleaned from Equation 8:
This result corresponds to Planck's exact solution of the problem (originally for Iz,I = Iz,I = 1) but is obtained here in a new and more practical form that shows impressively the relationship with the Henderson approximation. The index i signifies ions of one class, i.e., cations or anions. The mean mobilities are found, from Equations 3 and 11, to be given
NaCl
10-3
0.03 0.03 (1) 0.20 (1) 0.20 10-1 1.14 1.11 ( 2 ) 1 4.60. , . 4.60 12.11 10' 12.45 ( 3 ) lo2 22.45 23.13 (2) 33.72 34.52 (2) 103 -0.04 HCl -0.04 (1) -0.28 10-3 -0.32 (1) -1.73 -2.07 ( 3 ) 10'' -8.31 -9.40 (10) 1 -26.73.. . -26.77 -57.58 10' -52.84 (27) -94.06 lo2 -84.32 (7) -131.95 103 -118.81 (5) 0.02 NaOH 10-4 0.02 (1) 0.16 10-3 0.17 (1) 1.02 101.11(2) lo-' 5.66 (4) 5.27 1 19.35. . . 18.85 10' 43.54 (10) 44.33 76.42 loz 73.24 ( 5 ) 103 110.35 105.24 (4) a These diffusion potentials are generated in the aqueous diffusion layer ( 0 Q x Q d ) between the sample solution (at x = d ) and the electrolyte of the reference electrode (at x = 0 ) in cell 1. A mixed solution of KC1 and KNO, ( 4 : l ) is used as salt bridge or reference electrode solution. The mobility data used for the computations were obtained from equivalent ionic conductivities at 25 "C (18) according to Equation 15. The value in parentheses gives the number of iteration steps that are needed to come within c0.01 mV of the final result when starting with E D = 0 (see text). by the following relationship:
u-. = Z u c ( d ).etiFEo iR - C u ici ( 0 ) ZCi(d).&iFEDIRT - 2C i ( 0 )
(13)
Exceptions aside, these mobility parameters depend on ED. Thus, in contrast to the Henderson approach, the Planck solution does not generally yield the diffusion potential explicitly but has to be evaluated for ED by iterative methods. Calculation of Results and Discussion. An easy computation of Planck liquid-junction potentials is accessible due to the formalism given above. The procedure is the following: 1)Insert any arbitrary value of E D into Equation 13 and calculate first-order values for the parameters a, and LZ,. 2) Use these values in Equation 12 to calculate the next value for ED, etc. Some representative results are given in Table I. All the computations were performed using a pocket calculator (Hewlett-Packard HP-55). In most cases, only a few iteration steps are needed for a very close approximation to the final value (see Table I). ANALYTICAL CHEMISTRY, VOL. 49, NO. 6, MAY 1977
811
Table 11. Application of Equations 21-23 to Cation-SelectiveLiquid-Membrane Electrodes Type of Additional assumptions Selectivity-determining ion-exchange sites regarding the membraneu Results for E M factors References
RT
a,'
Lipophilic anions, R-, (no complexes with cations)
Steady-state in respect to R- (Boltzmann distribution) Exclusion of other anions
-_ F l n F
Negatively charged lipophilic ligands, S-, (neutral complexes with cations)
Exclusion of other anions Constant concentration of complexes (of total ligand) Complexation equilibrium at phase boundaries (K*,,: complex stabilities in water)
(1 - r ) -
Electrically neutral carrier ligands, S, (cationic 1:n complexes)
N o anion interference
km
(membrane solvent)
4,6,8, 26,27
RT Ck,a,' -In z mF ~k,a,"
-
RT
,
RT
z, F
RT
In
K*ms, kms, k m ,
Z: lema,"
z F
+r-ln
Zkmam'
In-
C K*mskmsam'
4,6,27
7
(both membrane solvent and ligand)
c K *msk,,a,"
xkmsa,,'
Constant concentration zmF xkmsams" of free carriers RT xK*,,kmSa,' Complexation equilibrium = -In z,F x K*m',Fkmsa," at phase boundaries
K*ms, kms (mainly ligand)
5,6A
17,27,28
For simplicity, the mobilities are assumed to be the same for comparable cationic forms. This procedure, for evident reasons, fails for Zci(0) = Zci(cl). Here, the logarithmic term in Equation 12 becomes zero. As a consequence, it must also hold that I Z m l a m + lzxlax= 0. Hence, using Equation 13, we may derive an explicit solution for this special type of liquid-junction potential which, as a matter of fact, corresponds to Goldman's equation (19) for the diffusion potential of biological membranes (see also (14,
system in cell 1 by a second ion-selective electrode that responds specifically to a different ion in the sample solution. Thereby, the activity of this species is introduced as a reference level. Such cells have been used, for example, for the monitoring of the Cl-/F- activity ratio on a fluorocarbonplant (24)or for the measurement of the Na+/K+ ratio in biomedical applications (25).
20)):
RT E , =-In F
Zumc,(0) Xu, c, ( d )
+ Zu,c,(d)
+ Xu,c,(O)
MEMBRANE POTENTIALS
(14)
for 1 . ~ ~=1 lz,l = 1 and Cci(0) = Cci(d). If this relation is applied to the simplest case, namely a liquid junction formed by two single electrolytes of the same concentration and with the same anion or cation, it further reduces to the well-known formula of Lewis and Sargent (21):
Theory. The membrane potential of ion-selective membranes is generally formed by two fundamental contributions, namely a membrane-internal diffusion potential and a boundary potential. Thus one may write:
EM =
@I1
-
The results in Table I demonstrate that there is a satisfactory agreement between the Planck theory and the Henderson approach as used in practice. Nevertheless deviations may exceed 10% (see e.g. values for HC1 as sample solution). Therefore, a rigorous application of the Planck equation is of practical interest since it may help to reduce uncertainties in the reference electrode potential. Unfortunately, no simple equivalent description is available for systems with more than two ion classes (12, 22). For a minimization of the liquid-junction potential in cell 1, a so-called "equitransferent solution" (23) (the exact condition is Zlz,lu,c,(O) = ~ ~ z , ~ u , c , ( or O ) u, = u,) has to be used as salt bridge, its concentration being much higher than that of the sample solution. An example is given in Table I and is based on the fact that u~ = 0.8 UCI + 0.2 UNO, (18). In some cases, EMF measurements can also be carried out on cells without liquid junction, i.e., without transference of ions. This is accomplished by replacing the reference electrode 812
ANALYTICAL CHEMISTRY, VOL. 49, NO. 6, MAY 1977
, (@(4 -@(())I
,
diffusion potential - @ I ) - (@(4 (16) boundary potential E, Here, the coordinates x = 0 and x = d refer to the boundaries
+ \($(O)
Here, the only parameters to be inserted are the equivalent conductivities of the two electrolyte solutions: A = A, + Ax with
=
@'I),
of the membrane (i.e., the diffusion layer) that are in contact with the sample solution (') and the internal solution ("). Accepting this notation, the diffusion potential can, in the framework of the present model, directly be described by the relations developed in the preceding section. The boundary potential is related to the ion-exchange processes at the phase boundaries between membrane and outside solutions. If a thermodynamic equilibrium is assumed to exist at each interface, the distribution of any species may be approximated as follows:
where a[ and a / are the respective activities of the outside solutions and ki is the partition coefficient (distribution constant), From this it is immediately seen that the boundary potential can be fully characterized by the equilibrium-distribution of one permeating ion across the membranmolution interfaces:
A more general result may be obtained from expressions of the type in Equation 17 by summarizing ions of the same ion class and rearranging:
EB
Zwikiai” Zwikiai’ - -R T RT In =z Ewici(d) Ewici(0) ziF
(19)
Here the symbol wi represents any additional weighting factor (e.g., wi = 1 or wi = ui). To obtain a useful expression for the total membrane potential, one has to combine appropriate terms for ED and EB so that all concentrations and potentials referring to the membrane phase are finally replaced by outside values. Whereas such a procedure is impracticable for more general cases, an adequate solution can be obtained from the Planck description. To this end, Equation 12 is rewritten in the form:
ED
=
RT
(1- r ) -In
RT +r-In--zxF where 7 =
7
E c (0)
zmF zcm(d) Ec, ( 0 ) %(d)
is the integral anionic transference number:
/z,lu,
+ /z,Iu,-
Z IE l,,
Now, Equation 20 can easily be combined with expressions of the type of Equation 19. The final solution for the membrane potential reads:
EM
=
+
RT
( 1 - 7) -In z, F
RT T -1nz,F
Z k , a,‘ Z k , a,
”
Ek,a,’ Ekxar”
The mean ionic mobilities, as entering into Equation 21, may be expressed as follows (see Equations 13 and 16-19):
The system of Equations 21-23 offers an implicit but welldefined solution for the membrane potential. Since evaluation for EM is not nearly as cumbersome as would be estimated (in analogy to the preceding calculation of ED),the use of this formalism in membrane biophysics is to be encouraged.
Discussion. The extended Planck model, which is given here for the first time, allows a rather universal description of the steady-state potentials arising on electroneutral liquid membranes. Equations 21-23 clearly demonstrate that the response of corresponding membrane electrodes may range from cation- to anion-sensitivity, reflecting the relative mobilities respectively permeabilities of participating ions. As is shown in Table 11, the selectivity of membranes towards cations can be enhanced by the incorporation of different types of ion-exchange sites. The results summarized in Table I1 are in agreement with earlier ones, as obtained from individual membrane models. A more detailed derivation will be given elsewhere (27).
LITERATURE CITED J. Koryta, “Ion-Selective Electrodes”, Cambridge University Press, Cambridge, London, New York, and Melbourne, 1975. K. Cammann, “Das Arbeiten mit lonenselektiven Elektroden”, Sprlnger-Verlag, Berlin, Heidelberg, and New York, 1973. G. Eisenman, Ed., “Glass Electrodes for Hydrogen and Other Catlons, Principles and Practice”, M. Dekker, New York, 1967. J. P. Sandbiom, G. Eisenman, and J. L. Walker, Jr., J. Phys. Chem., 71, 3862 (1967). S. M. Clanl, G. Elsenman, and G. Szabo, J. Membrane B H . , 1, 1 (1969). G. Eisenman, in “Ion Selective Electrodes”, R. A. Durst, Ed., Natl. Bur. Stand (,U.S.),Spec. Publ., 314, Washington, D.C., 1969. S.M. Clam, G. Eisenman, R. Laprade, and G. Szabo, in “Membranes-A Series of Advances”, Vol. 2, 0. Eisenman, Ed., M. Dekker, New York, 1973. H.R. Wuhrmann, W. E. Morf, and W. Simon, Hehr. Chim. Acta, 58, 1011 (1973). M. Planck, Ann. Phys. Chem., 39, 161 (1890); 40, 561 (1890). P. Henderson, Z . Phys. Chem., 58, 118 (1907); 83, 325 (1908). E. A. Guggenheim. J. Am. Cbem. Soc., 52, 1315 (1930). R. Schlogl, 2. Phys. Cbem. (Frankfurt am Main), 1, 305 (1954). D. A. MacInnes, “The Principles of Electrochemisby”, Dover, New York, 1961. N. Lakshminarayanaiah, “Transport Pherxxnena in Membranes”, Academic Press, New York and London, 1969. R. Gaboriaud, J. Chim. Pbys., 72, 347 (1975). P. l u g e r and B. Neumcke, In “Membranes-A Series of Advances”, Vol. 2, 0. Eisenman, Ed., M. Dekker, New York, 1973. W. E. Mod, P. Wuhrmann, and W. Simon, Anal. Chem., 48, 1031 (1976). “Handbook of Chemlstry and Physics”, 56th ed., Chemical Rubber Publishing Co., Cleveland, Ohio, 1975-1976, p D-153. D. E. Goldman, J. Gen. Physiol., 27, 37 (1943). R. P. Buck, Crit. Rev. Anal. Chem., 5 , 323 (1975). G. N. Lewis and L. W. Sargent, J. Am. Cbem. Soc., 31, 363 (1909). H. Pleijel, Z . Phys. Chem., 72, 1 (1910). K. V. Grove-Rasmussen, Acta Chem. Scand., 2, 937 (1948). P. 0. Kane, “Some On-Line Applications of Ion-Selective Electrodes”, paper presented at the InternationalReference and IowSelective Electrodes Conference, Newcastle upon Tyne, Jan. 7-9, 1976. P. Ch. Meier, M. Oehme, and W. Slmon, in preparation. R. P. Scholer and W. Simon, Helv. Cbim. Acta, 55, 1801 (1972). W. E. Morf and W. Simon, in “Ion-Selective Electrodes”, H. Freiser, Ed., in preparation. J. H. Boles and R. P. Buck, Anal. Cbem., 45, 2057 (1973).
RECEIVED for review October 7, 1976. Accepted February 7, 1977. This work was partly supported by the Swiss National Science Foundation.
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