1757
IMPEDANCE BEHAVIOR OF BIS(DIETHYLENETRIAMINE)COBALT(III)-(II)
Faradaic Admittance of the Bis(diethy1enetriamine)cobalt (111)-Bis(diethy1enetriarnine)cobalt (11) System
by Peter J. Sherwood and H. A. Laitinen Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Illinois (Received January 17, 1060)
61801
The impedance behavior of the bis(diethylenetriamine)cobalt(III)-bis(diethylenetriamine)cobalt (11) system at the dropping mercury electrode has been measured over the range 200 Ha to 20 KHz. The results could not be interpreted in terms of the classical model in which specific adsorption is ignored and separability of faradaicand nonfaradaic currents is assumed. The Senda-Dnlahay model was found inapplicable; the SluytersRehbach-Delahay model yields results consistent with the adsorption of the single species (the oxidant), a large charge-separation capacitance, and a rapid charge-transfer reaction. Limits of error in the evaluation of the adjustable parameters are discussed.
Introduction The faradaic admittance of the tris(ethy1enediamine)cobalt (III)-t ris (ethylenediamine) cobalt (II) system has been found’ to be anomalous, and has been interpreted in terms of specific adsorption of the reactants a t a mercury surface. I n the present study, another cobalt(II1)-cobalt(I1) system was chosen to see whether analogous behavior would be found and whether the electrode impedance could be interpreted in the light of recent t h e ~ r i e s . ~ - ~ The amine chosen for this study was diethylenetriamine (dn), a tridentate ligand. High formation 48) assured the constants (log &I1 = 14; log K P metal would be essentially entirely coordinated, even without added dn, or in the compact double layer, where the dn concentration could be considerably less than in bulk solution.
-
Theoretical Section The experiment considered is the determination of the equivalent impedance or admittance of an electrochemical cell at a given frequency, w , of sinusoidal potential variation. The sine wave alternating voltage is superposed onto a constant dc potential, not necessarily the equilibrium potential of the redox couple. The electrode reaction is ox ne --t red. The a priori inseparability of faradaic and nonfaradaic currents has been suggested by Delahay;2 based on his three fundamental equations, he later treated the problem of faradaic impedance.a Sluyters and coworkers4 have questioned Delahay’s being surface excesses; r = expansions of dI’i/’dt’s, ri% Fox 4- Fred. According to Timmer14 I’ may be expressed as a function of three independent variables, cox,Cred, and E, with the c’s representing concentrations
a t the electrode surface, and E the instantaneous potential of the indicator electrode
r
= r(cox,Cred,
E)
For systems with large rate constants, use of the Nernst equation reduces the number of independent variables by one, and we may write
or
depending on our choice of variables. If either ox or red is only weakly adsorbed at the electrode, eq 1 can be approximated by (as pointed out by a referee, these approximations are also valid a t saturation coverage, where (br/dci)j+i is also small) dt
dE cox dt
or
+
Equation 2a corresponds to eq l a with (br/bc0,). small and eq l b and 2b with (br/bcred)E small. In Delahay’s original treatment of the faradaic impedance, and (1) H.A. Laitinen and J. E. B. Randles, Trans. Faraday &e., 51, 54 (1955). (2) P.Delahay, J . Phys. Chem., 7 0 , 2373 (1966). (3) P. Delahay and G. Susbielles, ibid., 70, 3150 (1966). (4) B. Timmer, M. Sluyters-Rehbach, and J. H. Sluyters, J . Electroanal. Chem., 15, 343 (1967).
Volume 74s Number 8 April 16, 1070
PETERJ. SHERWOOD AND H. A. LAITINEN
1758
also in a paper by Sluyters-Rehbachj6 an assumption mathematically equivalent to eq 2 was made, namely
(3) I n neither of these papers was only one species assumed adsorbed; however, in a later paper by Timmer,4 impedance data for In(II1) in 1 M KCNS were found to fit a model in which the reduced form was not adsorbed (Fred = 0). It was also found in their analysis that the same results were obtained whether eq 3 or the more rigorous eq 1 was used in the derivation. From eq 2 that result is explicable by assuming, as did Timmer, that only one species is specifically adsorbed, so that the first term in eq 1 may be neglected. Assuming eq 3 is valid, it may be shown that
N
1 1
0
t
K = 125 pf /cm"
1
c:
s
and
200t
\
1
1
3
wherein Y' and Y'' are the real and imaginary components of the cell admittance after correction for the solution resistance. The parameters p and K are defined according to Sluyters-Rehbach6 by
w11:sec-1'2
Figure 1. Calculated W ~ / ~ / Y ' V S curves. .
The function w'/'/Y' may be written from eq 4 as drox K=-nF-+u dE Cox
Qred
u
Wva
- --
(nF-
Y'
)d;:d
with 8, the charge-transfer resistance, and u, the Warburg coefficient, given by
+ +2
p2 2p p + l U
(5)
+ o'/'Kp
The limiting value of p at zero frequency is zero, since p = p'w'/'. Proceeding from eq 5
8 = RT/nFi,
and uox
+ exp(-4>1 = + RT[1 -+ exp(4)l n2F22/2Dox(cbox + RT[1
n 2 F 2 a x ( c b o x
ured
dl'k'red)
From eq 6 it is evident that a plot of wl/l"/Y' vs. wl/ should have an intercept of 2a. Taking the derivative of eq 5 with respect to wl/' gives
+ 2)
w l ~ ' p ' ( p ' / u- 2K)(p'w1/' 1/2 1 U Kp'w]l
['++
dl/'Cbred)
+
as ul/'becomes small lim ul/z
The charge-transfer resistance and the Warburg coefficient are as defined by Randles;6 D is the diffusion coefficient, Ell2is the polarographic half-wave potential, cb is the bulk concentration, io is the exchange current and the other symbols have their usual density a t EDC, significance. The Journal of Physical Chemistry
--*
0
I)$(&[
0
=
1/a2 = O
so the limiting slope is zero, and u'/'/Y' cannot be linear with LO'/', as implied by Sluyters-Rehbach6 even in the case where K '= 0. To illustrate this point, (5) M. Sluyters-Rehbach, B. Tirnmer, and J. H. Sluyters, J . Eleclroanal. Chem., 15, 151 (1967). (6) J. E. B. Randles, Discuss. Faraday Soc., 1, 11 (1947).
IMPEDANCE BEHAVIOR OF BIS(DIETHYLENETRIAMINE)COBALT(III)-(II) Figure 1 shows plots of w’/’/Y’ us. w’/’ for p’ = 1.3 X 10-2, a = 120 ohm cm2/sec-1’2, K = 143 pF/cm2 (Figure la), and for p’ = 0.5 X a = 110 ohm cm2/sec-1/2, K = 125 pF/cm2 (Figure lb). These values for p ’ , a, and K are taken from Table I1 of Sluyters-Rehbach’s data for 0.5 m M Pb(I1) in 0.01 M HC1 and 1 M KC1. Equation 4 was used to calculate w1/’/Y’. The experimental data in ref 3 were obtained over the frequency range 320-4500 Hz; thus the smallest value of wl/* was 45. I n Figure l a , a straightline extrapolation to w l / ’ = 0 using only points with wl/’ 2 45 gives 2a = 233, while the actual function W ~ / ~ / aY ’sigmoid , curve, has an intercept of 240, as predicted by eq 6. Here the error of +1.7% is hardly enough to detect by comparison of half the intercept with a calculated from polarographic data. However, Figure l b shows a much worse case: the straightline intercept is 234, compared to the correct value of 220. This error of +6% is enough to cause quite a variation in the calculated charge separation parameters. Such a variation is evident from SluytersRehbach’s calculations of p’, 6, K , and cd from a single set of impedance data, but using different values for a ; thus, choosing a = 115, they calculated p’ = 0.008, 0 = 0.9, K = 124, cd = 27, but choosing a = 120, they obtainedp‘ = 0.013,B = 1.6, K = 143, c d = 41. To calculate a accurately from polarographic data requires precise evaluation of at least one of the diffusion coefficients; in this work Do, and Dred were determined from single drop current-time curves, to a precision of about 1%. The accuracy of these measurements is not directly verifiable, however. An alternative not used in this work is to treat a as unknown and find the best fit of the experimental data to a model with C d l , K , p‘, and a as parameters to be found. I n this connection it should be mentioned that as long as no present model completely accounts for observed faradaic impedance behavior, the Sluyters-Delahay model with three parameters is to be preferred t o the Senda-Delahay model’ which has six.
1759
employed for impedance measurements, using an external HP 241A oscillator (Hewlett-Packard, Palo Alto, Calif.) as source and a GR1232-A tuned amplifier (General Radio Co., West Concord, Mass.) as detector. The advantages of this transformer bridge over conventional ac Wheatstone bridges as previously used in this laboratory were lower noise pickup, due to the low impedance of the bridge transformers, and simpler grounding arrangements, obviating the need for Wagner earthing. Dc potentials were applied to the cell as shown in Figure 2; the dc blocking capacitors, C, were 250-pF electrolytic condensers, placed with opposite polarity to cancel any parasitic voltages. The inductor L was wound on a 2-em toroidal ferrite core and consisted of 21 turns of no. 18 wire; its function was to keep the 4.5-mHz oscillator in the birth detector from being shorted out by the bridge. Because of the low resistance of the bridge transformers-typically 0.06 ohm-it was unnecessary to correct for iR drop in the bridge circuit. The components shown in the dotted outline contributed a small but significant impedance, Z c o r r , to the total measured by the bridge; correction was effected by measuring Z c o r r alone with the Wayne-Kerr Q221 low impedance adapter. j X , o r r were made VS. freGraphs of Zcor, = R c o r r quency, f (at high frequencies), and l/f (at low frequencies) over the range used, 200 He to 20 kHz, and the smoothed curves used to obtain the appropriate corrections. At 1 kHz, Zcorr= 0.250 - l.09j ohms ( j = d=i). The time interval between drop birth and bridge balance was measured on the oscilloscope which was used as a null detector. The oscilloscope time base was calibrated immediately following each experiment. Drop birth was indicated by an overdamped 4.5-MHz oscillator circuit connected between the DME and counterelectrode, a large platinum gauze. The transformer bridge used measured parallel conductance, G,, and capacitance, C,. These were converted to series impedances through the relations real
+
Experimental Section Impedance measurements were made on solutions 0.0932 M dn as supconsisting of 0.98 M NaC104 porting electrolyte and various concentrations of Co(I1) and Co(II1). Co(II1) was added as a stock solution of C 0 ( d n ) ~ ( C l 0 ~while ) ~ , Co(I1) was generated by partial reduction of the Co(II1). A conventional cell was used; the temperature was held constant a t 25” by a water jacket. A large platinum gauze served as counter electrode, and a saturated sodium chloride calomel electrode (ssce) was connected by two salt bridges to the cell solution. The dropping mercury electrode used was all-Pyrex glass. A transformer bridge, the Wayne-Kerr B221 Universal Bridge (Wayne-Kerr Corp., Montclair, N. J.) was
+
imaginary I
1
From the real part of 2, the solution resistance was subwere tracted; real and imaginary components of Z,, also subtracted and both parts of 2 multiplied by drop area to obtain 2’ (real) and 2” (imaginary), both in (7) M.Senda and P.Delahay, J.Phys. Chem., 65, 1580 (1961).
Volume 74, Number 8 April 18, 1970
1760
PETER J. SHERWOOD AND H. A. LAITINEN
E (potentiometer) If rl ------1
r e f erencef ii
I
I
A
to transformer bridge
I I I I
I
I
1 I
1
I
I I
I I
I I
Figure 2. Cell connection.
ohm cm2. These were converted to parallel admittances by the relations
.
-1 L
y’ = (x’)2
+
(xf’)2’
yff
-11 IC
=
(d)2
+
(x”)2
in which ‘ indicates the real and ’ I the imaginary components. The area of the drop at balance was calculated by the procedure described by Nancollas and Vincent.8 The Warburg coefficient, u, was calculated from the polaro= 2.33 X Dox”Z/Dredl/2 = graphic data 1.010 (DOx1’?,as calculated from Koutecky’s eq for instantaneous ~ u r r e n t was , ~ found to increase about 2.501, between 3 and 5 sec after drop birth. The value given is for 2.5 sec after drop birth, the time a t which the impedance was measured.) and El/?= -431.5 mV os. ssce. The parameter p in eq 4 was obtained by writing that equation for any two frequencies and solving simultaneously for p’ = pw-’/’ and K . In this way p’, K values were computed for each of the n(n - 1)/2 pairs of the n points in a set of data. The values of p’ were then put into numerical order and divided into classes by the first two significant figures of its value. A histogram was made of the frequency of occurrence of each class. If the distribution of frequencies was anywhere near Gaussian looking, the median value was chosen. In cases where an abnormal looking histogram was found, an attempt was made to choose a value which was the median of a smaller subgroup within the histogram; i.e., if there were two clumps of recurring values, the median of the group with the most points was chosen. Sluyters-Rehbachs describes straight-line plots for evaluation of e, K , and Ca, using equations derived from eq 5 and 6. Thus
The Journal of Physical Chemistry
Plots of the left-hand side of eq 7 and 8 us. LJ, and of 1) gave two values each for 8, K , and eq 9 us. l / ( p
+
cd.
Results and Discussions Table I shows the electrode parameters calculated from experimental data such as those given in Table I1 for one set of conditions; additional data for other experimental conditions are available.*O The subscripts 0,1, and 2 refer to plots of eq 7, 8, and 9, respectively. It may be seen that there is good agreement between eo, el, and p’, ( = e); C d l and Cdz; and KO,K 2 , and K , the last being the best value calculated from eq 4 by the procedure described in the Experimental Section. Plotting Zr’ and Zf” us. w - l ” (Randles’ model6 for the faradaic impedance) did not give the predicted parallel straight lines; Zf’ was curved, and Zf” had a negative intercept. The Randles equivalent circuit is inapplicable in case of specific adsorption. Figure 3 shows the double-layer capacitance as a function of potential. The lower graph gives the capacitance for the supporting electrolyte, C d l r as measured in the conventional manner. The sharp rise in capacitance at potentials anodic of -350 mV is due to the pseudocapacitance associated with the oxidation of Hg to Hgz (dn)22+. This rise unfortunately obscures the usual hump near the electrocapillary maximum (8) G. H. Nancollas and C. A. Vincent, Electrochim. Acta, 10, 97 (1965). (9) J. Koutecky, quoted in J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press Inc., New York, N. Y., 1966, p 137. (10) P. Sherwood, Ph.D. thesis, University of Illinois, 1968.
IMPEDANCE BEHAVIOR OF BIS(DIETHYLENETRIAMINE)COBALT(III)-(II)
1761
Table I : Calculated Electrode Parameters for the Sluyters-Delahay Model
0.500 0.500 0.500 0.500 0.500 1.061 1.061 1.283 3.701 3.765
0 0 0 0 0 0 0 2.467 0 0
425 430 435 440 445 446 446 448 325 440
0.54 0.55 0.61 0.67 0.74 1.9 1.9 3.2 4.3 1.6
3.60 3.61 4.02 4.53 5.21 6.86 6.37 3.10 3.04 1.44
666.9 656.4 659.0 676.0 704.2 361.7 336.3 96.68 89.02 89.81
17.3 17.9 17.9 18.0 16.7 51.5 59.0 170 310 170
3.69 3.70 4.16 4.62 5.26 7.70 6.25 3.03 3.84 1.45
3.57 3.68 4.02 4.54 5.26 6.99 5.65 3.03 4.34 1.45
23.6 22.6 23.2 23.4 23.9 23.5 17.7 40.2 39.9 13.6
21.7 22.1 22.4 22.2 23.7 23.0 22.6 37.5 40.0 17.8
17.6 17.8 18.0 17.6 15.9 51.5 53.8 222 327 175
15.0 18.8 18.9 19.3 16.7 61.0 76.0 228 343 175
that adsorption of either species changes cd in the same (ecm) : an inflection point is barely present, at about direction. Figure 4 indicates that ox is adsorbed much - 520 mV. more strongly than red, as the cox us. AC plot is a smooth The upper graph in Figure 3 shows Cd, the doubleand the cr,d us. AC graph is not. The same set of + layer capacitance in the presence of 0.5 mM C ~ ( d n ) ~ ~curve, data is being plotted in both graphs, so that the points in the immediate vicinity of E,,,. There is a significant lowering of the capacitance in the presence of the redox are not independent. The evidence of this plot is not really conclusive; a better set of experiments would be couple, with the minimum occurring at -430 mV, very a series with constant cox and varying cred and vice close to El/%. Furthermore, the increase cathodic of versa. From Figure 4, lim AC h, - 15 pF/cm2, corre-430 mV is in opposition to a decrease in the capacitance of the supporting electrolyte. As the reduction in all probability involves adsorption of a transition state activated complex, it is not surprising that the minimum double-layer capacitance occurs near E I ~ ~ . From the theory of semi-infinite linear diffusion to a planar electrode one may calculate the concentrations of the Co(dn)22+and Co(dn)zS+at the electrode surface and compare them to the change in c d , taken as cd C d i = AC. Figure 4 shows such a comparison. If only one form, oxidant or reductant, were adsorbed, a plot of the surface concentration should be directly re0.5mM CoUll) + SUPPORTING ELECTROLYTE lated to the adsorption isotherm for the species adsorbed, while AC should vary randomly with the surface concentration of the other species. It is assumed 420 430 440 450 -E,mv vs SSCE Table 11: Data for 0.500 mM Co(II1) a t -440 mV Freq, Hz
10,000 8,000 6,000 4,000 2,000 1,000 900 800
Z’, ohm om2
Z”, ohm om2
Y’,
Y”,
(ohm oms) -1
(ohm oms) -1
0.0727 0.1072 0.1529 0.2160 0.4541 0.9439 1.0673 1.1770
0.4915 0.5955 0.7608 1.0861 1.9665 3.5244 3.8230 4.2388
0.2946 0.2929 0.2537 0.1760 0.1114 0.0709 0.0677 0,0608
700
1.3493
4.71%
0.0560
600 500 400 335 235
1.6068 1.9942 2.4538 2.9898 4.2953
5.3529 6.1734 7.3497 8.5158 11.1326
0,0514 0.0473 0,0408 0.0367 0.0301
1.9908 1.6264 1,2632 0.8856 0.4827 0.2647 0.2426 0.2190 0.1959 0.1713 0.1466 0.1224 0,1045 0.0781
I
-
I
I
0.0932& dn 098& NaC104 supporting electrolyte
400
600
-E,mv
800 1000 vs SSCE
1200
Figure 3. Double-layer capacitance. Volume 74, Number 8
April 16, 1970
1762
PETER J. SHERWOOD AND H. A. LAITINEN I
-5t
I
I
I
I
I
0
Acdt2 p f /cm -10 drop tme
O
0
0.5
I
I .5
+m u
\\ \ \
I 2
surface concentration of Codng, I
1
I
2
I )
400
I
I
600
BOO
I
1000
I
1200
\ I
1400
-E, rnv vs SSCE
Figure 5 . Electrocapillary curves.
I
0.5
I I
I
1.5
surface concentration of C o d n z T m Figure 4. Changes in
cd.
sponding to high concentrations of Co in solution (not necessarily full coverage of the electrode surface). In theory, it would be possible to find the adsorption isotherm by rfist assuming a dependence of the free energy of adsorption on potential and then attempting to fit the data to isotherm mode1s;ll however, in view of the limited number of data, and Parsons' arguments that charge is a more suitable electrical variable than potential, such a correlation will not be attempted here. A less sophisticated approach is discussed later in connection with the discussion on K . The ecm for the supporting electrolyte (0.0932 M dn and 0.98 M NaC104) was at -450 mV us. ssce, as determined from drop-time measurements. In the presence of 4.11 mM Co(III), the ecm shifts anodically to - 405 mV, indicating predominantly cation specific adsorption. These electrocapillary curves are shown in Figure 5 ; the two curves are displaced from each other along the ordinate, so the drop times are not comparable as to absolute value. The reason for this shift is that it was not possible to obtain reproducible drop times between replicate experiments, although curve shape was relatively invariant. Thus d2y/bE2 but not y was calculated from electrocapillary curves. The Journal
of
Physical Chelnistrtk
A derivation analogous to that of Mohilner's thermodynamic derivation for double-layer properties in the presence of a faradaic reactionlz was used by Timmer, Sluyters-Rehbach, and Sluyters4 to derive the following relation between the interfacial tension y and the double-layer parameters for the Sluyters-Delahay model
This equation was applied to a solution of 1.061 mM Co(II1) at -446 mV. The measurements were made a t 20". From a plot of dy/dE us. E, the slope (d2y/dEz) at -446 mV was found to be -13.87 pF/ om2, From Table I, C d = dq/dE = 23.25 bF/cm2, and K is 51.5 pF/cm2. From eq 10, with u o x / u = 0.6443 and u r e d / u = 0.3555, r was calculated to be 0.697 X mol/cm2. if a low total concentration of As Timmer points the electroactive species is used, it is reasonable to approximate the adsorption of said species by a linear isotherm, i.e. :=I
ri = kici
(11)
From the behavior of AC it was deduced that the oxidized species was more strongly adsorbed than the reduced; if F r e d is assumed to be negligible, and if 1_ dk,,_ 1 dcox _
cbox
that
0
I
I
005
0.10
015
exp + / ( I + s x p $3
Figure 6. Variation of K .
it may be shown18that
K
n2F2 exp d, RT (1 exp d,I3
= kox~box
I__
+
(13)
but with K = 51.5 pF/cm2, exp d, = 0.552, and c d = 23.3 pF/cm2, the right-hand side of eq 14 becomes 46.4 pF/cm2, while -dzy/dE2 is 13.9 pF/cm. It is likely that neither the linear isotherm (eq 11) nor the potential independence of k,, (eq 12) is true for the higher Co concentration. An estimate of the kinetic parameters is possible in principle from the variation of 0, the charge-transfer resistance, with EDc. From the well-known equation
with e = (nF/RT)(Edo- Ella). From our data we can show either eq 11 or eq 12 is applicable: accordexp e)3 a plot of log {n2F2/RTecox] ing to eq 13, a plot of K against (exp e)/(l vs. 4 has a slope q‘2.303 should be linear; with zero intercept and a slope of and an intercept of -log ksh at Eo. Because of the koxcbox(n2F2/RT). Figure 6 shows this plot for cbOx= large errors in e and a moderate error in +! the plot of 0.5 mM, cbred = 0. It is evident that not all the points log { n2F2/RTBcox] proved unworkable. From eq 15, fall on a line passing through the origin, a consequence if cox = Cred, the unknown CY vanishes. The data taken of the large error inherent in the calculation of K from O (EO = El,, RT/nF at -429.8 mV are closest to E the impedance data. Errors shown are 1% limits of In dl’z = -431.8 mV, with dl” = Dredl/a/Dox’/z = 0.99). error calculated from a standard propagation-of-errors Allowing for A(+) = 0.097, 0.249 5 cox 5 0.273 and treatment, using the following limits of error, X, esti0.229 5 Cred 5 0.254. Considering the function mated for the experimental variables from replicate &(CY, cox, C r e d ) defined by measurements: X(E) = 2 mV, X(El,,) = 0.5 mV, 4‘ = 10~Cox~--orCreda X(D,x’~a)/~ox”/” = I%, X(Dredl”)/Dredl/’ = 1.5%, X ( t ) / t = 1.501,, X(Gp)/Gp = 0.5%, X(C,)/C, = 0.5%, we may say that 0.229 2 t 5 0.273 (assuming 0 5 CY 5 X(Rsoln)/Rsoln = I % , A(Roorr)/Rcorr = 3%, X(Xcorr)/ 1). An analysis of the error involved gives 2.71 2 6’ 5 Xcorr = 3%. From the slope of 151 (+4, - 29) pF/cm2, 6.64. Use of eq 15 permits calculation of the limits of (based on lines passing through the origin), k,, is estierror for ksh, the heterogeneous first-order rate constant cm, or mated as 8.04 (f0.21, -1.5) X a t E”: 0.147 5 lGeh 5 0.429 cm/sec. The large lower the area “covered” by a single molecule is 414 i2. limit of ksh ensures that the redox process is reversible, At a surface concentration of 0.5 mM, then, Fox = justifying the use of the Nernst equation in writing eq 1. 0.4 X 10-Io mol/cm2, and 0.33 X 10-lo 5 Fox5 Some of the data were also analyzed according to the 0.41 X 10-lo. This value may be compared with theory of Senda and Delahay.’ I n view of the objecPox = 1.0 X 10-Io mol/cm found by Timmer4 for 0.4 tions of Sluyters et a1.14 to the high- and low-frequency mM In(II1) 1 M KCNS. It should be noted that for extrapolation method used by Baticle and Perdu,ls the the indium, as here for the cobalt, no abnormality was analysis was treated as a problem in nonlinear leastevident from the shape of the electrocapillary curve. squares fitting with Rt, R a o , R a r , c,,, Car, and c d as six According to the above assumptions, at a surface parameters to be determined. The fitting procedure concentration of 0.380 mM, corresponding to that existing at the electrode for 1.061 mM Co(II1) at -446 (13) G. S. Smith, T r a n s . Faraday Soc., 47, 63 (1951). mv, the value of Fox would be 0.304 X 10-IO mol/cm2. (14) M. Sluyters-Rehbach, B. Timmer, and J. A. Sluyters, J. ElecThat at least one of the assumptions inherent in eq 11 troanal. Chem., 19, 302 (1968). and 12 is invalid for this higher surface concentration is (16) A. M. Baticle and F. Perdu, ibid., 13, 364 (1967).
+
+
Volume 74, Number 8 April 16, 1070
HARMON M. GARFINKEL
1764 used was that outlined by Wolberg.lB I n many cases, the best-fit values for these parameters were inconclusive, because of large standard deviations in their estimated values. The large uncertainties are attributable to two features of the Senda-Delahay model which make it unsuitable for meaningful analysis of faradaic impedance data: too many adjustable parameters and a lack of physical justification.
For 0.5 mM Co(II1) over the potential range -425 to -445 mV, Rt was found roughly constant at 2.6 ohm cm2, and C d remained around 30 pF/cm2. Since Rt is expected to vary exponentially with E, the SendaDelahay model is inapplicable. The other four parameters varied irregularly with potential. (16) J. R.Wolberg, "Prediction Analysis," Van Nostrand-Reinhold Co., Inc., Princeton, N. J., 1967.
Transport Properties of Borosilicate Glass Membranes in Molten Salts by Harmon M. Garfinkel Research and Development Laboratories, Corning Glass Works, Corning, N e w York 14830 (Received October d7, 1969)
The transference numbers of potassium and sodium into Code 7740 borosilicate glass were determined as a function of bath composition and temperature. The mobility ratio U N & : U K is essentially independent of saltbath composition at 354'; the temperature coefficient of the mobility ratio was found to be 10.9 0.5 kcal/ mol. These results are in good agreement with those determined from emf studies and reported previously. The enthalpy change for potassium for sodium exchange in the borosilicate glass was estimated to be 3.2 f 0.9 kcal/mol; the change in entropy at 354" was estimated to be 4.4 f 0.9 eu. The effect of the two-phase nature of the borosilicate glass is discussed with respect to potassium for sodium and silver for sodium ion exchange.
*
Introduction
Experimental Section
In terms of the ion-exchange model for membrane potentials, the mobility ratio plays an important role in determining the electrode selectivity of a glass electrode.' Although equilibrium-sorption measurements indicated that Corning Code 7740 borosilicate glass is a two-phase system, the electrode behavior of the glass in fused sodium nitrate-potassium nitrate was fitted quite satisfactorily by the ion-exchange model without explicitly taking the two-phase nature of the glass into account.2 It was assumed that only the minor borosilicate phase was potentid determining, and this assumption allowed calculation of the mobility ratio U N a : U K as a function of temperature. In this work the transport numbers of sodium and potassium are determined from sodium nitrate-potassium nitrate melts into the borosilicate glass as a function of composition and temperature. The mobility ratios U N ~ : U Kwere then calculated from these results and found to be independent of composition and in good agreement with those values reported in ref 2. The significance of these results is discussed with respect to those reported by Keenan and Duewer3 on the transport numbers of silver and sodium into the borosilicate glass from binary melts.
The technique used was similar to that reported by Reenan and DuewerSs The transport numbers were determined by measuring the change in composition of a borosilicate glass bulb filled with a silver nitratesodium nitrate melt and immersed in a sodium nitratepotassium nitrate melt, after the passage of a known amount of current for a known amount of time. The transference cell consisted of a Vycor brand beaker fitted with a transite top, which had four holes drilled into it to position the borosilicate glass bulb, a 96% silica glass stirring rod, a 96% silica glass thermocouple well, and a large platinum electrode. The glass bulbs, which were blown at the end of borosilicate tubing 5 mm 0.d. X 3 mm i.d. X 12 cm long, were 30 mm in diameter with a wall thickness of 0.2-0.3 mm. The dc resistances of these bulbs varied from 5 to 7 kilohms a t 354". The internal solution was a mixture of silver nitrate-sodium nitrate (7 to 80 mol yo silver nitrate depending on the temperature) into which was immersed a silver-wire cathode. The external solution
T h e Journal of Phgsical Chemistry
(1) G. Eisenman, "Glass Electrodes for Hydrogen and Other Cations," Marcel Dekker, Ino., New York, N . Y., 1967, pp 133-173. (2) H.M. Garfinkel, J . Phya. Chem., 73, 1766 (1969). (3) A. G.Keenan and W. H . Duewer,
