data. One channel could be reserved to record voice descriptions of the sample. The digitally recorded data could be processed in the laboratory with the computer program just described. Off-site recording, therefore, need not compromise precision. The rotated cell and requisite equipment would be operated from a rechargeable battery pack. The nitrogen required for sparging could be supplied by controlled thermal decomposition of ammonium nitrite. The instrumentation could also be employed for field spectrometric or potentiometric measurements with very little modification. Note A d d e d in Proof. Bareco Ultraflex Amber Microcrystalline Wax (Petrolite Corp.), a Sonneborn type wax was tested. Electrodes prepared with this wax were stable for 4 to 6 weeks. They exhibited the same behavior on failure as that described above for the Sonneborn electrodes. See Figure 5 . Highly conducting silicon and germanium semiconductors were tested for use as replacement electrodes for graphite, but with very little success.
ANODE, a batch program written for the CDC 7600 computer, now exists. I t treats only single peak spectra, but could be modified to deconvolute overlapping double peaks. This program can be submitted remotely, thereby allowing off-site users to take advantage of a high-speed, low cost analysis package.
ACKNOWLEDGMENTS We thank E. K. Hyde and E. H. Huffman for their past support and J. M. Hollander for his recent support. Thanks also go to R. C. Fox of Chevron Research's Wax Department in Richmond, Calif., for his interest and for supplying test samples used in this study. We are also indebted to D. K. Roe of Portland State University, Ore., for his helpful suggestions. Received for review July 7, 1972. Accepted January 8, 1973. Work performed under the auspices of the U.S. Atomic Energy Commission.
Polarographic Applications of Multiparametric Curve Fitting Louis Meites and Leonard0 Lampugnani Department of Chemistry, Clarkson Coilege of Technology, Potsdam, A!. Y. 73676
Multiparametric curve fitting has been used for a number of purposes in dc polarography. These range from wave analysis through the evaluation of cell resistances from recorded polarograms and the calculation of the heights of ill-defined waves to the detection and resolution of overlapping waves having nearly, or even exactly, identical half-wave potentials. In some of these applications, it leads only to improvements of precision or convenience; in others, it enables the dc polarographer to surpass the as yet demonstrated capabilities of far more complicated electroanalytical techniques.
Multiparametric curve fitting ( 1 , 2) is a least-squares computational technique for selecting the values of any number of parameters in an equation (or set of equations) of any form in such a way as to obtain the best fit to a set of experimental data. Two programs have been devised, each amenable to execution on a minicomputer with modest capabilities. They have been applied to the evaluation of rate constants, equilibrium constants, and enthalpy and entropy changes for every one of the elementary steps in the complex mechanisms of oximation of carbonyl compounds from differential thermometric data ( 3 ) ;the calibration of apparatus for differential thermometry, thermometric titrations, and classical calorimetry ( 4 ); the evaluation of the dissociation constant of a weak electrolyte and its equivalent conductance a t infinite dilution from data on the concentration dependence of its equivalent conductance (2); the resolution of closely overlapping peaks on the ultraviolet absorption spectra of substituted benzaldehydes in alkaline solutions ( 5 ) ; and a number of others. This paper describes several applications of multipar-
ametric curve fitting to the current-potential data obtained in classical dc polarography.
EXPERIMENTAL Most of the polarograms were obtained with a recording polarograph designed and constructed by one of us in 1954. Those of the isomeric nitrobenzoic acids were obtained with a Sargent Model XVI Polarograph. Both instruments were carefully calibrated. Hcells (6) were used throughout, sometimes with saturated calomel and sometimes with silver-silver chloride (7) electrodes, hut all half-wave potentials reported here are referred to the SCE. A number of different capillaries were used; all were made from Corning marine barometer tubing, and each had a drop time between 3.9 and 5.6 sec on open circuit in the supporting electrolyte employed. Conventional (8) techniques of experimentation and measurement were carefully employed throughout. In all of the computations, the absolute, rather t h a n the relative, errors in the measured currents were assumed to be randomly distributed. Most of the computations were performed in EduBasic-25 on a Digital Equipment Corp. (Maynard, Mass.) P D P 8/I minicomputer operated in a 5-user configuration that provided 4096 words of user area for use in this work. A few were performed in Polybasic (Digital Equipment Corporation Users' Society, Maynard, Mass.), and a few others were done in Fortran-IV on an IBM 360/44 computer. The speed of execution increased in that order. Although revisions of the program initially published ( 1 ) have increased its speed by about a factor of 10, and although six- and even seven-parameter fits are quite feasible even with very limited computing facilities if computer time is no object, the use of a language at least as fast as Polybasic is certainly advisable for ( 1 ) T. Meitesand L. Meites, Taianta, 19, 1131 (1972). ( 2 ) L. Meites. J. Eiectroanai Chem., in press. (3) W. A . de Oliveria. M . S. Report, Clarkson College of
Technology, 1972. ( 4 ) L. Lampugnani and L . Meites, Thermochim. Acta, 5 , 351 (1973). (5) P. Zuman, W. J. Bover, and L. Meites, work in progress. ( 6 ) L. Meites and T. Meites, Ana/. Chem., 23, 1194 (1957). ( 7 ) L. Meites and S. A. Moros, Ana/. Chem., 31, 23 (1959). (8) L. Meites, "Polarographic Techniques," 2nd ed., Interscience, N e w York. N. Y . , 1965. A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 8 , J U L Y 1973
1317
the more complex of the applications described here. (Copies of the program, in both Basic and Fortran-IV, with detailed documentation and instructions for use, may be obtained from the Computing Laboratory of the Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676. A charge of $4.00 is made to cbver the cost of duplication and postage.)
RESULTS AND DISCUSSION Wave Analysis. Ordinarily the reversibility of a wave is gauged by constructing a plot of E d e against log i/(id - i) and evaluating its slope S', which is -59.15/n mV a t 25 "C for most reversible n-electron waves. If the process is found to be irreversible, the value of ana may generally be computed from the relation (9) S i = - 5 4 . 2 / a n a m V (25'
C)
(1)
provided that i and id have been measured a t the ends of the drop lives. Exceptions to these relations are well known but will not be further considered here; they include reversible processes that do not conform to the stoichiometry Ox + ne = Red or that involve a species having a constant activity, irreversible processes that involve two or more rate-determining steps or steps that are not pseudo-first-order, and some processes involving prior or subsequent homogeneous chemical reactions. Setting these exceptions aside, and taking into account the fact that the uncertainty in measuring the current a t any potential on a typical recorded polarogram considerably exceeds that in measuring the fraction of the bridge voltage being applied, it is appropriate t o rewrite the general equation for a polarographic wave (9, 10) in the form
and to assume that the potential E is error-free while the absolute errors in the current i are randomly distributed. The natural-log-plot slope S defined by this equation or its more recognizable variant
E = El,, - S In [ i / ( i d - i)]
(3)
is equal to 0.43429 times the familiar decadic-logarithm value S'. The ordinary case is that in which id is directly measurable; a variant in which it is not is discussed in a later section. When it is, values of the parameters E1 2 and S in Equation 2 may be obtained from a simple two-parameter fit. Nine polarograms of 0.2mF cadmium(I1) in 0.10F potassium nitrate were available and were analyzed in this way. Between 10 and 1 2 points over the range 0.05 5 i l i d 5 0.95 were used in each analysis. The average results, with their standard deviations, were E1 2 = -0.5778 f 0.0003 V 1's. SCE and S = 12.85 f 0.2 mV; these are in excellent agreement with the expected values, which are -0.5777 V (11, 12) and 12.84 mV, respectively. This does not challenge the capabilities of the technique and can be accomplished in other ways. It does, however, consume much less time than the manual construction of a log plot and it yields considerably more precise results. It is likely to be especially convenient when used in conjunction with a data-acquisition program that stores the currents a t the ends of successive drop lives directly as a polarogram is being scanned. Evaluation of the Heights of Ill-Defined Waves. A more substantial problem, and one that has not been satisfactorily solved in any other way, arises with a wave so (9) L. Meitesand Y . Israel, J. Amer. Chern. Soc., 83, 4903 (1961) (10) J. Heyrovsky and D. IlkoviE, Coiiect. Czech. Chern. Cornrnun.. 7, 198 (1935). (11) L. Meites. J . Amer. Chern. Soc.. 72, 2293 (1950). (12) J. J. Lingane, J. Amer. Chern. Soc.. 61, 2099 (1939)
1318
A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 8, JULY 1973
closely followed by another wave or by the final or initial current rise that its plateau is poorly defined or even invisible. Wave analysis has hardly been attempted with really ill-defined waves; even more important is that the more or less elaborate graphical constructions that have been recommended to the practical analyst attempting to measure the height of such a wave are usually only indifferently successful. By multiparametric curve fitting, however, all that is involved is a three-parameter fit of data on the rising part of the wave to Equation 2; the three unknown parameters are now id as well as E l l s and S. Four replicate polarograms of 0.1F sodium hydroxide solutions containing 0.9mM vanadium(V) were available; each had been recorded together with the residual-current curve for the supporting electrolyte. using a capillary for whicb the drop time T was 3.98 sec at -1.8 V. Values of i a t about a dozen potentials were obtained from each pair of curves over the range of potentials from -1.7 to -1.9 V; a t the latter potential the residual current was about a third of the total. Currents a t the ends of the drop lives were measured for convenience. The mean values obtained were i d = 4.77 ( * 0.03) PA, El/* = -1.786 ( * 0.004) V 1's. SCE, and S = 30.3 (f1.2) mV. In the hope of defining the ultimate limitations of the method, the residual-current curves were then discarded and the vanadium waves were analyzed as though they had been recorded alone. A typical curve is shown in Figure 1 (13). The portion of the curve preceding each wave was extrapolated to more negative potentials, and the extrapolated currents were subtracted from the measured ones to evaluate the sum of the current due to the reduction of vanadium(V) and that due to the reduction of the supporting electrolyte. Assuming the latter to be an exponential function of potential, the data were fitted to the six-parameter equation
id
' = 1 + exp[(E -
E~/P)/sI
+ 'i
exp[h(E'
- E ) ] (4)
The first term on the right-hand side gives the current arising from the reduction of vanadium(V); the second gives that arising from the reduction of sodium ion. In the second term i' is the current arising from reduction of sodium ion a t the arbitrary potential E ' , and the constant k is expected to equal FIRT, 38.9 V - l at 25 "C in this case. If the final current rise were due to the reduction of hydrogen ion or water, h would have a different value and one that might not be possible to predict because the product of an electrode reaction might cause the value of ana for that process to differ from the value deduced from a residual-current curve recorded in the absence of electroactive material. Computational time is decreased by taking k as known, but the situation in which it is not is the one of greatest concern. The average values obtained by these analyses were i d = 4.78 (f0.06) PA, Eli2 = - 1.788 (f0.006) V U S . SCE, S = 30.5 (f1.6) mV, i' = 0.228 (k 0.019) FA, k = 15.6 ( * 1.8) V-1, and E' = - 1 . 7 8 (f0.04) V cs SCE. The first three of these agree surprisingly well with the values obtained from three-parameter fits employing separately recorded residual-current curves. The value of h is fairly far from the one expected, probably because there were only a few points at which sodium-ion reduction accounted for a substantial fraction of the current. Including points at more negative potentials to improve the estimate of h would. however, be inadvisable except for very special purposes because the data on the rising part and plateau of the wave would then have less influence on the overall (13) Reference 8 Figure 3 16 p 153
goodness of fit. I t is interesting that such precision can be obtained in evaluating the height of a wave so badly defined, and no less interesting that it is so easy to circumvent the otherwise necessary assumption that the residual-current curve is unaffected by the presence of the electroactive substance and its reduction product. This is an assumption that has posed severe difficulties, especially in dealing with organic and other adsorbable materials. Corrections for Ohmic Potential Drops and Evaluation of Cell Resistances. Polarograms distorted by ohmic potential drops are obtained in most nonaqueous solvents. Correction has been effected by employing three-electrode (14, 15) or other (16) circuitry and has also been made algebraically in conjunction with direct or indirect measurements of cell resistances. One common way of evaluating the cell resistance is to record several polarograms of a well-behaved substance, such as thallium(1) or cadmium(II), a t different concentrations and computing A v 1 / 2 / ( 1 i d / 2 ) , where V1/2 is the half-wave voltage including the iR drop (17). Wave analysis is not ordinarily attempted in the face of a substantial cell resistance, although the evaluation of E1/2 by extrapolating a plot of VI against id to id = 0 is fairly common. The usual case is that in which id can be measured but where E1 2, S, and the cell resistance R are unknown. It is appropriate to combine Equation 2 with the Ohm’s law equation
E = V + i R
(5)
In the multiparametric curve-fitting routine this permits calculating the potential E a t each point from the measured current i and applied voltage V a t that point, the resistance R being one of three unknown parameters to be evaluated. The other two are of course E1t2 and S, whose relation to the measured value of i and the calculated value of E is described by Equation 2 . This is a straightforward three-parameter fit. However, the arbitrary choice of a set of values of R, E1.2, and S frequently produces spurious errors due to overflow or underflow when the fit is executed in a language that assigns only a limited number of bits to the exponent of a floating-point number, and care should be taken to guard against these. It is appropriate to set i equal to id if ( E - E 1 , 2 ) / S is more negative than, say, -10 or equal to zero if ( E - E 1 . 2 ) / S exceeds
+lo. Polarograms of O.lmF cadmium(I1) in 0 . F potassium nitrate, obtained with various known resistances in series with the dropping electrode, were analyzed in this way and the results are given in Table I. The smallest series resistance, 5000 ohms, was that across which the iR drop was recorded by the strip-chart recorder employed; larger values were obtained by means of a precision resistance box connected in series with the cell. In most cases the value of R exceeds the external resistance, and the average difference indicates that the internal resistance of the cell and dropping electrode was about 500 ohms. As would be expected, the relative precisions of E 1 2 and S deteriorate, but the values obtained for the last two of these are certainly not unsatisfactory. even when the iR drop a t the half-wave potential is nearly 600 mV. There is another imaginable case, in which the wave is not only distorted by the iR drop but is also so close to the final current rise that its diffusion current is not accessible to direct measurement. The analysis could then be performed in either of two ways. It might be based on (14) (15) (16) ( 1 7)
M . T. Kelieyetai.. A n a / . Chern.. 31, 1475 (1959). M . T . Kelleyet ai.. A n a l . Chern.. 32,1262 (1960) R. L. Pecsok and R . W. Farmer, Anal. Chem.. 28, 985 (1956) Reference 8, p 70.
-
-
1.5
1.9
EL,., volts vt. S.C.E. Polarogram of 0 . 9 m M vanadium(\/) in
Figure 1. hydroxide (73)
sodium
0.1F
Table I . Compensation for iR Drops and Evaluation of Circuit Resistancesa Series resistance, kohms
5 10 20
50 100 200
500 1000
€1
2.
PA
R, kohms
v vs
SCE
mV
1.141 1.149 1.144 1.151 1.147 1.145 1.150 1.148
5.65 10.23 20.59 51.02 99.89 201 .o 500.7 1001.2
-0.5774 -0.5778 -0,5779 -0.5776 -0.5783 -0.5772 -0.5781 -0.5762
12.91 12.80 12.96 12.73 12.50 12.56 12.01 12.67
id 9
s,
a The first column gives the value of the external resistance in series with the cell; the second gives the diffusion current (at the end of the drop life) measured on the plateau of the recorded wave and corrected for the separately recorded residual current: the last three give the values of R (which includes the resistances of the dropping electrode and the cell as well as the external resistance), € 1 . 2 , and S obtained from threeparameter fits to Equations 5 and 2.
values of i, so selected (as described in the preceding section) as to avoid incursion on the final current rise, and would then involve a four-parameter fit to obtain the values of id, R, E ~ Qand , S. Alternatively, the data might extend to potentials negative enough to include the plateau, and with it the beginning of the final current rise, and then a seven-parameter fit to Equations 5 and 4 would be necessary. We have simulated the former course by employing data for a number of the curves on which Table I is based, but without measuring i d and without A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 8, J U L Y 1973
1319
Table II. Evaluations of Circuit Resistances and Diffusion Currents and Wave Analysis of Ill-Defined Waves by Multiparametric Curve-Fitting Compensation for iR Dropsa (;/id)
R, kohms
max
0.5
0.75 0.9 0.95
207.4 203.8 202.9 201.7
€1
1.19 f 0.05 1.177 f 0.03 1.161 f 0 . 0 1 1 1.155 f 0.006
f 14.1 f 6.3 f 2.4 f 1.4
“The first column gives the largest current on the rising part of the wave used in the analysis, expressed as a fraction of the anticipated value of id; the remaining columns give the values of R, i d , € 1 , 2 , and S obtained from four-parameter fits to Equations 5 and 2. Each of these values is accom-
0.4 P
r 0
a . .
Q
- 0.4 -0.3
-0.5 E,
v
Schematic polarogram consisting of two overlap= -0.40 V, and S = 25.7 ping waves with id = 1 . 5 PA, mV, and id = 2.5 MA, El = -0.50 V , and S = 1 2 & mV, respectively; (6) best two-parameter fit to data on curve A , id being taken as 4 p A for t h e assumed single wave (calculated E , 2 = 0.469 V and S = 34.8 mV); (C) deviation-pattern-recognition plot of the difference between the current calculated from the best fit attainable to the single-wave hypothesis (curve 6 ) and the measured current (curve A ) against potential Figure 2. ( A )
including values of i t h a t exceeded various arbitrarily chosen fractions ( i / i d ) m a x of the “estimated” diffusion current. Table I1 gives the averages of the results thus obtained with four polarograms recorded with a series resistance of 2 X l o 5 ohms. The expected value of i d is the average, 1.148 f 0.002 pA, of the directly measured values recorded in Tahle 11. As would of course be expected, both the accuracy and the precision of the calculated values of i d improve steadily as ( i / i d ) m a x increases from 0.5 to 0.95; but even if it is only 0.5, they are well within acceptable limits. The calculated values of the other parameters likewise become both more accurate and more precise as the portion of the curve taken for analysis is made wider; but even with data that do not extend beyond the half-wave voltage, the results are quite accurate and precise enough for most work. 1320
21
v
vs SCE
ANALYTICAL CHEMISTRY, VOL. 45, N O .
8, JULY 1973
-0.579 -0.5784 -0.5782 -0.5775
f 0.003 f 0.0021 f 0.000~
f 0.0005
s,
mV
11.4 f 1.5 12.0 f 0.6 12.2 f 0.3 12.47 f 0.17
panied by its mean deviation, calculated from the results obtained with four different polarograms. Each polarogram was recorded with an external resistance of 200,000 ohms connected in series with the cell.
Resolution of Overlapping Waves. Deviation-Pattern Recognition. Once any of the fits so far described has been completed, it is a simple matter to secure a comparison of the values of i measured at each of the potentials employed and the corresponding values calculated from the best fit obtained. If the difference A i between these values is plotted against the potential (or voltage), there will be a random scatter of points around the potential axis if the polarogram actually contains only a single wave in the range of potentials analyzed. Figure 2 shows schematically what will be observed if there are actually two waves: the fit to the equation for a single wave then involves systematic errors, and the difference curve then has the characteristic shape of curve c . Figure 2 is exaggerated for clarity; Figure 3 is more realistic because it was obtained with a curve on which the existence of two separate waves could not be detected even on carefuljnspection. Heretofore the ability of the dc polarographer t o detect and resolve very closely overlapping waves has been severely limited; conventional log plots may suggest that two such waves are involved (18), but further analysis of them is so difficult and complex (29) that it is rarely undertaken. It is shown in the following sections that this abiIity is very greatly increased by the use of multiparametric curve fitting. It is especially noteworthy that the shape of the deviation-pattern-recognition plot is independent of such factors as the difference of half-wave potentials; the widths and amplitudes of its components vary, but its overall shape is uniquely characteristic of the cause of the deviations from the equation to which the fit was made. This is to say t h a t the shape shown in Figure 3 must result from the overlapping of two waves; reversible reduction to a n insoluble product or any other cause of deviation from Equation 2 would give a deviation pattern having a different shape. Resolution of Overlapping Waves. Techniques and Applications. Procedures for evaluating the individual heights of the closely overlapping waves of two different substances, assuming that the sum of their heights is directly measurable, have been devised by Frisque, Meloche, and Shain (20) and by Israel ( 2 1 ) . Both require exquisitely exact knowledge of the shapes of the individual waves under the experimental conditions employed, and therefore both necessitate renewed experimentation with pure solutions of the electroactive species individually if any of the experimental conditions is changed. A far more severe limitation is t h a t both require that the waves be due to different substances that can be studied individually; neither would be applicable to the resolution of two closely overlapping waves due to the reduction or oxida(18) V . Kargin, 0 . ManouSek, and P. Zuman, J Elecfroanai Chem., 12, 443 (1966). (19) I , Ruiic and M. Branica, J , Eiectroanai. Chem., 2 2 , 243 (1969). (20) A . Frisque. V . W. Meloche, and I . Shain, A n a / . Chem.. 26, 471 (1954). (21) Y . Israel, Taianta. 1 3 , 1113 (1966).
tion of a single substance (or of an equilibrium mixture of, say, an acid and its conjugate base). One of the chief advantages of multiparametric curve fitting is that it eliminates the need for such ancillary information. I t is necessary to assume that each of the overlapping waves obeys Equation 2 (or some other appropriate theoretical equation as dictated by the relevant thermodynamics or kinetics), but it is not necessary to know their half-wave potentials or any of their other characteristics. I t is therefore far more flexible than its predecessors. Taking id as the sum of the diffusion currents of the two overlapping waves and defining f as the fraction of that sum that is attributable to one of them, and making the single assumption that the currents are additive, Equation 2 becomes
f
1
+ exp[(E - Em,
0.04
I
0-N I T ROB EN LO I C PIC I D
I
-0.04 ~~
2)
/S,l
There are five parameters to be evaluated: f , E1 2,1, SI, E1 2 , 2 , and S Z . Several polarograms of mixtures of lead(I1) and thallium(1) in 1F hydrochloric acid containing 0.002% Triton X-100 were analyzed in this way. A typical polarogram is shown in Figure 4. Approximately 20 values of i were obtained from each curve, a t potentials equally spaced from the very foot of the combined wave to the beginning of its plateau. The initial estimates corresponded to -0.45 I E1 2 . 1 I-0.40, -0.50 I E1/2,2 I -0.45, 0.01 5 SII 0.02, 0.02 I Sa I0.03, and 0.25 5 f 5 0.75. Table I11 summarizes the results obtained and gives the expected values o f f calculated from the known concentrations of the ions and data obtained with standard solutions of them by using the same capillary. The relative error in the fraction of the total wave height that is assigned to the minor component is the quantity of most significance; this is 1.2% in the diffusion current of thallium(1) when this ion is responsible for 39% of the total wave height and 3.8% in the diffusion current of lead(I1) when this ion is responsible for 11%of the total wave height. These results are better than those that have been secured by any technique requiring lengthy prior calibration. The use of points equally spaced along the potential axis is arbitrary and supposes that there is no prior knowledge whatever about the value o f f and no way of guessing a t it from the polarogram. I t permits ordinary errors in measurements of the current at potentials where the current is due chiefly to the major component to exert unduly large effects on the values assigned to the parameters characterizing the minor component. Better results can be secured if the points are so chosen that their numbers are about equal in the regions where the two components predominate. This is illustrated by the last line of Table 111, which gives the results of a second fit to the same polarogram as the one that gave rise to the preceding line. That preliminary fit indicated that 10.5% of the total diffusion current was due to the reduction of lead(I1); for the second fit, the current was measured a t the end of each drop life over the range of potentials in which the current increased from 0.5 to 10% of the total diffusion current. then a t the end of every third drop life from the end of that range to the beginning of the plateau. This gave approximately equal numbers of points in these two groups, thereby weighting the data for lead(I1) much more heavily than in the preceding fit. The overall accuracy of the fit decreased marginally, as shown in the last column, chiefly because fewer points were employed, but the calculated
-0.3
-0.2
-0.4
- 0.6
-0.5
E Figure 3. Deviation-pattern-recognition plot for 0.071 m M onitrobenzoic acid in a Macllvaine buffer of p H 4.0 and ionic strength 1 .OM and containing 9 . 5 % ethanol and 0.0013% methylene blue The ordinate corresponds to i, - in,, where i, is the current calculated from Equation 2 and the parameters that gave the best fit of the data to that equation, while i,,, is the current measured experimentally at the same potential
Table I l l . Analyses of Mixtures of Lead(l1) and Thallium(1) by Polarography and Multiparametric Curve Fittinga Pb2+
Ti+
E1
E1 2.
f
v vs.
calcd
SCE
0.386 0.461 0.570 0.682 0.769 0.891 0.891
- 0.433 -0.423 -0.428 -0.437 0.439 - 0.444 -0.434
-
2,
S, mV
v vs. SCE
S, mV
t
14.2 12.9 13.4 13.1 12.6 13.9 13.0
-0.489 -0.476 -0.479 -0.474 -0.462 -0.477 -0.474
24.1 26.7 25.1 25.9 24.0 26.0 25.7
0.3905 0.4705 0.5632 0.6639 0.7834 0.8953 0.8901
u/il 2
0.005 0.010 0.004 0.006 0.004 0.003 0.004
The concentration of each of the metai ions in each of the solutions was about 0.2 (kO.1) mM; the expected values o f f [ = i d , , r I - / ( i d , i d , p b 2 + ) ] given in the first column were calculated from the exactly known concentrations and values of i d / C obtained with the same capillary under the same conditions. The next five columns give the values of the parameters obtained from five-parameter fits to Equation 6, and the last column gives the ratio a/il 2 of the standard deviation of a single measured current from the best fit to Equation 6 to the current at the "half-wave potential" of the combined wave.
+
fraction of the total diffusion current due to lead(I1) changed from 10.5% (expected: 10.9%) to 11.0%,a relative error of only 1%under moderately severe conditions. Even more extreme ratios could doubtless be handled provided that the polarogram was recorded with a span voltage small enough to provide a suitable number of points on the portion of the wave that was chiefly attributable to the minor constituent. The concordance of the computed values of E1,2 and S , as given in Table 111, with the expected ones demonstrates the general reliability of the technique. T o provide an example of what can be achieved in wave analysis, we conducted a brief study of the waves of the three isomeric nitrobenzoic acids under the conditions employed by Israel (21). We initially wished to compare the results of the present technique with those he obtained by an approach A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 8, JULY 1973
1321
Table I V . Characteristics of the Double Waves of the Nitrobenzoic Acids in Macllvaine Buffers of pH 4.0 and p = l M a
2
First wave E1
Isomer 0-
2 m-
ai .c
p-
I
E1 2 ,
2%
Concn, mM
vvs SCE
S, mV
0,0477 0.0626 0.0714 0,0477
-0.314 -0.312 -0.314 -0.279 -0.278 -0.282 -0.277
11.8 14.8 18.0 16.4 15.8 14.9 14.9 15.4 13.9 10.6
0.0626 0.100 0.0477 0.0626 0.100
-0.255 -0.261 -0.247
Second wave
v vs
S.
SCE
mV
f
-0.368 -0.367 -0.371 -0.323 -0.324 -0.303 -0.31 1 -0.303 -0.306 -0.284
24.2 20.9 21.3 13.9 12.5
0.177 0.297 0.321 0.580 0.579 0.437 0.198
22.0 22.1 13.3 15.1 18.9
0.558 0.462 0.222
a The value of f in the last column is defined as the ratio i , / ( i ~-il 2). where i j is the diffusion current of the jth wave, measured at the half-wave Dotential of that wave.
C I -(
5
I -0.65
1
Ed.,., V vs. S.C.E. Figure 4. Polarogram of a solution containing 0.16mM lead(l1) and 0.2mM thallium(1) in 1 F hydrochloric acid, with 0.002% Triton X-100 as maximum suppressor The expected value off is 0.461
based on matrix algebra. Polarograms were therefore recorded of each of the acids individually, using as a supporting electrolyte a MacIlvaine buffer having a p H of 4.0 and an ionic strength of 1.OM and containing 9.5% (v/v) of ethanol and 1.3 X 10-370 of methylene blue as a maximum suppressor. For each acid, the plateau was found to have a small positive slope, and this was taken into account by rewriting Equation 2 in the form
where k is the separately measurable slope, di/d(-E), of the plateau. Three-parameter fits were' performed to evaluate id (which may be defined as the diffusion current that would be measured a t the half-wave potential), E1:2, and S. When the results of these fits were employed in the deviation-pattern-recognition procedure described above, a plot resembling Figure 2 was obtained for each of the three acids. This unexpected result showed that a double wave is obtained for each of the acids individually under these conditions. It is probably this complexity that accounts for the limited success of Israel's matrix-algebra approach in analyzing mixtures of the acids. Polarograms of the individual acids a t different concentrations were analyzed by five-parameter fits to Equation 6, slightly modified to take the slope of the plateau into account, with the results shown in Table IV. A detailed interpretation of these results is not within the scope of the present work, but they may be ascribed to competition among the methylene blue, the nitrobenzoic acid, and the first reduction product of the latter in undergoing adsorption onto the drop surface. With the meta and para acids, the heights of the first waves are approximately independent of concentration, except for a small deviation a t the highest concentration of the latter; this is the char1322
ANALYTICAL CHEMISTRY, VOL. 45, NO. 8, J U L Y 1973
acteristic behavior of a wave whose height is limited by adsorption of the product of the electrode reaction. With the ortho acid, on the contrary, the height of the first wave becomes a larger fraction of the total as the concentration increases, which must reflect a complex adsorption process involving the relative rates of adsorption of the acid and the methylene blue. From the present viewpoint, what is most significant is the internal consistency of each of the variations involved even in the face of differences as small as 30 mV between the half-wave potentials. A final extreme case, and one that no previously proposed technique has been able to solve, arises when two overlapping waves have essentially identical half-wave potentials. In saturated hydroxylammonium chloride the half-wave potential of bismuth(II1) has been reported (22) to be -0.178 V us. SCE, and so has that of molybdenum(VI). A polarogram of a saturated solution of hydroxylammonium chloride containing O.lOmM bismuth(II1) and 0.03mM molybdenum(VI) was analyzed by performing a five-parameter fit to Equation 6. From the resulting value o f f , the diffusion current of bismuth(II1) was found to be 2% larger than the value measured from a separately recorded polarogram of a similar solution containing the same concentration of bismuth alone, while the diffusion current of molybdenum(VI) agreed within 5% with the expected value. The data for the mixture gave E 1 , 2 , 1 = -0.179 V, SI= 8.0 mV, E l I 2 , 2 = -0.178 V, and S2 = 65.2 mV. The values of E1 2 are in excellent agreement with those previously reported, while the previously reported values of E s I 4 - E 1 1 4 for bismuth(II1) and molybdenum(1V) correspond to S = 8.2 mV and 63.6 mV, respectively. The success of the technique in this extraordinary situation is a striking demonstration of its ability to provide information that is contained in experimental data but that is difficult or impossible to extract in other ways. It is well known (23, 24) that closely overlapping processes can be more easily detected on derivative polarograms or voltammograms than on conventional ones, though accurate analysis of the data is still difficult for a curve consisting of two very closely spaced peaks. It may be expected that applying multiparametric curve fitting to the curves obtained by derivative techniques, chronoamperometry with linearly varying potential, and other (22) R . H. Schlossei. B. S. Thesis, Polytechnic Institute of Brooklyn, 1962. (23) R . C. Rooney. J. Polarogr. Soc.. 9 (3), 45 (1963). ( 2 4 ) S. P. Perone and T. R . Mueller, Anal. Chem., 37,2 (1965).
techniques yielding peaks rather than waves might provide the ultimate in resolution. Gutknecht and Perone ( 2 5 ) attempted such fits to stationary-electrode voltammograms, employing arbitrarily chosen functions because solutions to the theoretical equations are not available in closed form. They concluded that their fits were, "on the average, about as good as the data," but did detect significant nonrandom errors, and they were unable to resolve peaks closer than about 32-35 mV. This is the sort of limitation that nonlinear regression with arbitrary functions may be expected to involve but that the use of theoretical functions tends to avoid. Another advantage of using theoretical functions is
that their parameters have physical significances whereas those in arbitrary functions do not. That the availability of a solution in closed form is not essential in multiparametric curve fitting will be shown in a later communication (26).
( 2 5 ) W . F. Gutknecht and S. P. Perone, Anal. Chem.. 4 2 , 906 (1970).
(26) B. H . Campbeli and L. Meites, work in progress
Received for review October 20, 1972. Accepted January 11, 1973. This work was supported by P H S Research Grant No. GM-16561 from the Institute of General Medical Sciences of the National Institutes of Health. It i s a pleasure to thank the Sational Science Foundation and the Eastman Kodak Company for Departmental grants that made possible the purchase of the computer system used.
Preparation and Properties of the Sulfate Ion Selective Membrane Electrode M. S. Mohan and G. A. Rechnitz' Department of Chemistry. State University of New York, Buffaio. N. Y . 74274
Detailed information is provided concerning the construction of sulfate selective membrane electrodes in terms of composition, membrane preparation, and electrode assembly. Special attention .is given to the effect of surface treatment on electrode response and to the attainment of optimum response characteristics by conditioning. The effect of other ions on sulfate response is evaluated.
In an earlier communication ( I ) we described a crystal membrane electrode with potentiometric response and selectivity to the sulfate ion. The electrode is based on a four-component mixture of PbS04, PbS, AgZS, and CuzS pressed into membrane form. Because of substantial interest in this electrode, we now present full details on the preparation and treatment of the electrode as well as the results of our more comprehensive investigation into the effect of experimental variables on electrode behavior. Preparation of sulfate electrodes in other laboratories should be facilitated by this information.
EXPERIMENTAL Apparatus and Reagents. Reagent grade chemicals were used without further purification. A Carver Model B laboratory press capable of attaining 25000 lb total load and a Perkin-Elmer evacuable KBr die (No. 186-0025) of 13-mm diameter were used to press the membrane pellets. Potential measurements were made with a Corning 104 digital electrometer in conjunction with a Honeywell Model 550 XY recorder. Solution temperature during the potential measurements was maintained a t 25 f 0.1 "C. All potential values reported are taken with respect to a n Orion 90-02 reference electrode.
1 Address
all correspondence to this author.
Rechnitz G H Fricke and M S Mohan, Anal Chem 1098 (1972)
(1) G A
44,
Preparation of the Membrane Constituents. Silver sulfide and lead sulfide were prepared by the slow addition of the metal nitrate solutions (1M) to a 20% excess of 0.3M sodium sulfide solution in a large beaker. The resulting precipitates were washed seven or eight times with cold water, twice with nearly boiling water, and, then, a few times with methanol. The precipitates were dried in a n oven a t 110 "C for 24 hr. Coprecipitated AgzS/ PbS mixtures were also prepared by this procedure, but were washed six t o seven times with cold water, twice with 0.1M "03, four times with hot water, a few times with acetone, twice with carbon disulfide, and then again several times with acetone. These complicated washing procedures are necessary to remove any unwanted reagent excess and to optimize the physical properties of the precipitate particles. Insufficient washing results in membranes having poor response or erratic behavior. Two methods were used to prepare the lead sulfate employed. In one method, P b S 0 4 was precipitated from neutral solutions ( = 0 . 2 M ) of Pb(N03)Z and NazS04, washed several times with cold water then with hot water, and dried overnight a t 100 "C.In the second method, 200 ml each of 0.5M HzS04 and 0.5M Pb(N03)Z were added simultaneously to 500 ml of boiling 0.1M HzS04. The precipitate was washed several times with cold water, hot water, and methanol and dried overnight a t 100 "C.
RESULTS AND DISCUSSION Effect of Varying Membrane Composition. As reported earlier ( I ) , the principal components of the membrane were P b S 0 4 , PbS, and AgZS. However, it was beneficial to also incorporate about 5 mol % CuzS into the membrane to improve its dynamic properties ( 2 ) . The effects of variations in membrane composition upon electrode performance were striking. Although all possible ratios of components were not s t u d i e d , the data of Table I show the effect of systematic compositional changes on electrode performance as expressed by the slope of response to changes in sulfate activity. At least 15 mol 7'0 Ag2S and 60 mol % Ag2S plus PbS must be present to obtain electrodes whose response approaches Nernstian ( 2 ) H Hirata and K (19711
Higashiyama Bull
Chem
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