AIter nating Current Polarography and Irreversible Processes Donald E. Smith1 and Thomas G . McCord Department of Chemistry, Northwestern University, Evanston, Ill. 60201 Timmer, Sluyters-Rehbach, and Sluyters have demonstrated that a finite, measurable, bindependent ac polarographic wave is obtained with irreversible processes, contrary to earlier beliefs. They considered the ubiquitouscase of irreversibility due to slow charge transfer. The present study is primarily an extension of the theoretical work of Timmer, Sluyters-Rehbach, and Sluyters. irreversibility is considered in a general context where any process which retards the reverse electrolysis step is relevant-e.g., irreversible chemical decomposition of the electrode reaction product. Results are presented which suggest that a finite, measurable, ac polarographic response is expected for irreversible processes in general, regardless of the origin of irreversibility. This conclusion is found to apply to both the linear and nonlinear aspects of the faradaic response.
TIMMER,SLUYTERS-REHBACH, AND SLUYTERS have recently established on the basis of theory and experiment (I, 2) that the ac polarographic wave of the so-called irreversible system (3) is not vanishingly small. They showed for the simple electrode reaction O+ne@R
(R1)
in the limit of very slow charge transfer that the ac polarographic response approaches a finite, measurable, k,-independent limiting value, contrary to popular notions. This conclusion has been reached independently in these laboratories and, because of its important consequences, we undertook a general theoretical study of the phenomena. Our investigations have encompassed not only the case of irreversibility due to slow charge transfer considered by Timmer, Sluyters-Rehbach, and Sluyters, but also cases of irreversibility induced by coupled chemical reactions. Both linear (fundamental harmonic) and nonlinear (second harmonic) components of the faradaic current have been examined. The aspects of our theoretical study which might be considered either refinements or generalizations of the incisive work of Timmer, Sluyters-Rehbach, and Sluyters are presented here.
tive in demonstrating theoretically the existence of a finite fundamental harmonic ac polarographic response for this mechanistic class, the work of Timmer, Sluyters-Rehbach, and Sluyters is subject to refinement since they utilized an approximate steady-state treatment of the dc process. The more rigorous Matsuda treatment ( 4 ) of the expanding plane boundary value problem for ac polarography can be applied with ease for the same purpose, which was our approach to examining this question. That the Matsuda approach was somewhat more rigorous was acknowledged by Timmer, Sluyters-Rehbach, and Sluyters who nevertheless felt that the steady-state calculation was a good approximation. We present here our theoretical work on the fundamental and second harmonic ac polarographic response for this class of irreversible systems to illustrate differences and similarities between equations based on the rigorous and approximate approaches and to demonstrate that the principle advanced by Timmer, Sluyters-Rehbach, and Sluyters is also applicable to the nonlinear aspects of the faradaic response. For simplicity, only the case in which the reduced form is initially absent from the solution will be considered. FUNDAMENTAL HARMONICCuRRENr. The general expression of Matsuda ( 4 ) for the ac polarographic wave of systems controlled by tbe rates of charge-transfer and/or diffusipn may @ written (5) (see below for notation definitions). Z(wt) =
Z,,,.F(Xtl’z)G(wlizX-l)sin (ut
+ #JJ
(1)
-
m
n=l
THEORY
The Case of Irreversibility Induced by Very Slow Heterogeneous Charge Transfer. The classical case of irreversibility is the one associated with very small k , values where a very negative potential (relative to the Eo value) must be applied to achieve a measurable reduction rate and vice versa for oxidation (3). Under such conditions, the rate of the reverse reaction is necessarily negligible at potentials where the forward reaction proceeds with a significant rate so that such processes have been classified irreversible. Although defini-
41 =
cot-’ (1
X = -ksf -(e DllZ
T)
+6
-ai
+ eBi)
(7)
To whom reprint inquiries should be addressed.
(9) (1) B. Timmer, M. Sluyters-Rehbach, and J. H. Sluyters, J. Electroanal. Chem., 14, 169 (1967). (2) B. Timmer, M. Sluyters-Rehbach, and J. H. Sluyters, Zbid., p 181. (3) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, p~ 76-83.
474
0
ANALYTICAL CHEMISTRY
(4) H. Matsuda, 2.Elekrrochem., 62,977 (1958). (5) D. E. Smith, in “Electroanalytical Chemistry,” A. J. Bard, Ed., Vol. 1, M. Dekker, Inc., New York, 1966, Chapter 1.
h
0 X
a
24C
3. u c z W
K 200 K 3 V
W
z & z
I60
K
W
2
120
0
z
0
r
080
I -I
2
5
040
2
Figure 1. Fundamental harmonic ac polarographic waves of irreversible systems calculated from various theoretical formulations Parameter values: n = 1, T = 25" C, A = 0.0350 cm2,A E = 5.00 mV, CO*= 1.00 X lO-aM, Do = D n = 1.00 X 10-~cm2sec-l, w = 100 sec-l, t = 10.0 sec, ks = 1.00 X 10-kn sec-l, CY = 0.500 ( a ) ____ Wave calculated from rigorous general formulation of Equations 1-13 (b) - - - - - Wave calculated from Equation 23 ( c ) -. - * -. - Wave calculated from Equation 27 ( d ) . , , . . . , . Wave calculated from general equations of Timmer, Sluyters-Rehbach, and Sluyters ( I ) Note: Special notation of waves b, c, and/or d is not visible at potentials where they merge with wave a
.f
D
(10)
= f0pfRU =
(11)
DopDza
=
0.8515 X 10-3(mt)2'3
(13)
It has been customary to employ the approximate expression for ~ ) ( X f l / ~ )suggested by Matsuda ( 4 ) L
7 (1.61 37r (1.13
e-aj
(12)
/3=l-a A
polarographic wave (6). Substituting Equation 15 in Equation 3 and invoking the relationships
>> 1 > e@j>> e > e1 ae-'I)(Xtl/2) >> 1
(16)
(1 7) which characterize irreversible conditions (only), one obtains F(Xt"2) =
+ + Xt1/2)2
when applying Equations 1-1 3. However, this expression becomes inaccurate for small k , values characterizing the irreversible processes of interest here. An obvious alternative is to employ the rigorous formulation of +(Atl/z) given by Equation 4. Unfortunately, the complexity of the series formulation of Equation 4 presents a serious obstacle in the way of our primary goal of reducing the general formulation to an analytical expression which is applicable under irreversible conditions. A more appropriate alternative appears to be application of the expression
which was shown to be a very precise approximation to Equation 4 over most of the potentials encompassed by the dc
n 2F2ACO*( w Do)'' 2AE Ire". =
(19)
RTe-j
Employing the relation 1.349kse- ' j t l / where
2 0 - l/2
= e-J
(21)
(6) D. E. Smith, T. G. McCord, andH. L. Hung, ANAL.CHEM., 39, 1149(1967). VOL. 40, NO, 3, MARCH 1968
b
475
Table I. Peak Potentials of Irreversible AC Polarographic Waves Calculated from Various Theoretical Formulations cmz sec-', t = 10 sec Values of relevant fixed parameters: n = 1, T = 25" C, Do = D E = 1.00 X [Edclpeak
CY
CY
CY
=
=
=
0.2
0.5
0.8
k , = 1.00 X lW5 cm sec-l k, = 1.00 X 10-6 cm sa-'
w = w = w = w =
100 1.00 100 1.00
x 104
k . = 1.00 X cm sec-l k , = 1.00 X 10-6 cm sec-l
0 = w = w = =
100 1.00 100 1.00
x x
104 104
k , = 1.00 X lW5 cm sa-' k , = 1.00 X 10-6
w = 100 = 1.00 w = 100 w = 1.00
x
104
cm sec-'
x 104
- &/2'
Calculated from general formulation Calculated from of Equations Equation 1-1 3 29 -0.804 -0.817 -0.948 -0.965 -1.099 -1.112 -1.245 -1.260
x 104
Calculated from Equation 30 -0.826 -0.974 -1.122 -1.270
-0.322 -0.380
-0.327 -0.386
-0.440
-0.445
-0.498
-0.504
-0.331 -0.390 -0.449 -0.508
-0.201 -0.237 -0.275 -0.311
-0.204 -0.241 -0.278 -0.315
-0.207 -0.244 -0.281 -0.318
Table II. Peak Magnitudes of Irrevenible Fundamental Harmonic AC Polarographic Waves Calculated from Various Theoretical Formulations M , D o = D R = 1.00 X Value of fixed parameters: n = 1, T = 25" C, A = 0.0350 cmz, AE = 5.00 mV, CO*= 1.00 x cm2sec-I, t = 10 sec Peak magnitude, microamps X 10 Calculated from general Calculated Calculated formulation Calculated from from of Equations from 1-13 Equation 33 Equation 32 Equation 34 0.902 0.901 0.889 1.133 w = 100 k, = 1.00 X 1.054 1.056 1.046 1.133 = 1.00 x 104 CY = 0.2 cm sec-l 0.902 0.901 0.889 1.133 w = 100 k , = 1.00 X 10-6 1.054 1.056 1.046 1.133 = 1.00 x 104 cm sec-1 a
CY
=
0.5
= 0.8
k , = 1.00 X loV5 cm sec-l k , = 1.00 X 1 0 - 8 cm sec-l
w = 100 = 1.00 w = 100 = 1.00
k , = 1.00 X cm sec-l k, = 1.00 X cm sec-I
w = = w = =
10-6
100 1.00 100 1.00
x 104 x 104 x 104 x 104
and algebraic rearrangement yields the result Z(ut) =
1 . 6 4 4 ~2FZA 1 CO*(WDO) "'A E RTcJ(l e1.09'J)[l (1 QeJ)211/2 sin [ut + cot-i(l + Q e J ) ]
+
+ +
Q = 1.907(~t)"~
(23) (24)
Equation 23 is a reasonably precise representation of the form assumed by Equations 1-13 in the limit of the irreversible process (see Figure 1). Although quantitative differences are apparent between Equation 23 and the corresponding result of Timmer, SluytersRehbach, and Sluyters (see Equation 20 of reference I ) , the disparities are often small and they are in complete agreement regarding the very important qualitative prediction that a measurable ac wave of k,-independent magnitude and shape will be observed with irreversible systems. In Equation 23 the parameter, k,, arises only in the potential-dependent quantity, J , so that the sole influence of k, will be in determining the position of the wave on the dc potential axis. The direct proportionality between the fundamental harmonic 476
ANALYTICAL CHEMISTRY
2.256 2.636 2.256 2.636
2.252 2.640 2.252 2.640
2.223 2.615 2.223 2.615
2.832 2.832 2.832 2.832
3.618 4.226 3.610 4.218
3.603 4.223 3.603 4.223
3.557 4.182 3.557 4.182
4.531 4.531 4.531 4.531
current magnitude and the charge transfer coefficient, C Y , is another important feature shared by Equation 23 and the result of Timmer, Sluyters-Rehbach, and Sluyters. An analytical expression for the peak potential and magnitude is difficult to ascertain from Equation 23. However, a readily differentiable simplification of Equation 23 can be obtained with the aid of the approximations
+ el.OglJ)
+ e-J [l + (1 + Q e J ) 2 ] 1 / 21 ~+ QeJ e-J(1
N
1
(25) (26)
which, when applied to Equation 23, yield Z(wt) E
1 . 6 4 4 ~'F2A 1 Co*(UDO) l / 2AE X RT(1 e-J)(l QeJ) sin [ut cot-l(l
+
+
+
+ QeJ)] (27)
One can show by differentiation, etc., that the Z(ut) function of Equation 27 reaches a maximum when e-J
=
Q1/2
(28)
Thus, the peak potential is given by
The analog of Equation 29 given by Timmer, SluytersRehbach, and Sluyters is a nearly identical relationship which, in our notation, may be written [Edclpeak
= El/;
1 .251k,t1/2 RT RT +In _ _ - ~ -In 2anF anF ( D112 )
Q (30)
Recalling the appropriate expression of the dc polarographic half-wave potential, Eli2, for irreversible systems (7), one also finds that
These results indicate that the ac polarographic peak potential with irreversible systems is a function of frequency with the peak shifting in a cathodic direction by 14.7/an mV per decade increase in frequency at 25” C and that it is displaced cathodically by a substantial amount from the dc polarographic half-wave potential. Although Equation 29 is based on the approximations of Equations 25 and 26, it predicts a peak potential very close to that obtained from numerical calculations based on Equation 23 or the general formulation of Equations 1-13. The same applies to Equation 30 of Timmer, Sluyters-Rehbach, and Sluyters. Results of sample calculations which demonstrate these points are given in Table I. An expression for the peak magnitude obtained on substituting Equation 28 in Equation 27 is
1 I ( w t ) 1 peak
1.644an~FZACO*( w D O l)/ 2AE RT(l Q1/z)z
+
or log r should be linear and of slope 0.0147/an (at 25°C) for this class of systems. Other bases for diagnosis and parameter calculation also are apparent in the foregoing expressions. The direct proportionality between peak current and t - l 1 2 suggested as a criteria for irreversible systems ( I ) should be applied with some care. Equation 33 suggests that a r 1 I 2dependence will not be precisely obeyed at lower frequencies since it is consistent with Equation 33 only when Q112>> 1. The same applies to the frequency independence of the peak current alleged by the results of Timmer, Sluyters-Rehbach, and Sluyters. Deviations of the peak current from the t-l12 and w o dependence suggested by Equation 33 are relatively minor, but should be experimentally detectable at low frequencies. Accordingly, one must recognize that such deviations do not necessarily imply that the system under investigation is following a mechanistic scheme different from the classical irreversible case. SECONDHARMONIC CURRENT.Employing the same degree of rigor used in obtaining Equations 1-13, one may show that Matsuda’s treatment may be extended to yield a general formulation of the second harmonic current given by (8) Z(2wt)
=
Z(2w)W(w) sin ( 2 o t
1 . 6 4 4 ~21F 2 A C ~ *D(o~) ” ~ A E R T Q I / Z (+~ Q - 0 . 5 4 5 ) [ 1 + ( 1 + Q l / 2 ) 2 ] 1 / 2
nFAE W(w) = __ 4RT (P2 + L2)”2
(32)
r
(33)
r 1
(7) L. Meites and Y.Israel, J. Am. Chem. SOC.,83,4903 (1961).
- .
6
1
L
Equations 32 and 33 may be contrasted to the expression
which in our notation is the peak current expression suggested by Timmer, Sluyters-Rehbach, and Sluyters (see statement above Equation 21 of their manuscript). Equation 34 corresponds to the high frequency limit of our more approximate formulation for the peak current, Equation 32. Table I1 compares results of peak current magnitudes for irreversible processes calculated from various expressions given above, while Figure 1 provides a similar comparison for the entire ac polarographic wave with a = 0.5. One sees that Equations 23 and 33 simulate with precision the results predicted by the general formulation, except at the anodic foot of the wave where Equation 23 becomes slightly inaccurate, probably because of errors introduced by the use of Equations 15 and 17 in the development of Equation 23. Our more approximate formulation (Equations 27 and 32) and those of Timmer, Sluyters-Rehbach, and Sluyters are generally less accurate. It is apparent from the foregoing relationships, not to mention the extensive theoretical and experimental work of Timmer, Sluyters-Rehbach, and Sluyters, that ac polarographic data can be most useful in the study of systems rendered irreversible by slow charge transfer. For example, Equation 31 indicates that a plot of [EdclPesk- Eli2 us. log w
+2)
where
while a more precise formulation results from utilizing Equation 23 in place of Equation 27: ~z(Wt)~gea= k
+
+ e-5
Gx
1 L
i
-
+(1 +
e)z -1
L
-I
Other quantities are defined either in the notation definitions or in relationships given above. As with the fundamental harmonic, these relationships may be reduced to expressions which are applicable tQ the irreversible process by applying Equations 15-17 and 21. One obtains Z(2w) =
2.325an2F2AC~*(wDo)1/2AE RTe-J(l el.OglJ)[l ( 1 QzeJ)2]112
+
Q Z=
z/?e
+ +
= 2.697(~r)~/~
(42) (43)
(8) T. G . McCord and D. E. Smith, ANAL.CHEM. 40,289 (1968). VOL 40, NO. 3, MARCH 196%
477
-
'
m
X
4.80
a
x
Y
4.00
!-
z w
E [II
3
3.20 W
zc a
5
2.40
W
5a 9
1.60
z
0
5
a
I 0 0.80
z
0 0 W
cn
Figure 2. Second harmonic ac polarographic waves of irreversible systems calculated from various theoretical formulations Parameter values: Same as Figure 1 (a) Wave calculated from rigorous general formulation of Equations 35-41 (b) . . - - - - - Wave calculated from formulation of Equations 35, 42-45. Not visible at potentials where it merges with wave a -. -. - * - * Wave calculated using Equations 35, 44, 45, 47. Not visible at potentials where it merges with wave a (6) , . . . . . . . Wave calculated using Equations, 35, 47-49. Not visible at potentials where it merges with wave c ~
P = a{l - 2[
1
1)
+2(1+ +QeJQeJIz
(44)
(45)
and J is given by Equation 22. Application of these relations together with Equations 35 and 37 gives the expression for the second harmonic wave of irreversible systems. Since k. appears only in the J parameter, it is apparent that these expressions predict a second harmonic wave whose shape and magnitude are independent of k,. Only the wave position is controlled by this parameter. The accuracy of Equations 42-46 in simulating the predictions of the general expression (Equations 35-41) is excellent as shown in Figure 2. One might suggest that, analogous to the fundamental harmonic case, application of Equations 25 and 26 will yield a reasonably accurate simplification of Equations 42-46. One would obtain Z(2w) =
2.325anZF2ACo*(wDo)'/211E RT(1 e-J)(l +QzeJ)
+
(1
478
ANALYTICAL CHEMISTRY
+ QeJP
(47)
L =
(1
2aQeJ QeJ>2
+
(49)
Unfortunately, this simplification leads to very serious error at potentials on the cathodic portion of the second harmonic wave as shown in Figure 2. This occurs because Equation 26 becomes somewhat inaccurate at negative potentials (eJ becomes small making QeJ insufficiently large to justify Equation 26). Although this yields an error which is not excessive in predicting the Z ( 2 w ) term, the form of the subsidiary functions, P and L , in the W(w) term amplify the inaccuracy of Equation 26. A reasonable compromise is to use Equation 47 for Z ( 2 w ) in conjunction with the more exact expressions of Equations 44 and 45 for P and L (see Figure 2). The shape which characterizes the predicted second harmonic wave of totally irreversible systems is apparent from Figure 2. Because of the cumbersome nature of even the least precise formulation given above, analytical expressions for the associated peak potential and magnitude have not been developed. However, numerical calculations based on either Equations 35-41 or Equations 42-46 show that the predicted second harmonic peak is found approximately 13.5/an mV anodic of the fundamental harmonic peak at all frequencies and is about a factor of an124 as large. The Case of Irreversibility Induced by Chemical Decomposition of the Electrolysis Product to an Electroinactive Form. Another important case where the electrode reaction cannot be reversed may occur when the well-known reaction scheme
k
O + n e S R + Y
(R2)
is operative. If the half-life of species R is short compared to the time scale of the electrochemical experiment, the electrode reaction may be considered chemically irreversible. In ac polarography, the basic requirement for chemical irreversibility is that the period of the applied alternating potential greatly exceeds the half-life associated with the chemical decomposition so that reoxidation of species R cannot contribute significantly to the alternating current. Since our primary purpose is to demonstrate that systems rendered irreversible due to chemical decomposition following charge transfer also give rise to measurable ac polarographic waves, we will consider the simplest case where the chemical reaction is first order. To simplify matters further and to focus attention on irreversibility which is purely chemically induced, the effects of charge transfer kinetics will be assumed negligible-Le., Nernstian conditions will be assumed to prevail in both the ac and dc sense. FUNDAMENTAL HARMONICCURRENT.For the scheme corresponding to Reaction R2 with the aforementioned restrictions, the expression for the ac polarographic wave within the framework of the expanding plane electrode model may be written (5, 9-11>. Z(wt)
=
17ev.F(e-jk1/2t1/2)G(g) sin (ut
+
Conditions for chemical irreversibility described above imply that
>> w
(59)
g
>> 1
(60)
so that Further, the positive shift in the potential of the dc wave attending the fast following chemical reaction (12-14) validates the relation e-j
> w and charge transfer kinetics have no effect on the response (equations accounting for charge transfer kinetic contributions are readily obtained-see references 5,9,and 16). Equation 76 predicts a drop-life and frequency-independent ac polarographic wave whose shape is identical to that of the diffusion-controlled or reversible wave (note that the term I,,,. of Equation 2 represents the amplitude of the diffusioncontrolled wave). Indeed, Equation 76 is identical to the expression for the reversible ac polarographic wave except that k , appears in place of w and the phase angle is zero rather than 45'. Since k , >> w is implicit in Equation 76, catalytic systems which obey this relation will yield fundamental harmonic ac polarographic waves which are larger than their diffusion-controlled counterpart. The proportionality between k c 1 l zand the alternating current amplitude parallels the identical behavior of the dc wave (17) for the conditions under consideration and represents a notable difference from the irreversible cases considered above where the wave magnitudes are independent of the relevant rate parameters. SECOND HARMONIC CURRENTS.The second harmonic wave with the catalytic mechanism and Nernstian conditions can be shown to obey the expression (IO)
n 3F3ACo*(wD~) l / ZAE2sinh Z(2wt) = 8R2T2cosh3
(i)
Tz2
+ Uz2
sin (2wt
+
$2)
X
(77)
where where $2
T
$1
= cot-'- U
cot-'
(- $)
(71)
The derivation of Equations 77-81 follows the same approach as for the fundamental harmonic theory and will be outlined elsewhere. Invoking Equation 73 leads to the result
Invoking the condition for chemical irreversibility, kc >> w
=
(73)
one obtains
n3F3ACo*(k,Do)l/2AE2sinh Z(2wt) =
X
(74)
u=o
sin (20t (75)
(16) J. R. Delmastro, Ph.D. Thesis, Northwestern University, Evanston, Ill., 1967. 480
ANALYTICAL CHEMISTRY
-
);
(82)
(17) J. Heyrovsky and J. Kuta, "Principles of Polarography," Academic Press, New York, 1966, pp 383-5.
Thus, as with the fundamental harmonic, one finds that the second harmonic wave of a catalytic process assumes the same appearance as the diffusion-controlled second harmonic wave (5, 18, 19) when Nernstian conditions prevail. When the chemical reaction is sufficiently rapid to negate the possibility of electrochemical oxidation of R,the expression for the second harmonic current (Equation 82) differs from the corresponding equation for the diffusion-controlled wave (5, 18, 19) only in the appearance of k , in place of w and the 45" difference in phase angles. Under these conditions, the second harmonic wave of a catalytic system will also exceed magnitude its diffusion-controlled counterpart.
= dc component of applied potential =
CONCLUSIONS
The foregoing discussion has considered the characteristics of the ac polarographic response under conditions where reversal of the electrode reaction is precluded during cycling of the applied potential. For three notably different sources of such irreversibility, an easily measurable fundamental and second harmonic alternating current was predicted (with the exception of the trivial case where (Y S 0 in the first example considered). The functional relationships for each class of irreversible wave suggested that their observation would be useful for mechanistic diagnosis and evaluation of kinetic parameters. Upon consideration of the implications of these results, it appears that they can be qualitatively generalized, leading one to conclude that a measurable ac polarographic response is expected in general for irreversible processes, regardless of the mechanistic scheme or faradaic component in question. NOTATION DEFINITIONS A
= electrode area
Di
diffusion coefficient of species i activity coefficient of species i = initial concentration of oxidized form = standard redox potential in European convention
fi
Ca*
EO
= =
(18) M. Senda and I. Tachi, Bull. G e m . SOC.Japan, 28,632 (1955). (19) D. E. Smith and W. H. Reinmuth, ANAL. CHEM.,33, 482 (1961).
amplitude of applied alternating potential reversible polarographic half-wave potential = irreversible polarographic half-wave potential = dc potential at peak of fundamental harmonic wave = Faraday's constant = ideal gas constant = absolute temperature = number of electrons transferred in heterogeneous charge transfer step = applied angular frequency = time = heterogeneous charge transfer rate constant at Eo = charge transfer coefficient = rate constants for homogeneous chemical reactions coupled to the charge transfer step = mercury flow rate in mg sec-l = Euler Gamma Function = fundamental harmonic faradaic alternating current = magnitude of peak fundamental harmonic faradaic alternating current = second harmonic faradaic alternating current = phase angle of fundamental harmonic faradaic alternating current relative to applied alternating potential = phase angle of second harmonic faradaic alternating current relative to applied alternating potential =
I I(wt) I peak Z(2at)
dl 42
ACKNOWLEDGMENT
The authors are grateful to Kathryn Bullock who aided with some of the calculations.
RECEIVED for review October 27, 1967. Accepted December 20, 1967. Work supported by National Science Foundation Grant GP-5778. One of the authors (T. G. McCord) is a NIH Graduate Fellow.
VOL 40, NO. 3, MARCH 1968
481
