If such a species or an analogous one were present at low water concentrations, peak B could be due to reduction of a pyridine molecule attached to Li(1) via a water molecule. The reduction would be more difficult (more negative and more irreversible) than that of pyridine directly attached to lithium, because the bridging water molecule would weaken the electron-withdrawing effect of Li(1) and electron attack on the -N=C==- bond would, hence, be more difficult. The direct reduction of Li(1) may occur via the pyr-HzO
complex, but the large water concentration dependence suggests that this is not so. However, without direct coulometric evidence as to the nature of peak B, direct reduction of Li(1) cannot be completely dismissed. RECEIVED for review March 24, 1969. Accepted September 2, 1969. The work described was supported in part by the Petroleum Research Fund of the American Chemical Society and the National Science Foundation.
Study of the Chemical Reaction Preceding Reduction of Cadmium Nitrilotriacetic Acid Complexes Using Stationary Electrode Polarography Mark S. Shuman and Irving S h a h Department of Chemistry, University of Wisconsin, Madison, Wis. 53706 The theory of stationary electrode polarography has been considered for the electrochemical system k/
k,, an
ne
Ai=?O*R;
Two parallel homogeneous reactions precede this charge transfer
A
F
ki
R
Cd(NTA)-
kb
whereA and-0 are in chemical equilibrium in the solution. 0 undergoes a reversible charge transfer reaction at the electrode and A undergoes an irreversible charge transfer reaction. A numerical method was employed to solve the integral equations obtained from the boundary value problem. The electrolysis mechanism of cadmium and its complex with nitrilotriacetic acid (NTA) was selected to test the theoretical calculations and to demonstrate the use of cyclic stationary electrode polarography for this reaction scheme. Extensive correlations were made between the experiment and the theory. The rate constant for the direct dissociation of Cd(NTA)- was too small to be determined by stationary electrode polarography, but an upper limit of 5 sec-’ could be assigned. The value obtained for a parallel acid 0.5 X lO5M-l sec-l in assisted dissociation was 4.1 0.1M acetate buffer and 1.OM KN03.
*
THE POLAROGRAPHIC characteristics of cadmium in buffered solutions containing excess nitrilotriacetic acid has been studied by several investigators (1-5). In the pH range 2.5 to 6.0, two polarographic waves are observed. Koryta (1-3) investigated the polarographic behavior of CdNTA solutions in detail and has shown that the first wave is the reversible reduction of “free” cadmium; that is, cadmium in the aquated form or complexed with the buffer. The second wave was found to be the direct reduction of the CdNTA complex. Koryta ( 2 ) has proposed the following mechanism to explain the observed polarographic behavior. The first wave involves the charge transfer step Cd2+
+ 2e ~t Cd(ama1)
(1)
(1) J. Koryta, Sbornik Meziriarod Polarog. Sjezdu Praze, 1st Congr., 1951 Part I, p 798. (2) J. Koryta, Collect. Czech. Chem. Commun., 24, 3057 (1959). (3) Ibid., p 2903. (4) K. Morinaga and T. Nomura, Nippon Kugaku Zusshi, 79, 200 (1958). ( 5 ) N. Tanaka, K. Ebata, and T. Takahari, Bull. Chem. SOC.Jap., 35, 1836 (1962). 1818
k-
Cd2+ 1
+ NTAa-
(2)
kl
Cd(NTA)-
+ H+ k-i=? Cd2++ HNTAZ-
(3)
2
The direct dissociation of CdNTA through Reaction 2 is slow. The much faster acid-assisted path through Reaction 3 involves a protonated complex intermediate. This acidassisted path causes the observed currents of the first wave to be pH dependent. The second wave involves the charge transfer step Cd(NTA)-
+ H+ + 2e
k,, an
Cd(ama1)
+ HNTA2-
(4)
Because a protonated complex participates in the charge transfer, the potential at which the second wave appears is pH dependent. The dissociation rate constants kl and kz have been estimated from polarographic data by several workers ( I , 2 , 4 , 5 ) using the theoretical results of Koutecky (6). The values that have been obtained for k, are in the range from 1 sec-l to 3 sec-1, and the values of kz are in the range from 6.5 X lo4M-'set-lto1.5 X 1OGM-lsec-l. A typical stationary electrode polarogram of a CdNTA solution is presented in Figure 1. The solution contained 1.0 x lO-3M Cd(II), 4.0 X 10+M NTA, and 0.1M acetate buffer, pH 3.88. The scan rate was 0.062 V/sec, and the electrode area was 0.112 cm2. The peaks marked I and I1 correspond to the first and second waves of the classical polarographic experiment. Wave I11 corresponds to the oxidation of the cadmium amalgam formed during the reduction process. Because there is no anodic counterpart of wave 11, the reduction of the CdNTA complex must involve an irreversible electron transfer reaction. The changes of peak currents and potentials of the cathodic waves that occur with (6) J. Koutecky, Collect. Czech. Chem. Commun., 18, 597 (1953).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
I
b
I
Here x is the distance from the electrode surface, t is the time, C A , Co, and C R are the concentrations of the substances A , 0, and R ; DA, DO, and DR are the respective diffusion coefficients, and K is the equilibrium concentration ratio of substance 0 to substance A . The function f ( E , t ) describes the surface concentrations of 0 and R as a function of time and defines the experiment as stationary electrode polarography, The surface concentrations are related to the potential of the electrode through the Nernst equation for the reversible R , and through the Eyring equation for charge transfer, 0 the irreversible charge transfer, A + R . After making the appropriate change of variables (7)) this boundary value problem was solved using a method described previously (8). The two integral equations that result have the form
I
L
a
0
t
\V Im
-*O.O -30.0
-0.5 -0.6 -0.7 -0.8 -0.9 Potential v s S C E
-1.0
l'
Figure 1. Experimental cyclic stationary electrode polarogram of CdNTA solution
esa,(at)
( I / ~ z Z >
changes in pH are the same as they are in the classical polarographic experiment. The stationary electrode polarographic technique had not been used prior to this study in any investigation of the reduction mechanism of CdNTA. Because of the wide time scale available in this method, additional kinetic information could be obtained from an investigation of the overall electrochemical behavior of CdNTA complexes. The theory of stationary electrode polarography was extended to provide the theoretical data for the second wave, and extensive correlations were made between the theory and experiment.
x(z)dz
+ cp(at)eu[~,x(at)la=
lt
( l / d c z ) cp(z)dz (14)
BOUNDARY VALUE PROBLEM
The overall reduction mechanism for CdNTA can be generalized by
Jo The dimensionless functions, x(at) and cp(at), in the integral equations are related to the fluxes of substances 0 and A by:
k., a n
Equation 5 represents the reversible reduction of cadmium preceded by the dissociation of the CdNTA complex. Because the direct dissociation of the complex (Equation 2) has been shown to be very slow, it is only necessary to consider the acid-assisted dissociation (Equation 3). Thus, k y is a pseudo first-order rate constant, and it includes the hydrogen ion concentration. Equation 6 is the direct irreversible reduction of the complex. The irreversible electron transfer process is characterized by k,, the standard heterogeneous rate constant, and a,the charge transfer coefficient. The Fick's law boundary value problem which describes the diffusion process to a plane electrode for the reaction scheme of Equations 5 and 6 is
+ kbco
(7)
+ k y c a - kbco
(8)
acA/at = DA(a2CA/dX2)- kyCA a c o i a t = Do(a2co/ax2) acR/at = t = 0, x
2
0:
= cA*;
&(a2cR/aX2) CO = cO*;
(9)
CR =
cR*
(zo);
C O * / ~ A=* K = ky/kb (10) t
> 0, x
+
m
: Ca * CA*; Co -+ Co*;
CR + CR* (11)
t > O , x = 0: DA(aCA/ax)
+ Do(eco/ax) f DR(dCR/dX) = 0
CA,Co, C R = f(E,t)
fA(t) = C* d n D a p ( a t )
(6)
A-R
(12)
(1 3)
where C*
=
Co*
+ CA*.
(1 7)
The total current is given by
it = nFALfo(t)
+ fa([)]
(1 8)
Other terms are the same as defined previously (9). The numerical method which was employed in solving these integral equations has been published previously (9). In the case considered here the dimensionless functions could not be separated and the equations had to be treated as simultaneous equations. A discussion of the numerical method, and a listing of the Fortran Program are available (10). THEORETICAL CORRELATIONS
The characteristics of the theoretical stationary electrode polarograms were determined from the numerical solutions of the integral equation. The effects of each theoretical parameter (k,, K , k,, an) on the shape and position of the stationary electrode polarograms were obtained by varying one of the theoretical parameters in the calculation while all the other parameters were held constant. (7) J. Koutecky and R. Brdicka, Collecf. Czech. Chem. Commun., 12, 337 (1947). (8) R. S. Nicholson and I. Shain, ANAL.CHEM., 37, 178 (1965). (9) Zbid.,36,706 (1964). (10) M. S. Shuman, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1966.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1819
I
0.4
c
A
.I
,--\*
.
!..’--
-0.41
I
I
100
0
-100
-200
(E-E,,,)n,
I
-300
-400
-500 -600
I
100
mV
0
-100
-200 -300 (€-€,,,In,
Figure 2. Theoretical stationary electrode polarogramsVariation of chemical rate constant. u = 1.0 V/sec; kf = 50 sW-1 500 S~C-’;
----
-k f
=
The results indicated that whenever the two waves are well separated, the characteristics are similar to those of the equivalent single step systems. Of course, any overlap of the waves complicates the interpretation, and the characteristics of the second wave are superimposed on those of the first. Because presenting a detailed analysis of all of the characteristics is not feasible, only the effects of varying k , and the scan rate are included here. The values of other parameters were typical of those encountered in the CdNTA system: K = 4.4 X k , = 2.5 X 10-5 cm/sec, D = 3.6 X 10F cmz/sec, a = 0.63, and n = 2. Other correlations are described elsewhere (IO). Effect of k p In general, the most important of the theoretical parameters is the rate constant for the preceding chemical reaction, because the value of k l determines the amount of 0 supplied to the electrode. Thus, the peak current of wave I increases with increasing kl because this additional material increases the current for the electrolysis of substance 0 (Figure 2). At the same time, the potential of wave I shifts to more cathodic potentials as the preceding chemical reaction approaches equilibrium. Conversely, wave I1 decreases in height as more of A is removed from the vicinity of the electrode by the chemical reaction. The potential of wave I1 is not affected by k , because this wave is due to a totally irreversible electron transfer process. Finally, wave I11 increases in height slightly as k , increases; that is, the total quantity of R formed during electrolysis increases slightly with k,. The potential of wave I11 moves in a cathodic direction because as the value of k , is increased, the value of kb also is increased ( K is constant). The effect is to shift the potential of wave I11 toward the potential at which the wave would appear if A and 0 were in rapid equilibrium-Le., toward cathodic potentials. Effect of Scan Rate. In stationary electrode polarography, the most important experimental parameter is the scan rate, because the time scale of the experiment can be varied to emphasize the effects of the kinetic interaction. Extensive theoretical calculations were carried out using values of the parameters selected to closely approximate the CdNTA experimental system. The values of the scan rate that were available for the experiment range from 0.001 to 1200 V/sec, and the theoretical curves presented in Figure 3 are typical of the results obtained. In these plots, the ordinate is proportional to i/dzand thus the curves are normalized with respect to mass transfer effects in order to emphasize the kinetic interactions. The general behavior of these curves with variation of the scan rate can be explained in terms of the 1820
0
-400
-500 -6OC
mV
Figure 3. Theoretical stationary electrode polarogramsVariation of scan rate. kf = 150 set-’; A, 0.001 V/sec; B, 0.1 V/sec; C, 1.0 V/sec; D,10.0 V/sec; E , 1000.0V/sec.
proposed mechanism. As the scan rate is increased, the peak current of wave I decreases since the time available for conversion of A to 0 decreases. The potential of wave 1 shifts in an anodic direction because the effect of the chemical reaction is diminished. The peak current for wave I1 increases with increasing scan rate because there is less time for A to be removed by the chemical reaction and consequently not as much is lost through reduction at more positive potentials. The potential at which wave I1 appears shifts to more cathodic values because the electron transfer reaction is totally irreversible. The peak current for wave I11 decreases because less R is formed before the switching potential is reached. The potential of wave I11 shifts to more anodic potentials and follows the potential of wave I. The range of values of the theoretical current functions that result when the scan rate is varied depends on the values of the theoretical parameters used in the calculations. Similarly, the magnitude of the shift along the potential axis is also a function of these parameters. In this work, the extent to which the experimental current-voltage curve changed shape and position with changes in voltage scan rate depended on the pH of the CdNTA solutions. However, to verify that the theory describes the CdNTA system, it was only necessary to make correlations at selected values of the experimental pH. The major emphasis involved the way in which each voltammetric wave changed with varying scan rate. EXPERIMENTAL
All experiments were made with a three-electrode controlled potential circuit. The circuit configuration, signal generators, detectors, and cells were similar to those described previously (11-13). A commercially available microburet assembly (Metrohm Micro-feeding apparatus E410, Metrohm Ltd., Switzerland) was used to form the hanging mercury drop electrode. The electrode area was typically 0.1 cm*. The uncompensated resistance was measured with a wave analyzer (Hewlett-Packard, Model 310A, HewlettPackard Inc., Palo Alto, Calif.) following the procedure suggested by Booman (14). The measured uncompensated resistance was 10 ohms which could be neglected in this work (15). (11) R. S. Nicholson and I. Shain, ANAL.CHEM., 37, 190 (1965). (12) W. L. Underkofler and I. Shain, ibid., 35, 1778 (1963). (13) G. S . Alberts, Ph.D. Thesis, University of Wisconsin, Madison, Wisconsin, 1963. (14) G. L. Booman and W. B. Holbrook, ANAL.CHEM., 37, 795 (1 965). (15) R. S . Nicholson, ibid., p 667.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
.0.1
-
-401,
,
,
,
0.01
0.I
1.0
IC
v, volt/1ec
0.01
0.1
1.0
IO
v, volt/r.c
Figure 4. Variation of the experimental current function with rate of voltage scan for wave I
Figure 5. Variation of Eplzwith scan rate for wave I Line, theory for k , = 540 sec-’, K = 1.8 X k , = 4.0 X cmz/sec, (Y = 0.63, n = 2. Points, expericmlsec, D = 3.6 X mental, pH 3.88
Lines, theory; points, experiment. A : solution IV, k , = cm/sec; E : solution 11, k , = 150 54 sec-1, k, = 4.0 X sec-1, k, = 2.0 x cm/sec; C: solution I, k , = 480 sec-I, k , = 8.0 X 10-6 cm/sec
Nitrilotriacetic acid (NTA) from several sources was used. An impurity which appeared to complex cadmium much more strongly than the NTA itself was eliminated by recrystallizing the NTA from a hot 1-mM cadmium solution and washing with water and finally ethyl alcohol. The NTA was considered sufficiently purified when a cadmium solution containing a hundred-fold excess of the acid gave no polarographic wave corresponding to the cadmium complexed with the impurity. A 0.2M stock solution of the acid was prepared and neutralized with sodium hydroxide to pH 7. The stock solution was standardized with copper using the procedure given by Ringbom (16). A standard cadmium stock solution was prepared by weighing cadmium carbonate (Merck, Reagent Grade), dissolving in dilute nitric acid, boiling to drive off the carbon dioxide, and diluting to volume. All other chemicals used in this work were reagent grade and were used without further purification. All experiments were carried out at 25 “C. THEORETICAL CORRELATIONS AND EXPERIMEhTAL VERIFICATION FOR THE CdNTA SYSTEhl Wave I. The theoretical calculations indicate that the peak-current function of wave I attains a maximum limiting value at small values of the scan rate. Whether this limiting value is reached experimentally depends on the pH of the experimental solution, because k , and K increase with decreasing values of pH. Thus, the current function-scan rate relation also depends on pH. In general the peakcurrent function decreases with increasing scan rate, and at high scan rates, the peak-current function reaches a limiting value that is dependent on the equilibrium concentration of “free” cadmium. The variation of the experimental current function ipi nFAZ/aDC* with scan rate over the range 0.015 to 6.20 Visec is shown in Figure 4. The points represent the average value obtained from six replicate experiments and the solid lines represent theoretical results from calculations using parameters closely approximating the experimental conditions. The cm2jsec) were obvalues of a (0.63) and D (3.6 X tained from the analysis of wave 11, and the values of K were determined from shifts in the polarographic half-wave po-
(16) A. J. Ringbom, “Complexation in Analytical Chemistry,” Interscience, New York, 1963.
0.11,
0.01
, 0.10
,
,
1.0
IO
,
, 100
1000
v , volt/s.c
Figure 6. Variation of the current function with scan rate for wave I1 Line, theory for k, = 15 sec-1, K = 4.0 X cm/sec, D = 3.6 X 10-6 cm*/sec,(Y = 0.63, n mental, pH 4.57
k, = 5.2 X Points, experi-
= 2.
tential at each acidity (see below). The largest value of the current function at any particular scan rate is observed for the smallest pH value in accordance with the theoretical results. The theoretical limiting value of the current function at low scan rates was reached experimentally at only one pH value. The limiting value at high scan rates was never attained. The potential at which the wave appears also assumes values between two limits depending on the scan rate. These limiting values are the same as those expected for the case of a simple preceding reaction coupled to a charge transfer reaction [Reference (9), Case 1111. The potential in this case is most conveniently measured at the half peak height since the peak potential becomes illdefined when the wave is drawn out as at high scan rates. The half peak potential (E,/z)of the current-voltage curve should precede the polarographic half-wave potential of the reversible Cd(I1) reduction by 28.0/n mV at high scan rates. Then, with decreasing scan rates, Epizshifts until it reaches a limiting value of -[(RT/nF) In K/(1 K ) - 28.0/n]mV cathodic of the polarographic half-wave potential. The variation of E,/z of wave I with scan rate is presented in Figure 5. The value of EP,z approaches the theoretically anticipated value of -38 mV at low scan rates. Epizshifts in an anodic direction with increasing scan rates as predicted by the theory. The quantitative agreement between experiment and theory is good. Wave XI. A comparison between experimental values of i , / n F A d = C * and theoretical values of the current function for wave I1 as the scan rate is varied is made in Figure 6.
+
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
0
1821
0.400
>
i-
E
1.0
0.1
I
10
v, voll/sce
Figure 7. Variation of the current function with scan rate for wave I11 Line, theory for k , = 540 sec-l, K = 1.8 X cm/sec, D = 3.6 X 10-6 cm2/sec,CY = 0.63, n mental, pH 3.88
k, = 4.0 X lo-' = 2.
Points, experi-
The height of wave I1 was measured from the current decay at constant potential of wave I(17). In spite of uncertainties introduced by this method of obtaining the base line, the experimental values are in reasonable agreement with the theory. The values of i,/nFAdaTC* are independent of scan rate for this set of experimental conditions when scan rates exceed 100 V/sec. The results of the theoretical calculation indicate that the variation of the peak potential of wave I1 with scan rate is identical to the variation expected for a simple totally irreversible charge transfer reaction ; that is, the peak potential should be a linear function of the scan rate. The experimental peak potential was measured as a function of scan rate over the range 0.03 to 100 V/sec using solutions at pH 4.57, 4.95, and 5.44. The average value of the slopes through the data points was 23 mV per decade of scan rate. This indicates that the value of an is 1.27 for this reduction process
Figure 8. Variation of (E, with scan rate for wave I11
volt/ses
- E I / z )+ (RT/nF)ln[K/(K+ l)]
Line, theory for same conditions as Figure 7. Points, experimental, pH 3.88
+
+
(RT/nF) In [K/(l K ) ] 28.5/n mV, which for this case is equal to +30/n mV. The value obtained here was $15 mV which agrees exactly with theory because the value of n is two for cadmium reduction. Cyclic Polarogram. A typical cyclic stationary electrode polarogram for a Cd(I1) solution with a 40-fold excess of nitrilotriacetic acid present in an acetate buffer is compared with a theoretical curve in Figure 9. This quantitative correlation between experiment and theory and the correlations made above indicate that the mechanism proposed describes the electrochemical behavior of CdNTA solutions over the entire range of scan rates, concentrations, and pH values investigated here. In addition, these correlations illustrate that stationary electrode polarography provides an extremely useful approach to the study of systems of this type. DISSOCIATION RATES OF CdNTA COMPLEXES
(9).
In making this correlation, care must be taken to assure that waves I and I1 are sufficiently separated. The theoretical calculations show that if the waves overlap, the experimental peak potential of wave I1 may be anodic of the actual peak for the reduction of the CdNTA species (10). Wave 111. The theoretical calculations indicate that the peak current and peak potential of wave I11 reach limiting values at low scan rates. The exact limiting value of the peak current function depends on the switching potential. The peak current of wave I11 as a function of scan rate is shown in Figure 7. Exact correlation between experiment and theory is difficult for this wave because the base line cannot be determined accurately. Normally, the current decay of the cathodic wave at constant potential is used as a base line for the measurements of anodic currents. However, because the electrode process that gives rise to wave I1 is irreversible and wave I11 is far anodic of the switching potential, another method had to be used. The base line of wave I11 was estimated by extrapolation of the current at the foot of the wave. In spite of this arbitrary and approximate method of measurement, the accuracy of the correlation is reasonable. The variation of peak potential with scan rate is presented in Figure 8. The potential has a constant value at very slow scan rates for this set of experimental conditions and moves toward more anodic potentials at high scan rates as predicted by the theory. The theoretical limiting value of ( E , - El,*) is (17) D. S. Polcyn and I. Shain, ANAL.CHEM., 38, 370 (1966). 1822
v,
A general quantitative application of the theory to the measurement of the CdNTA dissociation rate constants was precluded by the complexities of the mathematical solution. Nevertheless, it was found that when there was sufficient separation between waves I and 11, two methods of measurement were available for the estimation of k,. The first involved the measurement of the ratio of peak currents of waves I and 11. The second method involved the use of the results obtained for a simple preceding reaction (9). Both methods were evaluated as to their usefulness in determining the rate constant of the CdNTA dissociation reaction. In addition, the classical polarographic method was employed to estimate this rate constant so that comparisons could be made with the stationary electrode polarographic investigations. Estimation of the Concentration Ratio of CdNTA Complexes to All Other Cd Species. To calculate the dissociation rate of CdNTA complexes from electrochemical data, it was necessary to know the concentrations of the CdNTA complexes relative to the analytical concentration of cadmium in solution. Because the cadmium exists in the aquated form and in complexes with both the acetate buffer and the NTA, calculation of the relative concentrations from published values of equilibrium constants was unsatisfactory. Such a calculation would require an accurate knowledge of the three acid dissociation constants for NTA, the formation constants of all possible NTA complexes with Cd, and the formation constants of all acetate complexes with Cd. A simple polarographic method for obtaining the relative
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
50
0
-50
-100
-150
-200
-250
-300
I
mV
I
Figure 9. Cyclic stationary electrode polarogram of 1.0 X 10-3M Cd(I1) in 0.040M NTA, 0.1M acetate buffer, pH 3.37, 1.OM KN03 Line, theory for k , = 145 s e c - 1 , K = 4.4 X cm/sec, D = 3.6 X 10" cm2/sec, LY = 0.63, n Points, experimental
=
k, = 1.6 X 2, L' = 1.6 V/sec.
concentrations of the complexes has been developed by Koryta ( 3 ) and by Buck (18). Buck and Koryta derived the equation E1/~(oornp1er) = El/Z(buffer)
-
(RT/nF)In (1
+ K ) / K - (RT/nF)In (i&)
(19)
In this equation E l ~ f ( e o m p l e x ) is the half wave potential of the metal in a buffered solution containing the ligand, &P(buffer) is the half wave potential of the metal in the buffer solution only, K is the ratio of the equilibrium concentration of CdNTA complex to all other Cd forms in the bulk of solution, ik is the value of the limiting current for the kinetic wave, and id is the value of the limiting current that would be observed if the reduction were diffusion controlled. This equation kb)t >> 1 gives reliable results only when K < 1 and ( k , where k , and k b are the rate constants in Equation 5 , and t is the drop time. That is, the chemical equilibrium must favor substance A and half-lives of the chemical reactions must be much shorter than the drop life of the DME. These requirements were met under all experimental conditions used in this investigation. The value of K was computed from Equation 19 after the values of the other terms had been obtained from experiment. In the case of CdNTA complexes, id was conveniently obtained by measuring the total height of the polarogram because the total height is proportional to the analytical concentration of Cd(I1). The value of K for each solution composition used in this investigation was determined by this method (Table I). Calculation of K from published values of the various equilibrium constants results in values of K that are three to eight times larger than those presented in Table I. Such large values of K are inconsistent with the results of all electrochemical experiments performed in this work. Furthermore, the variation of the values o f K with pH observed in this work is consistent with the known changes in composition of the HNTA and H2NTA species, as determined from the available acid dissociation constants. Determination of the Dissociation Rate Constant from the Ratio of Peak Currents. As the rate of the preceding chemical reaction increases relative to the scan rate, the height of wave I increases and the height of wave I1 decreases. Therefore, a plot of the peak current function of wave I divided by the peak current function of wave I1 against I/a (where I = k f k b ) increases with increasing [/a. A theoretical working
+
+
IO
IO'
I /a
IO'
Figure 10. Ratio of peak current of wave I to peak current of wave I1 (normalized with respect to l / Z ) as a function of I/a Lines, theory; points, experimental, for A : K = 4.4 X lo+, pH = 3.37; E: K = 1.8 X lo+, pH = 3.88
curve may be constructed and the dissociation rate of the CdNTA complex determined by a point-to-point comparison between experimental data and the working curve. A working curve must be constructed for each value of the equilibrium constant. The current due to the second charge transfer process is given by Equation 18, or the values of cp(ar) can be normalized cp(br) =
4%cp(at)
(20)
and the current expressed in terms similar to the form originally used for an irreversible charge transfer reaction (9)
This use of the normalized current function makes the ratio independent of n and cy. A difficulty arises when this method is used for estimating the dissociation rate constant of the CdNTA complex. The two waves often merge when low scan rates are used or large reaction rates are encountered. Accurate measurement of the second peak current under these conditions is very difficult. The method is most reliable whenever the two waves are widely separated. But whenever this occurs, it is more con-
Table I. Concentration Ratio of Solutions Employed in the Study of CdNTA Complexes
Solutiona I I1 111
IV V VI VII"
VI11 IX
PH 2.92 3.37 3.69 3.88 3.99 3.99 3.99 4.57 4.95
Re1 av dev Z
K* 5.0 x 4.4 x 2.2 x 1.8 x 1.4 x 1.3 x 6.8 x 4.0 x 3.5 x
10-2 10-2 10-2
10-2 10-3 10-3
5.0 4.5 3.0 2.0 1.4 4.0 8.0 1.5 2.5
Solutions contained 1.0 X 10-3M Cd(II), 4.0 X 10-2M NTA, 0.1M sodium acetate, 1.OM KNO3; and were adjusted to the desired pH with acetic acid. * Average value of five separate determinations corresponding to five different drop times. 8.0 X 10-2MNTA. (I
(18) R . P. Buck, J . Electroanal. Chem., 5, 295 (1963). ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1823
I
I.o
0.I
4”Figure 12. Variation of (E,,z - Eliz)nwith v& for wave 1
Figure 11. Variation of i d / i k as a function of A , solution IV; B, solution 11
+.
Line, theory [Reference (9)] for K = 2.2 X 1 0 1 ; points, experimental for solution I1
venient to consider wave I as a simple preceding wave and employ the results of Reference (9). In Figure 10, comparisons between working curves and experimental data are made for two sets of experimental conditions. When there is a wide separation in the two waves, the agreement between the theoretical working curve and the experimental points is reasonable (Curve B). However, when the two waves are very close together, agreement between the working curve and experimental points is poor (Curve A , large l/a). This method of estimating the rate constant was abandoned in favor of a more reliable method that involved only wave I. Dissociation Rate Constant from the First Wave Data, Two methods were available for the determination of the dissociation rate constant employing data only from wave I. The first method involved using the empirical equation presented previously [Equation 73, in Reference (9)], i&
= 1.02
+ 0.471 & / K d i
(22)
where ik is the kinetic peak current, and id is the current expected if the process were diffusion controlled. A plot of experimental values of i d / i k against the square root of the scan rate is linear with a slope of 0.471(dnF/RT) ( l / K d i ) . Thus, the measurement of this slope and knowledge of K provides a means of calculating k,. A plot of this type was constructed for each solution by performing experiments in which the scan rate was varied from about 30 mV/sec to about 10 V/sec. The experimental peak current of wave I was taken as the kinetic current i k . The value of the diffusion current, i d , was estimated from the height of the second wave at large values of the scan rate where the preceding reaction has little effect and the current is due only to the diffusion of the material to the electrode surface. Unfortunately, wave I1 involves an irreversible electron transfer process, whereas the diffusion current in Equation 22 is based on a reversible charge transfer reaction. However, a correction can be made to take this into account by considering that the peak current due to a reversible charge transfer is (9)
and the current due to a totally irreversible charge transfer reaction is (ip)irrea =
n~~~*dZ&(br)
Using a value of 0.63 for 01 in Equation 25 gives ( i p ) r e o / ( i p ) i l r e , equal to 1.13. Another correction still must be made, however. At high scan rates, when there is little time for dissociation of the complex, the current for wave I1 is a function of the equilibrium concentration of the complex. Thus, the calculated ratio ( i p ) r e a / ( i p ) i r r e o must be multiplied by the factor l/(l K ) to take this into account (10). The resulting value of id, the mean of forty individual experiments normalized with respect to scan rate, was 79.0 pamp/(V/sec)1/2. Substitution of this value into Equation 23 yielded a value of the apparent diffusion coefficient of 3.6 X 10-8 cmz/sec in 1M KNOI and 0.1M acetate buffer at 25 “C. Cizek et. al. (19) estimated the apparent diffusion coefficient of CdNTA at 25 “C in 0.1MKN08to be 5.3 X cm2/secby comparison of the polarographic diffusion current to the diffusion current of uncomplexed cadmium. The lower value obtained in this work is expected because it pertains to a solution of much higher ionic strength (20). TWO examples of id/ik plots as a function of are presented in Figure 1 1 ; the dissociation rate constant of the CdNTA complex was calculated from the slope of these plots. The second method of using data from wave I to calculate the dissociation rate constant can be applied at high scan rates, where id/ik is no longer a linear function of At high scan rates the peak current of wave I is dependent only on the rate of the homogeneous chemical reaction and hence independent of the scan rate. Under these circumstances, the
+
6
4.
(19) J. Cizek, J. Koryta, and J. Koutecky, Collect. Czech. Chem. Commun., 24, 3844 (1959). (20) I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed.,
Interscience, New York, 1952. 1824
(24)
where x(ar) has the value 0.446 and ~ ( b tthe ) value 0.496. Dividing Equation 23 by 24 results in
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
Table 11. Determination of Dissociation Rate Constant of 1:l CdNTA Complexes from Slope of i& us. 6 P l o t s Solution
I I1 I11 IV V
VI VI1
Slope 0.842 1.60 2.90 4.20 5.05 5.21 7.43
1IK‘d-b 0.200 0.386 0.700 1.01 1.21 1.25 1.79
d
rate constant can be calculated directly from the magnitude of the current of wave I which is proportional to the rate of the preceding homogeneous reaction (21). ik = n F A C * d z K f i
(26)
The rate constants obtained by these two methods are presented in Tables I1 and 111. The discrepancy between the two sets of values may be related to the procedure used for estimating id in the first method, and/or to an uncertainty in the diffusion coefficient used in the second method. Estimation of K from Stationary Electrode Polarography. Although the theoretical treatment of a preceding reaction provides no explicit method for separating the rate constants, kf and kb, from the equilibrium constant, K , the theory does provide a means by which the value of K may be determined from experimental data once the value of l / / K d is determined. To determine K , values of Eplzare determined experimentally as a function of scan rate, and a plot of values of (Ep12- E& us. l / a / K d r i s compared to the theoretical working curve of (Epiz- El& - (RT/F) In [K/(1 K)] us. ‘ d u / K d T [Reference (9), Figure 51. EllZis the polarographic half-wave potential in the absence of the ligand. The value of K is obtained from the intercept on the potential axis. Using this approach (Figure 12), the value of K obtained agreed with the value determined from the polarographic data to within 10%. Because the value of Eplzis difficult to determine accurately, this method does not seem as useful as the polarographic approach. However, improvements in data acquisition techniques would make this method more useful.
+
EVALUATION OF RESULTS The apparent pseudo first order rate constants for the dissociation of the 1 :1 CdNTA complex obtained by stationary electrode polarography (Tables I1 and 111) were plotted as a function of hydrogen ion concentration, and the slope of the line indicated that the second-order rate constant kz in Equation 3 has a value of 4.1 f 0.5 X 105M-l sec-l. The rate constant of the back reaction k-2 was estimated from the equilibrium constant and was approximately 105M-1 sec-1. The rate constant kl in Equation 2 was too small to be measured directly by the electrochemical methods employed here. However, its value could be estimated to be less than 5 sec-l from the values of the apparent rate constants calculated in the higher pH range. These rate constants compare well with those obtained by Rabenstein and Kula (22) from NMR (21) J. M. Saveant and E. Vianello, Electrochim. Acta, 8, 905 (1 963). (22) R. J. Kula and D. L. Rabenstein, J. Amer. Chem. Soc., 91, 2493 (1969).
k x 100 58.9 64.9 55.0 59.0 61.5 82.0
k f , sec-1 476 147 90.5 53.5 48.2 46.5 45.7
Re1 av dev of computed kf, 10.0 9.5 8.0 7.0 6.5 9.0 13.0
z
Table 111. Determination of Dissociation Rate of 1 :1 CdNTA Complexes Using Equation 27 Re1 av dev of Solution ik, PA kf, sec-1 computed kf, V 13.9“ 33 6.5 VI 14.1b 36 10.5 VI1 10.5b 38 1 1 .o VI11 5.0” 15 4.0 Average of 12 experiments. b Average of 6 experiments. Average of 8 experiments.
z
4
line broadening. Those authors used higher ionic strength solutions and obtained a value for kz of 2.9 f 0.5 X lO5M-l sec-l. This lower value of kz at higher ionic strength is consistent with the trend of decreased rate with increased ionic strengths for reactions between ions of opposite charge (23). The rate constant kz was also determined using DME polarography in an attempt to compare the two methods. In general, the polarographic values were lower than those obtained with stationary electrode polarography, with the largest deviations observed in the more acidic solutions. These anomalous results were caused by using the normal procedures of analyzing the polarographic data (6, 24, 25) in which it is assumed that K
