CONCLUSIONS The present work demonstrates that linear sweep phase-selective second harmonic ac voltammetry should provide one of the most sensitive electroanalytical techniques for the determination of species exhibiting reversible or close to reversible electrode processes. For the reversible class of electrode process, the equations describing the voltammograms at stationary electrodes are essentially the same as those obtained under dropping mercury electrode (polarographic) conditions. Thus, the technique can readily be used in a systematic fashion like second har-
monic ac polarography. No restrictions, other than AEwt u t , are placed on the use of fast scan rates as charging current contributions are negligible and the resolution is somewhat better than derivative dc linear sweep techniques. The readout of the current-voltage curve and the nature of the experiment is most suitable for automation of the experiment and this aspect of the technique is currently under investigation.
>>
RECEIVED for review October 15, 1973. Accepted February 20, 1974.
Theoretical Treatment of the Selectivity and Detection Limit of Silver Compound Membrane Electrodes Werner E. Morf, Gunter Kahr, and Wilhelm Simon Laboratorium fur Organische Chemie der Eidgenossischen lechnischen Hochschule, Zurich, Switzerland
Based upon a universal integral equation for the steadystate EMF of an electrochemical cell containing any type of ion-selective membrane electrode, equations are derived that describe the selectivity behavior and the detection limits of different solid-state membrane electrodes. Using published values for solubility products and complex stability constants, the selectivities computed for silver compound electrodes that respond to anions, cations, as well as neutral complex forming species, are in perfect agreement with measured data. The detection limit is either dictated by the solubility of the membrane material or given by the activity of the silver defects in the membrane surface, whichever is larger. In consequence, anomalies in the response to I - and S'- ions with extremely high slopes have to be expected: this limits the useful activity range of the respective silver compound electrodes when using unbuffered sample solutions.
During recent years, solid-state membrane electrodes selective towards cations, anions, as well as other species forming complexes with membrane materials have been used for a variety of analytical applications (1-5). Here, a knowledge of different parameters, especially selectivities and detection limits, is of utmost importance. Corresponding theoretical treatments have been restricted to rather limited cases such as the anion selectivity of silver halide (6-11) and LaF3 (7, 8) membrane electrodes, and the detection limits of AgCl (8, 12) and LaF3 (12) solid(1) R . P. Buck, Anal. Chem., 44, 270R (1972). (2) J . Koryta. Anal. Chim. Acta, 61, 329 (1972). (3) G. J . Moody and J . D. R. Thomas, "Selective Ion Sensitive Electrodes,'' Merrow Publishing Co., Watford, Herts, England, 1971, (4) R. A. Durst, Ed., ''Ion-Selective Electrodes," National Bureau of Standards, Spec. Publ. 314, Washington, D.C., 1969. ( 5 ) E. Pungor, Pure Appl. Chem., 27, in press. (6) E. Pungor and K. Toth, Analyst ( L o n d o n ) , 95, 625 (1970); Hung. S o . Instrum., 18, 1 (1970). (7) J. W. Ross, Jr., in Ref. ( 4 ) . (8) R. P. Buck, Ana/. Chem., 40, 1432 (1968). (9) A. K . Covington, in Ref. ( 4 ) . (10) W. Jaenicke, Z. Elektrochem., 55, 648 (1951); W.Jaenicke and M . Haase. Z. Elektrochem., 63, 521 (1959). (11) G. P. Bound, B. Fleet, H. von Storp. and D. H. Evans, Ana/. Chem., 45, 788 (1973). (12) J. Havas. IUPAC International Symposium on Selective Ion-Sensitive Electrodes, April 9-12, 1973, Cardiff, England.
1538
state electrodes. The object of the work reported here is the presentation of generally applicable equations that describe the selectivities and detection limits of different solid-state membrane electrodes.
THEORY Based upon a set of clearly specified assumptions (131.9, a universal integral equation was derived (13) which describes the steady-state EMF of an electrochemical cell containing any type of ion-selective membrane electrode. In the application of this equation to homogeneous solidstate membranes, contributions to the EMF of the cell due to diffusion potentials within the membrane may be neglected (13)[see also (8)] and one obtains: EMF
=
E,,
RT +I 2 ,F
a,'a,(d)
n a m
where a,(O), a , ( d ) = activity of ion I*, on the membrane surface contacting the sample and reference solution, respectively: For homogeneous membranes: a,(O) = a,(d). a,', a," = activity in the boundary of solution contacting the membrane on the sample and reference side, respectively. These boundaries are in equilibrium with the membrane phase, which must not necessarily hold for the bulk of the sample (activity a,) and reference solution. For a given reference system, a," is constant throughout. Since membranes prepared from silver compounds Ag,X are of special analytical significance, they will be treated in detail here. Equation 1 therefore may be reduced to: EMF
=
E4g0
RT +T -lnaAg'
(2)
Using the solubility product L A , ~ , x :
LA,:, = a.Ag%' Equation 2 can be rewritten in the form:
(3)
(13) H.-R. Wuhrmann, W . E. Morf, and W. Simon, Helv. Chlm. Acta, 56, 1011 (1973). (14) W. E. Morf. D. Arnrnann, E. Pretsch. and W. Simon, Pure Appl. Chem., 36, 421 (1973). (15) G. Eisenman, in Ref. ( 4 ) .
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 11, S E P T E M B E R 1974
E MI
[mvl
MEMBRANE
ApCI
600 -
Exo-
RT
2
l n a d (4)
which holds for sample solutions containing as anions only X2-with an activity ax' a t the boundary contacting the membrane. Lower Detection Limit. The intrinsic lower detection limit, as dictated by the membrane material, is related to the minimal activity a . 4 g , m l n ' which is given by the dissolution of the membrane as well as the silver defect concentration a t the membrane surface (16, 17).A successful rationalization of all experimental results (17) (see below) is achieved by postulating a distribution of silver ions between interstitial sites [mainly Frenkel defects ( I S ) ] and the solution at the membrane surface. The defect activity CY at the membrane surface is roughly constant for a given
500
-
LOO
-
300 200 IW0-
Table I . Numerical Values Used in the Computation (activities i n mole liter-1) Solubility product (25 " C ) -log LAmZ
(1%
Compound
Ag,S AgBr AgSCN AgCl AgOH (''.Ag>O
"
-t ',/?HyO)
5 . 5 (est.). ( 1 7 ) 6 . 0 (est.) (17) 6.3 (211
48.54 16.08 12.30 11.92 9.75 7.68
Complex
Stability constant (25 " C ) 1% P n
Ag(CN)zA g ( S 2 0 ~ ) -2~ Ag(SC("d!)2' Ag(SOS)?'Ag("3)z HgI HgIz HgL HgI4'-
1 8 . 7 5 (20) 1 3 . 2 (20) 12 (est.) (17) 8 . 4 5 (20) 7 . 2 (20) 12.87 (20) 2 3 . 8 2 (20) 2 7 . 6 0 (20) 2 9 . 8 3 (20)
Defect activity (25 " C ) -log a
8.8
(21)
+
+
The same value is applicable for pressed bodies of AgiS mixed with silver halides (I7).
set of experimental parameters but may be changed by a different conditioning of the membrane [see (16)J.Assuming a constant value of CY for a given silver compound as membrane material, the following relationship is acceptable to describe the deviations in activities between the boundary ( a ' ) and the bulk ( a ) of sample solution: a\-'
- a,:
=
Aa,'
- a,)
+
(5)
CK
Silcer Ion Response By combining Equations 3 and 5 and restricting to a sample solution of a silver salt containing anions which do not interfere with the membrane material ( a y = 0), it follows: a4t/ +'
- a+,,"(u,,
+ a ) - ZL,,\
=0
(6)
To obtain an explicit expression for a ~ and ~ therefore ' the EMF response of the cell according to Equation 2, the following cases will be explored: Case 1. Membrane Material AgX ( z = 1); L A p X >> a 2 . This situation is applicable, for instance, to AgCl (see Table I). Equations 2 and 6 give a function which is in agreement with one derived by Buck (8) and Havas (12) and is similar to an equation used by Pungor (6): (16) G. Kahr, W . E. Morf, and W . Simon, Anal. Lett., in preparation. (17) G . Kahr. Dtss. ETH Zurich Nr. 4927, Zurich, 1972. (18) F . A Kroger, "The Chemistry o f Imperfect Crystals," North-Holland Publishing Co.. Amsterdam, The Netherlands, 1964. (19) "Handbook of Chemistry and Physics." 52th ed., Chemical Rubber Publishing Co., Cleveland, Ohio, 1970. (20) L. R . Sillen and A. E. Martell, "Stability Constants of Metal-Ion Complexes." Spec. Pub/. No. 17, The Chemical Society, London, 1964. (21) R . Matejec. H. D. Meissner, and E. Moisar, in "Progress in Surface and Membrane Science," J. F. Danielli. M. D. Rosenberg. and D. A . Cadenhead. Ed.. Academic Press, New York and London, 1973.
According to Equation 7, the so-called Nernstian response ~ shown ~ in Figure 1( 1 7 , 2 2 ) . holds only for aAg>> v ' T ; Aas Case 2. Membrane Material AgzX; >> z L . A ~ ~ x . This situation holds, for instance, for AgI or Ag2S (see Table I). Equation 2 becomes:
EMF
=
EAgo-t
RT
ln(aAg 4- a )
(8)
As shown in Figures 2 and 3, a Nernstian response is observed only for uAg >> cy (see Table I). Here, the detection , ~ by ~ CY ~(defect ' activity of the memlimit a ~ ~ is given brane material), whereas in case 1 the corresponding acX of the membrane matetivity becomes V / Z A ~ (solubility rial). A n i o n Response. For sample solutions containing only the anion Xz- inherent to the membrane as well as cations not interfering with the membrane material, one obtains:
In analogy to the treatment of the cation response, cases 1 and 2 mentioned above are discussed. 1) For the anion response of membrane materials such as AgCl, the following equation is derived:
(22) J. Bagg and G. A. Rechnitz. Anal. Chem., 45, 271 (1973)
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
1539
EMF I
[mvl
I
1
MEMBRANE
I
Agl
lmvl UEYBRANE
600 -
4q25
LOO-
200 -
0-
-2w-
-6
-7
-5
-1
-3
2
-1
0
log a
Figure 2. EMF response of a silver lodlde membrane electrode to A g + and I Calculated solid lines Experimental Ag+ (0) I - ( 0 )
Figure 1 [see also (S)] confirms the symmetric response to cations and anions as demanded by Equations 7 and 10. 2) In contrast to l), the anion response of membrane materials such as AgI and AgzS becomes more complex:
-7
-6
-5
-I
-3
-2
-I
0
lcqa
Figure 3. EMF response of a silver sulfide membrane electrode to Ag+ and S z Calculated: solid lines Experlrnental Ag+ ( 0 ) S2: (0)
As a reasonable steady-state approximation, the following relation holds:
(13) In similarity to Equation 5, the activity aAg' of the silver ions formed according to Equation 12 becomes:
for a,
=
for a,
CY
-: z
EMF = ExO
< CY-: EMF z
=
RT -z F In( L A g 2 s / z z ) 1 fz+lb
E4g0
RT +F ln(a - za,)
=
(llc)
According to Equations l l a - l l c , a sudden increase in the EMF with decreasing activity a x must occur at ax = Ly/z Theoretically, slopes in the EMF response of up to 55 and 1010 times the Nernstian slope (25 "C) have to be expected in this region for AgI and AgzS, respectively. This means, of course, that the EMF may become rather unstable near a x -- a / z A careful study of the EMF response of different membrane materials finally confirmed this expectation [Figures 2 and 3, see also (16)].Since CY is about to 10-7M (Table I) for membranes in practical use, the detection limits must be in the same range. As mentioned above, the activity of silver defect ions a t the membrane surface depends to a certain degree on the conditioning of the membrane. Thus, the continuous use of AgI membranes in iodide solutions may lead to a slight reduction of CY and the corresponding EMF response curve may become rather similar to the type found for AgCl membranes ( c f , Figures 1 and 2 ) . On the other hand, statements referring to a AgzS membrane electrode such as "The electrode will respond to concentrations between 1 and lO-?M in total silver or sulfide, or to free sulfide ion activities down to 10-17M or below in the presence of sulfide complexes, and to free silver activities to 10-25M in the presence of complexed silver ion" (23) are somewhat misleading. Response of Silver Halide Membranes to Different Cations. One of the few cations which reacts with silver compound membrane materials is Hgz+. The overall reaction with silver halides AgX may be described as follows: Hg2+ nAgX === HgXnL-" nAg+ (12)
+
(23) R A Durst in Ref ( 4 )
1540
+
= CnaHgx,'
QAg'
+a
(14)
With the expressions for the complex formation and membrane solubility a t the phase boundary: ang);,' = Pnanglaxtn ( 15) a.4,'ax' = LAps
(16)
one obtains an implicit function and, therefore, numerical values for aAg':
(17) Since the weighting factor of C L H ~in Equation 17 is identical to the average number ri. of halide ions coordinated to the Hg2+ ions at the phase boundary (mean degree of complex formation), Equations 2 and 16 may be combined:
EMF
=
EA:
RT +F
ln(ka,,
+ a)
(18)
In Figure 4, the computed response (Equations 17 and 18; Table I) of a AgI membrane electrode to variable activities of Hg2+ is compared to experimental values. Obviously, there is both experimental and theoretical evidence (Figure 4) that the response of such sensors is neither Nernstian nor linear. In the lower activity range of Hg2+ to lO-5M), the response curve may very crudely be approximated by a straight line: the calculated slope, however, is only about 48 mV (25 "C). At high activities (above 10-1M) the slope of the calculated response curve approximates an asymptotic value of 29.6 mV (25 "C). Response of Silver Halide Membranes to Different Anions. Membranes consisting of pure silver halides AgX or of AgX mixed with Ag2S (7) may be used for the determination of anions Yz- forming sparingly soluble salts AgzY with Ag+. Such a determination of anions Yz- is
ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 1 1 , SEPTEMBER 1974
I
EMF
MEMBRANE
ApI
500 1
1
EXP
log K';!
5 9 2 mV
,'
296mV
400
j
p.id
0
XN-
-7
-6
-5
-b
-1
-2
-I
0
log a
Figure 4. EMF response of a silver iodide membrane electrode
-10
to Hg*+
-0
-6
-b
-2
0
2
4
6
8
I4 I- : !K
Calculated solid lines Experimental circles
where KxyPot is given by Equation 22 with z = 1. In contrast to case a, where either X - or Y- determines the EMF of the cell, both anions contribute simultaneously in case b. A typical example is the system AgC1-AgBr (8, 24). Response of Silver Halide Membranes to Different Ligands. In contrast to the previous chapter, negatively charged ligands as well as neutral species forming discrete complexes with Agf are considered here. Since such ligands react with the membrane material according to:
nPEquation 2 leads to:
Eyo-
RT t~
+ AgX
+===
AgL,,'-"' 4- X-
i26)
forming X - , a detection of these ligands becomes possible (2). For a sample solution containing the ligand L"- and possibly anions X - a t an activity ar, and a x , respectively, the approximation holds: lna, (20)
The selectivity factor K X y P O t is defined by: =
Y
CALC
Figure 5. Comparison of the experimental and the calculated anion selectivity of different silver halide membrane electrodes
possible only if there is an equilibrium between the boundary of the sample solution and AgzY deposited on the membrane. Two types of such AgzY coatings are discussed here: a ) Single Phase Ag,Y Covering Completely t h e AgX Phase. Such a phase is formed by contacting an AgX membrane with a sample solution containing Yz- only a t an activity ay well above the detection limit. Although additional potentials may develop between this new phase and the AgX phase, such contributions are neglected in the following treatment. IJsing the solubility product:
EMF
I2
RT Exo - 7 ln(KxuPO'aYl*)
(21)
and becomes:
The agreement between Equation 22 and experimental results is perfect [see Figure 5 and Pungor et al. (S)]. For sample solutions containing both X - and Yz-, Equation 20 describes the E M F up to an activity ax (7, 8 ): (23)
aLt
+ Cna4gL,'= a L
(27)
A somewhat more detailed treatment will be given in the Appendix. For sufficiently high activities al, or a,,', the complexed silver ions dominate relative to free AgT at the phase boundary and therefore Equation 5 is applicable in the form:
In addition, the solubility product (Equation 16) and the description of the complex formation: =
13,a,,'a,.'"
( 29)
are used in the following treatment. Since Agf generally forms linear 1:2 complexes ( n = 2 in Equations 26-29) the following quadratic equation in a.4p' may be derived:
A t higher activities a x , the phase Ag,Y is no longer existent and the EMF is given by:
EMF
=
Exo -
RT
F
lna,
(24)
Such a behavior is observed when contacting an AgBr membrane with a sample solution containing Br- and S C N - (7, 8). b ) Mixed Phase AgY-AgX (8, 2 4 ) . A detailed treatment has been given elsewhere (8, 23). For two monovalent anions X - and Y - at activities a~ and a y well above the detection limits, these models lead to:
EMF
=
RT EXO- - ln(ax
+ KXyPoray)
(241 W Jaenicke, Z. Elektrochem., 5 7 , 843 (1953)
(25)
LA,\ = 0 ( 3 0 ) This equation may be simplified for the following two cases: a ) S a m p l e Solution Containing Ligand L I ' - Only (au = 0). Equations 4, 16, and 30 combine to:
with K = P?L.A,S (32) The parameter K stands for the relative complex formation constant related to Equation 26 with n = 2. In Figure 6,
ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 11, S E P T E M B E R 1974
1541
..
EMF - E ,
1000-
1000-
-
900 -
900
BW
-
700
-
€00
-
500 4W,
01 -10
,
,
. ,
-8
-6
-1
,
, -2
,
,
,
0
2
,
,
,
4
,
04
6
-9
,
,
,
-8
-6
I
,
,
I
-4
-2
, 2
0
4r
6
Comparison of the calculated (solid and dashed lines) and the experimental (circles) E M F response of different ligand/ membrane combinations. The ligand activities a L are l o - ' , and 10-3M for A , 8,and C, respectively. Dashed lines denote silver sulfide membranes
Figure 6.
the relative EMF of a given sensor (EMF-Exo) is plotted as a function of K for different activities ar, (for AgzS membranes, see later). The surprisingly good agreement corroborates the basic assumptions made. Three major parts are discerned in Figures 6a-6c. Region 1. (right hand side in Figures 6a-6c): 4~ >> 1. Equation 31 simplifies to:
(33) The term YZ aI, is due to the fact that the complexes AgL21-2y are predominant a t the phase boundary and therefore a formation of ax' = 1h U L (Equation 26) has to be expected. For such values of K , the EMF of the corresponding electrochemical cells is independent of K (horizontal lines in Figures 6a-6c) and the linear range of the response of the electrode system to the ligand activity in ) a slope of -59.16 the sample solution (e.g., a ( , ~ - shows mV (25 "C). Region 2 (center part in Figures 6a-6c): 1 >> 4K >> (a/ a L ) 2 . Equation 31 is reduced to:
EMF
=
E,[) -
RT
ln(ljjla,,)
(34)
In this case, the free ligands L"- dominate at the phase boundary, the activity of the anions formed (Equation 26) is, however, larger than the defect concentration cy. In similarity to region 1, the EMF response lfas a slope of -59.16 mV (25 "C). In contrast to the previous case, the EMF now depends both on the ligand activity as well as the complex formation behavior of the ligand. The center portions in Figures 6a-6c, therefore, have a slope of -59.16 mV (25 "C) (see Equation 34). Region 3 (left hand side in Figures 6a-6c): 4~
