Polarographic Theory for an ECE Mechanism Application to Reduction of p-Nitrosophenol RICHARD S. NICHOLSON, JANET MAE WILSON, and MICHAEL L. OLMSTEAD Chemistry Department, Michigan State University, East Lansing, Mich.
b Polarographic theory was developed for the case in which a homogeneous chemical reaction is interposed between two electron transfer reactions. Results were obtained by numerical solution of the appropriate integral equation for diffusion to an expanding plane electrode. These results provide a simple correlation between polarographic limiting current and the rate of the interposed chemical reaction. Application of the theory is illustrated for reduction of p-nitrosophenol where the chemical reaction is shown to involve acid-base catalyzed dehydration of p-hydroxylaminophenol.
chemical step was rapid and reversible. Even then only an approximate solution was obtained. Usual polarographic conditions are assumed, and further it is assumed that substance C is more easily reduced than substance A . Fick second law equations for an expanding plane electrode are applied to substances A , B , and C:
T
Initial and boundary conditions are:
HE
ECE electrolysis mechanisms k
me
=
C B
+ cc
f#l = Csexp(kt) = .tu3
(12) (13) (14)
(3/7)t"3 (15) The boundary value problem now simplifies to y
=
me
A-B-C-D
have been the subject of considerable interest recently. Theoretical treatments have appeared for chronopotentiometry ( I % ) , potentiostatic electrolysis a t a stationary electrode ( I ) , and cyclic voltammetry ( 7 ) . Very recently Herman and Bard (4) have developed the theory of cyclic chronopotentiometry for this mechanism. Although all of these methods are suitable for a wider range of rate constants than the conventional polarographic experiment, the latter is easiest to apply experimentally, instrumentation is simple and often more readily available, and treatment of data is straightforward. Thus, it would seem that where applicable the polarographic method is to be preferred for measurement of k . Surprisingly, however, no rigorous mathematical treatment has appeared for polarography. Thus, it seemed desirable to develop polarographic theory, and such a treatment is reported here. Reduction of p-nitrosophenol was used to test the theoretical results. At the same time knowledge of the reduction mechanism of p-nitrosophenol was extended by measuring IC as a function of PH. THEORY
A theoretical treatment of System I has been attempted previously ( I I ) , but only under conditions where the 542
*
ANALYTICAL CHEMISTRY
t = 0, x
20 CA =
c*
CB = CC = 0 t > O,x+
(4) (5)
m C A
-b
c*
CB * C C + 0
(6) (7)
t>O,x = 0
CA
=
0
(8)
cc
=
0
(9)
The equation for the total polarographic current is: ik
= FA
[nlD
Equations 16, 17, and 18 can be converted from differential to integral form with the aid of the convolution theorem from operational calculus (2) :
(2) + 2-0
There ib is the kinetic limiting current, F is the Faraday, and A is the electrode area, which for the dropping mercury electrode is dependent on time. To use Equation 11, it is necessary to solve the preceding boundary value problem for the surface fluxes of substances A and C. Solution of this problem is facilitated by the following changes of variable
where
With Equation 23, Equation 26 can be solved directly. The result in terms
t
of the original time and distance variables is
2.0 Figure 1. Working curve for calculation of rate constant
Equation 30 will be recognized as a form of the Ilkovic equation. Equations 27 and 28 can be combined with Equation 30 and the appropriate boundary conditions to yield a single integral equation 1/
., .., .
---
Plane electrode Spherical electrode Expanding plane electrode
I.o
e - - k ~)wff a - w71df.
I
I
-1.2 -0.8 -0.4
= r -
I
I
I
0
0.4
0.8
I
1.2
Log ( k t )
The function gc(y), which is the solution of Equation 31, is defined as
Solution of Equation 31 for g&) provides the complete solution for the problem. Solution of Integral Equation. The definite integral on the left hand side of Equation 31 cannot be expressed in closed form in terms of elementary functions; however, it can be evaluated in terms of a series expansion. This provides the possibility of a series solution to Equation 31. The result is
measurement of ratios of kinetic to diffusion controlled currents, a new function was defined
where id = i k when k = 0. (In writing Equation 34 it has been assumed that n1 = n2. This is the case for p-nitrosophenol, and appears to be fairly general for ECE mechanisms.) Equation 34 together with the change of variable { =
k($)“‘
(35)
yields
Figure 1 makes a convenient working curve from which values of k can be determined from experimental values of &/id. Normally an un-damped polarograph would be used and maximum currents during drop-life would be employed for the values of ik and id. The parameter determined would then be kt,,, where t,, is the drop-time. Calculation of k is straight-forward since tmax can be measured easily. It is of interest to note the effect of electrode geometry on the working curve of Figure 1. Thus, Figure 1 also includes a working curve calculated by the once common practice of taking theories for plane or spherical electrodes and multiplying by the empirical constant 1/7/3. Theory for the plane electrode was taken from the work of Alberts and Shain ( I ) . For the spherical electrode, data of Alberts and Shain also were used for experimental parameters typical of the p-nitrosophenol system. I t is interesting to note that
(33) There r remesents the gamma function (S), and oiher terms have been defined previously. Equations 11, 30, 32, and 33 provide a complete solution to the problem. Although the series of Equation 33 is strictly convergent for all values of kt-e.g., by application of the ratio test (20)-the convergence is extremely slow for large values of kt, and the use of a computer to evaluate Equation 31 is almost essential. Even then rounding errors become important, and the results are uncertain. Although accelerated convergence methods could be used-e.g., the Epsilon Algorithm @)-these introduce considerable uncertainty into the results, and we have had only limited success in applying them. Because of these difficulties an alternate solution to the problem was sought, namely the direct numerical solution of Equation 31. Because the most common and convenient way to evaluate polarographic data involves
Table 1.
Equation 36 is completely dimensionless and the solution for h(kt) yields directly a working curve of &/id as a function of kt. Equation 36 was solved numerically with a method described previously (9), and the definite integral on the left hand side was evaluated with the aid of Simpson’s Rule (IO). Results agreed well with Equation 33 for the region where convergence was not too slow. Results of Calculations. The results of the numerical calculations are summarized in Figure 1 (solid curve) where h(kt) is plotted against the logarithm of kt. Values from which this figure was constructed are in Table I, and are believed accurate to better than 0.5%.
Data for Construction of Working Curve
kt
ik/id
0.000 0.025 0,050 0.100 0.150 0.200 0.250 0.300 0.400 0.500 0.650 0.800 1.ooo 1.250 1.600 2.000 2.500 3.200 4.000 5.000 6.300 8.000 10.000 16.000 25.000 30.000
1,000 1.012 1.024 1.048 1.071 1,093 1.114 1.135 1.173 1.209 1.253 1.298 1,351 1.408 1.474 i ,534 1.591 1,648 1.694 1.732 1.766 1.795 1.818 1.858 1.888 1.899
m
2.000
VOL. 38, NO. 4, APRIL 1966
543
this procedure would give rate constants considerably different (smaller) from those calculated for an expanding plane electrode. (This is not surprising, even in the case of the spherical electrode, because for times of the order of tmax, the most important consideration is convection rather than sphericity.) Moreover, the magnitude of the difference is a function of &/id, so that the error ink would not be constant. APPLICATION TO REDUCTION OF p-NITROSOPHENOL
Reduction of p-nitrosophenol has been studied extensively, and the following mechanism has been established for p H 4.8 in acetate buffer
I
I
OH
OH
H\ N
There k,,, is the apparent value of the rate constant (pseudo first order) for the dehydration. All previous quantitative experiments have been limited to the above set of experimental conditions (1,4,8). If the mechanism represented by System I1 is correct, and does not change with pH, then the pH dependence of k,,, should be consistent with the mechanism for dehydration of p-hydroxylaminophenol. This process would be expected to be acid-base catalyzed, and therefore variation of k,,, with p H should follow acid-base catalysis theory. To test this hypothesis, and a t the same time evaluate the theory presented above, k,,, was measured as a function of pH. EXPERIMENTAL
Apparatus. Preliminary experiments were performed with a Sargent Model X X I Polarograph and an Hcell using standard polarographic procedures. However, in some cases the rate of potential scan of the Model XXI was too high t o give good resolution of the polarographic waves over the potential range of interest. Thus, most of the data reported here were obtained with an electronic potentiostat and function generator described previously for use with cyclic voltammetry (6). The recording device was a Sargent Model SR recorder with 1second pen response. No damping was used. pH was measured with a Beckmann Model G pH meter and a Beckmann Type E-2 glass electrode. The pH 544
ANALYTICAL CHEMISTRY
I
I
t
I
o.2
I
t
I
I
I
1
2
3
4
5
6
I
PH
Figure 2. Variation with pH of rate constant for dehydration
meter was calibrated with standard Beckmann buffers over the pH range used. All measurements were made in a constant temperature room at ambient temperatures of 23' t o 25' C. Materials. -411 materials were reagent grade used without further purification, except the p-nitrosophenol. p-Nitrosophenol was prepared from the sodium salt in the manner described by Alberts and Shain ( I ) . Solutions were prepared with Britton-Robinson buffers and contained in addition: 0.00570 gelatin, 20% ethanol, and 0.1M potassium nitrate (supporting electrolyte). Although no attempt was made to keep ionic strength strictly constant, the ionic strength is determined primarily by the potassium nitrate and therefore was nearly constant, RESULTS AND DISCUSSION
Polarographic Behavior. Polarograms were recorded in the p H range 1 t o 10, and results were essentially those summarized by Alberts and Shain (1). Solutions of pH 5 or greater were stable (in terms of polarographic behavior) for periods of several days or more. Acidic solutions were considerably less stable, as evidenced by decreased limiting currents and slight shifts of half-wave potentials. The exact cause of this was not established. However, it was observed that in acid media (pH 1-2) the ethanol in the supporting electrolyte tended to react slowly with the acetic acid in the buffer to form ethyl acetate. Nevertheless, the deterioration of acid solutions was slow enough that reproducible results could be obtained by recording polarograms immediately after preparation of the test solution. This procedure was checked by preparing blank solutions a t pH 1 (both with and without acetic acid present), putting a measured volume of the blank solution in the polarographic cell, deaerating, setting the initial potential of the DhIE in the limiting current of the wave of interest, and then adding a measured small
volume of p-nitrosophenol from a stock neutral solution. Results obtained in this way were no different than results obtained by running polarograms on freshly prepared solutions. At low pH (1-2) and anodic potentials the shape of the single i-t curves for the polarograms indicated adsorption phenomena, even in the presence of 0.005% gelatin, although maxima were never observed. However, a t the more negative potentials at which all measurements were made, there was no indication of adsorption. Determination of Rate Constant. At p H less than about 5, both waves of p - nitrosophenol are resolved. Therefore, in this range i, was taken as the maximum current for the first wave, and i d was taken as one half the maximum current for the second wave. This method of obtaining i d has the advantage that it automatically compensates for any changes with p H of diffusion coefficients that might occur. Currents for the second wave were corrected to the drop-time of the first wave through use of the Ilkovic equation. At pH 6 the value of i d was the one obtained a t p H 5. ilt pH greater than 6 the kinetic step was too rapid to measure polarographically. Values of kt,,, were read directly from the working curve of Figure 1. Drop-times were measured at the appropriate potentials from an average of 30 drops. Dependence of Rate Constant on pH. Values of k,,, were measured in the manner just described over the p H range 1 to 6. Results are summarized in Figure 2 where k,,, is plotted as a function of pH. -41though reproducibility of polarograms was quite good, a small uncertainty in i k / i d results in a fairly large uncertainty in k,,, (see Figure 1). Error levels in Figure 2 are assigned on this basis for a 2% uncertainty in &/id. Although this value is reasonable at low pH, it may be overly optimistic a t high p H where the second wave is not well defined. The value of k,,, a t pH 4.8 is of special interest because the system already has been studied extensively a t this pH. The value from Figure 2 is 0.4 set.-' which is in fairly good agreement with the value of 0.6 sec.-l obtained previously. (Previous measurements were in a different buffer system with different ionic strength, and an alcohol content of 19% rather than the 20% used here. Thus, small differences in k,,, are not surprising.) This result serves as a check of the theory presented here. In addition, it indicates the good accuracy which can be obtained polarographically in spite of the fact the mass transport processes are not as well defined as with stationary electrodes.
The variation of k,,, with p H in Figure 2 is qualitatively that expected for an acid-base catalyzed reaction. For System I1 in acid media dehydration might be expected to follow a mechanism of the type
0 H ‘ H ‘04 H’ N ‘’
H
N’
H
There the first step is assumed to be in equilibrium (with equilibrium constant ICE+), and the rate determining step involves kE+. A similar mechanism can be suggested for base catalysis. Thus, k,,,, the rate constant measured polarographically in buffered solutions, should be a measure of the overall rate of reactions of Type 111, and as a result k,,, should be of the form
LITERATURE CITED
(1) Alberts, G. S.,Shain, I., ANAL.CHEW.
There k , is the self-dehydration rate constant in the absence of catalysis, and other constants already have been defined. The data of Figure 2 are in agreement with this concept, since the dashed line is a plot of Equation 35 with k H + K H + = 4.45 M-l-sec-l, k O ~ - K 0 n -= 4.49 X 107LM-1-sec.-1, and k , = 0.36 sec.-l The polarographic behavior of pnitrosophenol, therefore, is consistent with the overall mechanism of System I1 where the mechanism of the dehydration is that indicated by System I11 (and the inferred system for base catalysis). While these results serve to establish further the mechanism for reduction of p-nitrosophenol, certain features of the overall reaction remain uncertain. These include the adsorption phenomena reported above, effects of chemical reactions preceding the first charge transfer, and the heterogeneous rates of the individual electrode reactions. All of these processes should be amenable to study by application of modern electroanalytical techniques, such as stationary electrode polarography (619)’
35,1859 (1963). (2) Churyfiill, R. V., “Operational Mathematics, 2nd. ed., McGraw-Hill, New York. 1958. (3) Davis, P. J., “Hydbook of illathematical Functions, M. Abramowitz and I. Stegun, eds., p. 253, National Bureau of Standards, Washington, D. C., 1964. (4) Herman, H. B., Bard, A. J., J. Phys. Chem. 70,396 (1966). (5) MacDonald, R. J., J. A p p . Phys. 35, 3034 (1964). (6) Nicholson, R. S., ANAL.CHEW 37, 1351(1965). (7)lv~cholson,R. S.,Shain, I., Ibid., p. IIO.
(8) Zbid., p. 190. (9) Ibid., 36,706 (1964). (10) Sherwood, G. E. F., Taylor, A. E., “Calculus,” PrenticeHall, New York, 19.54. ~ . . ~ (11) Tachi, I., S,enda,
M.,“Advances in Polarography, I. Longmuir, ed., p. 454, Pergamon Press, Xew York, 1960. (12) Testa, A. C., Reininuth, W. H., ANAL.CHEM.33, 1320 (1961). RECEIVED for review December 2, 1965. Accepted February 11, 1966. Presented in part at the Division of Physical Chemistry, 151st Meeting ACS, Pittsburgh, Pa., March 1966. Work supported by funds received from the National ScienceFoundation, under Grant No. GP-3830. During the summer of 1965, J. bl. Wilson was an NSF .Undergraduate Research Participant a t Michigan State University.
Se pa ration of Lead(II) from Bismuth(Ill), T ha IIium(Ill), Ca d mium(II), M erc ury(II), Go Id(III), PIa ti num(IV), Palladium(Il), and Other Elements by Anion Excha nge Chroma tog ra p hy F. W. E. STRELOW and F. V O N S. TOERIEN National Chemical Reseorch laboratory, Council for Scientific and Industrial Research, Pretoria, South Africa
b Lead and other elements are absorbed from between 0.1 and 4.ON hydrobromic acid solution on a column of A G l - X 8 anion exchange resin in the bromide form. The following elements are eluted with 0.1N HBr: U(VI), Th(lV), Zr(lV), Hf(lV), Ti(IV), Sc(lll), Y(III), La(”, and the rare earths, AI(III), Ga(lll), In(lll), Fe(lll), Be(ll), Mg(ll), Ca(ll), Sr(ll), Ba(ll), Zn(ll), Mn(ll), Co(ll), Cu(ll), Ni(ll), Cr(lll), Sb(lll), Ge(lV), Li(l), Na(l), K(I), Pb(l), and Cs(l). Then lead i s eluted selectively with 0.30N H N 0 3 plus 0.025N HBr, and after evaporation of the acid it can be determined b y EDTA titration or b y the mass spectrometric isotope dilution method. Bi(lll), TI(IIl), Cd(ll), Hg(ll), Au(lll), Pt(lV), and Pd(ll) are retained b y the column quantitatively. No element which could interfere with the EDTA titration accompanies lead.
T
of lead from other elements by ion exchange chromatography has received some attention in recent years. Among the most selective methods which have been described are the cation exchange procedure of Fritz (2, 3) which uses hydrobromic acid as eluent, and the anion exchange procedure of Korkisch (5) in nitric acidtetrahydrofuran media. Unfortunately, in the first lead is eluted preferentially, while most of the common bulk elements remain absorbed. This limits the amount of material which can be handled and makes the method less suitable for the separation of small amounts of lead from large amounts of such elements as Al(III), Fe(III), Ca(II), Mg(II), Cu(II), and U(V1). I n the second, the rare earths which are main constituents of many radioactive ores are among the few elements that HE SEPARATION
accompany lead. Furthermore, some elements with tendencies t o the formation of nitrate complexes, such as Sr(II), Ba(II), Hg(II), and Au(III), have not been investigated by Korkisch. The fairly high distribution coefficient Kd = amount of element in resin phase X amount of element in solution grams of dry resin ml . of solution for Ca(I1) suggests that Ba(I1) and Sr(I1) are also likely t o accompany lead. The classical anion exchange separation of lead in hydrochloric acid based on the work of Kraus (7, 8) is probably the most selective of all the methods available; but the low maximum value of the distribution coefficient of lead a t VOL. 38,
NO. 4,
APRIL 1966
545
