within 3-4x. Figures 10 and 11 show typical fundamental and second harmonic ac polarograms of 0.95 m M colchicine in 1.9 m M TEAP. The fundamental harmonic peak in Figure 10 is of nearly diffusion-controlled shape (half-width = 93 mV) and the slight deviations appear attiibutable to the following chemical reaction (17, 32). The second harmonic wave is quite asymmetric, but this is expected when a following reaction is operative (17, 32). This substantial chemical perturbation on the second harmonic wave does complicate intzrpretation of the data with regard to whether the potentiostat is accurately controlling potential. Unlike the simpler nitrobenzcne system, one cannot rely heavily on comparison of absolute wave shape with theory without a detailed study sufficient to elucidate the chemical rate parameters. However, the fact that the second harmonic wave shape and magnitude is nearly unchanged (k3-4%) from 0.1M to 0.002M TEAP (roughly a 20-fold change in ohmic resistance accompanies this concentration change) indicates that ohmic resistance effects are negligible and the characteristics of the wave are controllzd by normal “faradaic rate processes.” The fact the ac polarographic responses of the nitrobenzene and colchicine systems show no evidence of effects attributable to iR drop with very low supporting electrolyte concentrations is quite encouraging and indicates considerable scope for the positive feedback approach. No attempt has been made as yet to extend these measurements in very high resistance media to higher frequencies. However, since the results of the other measurements cited here indicate that theoretical estimates of potentiostat bandpass (12) are probably on the conservative side, it appears likely that frequencies
over 100 Hz will be accessible as indicated by Table 111 of reference (12). These results in aprotic organic solvents seem to establish that application of positive feedback iR compensation will open new vistas for work in moderate- and high-resistance media. Measurements in the kilohertz range are possible with moderate resistance solvent-supporting electrolyte systems and, with care, one may at least approach the kilohertz realm with high-resistance systems. Another recent study has demonstrated the capability of effecting measurements in absence of supporting electrolyte at dc and very low frequencies (40). The possibility of effecting electrochemical relaxation measurements with very low supporting electrolyte concentrations demands that conventional concepts regarding solution conditions necessary for voltammetric measurements must be reassessed. This capability also suggests numerous unique and interesting applications among which are studies of double-layer effects with the expanded diffuse double-layer characterizing low supporting electrolyte concentrations where some unusual effects have already been revealed (41-43, and trace analysis work under similar conditions where electrolyte purity will prove less crucial. RECEIVED for review January 25, 1968. Accepted May 29, 1968. Work supported by National Science Foundation Grant GP-5778. (40) (41) (42) (43)
D. A. Hall and P. J. Elving, Electrochim. Acta, 12,1363 (1967). F. C. Anson, J . Phys. Chem., 71, 3605 (1967). P. Delahay and G. G. Susbielles, ibid., 70, 3150 (1966). G. C. Barker, J. Electroanal. Chem., 12,495 (1966).
Theory of Potential Distribution and Response of Solid State Membrane Electrodes 1. Zero Current R. P. Buck Department of Chemistry, Unicersity of North Carolina, Chapel Hill, N . C . 27514 Rapid, reversible interfacial ion transfer processes at solid state membrane electrodes permit application of the electrochemical potential concept for the description of interface Galvani potential differences. The Nernst-Planck equations applied to mobile defects allow a description of the internal diffusion potential. A general theory of potential distribution at zero current i s offered to interpret responses of solid state membrane electrodes. Deviations from Nernstian response are discussed on two models: solid bulkmobile interferences and bulk-immobile but surface absorbing interferences. Connections between crystalline membranes and electrodes of the second kind are presented.
A DISCOVERY of great practical consequence and some theoretical importance is the development of ideal Nernstian membrane potentials by water-insoluble crystalline substances when used to separate solutions containing component cations and anions of the membrane material ( I ) . Usefulness of the effect is already apparent in a variety of commercially avail(1) M. S. Frant and J . W . Ross, Science, 154, 1553 (1966).
1432
ANALYTICAL CHEMISTRY
able selective ion electrodes (2). This discovery, together with a comparable effect using water-rejecting organic liquid ion exchanging barriers ( 3 ) , represent the first significant new discoveries in potentiometry, a field long considered dormant, for perhaps half a century. The importance resides in part from the possibility of new selective ion electrodes sensitive to ions which cannot be measured as well by electrodes of the first, second, or third kinds. Second, the phenomena provide theoretically interesting connections between fixed-site ion exchange membranes, mobile-site membranes, and crystalline ionic solids. There is a connection between inorganic solid membranes and corresponding electrodes of the second kind which provides an appealing point of view on the characteristics of solid materials needed to produce Nernstian response as membranes. On the basis of an elementary analysis it will be shown that electrodes of the second kind and the membrane (2) Orion Research, Inc., Bulletins 94-09, 94-17, 94-35, 94-53, 94-16; 11 Blackstone Street, Cambridge, Mass. (3) J. W. Ross, Science, 156, 1378 (1967).
cell made from the same salt differ only in the form of ohmic contact at one interface. Salts which function well as part of electrodes of the second kind may not be suitable in membrane applications. In contrast, when electrodes of the first kind fail because of extreme irreversibility by virtue of slow electron transfer, or surface film inhibition, an insoluble salt of the corresponding metal may still be satisfactory in membrane applications.
La(II1) and F- in equilibrium. To avoid dissolution of the membrane, the solutions are equilibrated with powdered AgCl or LaF3 in advance of placing them in the cell. The solutions are presumed to be free from cations and anions which would form compounds more insoluble than AgCl or LaF3. For each interface at which one or more ionic species exists in equilibrium in the two phases, the electrochemical potential per mole of each ionic species p, is equal in each phase. The electrochemical potential in any phase has the form at constant temperature and pressure
MEMBRANE CELLS
A membrane electrode is a portion of a membrane cell and consists of a membrane, a solution called the inner or filling solution, ‘and an internal, reversible half cell or probe. The membrane cell contains the membrane electrode dipped into a second solution, usually the solution under investigation, and a second half cell probe. Ion exchanging membrane cells have been described in detail in recent books (4, 5). Both junction cells and junction-free cells are important in the discussion of crystalline membrane cells. These can be written, for example,
pi =
pco
+ RTlna( + ziFp
(4)
where p,’ is the ionic standard chemical potential; a, is the ionic activity; ziis the ionic charge with sign; p is the inner potential; R, T, and F are the gas constant, absolute temperature, and the Faraday, respectively. Analysis of the interfacial potentials at zero current follows conveniently by application of the electrochemical potential equality at the various interfaces. Using a saturated calomel half cell as an example, by equating pHg+in Hg and pHg+ in HgzC12, ( D H ~ - ~ H ~ is~ established; c I ~ by equating PCI-
Junction Cell I
HglHgzCIdKCl (sat’d)
II
Solution XI extending extending negatively membrane positively 11 KC1 (sat’d) Hg2Cl2 Hg from - (thicXness j from x = -L 2L ‘x =L
1
I
I
Junctionless Cell I1
Solution I AglAgCll containing C1-
Solution I1
1 membrane 1 containing C1-
(2)
lAgCllAg
General Form
I
External half cell or
I I Test solution I
where Solution I1 and the right-hand half cell are the internal filling solution and inner probe. For convenience, equations are written such that the measured potential is that of the electrons in the wire lead to the inner probe minus that in the wire lead of the same metal to the left-hand or external probe. Distances through the membrane are measured positively from left to right with zero of coordinates at the central plane of the membrane as shown in Figure 1. The aim of the following calculations is a description of various potentials which sum through the cell to give the measured potential AVc at zero current. More generally, in Part 11, an arbitrary applied potential AVc, distributed across the cell, will correspond to the potential of zero current plus potential drops associated with the flow of current. An analysis of potentials within and between phases is required. The former, diffusion potentials, are evaluated by use of the Nernst-Planck equations, while the interfacial potentials are best treated by means of the electrochemical potential concept. The inner or Galvani potential difference between phases is not a measurable quantity except between phases of identical composition. This limitation arises from the thermodynamic indefiniteness of processes involving transport of a single kind of charged species between phases. However, interfacial and diffusion potentials can be written and summed throughout systems to yield physically-significant, measurable potentials related to thermodynamicallyaccessible properties of components. Consider a two-compartment cell with junctions such as Cell I in which a AgCl or LaF3 membrane is selected. Solutions I and I1 contain unequal activities of Ag+ and C1- or (4) F. Helfferich, “Ion Exchange,” McGraw-Hill, New York, 1962, Chapter 8. ( 5 ) J. A. Marinsky, Ed., “Ion Exchange,” Vol. 1, Marcel Dekker, New York, 1966, Chapter 1.
Membrane electrode
(3) in HgzClz and ~ C I -in saturated KC1, p(HgZCl2)- (P (saturated KC1) is established. Thus the inner potential drop through the half cells has the form cp(Hg) - q(sat’d KCl) = 1
F [/.kg+o in sol’n -
pHg+O
in Hg]
+
This expression involves single ion standard chemical potentials and an individual ion activity. Despite a limitation on quantitative evaluation, directional changes in the potential due to salt activity changes can be computed. A more definitive evaluation can be made by the conventional referencing of half cells against the standard hydrogen electrode. Then the measured cell potential becomes independent of single ion activities and the unknown first term Equation 5 is assigned the value given in tables 0.799 volt (6). Potential drops $> through the liquid junctions are minimized by the use of double junctions made up of chloride-free components of comparable ionic mobilities. The sum of the potentials across the cell at zero current must be equal to the potential applied by the potentiometer, AV,. Consequently,
+
AVc = cp(so1’n 11)
- p(so1’n I) +
(6) W. M. Latimer, “The Oxidation States of the Elements and their Potentials in Aqueous Solutions,” Prentice-Hall,New York, 1938, p. 296. VOL. 40, NO. 10, AUGUST 1968
1433
A.
Equilibrium solubility 0 1 zero ionic strenglh
c
>
c
f
-
C 3. ._ c
10-3
10-2
C
10-5
10-4
Activity of silver or chloride ions
*a
6
Figure 2. Test solution interface potential or the measured cell potential for a junction cell using a silver chloride membrane Activity coefficient changes due to the presence of inert salts shift the entire curve to the left by increasing the solubility product. Reference solution contains constant activities of excess silver or chloride. Curves calculated according to Equation 10a and b. Potential increases positive downward. I
I
-L
=
I
0 Distance X -
Curve A . Condition of zero current corresponding to a.d-L)(> a.dL) or a 4 - L ) ) < UA(L) Curve B. Condition of finite negative current flow in the positive direction or positive current flow in the negative direction +(x) increases positive downward
where the chloride activities are bulk values at the probe surfaces. Obviously the potentials of the identical reference electrodes in Cell I cancel out the measured cell potential. For Cell I1 to which Ag+ or C1- has been added in both Solutions I and 11, the expression holds with deletion of the junction terms. Thus, junction cell potentials are nearly independent of the reference probes for identical half cells. At worst, nonidentical half cells contribute a constant "off set" potential to the measured membrane cell potential. The half cells used in junctionless cells may contribute activity-dependent potentials to yield an overall apparent subNernst or super-Nernst response of the entire cell with respect to the membrane potential. Examples will be given latter. Junction cells are more desirable because the response is solely dependent upon the membrane potential and the small junction terms. However, there is the ever-present possibility that the half-cell materials and the membrane material may react metathetically on long standing to destroy either the membrane or one component of the probe. Junctionless cells require control of the activity of an ion in the test solution which may be impractical in a given experimental situation. The measured cell potential for either Cell I or I1 depends upon the potential difference across the insoluble salt membrane. The latter can be described in general terms by rewriting p(L) - p ( - L ) as =
- Cp(01
[ d L >- d-L>1 1434
ANALYTICAL CHEMISTRY
+ A+(diffusion) + Ap(interface at -L)
Figure 1. Potential distributions at reversible homogeneous ion conducting membranes
dL) - d - L )
Acp(interface at L )
+L
+
+ [+(-L) - d - 0 1
where super bars are potentials inside the solid phase beyond the interior diffuse double layer regions. The membrane potential p(L) - p(-L) can be reduced to three terms. The magnitudes can be evaluated for typical membranes, The following derivations are based upon the assumptions of rapid, reversible-interface ion-transfer processes and low net current flow. The latter is determined in null measurements by the impedance of the measuring device. Rapid, reversible-interface processes imply a high-exchange-current density over a wide solution-activity range. At zero current, values of p are constant throughout the solution phases beyond the limits of the diffuse double layers at the membrane surfaces x = + L . EQUILIBRIUM INTERFACE POTENTIALS
The interfacial potential between a pure solid and a solution containing ions of the same solid (among others) is not unambiguously defined because of the appearance of single ion standard chemical potentials and single ion activities. There is no reference half cell by which single ion activities can be removed, nor is there a means for establishing by convention a value for the unknown constants. Fortunately, inasmuch as a membrane is used to separate two solutions, the unknown parameters cancel in the calculation of p(L) - cp(--L). An example is the silver chloride membrane against solutions containing activities U A ~ + and ucl- and inert salt. Thus at x = L, by equating chemical potentials (pAg+O
+ pc1-O) -
(pLng+O
+
pc1-O) sol" I1
= K B P
RT In 7(8) aAgCl
and by equating electrochemical potentials, the interface potential is found to be
1
(74
(7b)
= - [pci-"(L)
zF
]
- pc1-O - RTln aci-(L)
(9)
where z is the absolute ionic charge equal to one in this example. In view of Equation 8, the expressions on the right in Equation 9 are identical. Because the standard chemical potential terms are taken to be space-independent, and constant at fixed temperature, the interfacial potentials are determined within an additive constant by the single ion activities in solution and in the crystalline membrane, beyond the space charge region. The range of interface potential values is limited by the solubility of the membrane material. Consequently, for ~ the ponominal excess silver ion concentrations C A+(L), tential is
- +w = const.
2aAg + RT - In F Y A ~ + [ C A ~+ + ((c~g+’(L) L) +4 K s p / ~ ~ ~ ) ” ~ l +
(1Oa)
for C A ~ ~ 2L 1/KB,/r ) and for nominal excess chloride CCI -(L) p(L)
- +(L) = const. + RT -X F In
+
~ A ~ + [ C C I - ( L(Ccl-*(L) ) 2K,,
+ 4 K & * 2 ) ” 2 1 ~ ~ ~ -(lob)
for CCI i L ) 1KBpIY* These equations are illustrated in Figure 2. Both expressions yield the same value for a solution of inert salt saturated with AgCl.
The defect concentration is proportional to the square root of the quantity total concentration of silver (?A, multiplied by concentration of possible defect sites (9). Furthermore, the number of possible defect sites is also proportional to the total silver concentration. It is then reasonable that the right side of Equation 13 can be expressed
An analogous derivation leading to the same result can be made for a crystal whose equilibrium defect concentration of ion pairs arises by vacancies via a Wagner-Schottky mechanism. A somewhat more complicated problem arises when both defect anion and cation species contribute to the charge transport process and therefore to the diffusion potential. Lead iodide membranes show this behavior (10). In this case electroneutrality requires the presence of equal numbers of cation and anion vacancies whose interactions may leave them essentially paired or virtually dissociated. The motion of bulk cations and anions via vacancies can be viewed as a free diffusion problem analogous to a liquid junction between solutions of the same species at different concentrations or the polarographic problem in the absence of supporting electrolyte. Applying Nernst-Planck equations to both cations and anions and invoking electroneutrality and zero current (zero net flux), the potential gradient is
INTERNAL DWFUSION POTENTIAL
The diffusion potential within a solid depends upon the concentrations and mobilities of the current carrying species. In AgCl where the current is carried almost exclusively by interstitial Ag+ (7), the cation vacancies are essentially uniformly distributed in the bulk except at the surfaces where there is a piling up of interstitials or vacancies to create the inner part of a diffuse double layer (8). The interstitials diffuse in a sea of negative charge subject to electroneutrality in any volume large compared with the volume of a single ion. The flux of silver ions is described phenomenologically by the Nernst-Planck equation
where J is flux in moles/cm2 sec, (?A,+’ is interstitial silver in moles/cm3, aAg+’is mobility in moles/cm2 sec. joule, p ~ ~ ’+is’ the chemical potential in joules/mole, R the gas constant in joules/mole degree, T, the absolute temperature, z the charge in equivalents/mole, F the Faraday in coulombs/equivalent. The mobility is assumed to be a function only of temperature; and the standard chemical potential and ionic activity coefficient are also assumed to be independent of distance. At zero current (zero flux)
Integration of the potential gradient across the interior of the crystal gives (7) E. Koch and C . Wagner, 2.Phys. Chem. (B), 38,295 (1937). (8) T. B. Grimley, Proc. Roy. SOC.(A), 201,40 (1950).
where
+
For the lead halides, and - refer to cations and anions with absolute ion charges z+ = 2, 2- = 1. Integration across the material gives an expression of the same form as Equation 14.
Obviously at zero current, the diffusion potential is zero for pure crystalline membranes. This derivation in this context is not important and the final result could have been obtained by the thermodynamic argument used by Dole (11) for glass membranes. However, in the interpretation of electrode errors due to ions forming very insoluble compounds with membrane species, the diffusion potential for homogeneous mixed crystals can take non-zero values. TOTAL MEMBRANE POTENTIAL
Regardless of the distribution of current carrying species in a pure solid membrane, the measured cell potential A V c ’ , ignoring junctions and reference electrode potentials, is the difference of the interfacial potentials provided the interface processes are rapid and reversible. Thus for a silver chloride membrane (9) N. F. Mott and R. W. Gurney, “Electronic Processes in Ionic Crystals,” 2d ed., Dover Publishers, New York, 1964, p 31. (10) W. Jost, “Diffusion and Chemical Reactions in Solid Materials,” Dresden, 1937, p 80. (11) M. Dole, “The Glass Electrode,” Wiley, New York, 1941, pp 259--260. VOL. 40, NO. 10, AUGUST 1968
e
1435
or
for the cases of internal and external solutions containing excess Ag+ or excess C1-. Two other combinations are possible by referring to Equations 10a,b for cases in which the silver is in excess on one side, but chloride predominates on the other. Depending upon the type of cell in which the membrane is used, junction terms or noncancelling reference probe potentials must be added to obtain the measured potential. Whenever C A ~>> + 22/K.,/r*or the corresponding case of excess chloride, CCI->> 2 4 K B P / 7 * , Equations 17a and b reduce to
recently for AgBr membranes by Ross (12). If the ratio of activities of an interfering species to that of the chloride is less than the ratio of the solubility products, then the interference can be safely ignored. At the other extreme, the electrode responds chiefly to the interference activity. The quantitative description of interferences is similar to the theory of biionic potentials at ion exchange membrane electrodes. However, at crystalline membrane electrodes there are conceivably at least two mechanisms of interference response depending upon the miscibility of the salt formed by the interference and the membrane substrate. The biionic membrane potential equation is related to the response expected when the interfering salt is miscible in all proportions with the membrane. Complete miscibility is equivalent to the requirement imposed during the derivation of the biionic potential that the ion exchange membrane can exist in all possible conditions of counter ion concentration between pure species 1 and pure species 2. The hydroxide interference in the measurement of fluoride activity by means of a LaF8 membrane is an example where the complete miscibility approach is appropriate. Letting anion 1 with mobility f i represent ~ fluoride, and anion 2 with mobility fiz represent hydroxide (La3+ is virtually immobile) (13), the steady state diffusion potential is
or
provided the concentration of mobile fluoride and hydroxide defect species remains proportional to the bulk species concentration. The interfacial potential at x = L for reversible fluoride and hydroxide exchange is
The general expression for the reversible membrane potential for z+ - z- membranes is quite complicated. However, when Z+ = 1z-I = z , the expressions corresponding to Equations 17a and b are
A simple mixed crystal formation reaction of the type
AVc'
=
p(L)
-
CC-
p(-L)
SA
+ Az-
+
S A 2
+ AI-
is assumed where S may be LaFz+. The equilibrium constant has the form
+ (Cz- + 4 K s p / ~ ~ ' ) ' " ] Y- at L + (Cz- + 4 K s p / ~ * 2 ) 1 ~7-2 at -L ]
+
(19b) For the cases C+ or C- in excess, respectively, the measured cell potential is given by Equations 6a or b when Equation 19 is substituted for the membrane contribution. Apparent super- and sub-Nernstian responses with respect to the membrane potential can occur only with junctionless cells. Cell 11 using a membrane permselective to divalent cations such as Ca2f or Cuz+will not yield a plot of AVc us. log ucaz+ or log ucuzC with a slope of approximately 30 mV at 25" when solution I contains CaCh or CuClz. Instead a slope of 3/2 (2.303 RT/F) or approximately 90 mV will be observed (3). In an extreme example of an AgCl membrane in cell 11, the potential is independent of the solution activities.
62, the diffusion By recognizing the constancy of the sum potential can be expressed in terms of the site S concentration, K and solution activities. Upon adding in the two interface potentials, the measured cell potential in solutions of anions at activities al and u2,excluding references and junction potentials is r
1
INTERFERENCES AND BIIONIC POTENTIALS
When a membrane is exposed to solutions containing ions forming very insoluble compounds with the component ions of the membrane-e.g. iodide, bromide, and sulfide in the case of a AgCl membrane, these ions will affect the measured membrane potential and constitute interferences as shown 1436
ANALYTICAL
CHEMISTRY
(12) J. W. Ross, Paper #112, Pittsburgh Conference on Analytical
Chemistry and Applied Spectroscopy, Cleveland, Ohio, March 1968. (13) A. Sher, R. Solomon, K. Lee, and M. W. Muller, Phys. Reu. 144(2), 593 (1966).
This equation, which has the form of the Nicolski equation for glass membrane electrodes (14), applies to - ideal membranes with constant internal ionic mobilities, u1 and &, and constant internal activity coefficients, T1 and T2. Equation 22 is independent of time after establishment of surface equilibrium for fixed site membranes. The term in Equation 22b multiplying a2 can best be considered an empirical sensitivity or selectivity coefficient. For a constant internal solution, Equation 22 can be written AVc
=
-
RT
7In [al(-L) + Ks,,,a2(-L)] + const.
I\
iodide Activilies
-+-
10-7
k0
millivolts
IO-$ 10-10
‘ 5 si
If ~
.E
(22c)
When the activity coefficients depend upon the internal ion concentration to a power, an empirical form due to Karreman and Eisenman (1.5) may be useful. The factors multiplying the activities in Equation 22a differ from one species to the other, so that the selectivity coefficient may be greater or less than unity in Equation 22b. The species with the larger factor, say species 1, dominates the response over the wider range of activities. Whenever
Equation 22b reduces to
At lower values of al, species 2 constitutes an interference to the linear logarithmic response to activity of species 1 as given in Equation 24. For the lanthanum fluoride electrode, the selectivity coefficient for hydroxide ion interference is about 0.015. Thus, 10-jM or greater fluoride can be measured unambiguously provided the pH is kept below 9 to 10. Although experimental data (2).obey Equation 22a, there is no evidence that the mechanism conforms to this model. An adsorption model discussed below yields an equation of the same form. For immiscible salts such as AgCl and AgI, a different analysis is necessary to explain the interference of I- on measurements of acl- with a AgCl membrane. For all acl - in the test solution ( x = -15) such that
10
10-1
10-2
10-3
10-4
10-5
Chloride o c l i v i t i e s in test solution
Figure 3. Potential response influenced by interferences according to two models Potential corresponds to measured cell potential for a junction cell using a silver chloride membrane with variable activity chloride and constant values of iodide activity in the test solution Solid curves. Response to acl-(-L) by Equation 18b. Response to aI-(-L) by Equation 26 for pure immiscible phase model. Dashed curves. Response to iodide according to Equation 29b by an adsorption model. Potential increases positive downward,
ter has recently been reported for SCN- interference at a AgBr membrane (12). The diffusion potential in each pure phase is zero; however, potential drops occur at AgC1-AgI interfaces due to differing Ag+ activities in the two solids. These solid-solid potentials are cancelled by the compensating potential differences between Ag+ in solution at ( - L ) and in the AgI film. This result follows from the individual expressions for the electric potential by recognizing the constancy of the electrochemical potentials of Ag+ in equilibrium at each interface. Experimentally, the predicted abrupt .behavior does not occur because the formation of a AgI film of unit activity probably occurs by successive layering of AgI until unit activity is achieved. An appropriate expression can be derived by assuming a mixed adsorption isotherm for I- and C1- on AgCl. The solid surface concentrations of I- and C1- are ar-ycl- -
co 6,- = acl -TI -
the response given by Equation 17b or 18b is Nernstian in a,---. For lower chloride activities in the test solution containing constant U I -(-L),
a1-rc1-
l + A -
ac1-41-
AV,‘ =
This response is illustrated in Figure 3 for constant conditions at side L. Upon decreasing chloride activity, the potentials follows Equation 18b, then changes abruptly to a constant value given by Equation 26, determined by the iodide activity. This behavior results from the immediate formation of AgI on the surface of the AgCl membrane at an activity of chloride given by the equality in Equation 25. Behavior of this charac(14) B. P. Nicolski, Zhur. Fi:. Khim.,10,495 (1937). (15) G. Karreman and G. Eisenman, Bull. Math. Biophys., 24, 413 (1962).
60- =
1c-a1- I C 1 -
where 6, is the total number of sites equal to the Ag+ concentration of the solid. By summing potentials across the membrane, the cell potential excluding junctions and reference electrodes is found to be RT AV,’ = F
a1
-(L)
The interior solution will contain only the ion common to the membrane material so that the measured potential can be simply expressed, VOL. 40, NO. 10, AUGUST 1968
0
1437
Avc
RT
=
-In [ a c ~ - ( - L >+ Kaen8, m-(-L)] F
=
- -- F In [acl-(-L)
RT
+
+ const.
Kasens.ad-L)1 t
+ const. (29b)
where the ith sensitivity constant is chiefly determined by the ratio of the solubility product of the membrane material divided by the solubility product of the silver salt of the interference. This equation can be easily put in an equivalent form for other membranes. It should be noted that these equations have the same form as Equation 22 when the mobilities of the anions are set equal to each other. Also Equation 28 reduces to Equation 26 for the appropriate relation of extant iodide to chloride activities on side x = -15. SOURCES OF NON-NERNSTIAN RESPONSE
Nernstian response of selective ion membrane electrodes is that given by Equations 18a,b. Deviations can be considered as apparent or real. Apparent spurious results may occur for solutions containing ions which form soluble complexes with one or more of the ions constituting the membrane. However, until the membrane dissolves, the electrode response faithfully corresponds to the existing low level of species provided the surface exchange rate remains rapid. Similarly, solutions of precipitants forming salts more insoluble than the membrane itself will ultimately destroy the membrane by metathesis. But in the long-lived transient state, the potential will be given by equations of the form of Equation 17. Apparent non-Nernstian response of junctionless cells has already been mentioned. Most frequently, non-Nernstian response of the membrane electrode in solutions free from interferences--e.g., less than 2.303 RT/zF millivolts per decade activity increment-occurs because of gross imperfections in the membrane or leaks around the membrane. Whenever leaks are present such that co-ions are able to move from one solution to the other, the potential change per decade is reduced. Ultimately for large holes, the response is that of a liquid junction. Another failure of Nernstian response is expected in dilute solutions approximating the solubility of the membrane material as demonstrated in Figure 2. Oxidizing agents which attack crystalline membranes by removing one of the component ionic species, will change the ionic activities at the membrane interface and therefore cause spurious potential measurements. Because these processes are often slow--e.g., Cr2072-attack on AgBr (12)-they can be treated quantitatively by the same methods used to evaluate potential shifts due to the passage of current through crystalline membranes, These effects are discussed in detail in the following paper. RELATION OF CRYSTALLINE MEMBRANES TO ELECTRODES OF THE SECOND KIND
Electrodes of the second kind--e.g., Ag/AgCl-are made from a metal and its insoluble salt, often without regard to the extent of coverage of the metal by the salt. In an extreme case, the solution containing chloride is simply saturated with respect to solid AgCl and a silver wire is dipped into the solution. This example is mentioned to point out that interfacial processes essential for membrane electrodes may not necessarily be involved in electrodes of the second kind made from the same salt used as a membrane. Rather, 1438
ANALYTICAL CHEMISTRY
the electrode may respond as an electrode of the first kind to the equilibrium cation activity. Although adherent films are desired on the metal substrate, there is no proof that the film is free from pores which admit solution to the metal surface. Consequently, the mechanism of the operation of a given electrode of the second kind may not be identical with that for a membrane made from the corresponding salt. It is likely that electrodes in which the metal is entirely buried in the salt will function as an electrode of the second kind through the mechanism of reversible ion exchange at the solution-salt interface. However, in addition to rapid surface equilibrium at the solution interface and acceptable ionic conductivity within the salt, a further stringent requirement occurs at the salt-metal interface: rapid, reversible ion exchange between the metal and its salt. When this condition is met, equating the electrochemical potential of the ion in the metal to that of the ion in the salt is permissible. The derivation of Equation 5 makes use of this assumption. The feasibility of rapid ion exchange of a common ion between two solid phases depends upon the energy barriers. For example, the energies of removal of Ag+ from AgCl and Ag are nearly equal: 4.66 eV for the first process by a MottLittleton calculation (16), while Klein and Lange (17) give 5.86 eV for silver ion removal from Ag metal. It is not unreasonable that the solid-solid surface exchange may occur with the same facility as the reversible Ag-Age (solution) reaction. Alternatively, at thermodynamic equilibrium, a metal salt in contact with its metal must contain some, albeit small, quantity of the metal as positive ions and electrons in the conduction band of the salt (18). In this case electrochemical equilibrium may be maintained by electron transfer between metal and salt. Examples of reversible membrane electrodes using metallic backing occur in the literature (19-21). These are glass membrane electrodes backed with mercury, copper, silver, or platinum. Response to pH is normal at the external interface, while the metal serves as an external contact to the interior membrane interface and provides its own lead to the measuring circuit by replacing the normal internal reference electrode. Presumably the exchange between metal ions in the metal and in the ion exchanging glass surface is rapid and reversible, although no experimental studies of the mechanism have been reported. LIST OF COMMON SYMBOLS
ionic or molecular activity in solution ionic or molecular activity in solid Ci ionic concentration in solution Ci = ionic concentration in solid Co = fixed site concentration in solid F = Faraday in coulombs/equivalent yi = molar ionic activity coefficient in solution r., = mean molar ionic activity coefficient in solution 7 % = molar ionic activity coefficient in solid Ji = ionic flux in moles/cm2 second K = interface equilibrium exchange constant ai
ai
= = =
(16) N . F. Mott and M. J. Littleton, Trans. Faraday SOC.,34,485 (1938). (17) 0. Klein and E. Lange, Z . Elektroclzem., 44, 562 (1938). (18) N. F. Mott and R. W. Gurney, “Electronic Processes in Ionic Crystals,” 2nd ed., Dover Publishers, New York, pp 251, 259, 260. (19) M. R. Thompson, Bur. Std. J . Res., 9,833 (1932). (20) P. A. Kryukov and A. A. Kryukov, Russ. Patent 51,509 (July 31, 1937). (21) H. Bender and D. J. Pye, U. S. Patent 2,177,596 (May 17, 1938).
+
response sensitivity defined in Equation 22
R T
solubility product
fii
distance corresponding to membrane interface at the internal solution distance corresponding to membrane interface at the external or test solution denotes mobile interstitial or vacancy species electrochemical potential in joules/mole standard ionic solution species chemical potential in joules/mole standard ionic solid state species chemical potential in joules/mole inner potential of the solution phase in ordinary volts = inner potential of the solid phase in ordinary volts
= gas constant in joules/mole degree = absolute temperature
= ionic mobility in solid, moles/cm2 second joule AVc = entire membrane cell potential x = distance in membrane; center x = 0 = ionic charge with sign in equivalents/mole zi z = absolute ionic charge in equivalents/mole
RECEIVED for review April 2, 1968. Accepted June 3, 1968. Presented in part at the Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, Ohio, March 1968. Work supported by the University of North Carolina Materials Research Center under Contract SD-100 with the Advanced Research Projects Agency.
Theory of Potential Distribution and Response of Solid State Membrane Electrodes II. Non-Zero Current R. P. Buck Department of Chemistry, University of North Carolina, Chapel Hill, N . C . 27514 Under current drain, due to the finite input impedance of measuring devices, potential measurements on solid state membrane electrodes are subject to a variety of distortions. A general theory of slow ion transport processes has been developed to give quantitative descriptions of activation and concentration polarization overpotentials and internal iR drops. Values of the bulk membrane ionic mobilities, first order surface rate constants and Nernst layer thickness are assigned to minimize potential distortions for a given current density. The equivalent theory of potential distortion due to attack of the membrane by oxidizing and reducing agents is presented. Potential measurements at membranes under redox reagent attack offers a method for studying rates of heterogeneous redox processes.
IN the foregoing paper ( I ) , the assumption of rapid, reversible interfacial ion exchange or ion transfer reactions implies that only interfacial and diffusion potentials contribute to the membrane potential. At finite current or under an applied low voltage step, the excess potential, over and above the potential of zero current, may appear exclusively as an internal iR drop within the membrane and in the solution between the probes, provided certain experimental conditions are satisfied. At low current densities with vigorous stirring, the potential drops through the solutions will be ohmic because the concentrations outside the double layer remain space- and timeindependent. Uniformity of concentration with current flow is achieved by restricting current densities (amps/cm*) drawn by the measuring apparatus to values less than approximately x electroactive species concentration in moles per liter. Potential drops in the reference probes with current flow can be made negligible by using low impedance, high area electrodes whose current-voltage curves are essentially delta functions on a fixed potential. Junction potentials, which -
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(1) R . P. Buck, ANAL.CHEM., 40, 1432 (1968).
are time- and current-dependent, are serious contributions to uncertainties in potential measurements at zero current, Chemical attack of the membrane materials, by an oxidant or reductant, can also distort the measured potential at zero current, as will be shown subsequently. IA addition to the possibility of concentration polarization with its attendant potential drop Apep and the creation of iR drops, passage of current by the measuring device may produce potential errors when the ion transfer reactions are not rapid or reversible. Incremental potential drops for the general case of slow ion transfer reactions, Apk, accompanied by concentration polarization and iR drop conspire to shift the measured cell potential AV, positively for negative current flow as shown in Figure 1. The opposite shift occurs for positive currents. In the present paper, the magnitudes of these potential distortions are developed quantitatively. Approximate minimum numerical values are established for the ion transfer rate constant and the membrane conductivity. A maximum value for the measuring current is established to reduce the error in the measured cell potential to negligible levels. POTENTIAL INCREMENTS FROM SLOW INTERFACE REACTIONS
The reversible interface reaction model, while suitable for most ion exchange membranes, cannot be generally valid, and the idea of slow, potential-dependent interfacial ion transfers may be needed to interpret lack of response of some crystalline membranes. The following ad hoc theory is based on the corresponding treatment of slow, potential-dependent electron transfer interfacial processes. The membrane material is assumed to be stoichiometric and a pure ionic conductor. The membrane potential arising from applied current (or applied voltage) must be less than the band gap of the salt so that electronic conductivity remains small compared with ionic conductivity. High apVOL. 40, NO. 10, AUGUST 1968
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