Second Harmonic AC Polarography. Theoretical Predictions for Systems with First-Order Chemical Reactions Following the Charge Transfer Step Thomas G . McCordl and Donald E. Smith2 Department of Chemistry, Northwestern University, Evanston, 111. 60201 Theoretical predictions for the second harmonic ac polarographic response of systems involving rate control by diffusion, first-order following chemical reactions, and/or heterogeneous charge transfer kinetics are presented and discussed. The theoretical formulation is based on a rigorous solution of the expanding plane boundary value problem by the Matsuda method. The calculated second harmonic behavior, including both amplitude and phase characteristics, is examined for a wide range of kinetic conditions encompassing both nernstian and non-nernstian behavior. Characteristics of the second harmonic response which make its observation eminently useful in quantitatively studying the kinetics of homogeneous chemical reactions following heterogeneous charge transfer are made apparent. Approaches to data analysis are recommended.
THEELECTRODE REACTION MECHANISM involving a first-order homogeneous chemical reaction following the heterogeneous charge transfer step, kz
0
+ ne*R ekl Y
(R1)
is widely recognized as a notably ubiquitious pathway among electrochemical processes. Considerable literature has evolved in this decade which is concerned with this mechanism from either an experimental (1-13) or theoretical (5-30) viewpoint. For the case of aromatic hydrocarbons the existence of numerous mechanisms of this type and simple variations thereof has been particularly well documented (see reference 31 for review) and, although less detailed evidence is presently available, the situation appears to be similar in the inorganic and organometallic realms (see reference 2 for review of inorganics and reference 32 for relevant data on organometallics). Accordingly, a quantitative understanding of the influence of the following chemical reaction on electrochemical observables is essential to enable meaningful interpretation of much electrochemical data, as well as to facilitate adaptation of electrochemical techniques to kinetic measurements of such chemical reactions. The ac polarographic approach represents one widely applied electrochemical method whose potential advantages in the study of systems following Mechanism R1 have been made evident for the case of the fundamental harmonic response (8, 9, 16, 18, 23, 30). However, for this mechanism little attention has been given nonlinear characteristics of the ac polarographic response of which the second harmonic is one of the most conveniently observed and widely studied (23, 33-39). Recently, a general theory was presented for the second harmonic ac polarographic wave of systems involving rate 1 NIH Graduate Fellow, 1967-68; present address, General Electric Corp., Materials and Processes Laboratory, Schenectady, N. Y., 12305. * To whom correspondence should be addressed.
control by diffusion, a single heterogeneous charge transfer step, and any number or type of coupled first-order homogeneous chemical reactions (40). The theory was developed within the framework of the reasonably precise expanding (1) A. C. Testa and W. H. Reinmuth, ANAL.CHEM.,32, 1512 (1960). (2) A. A. Vlcek, Collect. Czech. Chem. Commun., 25, 668 (1960). (3) W. Jaenicke and H. Hoffman, 2.Elektrochem., 66, 808 (1962). (4) W. Jaenicke and H. Hoffman, ibid., p 814. (5) H. B. Herman and A. J. Bard, ANAL.CHEM., 36, 511 (1964). (6) D. Hawley and R. N. Adams, J. Electroanal. Chem., 10, 376 (1965). (7) L. B. Anderson and C. N. Reilley, ibid., p 538. (8) G. H. Aylward and J. W. Hayes, ANAL.CHEM., 37, 195 (1965). (9) G. H. Aylward and J. W. Hayes, ibid., p 197. (10) W. M. Schwarz and I. Shain, J. Phys. Chem., 70, 845 (1966). (11) S. P. Perone and W. J. Kretlow, ANAL,CHEM., 38, 1760 (1966). (12) L. K. J. Tong, K. Liang, and W. R. Ruby, J. Electroanal. Chem., 13,245 (1967). (13) P. A. Malachesky, K. B. Prater, G. Petrie and R. N. Adams, ibid., 16, 41 (1968). (14) 0. Dracka, Collect. Czech. Chem. Commun., 25, 338 (1960). (15) C. Furlani and G. Morpurgo, J. Electroanal. Chem., 1, 351 ( 1960). (16) D. E. Smith, ANAL.CHEM., 35, 602 (1963). (17) J. M. Saveant and E. Vianello, Compt. Rend., 256, 2597 (1963). (18) G. H. Aylward, J. W. Hayes and R. Tamamushi in “Pro-
ceedings of the First Australian Conference on Electrochemistry 1963,” J. A. Friend and F. Gutmann, Eds., Pergamon Press, Oxford, 1964, pp 323-31. (19) C. M. Groden, G. H. Aylward, and J. W. Hayes, Aust. J. Chem., 17, 16 (1964). (20) R. S. Nicholson and I. Shain, ANAL.CHEM.,36, 706 (1964). (21) R. Koopman, Ber. Bunsenges. Ges., 70, 121 (1966). (22) R. S. Nicholson, ANAL.CHEM., 38, 1406 (1966). (23) D. E. Smith in “Electroanalytical Chemistry,” Vol. 1, A. J. Bard, Ed., Marcel Dekker, Inc., New York, N. Y., 1966, Chap. 1 . (24) W. J. Albery and S . Bruckenstein, Trans. Faraday SOC.,62,
1946 (1966). (25) 0. Dracka, Collect. Czech. Chem. Commun., 32, 3987 (1967). (26) M. L. Olmstead and R. S . Nicholson, J. Electroanal. Chem., 14, 133 (1967). (27) J. H. Christie, ibid., 13, 79 (1967). (28) W. J. Albery, Trans. Faraday SOC.,63, 1771 (1967). (29) J. M. Saveant, Electrochim. Acta, 12,999 (1967). (30) T. G. McCord, H. L. Hung and D. E. Smith, J. Electroanal. Chem., 21, 5 (1969). (31) M. E. Peover in “Electroanalytical Chemistry,” Vol. 2, A. J. Bard, Ed., Marcel Dekker, Inc., New York, N. Y., 1967, Chap. 1. (32) R. E. Dessy and R. L. Pohl, J. Amer. Chem. SOC.,90, 2005 (1968) and all earlier papers in this series. (33) T. G. McCord, E. R. Brown and D. E. Smith, ANAL.CHEM., 38, 1615 (1966). (34) H. H. Bauer, J. Electroanal. Chenr., 1,256 (1960). (35) D. E. Smith and W. H. Reinmuth, ANAL.CHEM.,33, 482 (1961). (36) J. Paynter, Ph.D. Thesis, Columbia University, New York, N. Y., 1964. (37) H. H. Bauer and D. C. S . Foo, Ausr. J . Chem., 19,1103 (1966). (38) T. G. McCord, and D. E, Smith, ANAL.CHEM., 40,289 (1968). (39) T. G. McCord and D. E. Smith, ibid., 41, 131 (1969). (40) T. G. McCord and D. E. Smith, ibid., 40, 1967 (1968). VOL. 41, NO. 11, SEPTEMBER 1969
1423
plane model of the dropping mercury electrode (41) and invokes only minor restrictions on the magnitudes of the relevant rate parameters. It provides the theoretical formulation for Mechanism R1 as a special case. The present discussion is concerned with results of a detailed study of predictions of this theory for the second harmonic ac polarographic response with the mechanism in question. The primary goals of this effort were: to determine on a theoretical basis whether second harmonic studies of systems following Mechanism R1 would prove advantageous; to provide a reasonably accurate theory which would enable quantitative evaluation of the kinetic parameters associated with Mechanism R1 from second harmonic data; to develop guidelines as to the most useful second harmonic observables for kinetic measurements ; and to elucidate useful and distinct qualitative features of the second harmonic response with Mechanism R1, through which its existence might readily be recognized. THEORY
General Considerations. Specialization of the general theory for the second harmonic ac polarographic response (40) to the case of Mechanism R1 yields the following theoretical formulation Z(2wt) = 2(2w)W(w) sin (2wt Z(2W)
=
+ 42)
(1)
nzF2ACo*(2wDo)112AE F(OG(2w) 4RT cosh2(
G(2w) =
(2)
+)
[
2
v 2 2
+ UZZ
nFAE W(w) = -(PZ 4RT
]‘I2
(3)
+ L2)’/2
(4)
(24)
h8fxa
(25)
f
= cot-‘
42
D = DoaDRa =
p = 1 - a
(7)
vz = -
1
(1
+ 1
h2)”2
+h
+ h2
I+ u 2
& .!-+ 1) l+K
L = (VZ
+ UZ)
r u ~2w”’z1 1+p - P U T
(8)
(lo)
(41) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, N. Y., 1966, pp 77-83. 1424
0
ANALYTICAL CHEMISTRY
(26)
Notation definitions are given below. The foregoing theoretical relationships account for the influence on the second harmonic response of three rate processes; diffusion, heterogeneous charge transfer, and the following homogeneous chemical reaction. The only restrictions imposed on the kinetic parameters are that kt 2 10 and DE = D y . The significance of key terms such as Z(2w), W(w),F(t), etc. has been discussed earlier (23, 30, 38, 40). The potential dependent parameter employed in the above relationships, j , refers the applied dc potential Edo to the quantity Eli< which is the reversible (diffusion-controlled) dc polarographic half-wave potential and ac polarographic (fundamental harmonic) peak potential in absence of the following chemical reaction. However, for a wide range of kinetic situations involving Mechanism R1, E l / ; is not the most rational reference potential choice. When the equilibrium constant K is small and the chemical reaction is reasonably rapid, the ac polarographic wave is located at potentials far-removed from E1/2’. Except when the following chemical reaction is irreversible, a more rational choice of potential dependent parameter is
which employs as the reference potential the dc polarographic half-wave potential in the presence of the chemical equilibrium E1/zrc,where
Table I. Special Cases of General Theory Special case Mathematical condition (1) Chemical equilibrium exists in the dc sense xF > 1) and/or kinetically inoperative ( k -+ a), the theoretical expression reduces to one characteristic of the simple quasi-reversible case (40) (rate control by diffusion and heterogeneous charge transfer only). Table I lists additional special situations, together with the relevant mathematical conditions, which allow one to simplify the general theory. With the aid of the mathematical relationships of Table I and algebraic manipulation, the above general theory is reduced readily to simplified formulations appropriate to each special situation. Considerable reduction in algebraic complexity of the theoretical formulation often results as, for example, was shown for the case where Conditions 3, 4, 5 , and 6 apply (42). Table I is essentially the same as the corresponding table of mathematical simplifications given for the fundamental harmonic case (30). The only difference is the replacement of the parameter g by the parameter h in certain relationships. Consequently, we refer the reader to the previous work (30) for additional discussion of the interpretation and implications of Table I. RESULTS AND DISCUSSION
An investigation of the predictions of the foregoing theoretical relationships has been carried out with the aid of a Control Data Corp. Model 3400 digital computer. Computer readout was provided in an analog form with the aid of a Calcomp Model 565 digital incremental plotter, as well as in the usual digital form. All calculations were based on the general theory defined by Equations 1-26. The FORTRAN program used for this purpose is available on request. The most significant aspects of the predicted second harmonic response are illustrated in Figures 1-14. Among the types of plots represented in these figures are three different forms of experimentally accessible (23, 43, 44) second harmonic ac polarograms: total current amplitude us. dc potential (conventional second harmonic ac polarogram), phase angle us. dc potential (second harmonic phase angle polarogram), and the complex plane response (23) us. dc potential (second harmonic complex plane polarogram). The direction of change and the magnitude of the independent variable, Edo,is indicated by markers in the complex plane polarograms which simultaneously represent changes in three variables. Implicit in the complex plane polarograms are still other types of experimentally feasible forms of data readout-e.g., plots of the real or imaginary second harmonic current components us. dc potential [phase-selective second harmonic ac polarograms (43)l. Nernstian Conditions. The case where ks is sufficiently large that electrochemical equilibrium (nernstian conditions) prevails in both the dc and ac time scale deserves special attention. It represents a special case which is likely to be (43) D. E. Smith, ANAL.CHEM., 35, 1811 (1963). (44) D. E. Smith, unpublished work, Northwestern University, Evanston, Ill., 1968.
1426
ANALYTICAL CHEMISTRY
encountered frequently in experimental investigations at low and intermediate frequencies. This case is particularly relevant to large classes of aromatic organic and organometallic species which are known to undergo facile oneelectron transfer processes in aprotic solvents (31, 32). The nernstian case represents the ideal case to the chemical kineticist whose interests lie primarily in the homogeneous chemical reaction rate. Only the chemical reaction and diffusion are rate-determining and a notable simplification of the mathematical description results which facilitates data analysis. In addition, the nernstian case is important from a strictly pedagogical viewpoint because the predicted effects of the homogeneous chemical reaction on the second harmonic response are more readily demonstrated and recognized when the wave is uncomplicated by charge transfer kinetic contributions. The various effects of a reversible following chemical reaction on the second harmonic response are illustrated in Figures 1 and 2. The curves show the effect of the rate parameter k on current amplitude, phase angle, and complex plane polarograms for three values of the equilibrium constant K ( K = 1, lO-l, All other parameters are held constant and are of typical magnitude. One should note that the frequency considered ( w = lo3, frequency = 159 Hz) is at least two orders of magnitude below the upper limit at which second harmonic measurements can be made with modern instrumentation (45). consequently, the applied frequency value must be considered on the low side of the accessible frequency spectrum. An important frame of reference to apply in evaluating these results is the well-known second harmonic response with a strictly diffusion-controlled process. In this limiting case the current amplitude polarogram is characterized by two symmetrical peaks separated by a zero-current minimum. The phase angle polarogram is a step function where the phase angle' value jumps discontinuously from -45" (anodic potentials) to f135" (cathodic potentials) at the E d 0 value where the current amplitude becomes zero. The diffusioncontrolled complex plane polarogram is a straight line of -1 slope. For the nernstian case under consideration, diffusion-controlled conditions arise either when the following chemical reaction is reversible and sufficiently rapid that chemical equilibrium exists in the dc and ac sense-Le., Conditions 1 and 2 of Table I apply-or when the chemical reaction is effectively inoperative ( k -+ 0 or K >> 1). Polarograms for the largest k-value in each of the three sets of curves ( K = l,lO-I, lo-') in Figures 1 and 2 correspond to the former diffusion-controlled situation (chemical equilibrium). In this case the current amplitude becomes zero and the phase angle transition occurs at the potential Edc = (see Equation 28). The diffusion-controlled responses for an inoperative chemical reaction are not shown, but are identical to the polarograms for the chemical equilibrium case except that they are centered about the potential Ede = El/;, rather than Ed== With these considerations in mind, the relatively profound and varied influences of the following chemical reaction on the second harmonic response become apparent from perusal of Figures 1 and 2. As one would expect, some of these effects are analogous to those observed with the corresponding fundamental harmonic response (30), For example, inspection of the peak currents as a function of the kinetic (45) E. R. Brown, D. E. Smith and G. L. Booman, ANAL.CHEM., 40, 1411 (1968).
I.I *IO0
-
1
i il O-
-
l -100
,
o
0+IO0
1-c
l
-
-100 l
I
Edc,- E{ (milliwlltl
t025-
T
-
2.
.1
+
50 0 -25
o
+IO0 0
l
1-F
i
+0.21
v e
H
p8
t2w
.1
50 0
-25
o.oo~
000-
&
z U I
-025t
-025-
1
I -0 25
+ O 25
0
-0 25
0 00
REAL COMPONENT (pomp4
+o 2 1
REAL COMWNENT (pomprl
Figure 1. Second harmonic ac polarographic response us. dc potential with nernstian system A, D-conventional second harmonic ac polarograms B, E-second harmonic phase angle polarogram C, F-second harmonic complex plane polarogram Parameter values: n = 1, T = 298 O K , A = 0.035 cm2, A E = 5.00 mV., C*, = 1.00 X 10-3M,Do = D R 1.00 x 10-6 cm*sec-1, k, = a,w = 1.00 X 103 sec-l,f = 1, k-values (kl k2 in sec-1) shown on figure
A , B, C-K D , E, F-K
12.0 SW = 1.00, I = 0.100, t = 6.00 SIX
parameter k (see Figures l A , 10, 2A, 20) shows that a plot of peak current magnitude (either peak) vs. log k will exhibit a minimum whose depth and position on the log k coordinate depend notably on the chemical equilibrium constant K. The smaller the K value, the deeper the minimum and the larger the log k value corresponding to the minimum. The existence of a minimum in the peak current -log k profile is a necessary consequence of the obvious fact that, for a reversible following chemical reaction, the chemical kinetic contribution must reach a maximum (minimum peak current) at some value intermediate between the k + 0 and k -+ limits where diffusion-control prevails. The influences of the K value are simply special manifestations of 0)
+
=
the more general principle that, holding other factors constant, smaller equilibrium constants accentuate chemical kinetic effects. This idea applies to voltammetric techniques in general. Other aspects of the second harmonic response shown in Figures 1 and 2, such as wave symmetry, shapes of the phase angle and complex plane polarograms, etc., also are illustrative of this profound influence of the following chemical reaction equilibrium state. The enhancement of the chemical kinetic effect attending a reduction in K is sufficient that when K is reduced to K = 0.01 (Figure 2), a detectable chemical kinetic contribution is observed even for k = log sec-l as seen in Figures 2A-2C. Considering that the w value in question is only lo3 sec-1, it becomes apparent that, for larger o VOL. 41,NO. 11, SEPTEMBER 1969
1427
1
I
2-1
1 1 3
.'..,
2-D
045
+zoo
+IO0
+0251
l
o
0
+IO0
t200
l
t
-1
!I P
-io0
+0.08
'
Y
-
8
s
o
o
-025
I
I
-0.25
1
0 REAL COMPONENT
I
+0.25
(pawl
I-0.08 !
I
I
0 +0.08 REAL COMPONENT (pampa)
1
+0.16
Figure 2. Second harmonic ac polarographic response us. dc potential with nernstian system All parameters and notation same as Figure 1, except K = 1.00 X
t = 6.00 see, and k-values given on figure
values within the experimentally accessible realm-e.g., w = 106 sec- L e v e n reactions proceeding at collision-controlled rates-e.g., k = 1011-101* sec-]-(30) should manifest themselves kinetically on the second harmonic response. If the electrical double-layer effects anticipated for such fast reactions can be accounted for (46,47), then kinetic parameters of the fastest homogeneous solution reactions following charge transfer should be accessible via the second harmonic ac polarographic experiment, provided K 5 0.01. One of the more obvious and diagnostically useful chemical kinetic effects illustrated in Figures 1 and 2 is the loss of the current amplitude polarogram symmetry as the chemical kinetic contribution becomes more important. Although
kinetic influence of the follow-up chemical reaction suppresses both peaks of the current amplitude polarogram, the cathodic peak is attenuated significantly more than the anodic peak, yielding a grossly unsymmetrical polarogram under many circumstances. With sufficientlysmall K values, the cathodic peak can be suppressed to the point that it becomes merely a shoulder on the larger anodic peak-e.g., Figure 2 0 for k = 103 and 104 sec-1. It should be noted that with preceding chemical reactions the opposite effect is observed-i.e., the anodic peak is smaller (48, 49). The appearance of an asymmetric second harmonic ac polarogram characterized by a much larger anodic peak proves to be indicative of a variety of mechanisms involving a follow-up chemical reaction lead-
(46) R. R. Schroeder, Ph.D. Thesis, The University of Wisconsin, Madison, Wis., 1967. (47) H. W. Nurnberg in "Polarography 1964," G. J. Hills, Ed., Macmillan, London, 1966, p 149.
(48) J. Paynter, Ph.D. Thesis, Columbia University, New York, N. Y.,1964. (49) T. G. McCord and D. E. Smith, unpublished work, Northwestern University, Evanston, Ill., 1968.
1428
ANALYTICAL CHEMISTRY
ing to an electroinactive species (including second-order decompositions, etc.) (50). The observation of a second harmonic current amplitude polarogram of this type is reasonably definitive evidence for a homogeneous chemical reaction following the electrode reaction, although it is not proof that the simple single-step, first-order chemical reaction of Mechanism R1 is operative. More careful, quantitative data analysis is required to establish this. The fact that the cathodic peak is more sensitive to chemical kinetic effects clearly suggests that the best estimates of the chemical kinetic parameters from current measurements usually will be obtained from measurements of the cathodic peak current. The fact that second harmonic currents are more sensitive to effects of the following chemical reaction at cathodic potentials must be viewed as somewhat unusual and interesting, considering the fact that with many voltammetric responsese.g., dc polarographic currents, fundamental harmonic ac polarographic currents, and phase angles-greater sensitivity to such chemical kinetic effects is found on the anodic side of the wave. Indeed, Figures 1 and 2 show that the second harmonic phase angle tends to exhibit the latter normal type of behavior-Le., phase angle deviations from the diffusioncontrolled limit induced by the follow-up chemical reaction are greater at the anodic extremity of the wave than vice versa. The unusual behavior of the current amplitude response arises from the interaction of the Z(2w) and W(w) components of the current amplitude expression (see Equation 1, 2, 3 , etc.). The Z(2w) term manifests the influence of the normal faradaic impedance at the second harmonic frequency (40). It varies as a function Ed, in a manner identical to the fundamental harmonic response at the frequency 2w. The W ( w ) function manifests faradaic nonlinearity at the applied frequency w (40). With nernstian systems, its shape as a function of Ed, closely corresponds to the hyperbolic tangent function for klw < 1, with slight to moderate distortions of this function arising for klw -.,1. Calculations show that under conditions where the chemical kinetic effect is significant, the peak of the Z(2w) function is located at dc potentials considerably anodic of the W ( w ) function minimum. As a result, even if both Z(2w) and W ( w ) are nearly symmetrical with respect to Ed, (often true with nernstian systems), their points of symmetry on the Ed,coordinate are displaced with respect to one another and the Z ( 2 w ) . W(w) product function is grossly unsymmetrical. Further, the largest values of this product occur in the vicinity of the peak Z(2w) value-Le., at anodic potentials. One of the most interesting and important features of the predicted second harmonic current amplitude polarogram with nernstian systems is the behavior of the minimum between the peaks. Perusal of Figures 1 and 2 shows that under many conditions the current magnitude at the minimum is zero or nearly zero. Furthermore, whenever it is characterized by a null current signal, the minimum occurs either at Edo = Eliz' or E
