Calculations Concerning the Measurability of the Potential Dependence of the Charge Transfer Coefficient by Direct Current Polarography Leon N. Klatt and David R. Lewis Department of Chemistry, University of Georgia, Athens, Ga. 30602
Theoretical considerations concerning electron transfer reactions predict a linear dependence of the electrochemical transfer coefficient, a , upon the electrode potential. However, attempts to experimentally verify this prediction have been inconclusive. This study examines the effect that a potential dependent transfer coefficient should have upon the dc polarographic current-voltage curve. For quasi-reversible systems ( k s > 5 X cm/sec) the current-voltage curve differs only slightly from the one calculated using a potential independent transfer coefficient. I n the case of totally irreversible charge transfer reactions ( k s < 5 X c m / sec), large effects are seen in the i vs. E curves. The € 7 1 2 is shifted negatively with respect to the corresponding E1/2 of the i vs. E curve with a constant a ; and the approach of the current to the diffusion limited value is very slow. Also, the i vs. E curve has an inflection point positive of the half-wave potential (assuming a reduction process). Although these differences seem significant, the currentvoltage curve can be simulated if one considers a (independent of potential) and k s as adjustable parameters; thereby making the shape of the i vs. E curve as the sole criterion for the determination of a's potential dependence rather useless. An important observation from the results of this work is that the value of a required for the simulation decreases to small values as the rate constant decreases. I f a is dependent upon potential, this trend should be observed experimentally for chemical systems with k , ranging from to cm/sec.
During the past two decades, theoretical and experimental studies of heterogeneous electron transfer reactions have received considerable attention by electrochemists. Infinite series solutions to the difficult boundary value problem describing the polarographic experiment have been developed by Koutecky (1-5) and Matsuda and Ayabe (6, 7). Qualitative discussions and quantitative treatments of reaction schemes and the chemical nature of the activated complex have been presented by Marcus (8, 9) and Dogonadze ( I O ) . Bimolecular reactions involving the transfer of atoms or groups of atoms require strong interactions of the electronic orbitals of the reactants in the activated state in order to effect the transfer of nuclei. However, this degree
of interaction is not necessary for an electron transfer (8, 9). Small electronic interactions are sufficient to permit this reaction. Furthermore, in order for the probability of product formation to become large enough to represent a significant reaction rate, the small overlap must be accompanied by changes in the translational, vibrational, and rotational coordinates of the reactant particles to produce the nuclear configuration of the activated state. This involves not only changes in configuration of reactant molecule atomic structure, but also changes in the solvent sheath and in neighboring solute molecules. A heterogeneous electron transfer reaction can be described in the following manner. As a reactant approaches the electrode, its electronic, atomic, and solvent sheath configurations are in a stable state corresponding to the solvated reactant. Similarly, we find the distribution of electrons in the energy levels of the electrode to be that predicted by Fermi-Dirac statistics. As further approach occurs (movement along the reaction hypersurface toward the classical activated complex state), the solvent sheath fluctuates, allowing a small overlap of the reactant electronic orbitals with the electronic levels of the electrode. The system has entered the initial activated complex state which is characterized by the atomic and electronic configurations of the reactant and by the solvent sheath configuration of the activated complex. The system can revert back to a complete reactant configuration or proceed by an electron transfer to a second activated complex state. This second complex is characterized by the atomic configuration of the reactant with the electronic configuration of the product and no change in the orientation of the solvent sheath. The system is again free to revert to a pure reactant state uia the reverse electron transfer or may continue toward reaction completion by atomic fluctuation to a configuration of that of the product. This change in atomic structure, coupled with movement away from the electrode surface and reorganization of the solvent sheath to a stable configuration for the product, completes the charge transfer reaction. This reaction scheme involving the overall conversion of the reactant 0 to the product R k
O + e + R is represented by 0
J. Koutecky, Collect. Czech. Chern. Cornrnun., 18, 597 (1953). lbid.. 19, 1093 (1954). /bid.. 20, 116 (1955). /bid.. 21, 1056 (1956). J. Weber and J. Koutecky, ibid., 20, 980 (1955) t i Matsuda and Y . Ayabe. Bull. Chern. SOC.Jap., 28,422 (1955). lbld.. 29, 134 (1956). R. A . Marcus, J. Chern. Phys., 24, 966 (1956) lbid.. 43, 679 ( 1965) R. R. Dogonadze, "Theory of Molecular Electrode Kinetics," in "Reactions of Molecules at Electrodes," N. S. Hush, Ed., Wiley-lnterscience, New York, N.Y., 1971, pp 135-227.
k
k-
x* i:
X * + e & X k-.
ir
X - R with k, = k , / [ l
+ (1 +
The two activated complex configurations, X* and X, constitute the classical activated complex. They are separate and distinct because of the small electronic overlap and subsequent low probability of electron transfer. If this overlap and probability of transfer had been large, an electron would have moved nearly instantaneously, and distinction of the separate states of the activated complex would not have been necessary. From a statistical mechanical approach to this electron transfer mechanism, Marcus (9) has predicted a linear potential dependence for the transfer coefficient, a. Attempts to verify this prediction in the heterogeneous case have been the subject of two recent studies (11, 22). Parsons and Passeron (12) polarographically studied the charge transfer kinetics of the Cr(II)/Cr(III) system in perchlorate media, and after correcting for double layer effects reported quantitative conformance to Marcus' prediction. Anson et al. ( 1 1 ) attempted to extend the work of Parsons and Passeron by examining the effect that the supporting electrolyte may have upon the potential dependence of alpha. However, they were unable to duplicate the above work and concluded that propagation of experimental errors in both polarographic measurement and in theoretical conversion of polarographic data to rate data were responsible for the failure of their attempts. The object of the study reported here was to elucidate this problem by calculating error free polarograms with both potential dependent and independent transfer coefficients followed by detail analysis aimed a t ascertaining unique features of the current-voltage curve which would permit one to differentiate between the two cases. The results permit us to define the type of polarographic correlation one should observe if the'dependence of a upon potential, as predicted by Marcus, can indeed be detected polarographically. CALCULATIONS A N D RESULTS In the discussion that follows, we will consider that the reaction, kt
O+ne-;-tR ii
(1)
occurs as a reduction a t the dropping mercury electrode. However, the results apply equally well to oxidations. A complete definition of symbols is found in Appendix I. Koutecky (1-5) has shown the relationship between the instantaneous current for a slow charge transfer reaction, i, and the theoretical instantaneous reversible current, ire,, to be i/Oe, =
F(Y)
where ( 1 3 )
with the polarographic diffusion current, id, being given by the numerator of Equation 2. F ( y ) is defined by an infinite series as follows, (3)
\'
P,=
r(+) y is determined by the kinetic parameters, h, and a , and electrode potential E by the Equations 6 through 8. y =
qm + h) 1% hi
Upon making the assumption that Do = 6-8 yield: y =
E{
T k , exp -&-(E
[
-
E")]
DR,
Equations
+
n F RT
exp (1 - cu)"-(E
- E"')
Using Equations 1 through 5 and 9, a current-voltage curve for a system characterized by a slow charge transfer reaction defined by particular values of a and k , can be generated. From Marcus' (9) expression for the free energy of activation of a heterogeneous electron transfer reaction, it can be shown that the transfer coefficient for reaction 1 can be written as (assuming n = 1): CY =
0.5
E - EO') + F ( 8AG,'
where AGRf is the molar free energy of activation for the system a t E = EO'. A reasonable estimate of A G R Z can be obtained from Marcus' (9) expression for the forward heterogeneous electron transfer rate constant, ill,
where K is a transition probability, normally one for an adiabatic reaction (a system that remains on the same potential energy nuclear configuration hypersurface throughout the course of the reaction), ( is the ratio of the mean square deviation of the distance between the ion and electrode to the mean square deviation of the perpendicular distance from the reaction hypersurface, also normally close to unity. Zbet is the heterogeneous collision number for a reactant species of mass m aand is given by
Assuming that f = 1,
which a t E = EO' reduces to
with ( 1 1 ) F. C. Anson, N. Rathjen, and R. Frisbee, J. Electrochem. SOC., 117,477 (1970). (12) R. Parsons and E. Passeron, J. Electroanal. Chem., 12, 524 (1966). (13) P. Delahay, "New Instrumental Methods in Electrochemistry," Interscience, New York, N . Y . , 1954, p 55.
Upon substitution of Equation 14 into Equation 10, the following expression for the potential dependence of the transfer coefficient as a function only of the standard heterogeneous rate constant is obtained
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 1, J A N U A R Y 1974
25
-0.1
00
-0.2
-0.3
-0.1
( E - E ~ ) VOLTS
Figure
1.
Polarograms characterized by charge transfer kinetics cm/sec
cy
= g ( E ) as defined by Equation 15
=
0.5
F ( E - Eo') + 8RT In (Zhet/hs)
(15)
(16) which for y 2 2.0 yields F ( y ) values with errors of only 0.1%. The F ( y ) array, along with the corresponding ire, values calculated from the overpotential array then gives rise to an i array. For convenience, the i values were converted to current ratios (i/id) and plotted as a function of ( E - EO') to represent the theoretical polarogram. All calculations were carried out assuming a reactant mass of cm2/sec, m = 1.0 mg/sec, time 100 amu, DO = 6.0 x t = 4.0 sec., and n = 1. Meites (15) has shown, following Koutecky's work (3-6), that a totally irreversible polarogram with constant transfer coefficient is described by the following relationship,
Nn
(F) (17)
,F
for the range 0.1 5 i/id I0.94. Equation 17 is also obtainable from Equation 16 and Equation 9 neglecting the second exponential. This relationship provides a convenient method for calculating kinetic parameters, cy and h,, from the shape and position of a totally irreversible polarogram, respectively. Another method of computing these kinetic parameters involves back calculating from i/irev us. E data to y us. E data along the lines of the Koutecky method outlined (14) D. E. S m i t h , T. G . McCord, and H . L. Hung, Anal. Chern., 39, 1149 (1 967). (15) L. Meites and Y . Israel, J . Arner. Chem. SOC., 83,4903 (1961). 26
13
Figure 2. Polarograms characterized by charge transfer kinetics cm/sec fork, = 5 X -A-,
A computer program was written to calculate a fiftypoint, idealized error-free polarogram from the above relationships. An array of overpotentials, calculated to produce a polarographic current which varied from i/id = 0.1 to 0.9, was used to generate a corresponding array of y values for each set of kinetic parameters studied. The value of cy was either set to a constant or was calculated as a function of potential. From these y values, the corresponding F ( y ) array was obtained via Equations 3-5 for y < 2.5. Due to the numerical instability of Equations 3-5 for y I2.5 ( 5 ) , F ( y ) for y 1 2.5 were calculated using the approximate function (14),
2.110 RT log
-0.9
IE-EO') VOLTS
fork, = 5 X
- A - , N = 0.50;-0-, 01
.0.8
a = 0.50;- 0 - , a = g ( E ) as defined by Equation 15
above. Then, assuming the contribution of the back reaction to be negligible (which is equivalent to equating i/irev to i/id), one may plot, from Equation 9,
and evaluate cy and k , from the slope and intercept, respectively. Error free polarograms were computed for heterogeneous rate constants ranging from 5 x 10-3 cm/sec through 5 x 10-8 cm/sec, with cy = 0.50 and cy = g ( E ) defined by Equation 15 for each of the rate constants considered. (In the cases where cy is assumed to be independent of potential, a value of 0.5 was selected because it is most rdpresentative of typical chemical systems (It?), and is the value predicted by Marcus a t E = EO'.) Differences between the cy = g ( E ) and cy = 0.50 cases became discernible at k , N 2 x 10-4 cm/sec, with a 4.8-mV negative shift in the half-wave potential of the cy = g(Ej curve from that of the cy = 0.50 curve. This negative shift of the cy = g ( E ) half-wave potential increases as k , decreases. Comparison of the i/id us. E and i/ireb us. E curves indicates that the shape of the current-voltage curves for both cy = 0.5 and cy = g ( E ) are influenced by the back reaction for systems with k , > 2 x 10-5 cm/sec. The largest relative deviation between the i/id us. E and i/irev us. E cases occurs at the most positive potentials and is 2.8% at h, = 2 x cm/sec, but decreases to 0.4% a t h, = 7 x cm/sec. These differences for the quasi-reversible systems are small and probably will not be detectable experimentally. This study then concentrated on the effects of potential dependence of the transfer coefficient for polarographic systems with standard heterogeneous rate constants ranging from 5 x to 5 x cm/sec. The larger rate constant was chosen to coincide with the recent investigations of Parsons et al. (12) and Anson et al. ( I I ) , and the rates studied decreased from this upper limit to avoid unwanted back reaction contributions to the polarographic curves. Error free polarograms were computed for rate concm/sec, with investigation of stants of 5 x 10-5-5 x both the cy = 0.5 and cy = g ( E ) cases. Comparisons of current-voltage curves obtained for the two types of transfer coefficients showed an increasing disparity with decreasing rate constant (Figures 1 and 2 ) . As shown in Figures 1 and 2 , as well as noted for the quasi-reversible systems, the half-wave potentials for the cy = g ( E ) polarograms (16) N Tanaka and R T a m a m u s h i , Electrochim Acta, 9,9b3 (1964)
A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 1 , J A N U A R Y 1974
0.0
- 0.I -
-
-o
-0.2
Y
I
w
I
-03
-0.0
-0.2
-0.1
-0 4 -I 0
-0.5
0 LOG
0.5
Figure 4.
[Ibi]
and
Analysts of polarograms in Figure 1 according to Equation 1 7 Figure 3. -A-,
(E-EOO
1.0
a = 0.50; -0-. a = g(E) as defined by Equation 15
-0.3
VOLTS
Comparison of polarogram calculated for cy = g ( E ) with polarogram calculated for a e f f e c t i v e and k e f f e c -
ks,tr"e
tive
a = g ( E ) as defined by Equation 15. k S = 5 X cm/sec. -0-, a = 0.396,k, = 6.89 X cm/sec as evaluated from the slope -A-,
and intercept of plot shown in Figure 3
occur at potentials negative of the a = 0.5 case. This difference increases as k , decreases such that at k , = 5 x 10-8 cm/sec 178 mV separates the respective half-wave potentials. Using a numerical differentiation scheme to calculate the inflection points of the a = g ( E ) polarograms, these inflection points were compared with the classical E1/2 values. For all cases in which the transfer coefficient was constant, the calculated inflection point agreed with the half-wave potential well within the limits of the numerical method used, and this agreement was expected on the basis of Equation 17. However, this correspondence was not observed for the polarograms which included the potential dependence of the transfer coefficient. Minimum difference was observed in the 12, = 5 x cm/sec case with a deviation of the inflection point from El,2 of 7.5 mV positive. A deviation of 38.3 mV positive was observed in the k , = 5 x cm/sec case. Shape analyses for the curves shown in Figures 1 and 2 and additional cases were calculated using a linear least squares fit to Equation 17. The cases with a = 0.5 gave virtually identical shape analysis plots, showing a small curvature well within the limits of the method (15). Calculated kinetic parameters from these curve fits were in good agreement with the values used to generate the polarograms. However, the same analyses for the cy = g ( E ) cases showed downward concave curvature (Figure 3 ) . Maximum deviation from linearity occurred a t i/id N 0.95 and ranged from -10 mV for k , = 5 x cm/sec to 71 mV for the 5 x cm/sec case. The effective transfer coefficients and rate constants calculated from the least squares fits of the current-voltage curve to Equation 17 for a = g ( E ) cases varied from a = 0.396, k , = 6.89 x cm/sec to a = 0.196, k , = 4.28 x 10-6 cm/sec when the true value of k , was 5 x 10-5 and 5 x 10-8 cm/sec, respectively. These effective kinetic parameters further indicate the increase in disparity of the a = g ( E ) polarograms from the a = 0.5 curves with decreasing rate constant. Even though the linear coefficient for the potential dependence of cy becomes greater with increasing rate constant (Equation 15), the net effect of this dependence appears to be minimal for large rate constants and grows rapidly with decreasing rate constants. The above computations have presented sufficient evidence to show that a = g ( E ) polarograms differ both in shape and position from a = 0.5 polarograms. However,
-0.1
-0.8
-0.9
(E-&)
Figure 5.
and
k,,true
-1.3
VOLTS
Comparison of polarogram calculated for a = g ( E ) with polarogram calculated for a e f f e c t i v e and k e f f e c -
tive
cm/sec. (Y = g ( E ) as defined by Equation 15. k, = 5 X -0-, aeffective = 0.196,keffective = 4.28 X cm/sec as evalu-A-,
ated from the slope and intercept of plots similar to those shown in Figure 3
these results were not able to answer our initial question of the polarographic measurability of a potential dependent transfer coefficient. To answer this, it was necessary to ascertain whether one could calculate a polarogram with constant a which would come close to duplicating the curve resulting from a potentially dependent transfer coefficient. Idealized, error free polarograms with constant transfer coefficients were computed for each of the rate constants considered above. The kinetic parameters were chosen by exhausting all possible combinations of aaverage, the mean value of cy in the potentially dependent cases; C Y e f f e c t l v e , the value of the transfer coefficient calculated from the analyses of the polarograms according to Equation 17 for the a = g ( E ) cases; ktrue,the actual value of k , generating the (Y = g(E) curves above; and keffectlve,the effective value of the rate constant calculated from the analyses of the polarograms according to Equation 17 for the above a = g ( E ) cases. In all cases studied, the polarograms with and rate constant k e f f e c t l v e transfer coefficient of aeffectlve came closest to approximating the a = g ( E ) polarograms. Greatest similarity was exhibited at the largest rate con-
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1, J A N U A R Y 1974 * 27
(E-EOO
Figure 6.
VOLTS
Comparison of curvature in shape analysis plot
-X-,
a = g ( E ) , k, = 5
X
a =
aeffective and k ,
= kerrective,including back reaction. -A- a =
g ( E ) ,k, = 5 X
cm/sec, neglectingback reaction. -0-,
cm/sec, including back reaction
stant, where the two curves are virtually indistinguishable, and least similarity was shown at the smallest rate constant (Figures 4 and 5). It is interesting to note that as the rate constant decreases, a e f decreases more rapidly than aaverage. The closest agreement occurs a t the = 0.448 and aeffectlve largest rate constant, aaverage 0.396 and the point of poorest agreement a t the smallest rate constant considered, aaVerage = 0.346, aeffectlve 0.196. In order to ascertain the comparative sensitivities of Equation 17 and Equation 18 of analyzing the shapes of polarograms characterized by slow charge transfer reactions, linear least squares fits to Equation 18 were calculated for the above polarograms. All fits for constant a values showed linear behavior near the tops of the polarograms, but a concave downward curvature was present at the more positive potentials of the largest rate constant case. Maximum deviation observed for the k , = 5 x cm/sec case was 5 mV at i/id 0.1, and reduced to 0.8 mV at the same current ratio in the k , = 2 x cm/sec case. Essentially linear behavior was observed for all smaller rate constants. Upward concave curvature was observed in the cases which included a potentially dependent transfer coefficient. Maximum deviation from the linear least squares fit for k , = 5 x 10-5 cm/sec was 9 mV 0.95, increasing to 7 3 mV at the same current at ijid ratio for the 5 x cm/sec case.
-
-
CONCLUSIONS Heretofore, electrochemists ( I 7), specifically polarographers (11, 12), have been attempting to verify Marcus’ prediction of the potential dependence of the transfer coefficient by investigating systems with comparatively large rate constants. The studies were done with these systems because the predicted potential dependence of a is greatest for large 12,. However, it is the conclusion of this study that optimal conditions for the verification or disproof of this aspect of Marcus’ theory exist a t smaller rate constants, a t least in the dc polarographic case. From our study of polarographic shape analyses for calculated polarograms characterized by slow charge transfer reaction, it is evident that large deviations from linearity should be observed in experimental systems with small k , values. This phenomenon is not caused solely by the magnitude of the potential coefficient of a , which decreases with decreasing k , , but rather by increases in both the (17) D. M. Mohilner, J. Phys. Chem., 73, 2652 (1969).
28
magnitude and range of negative overpotential necessary to overcome the slower charge transfer kinetics a t the smaller rate constants. This overpotential increases negatively at a faster rate with decreasing lz, than the corresponding decrease in the linear potential coefficient of a. The net effect is an increase in deviation from linear behavior of the shape analysis plots for systems with decreasing rate constants. Studies at the faster rate constants (2 x cm/sec and larger) are undesirable for yet another reason. Not only is the expected curvature due to a = g ( E ) small, but also the contribution of the back reaction near the foot of the polarographic wave causes curvature in the opposite direction to that expected for a = g ( E ) behavior. The resulting shape analysis plot deviates ffim linearity much less than would be expected from curvature due solely to a = g ( E ) . This effect is illustrated in Figure 6, which shows shape analyses for ( a ) a polarogram calculated with a = g ( E ) , but neglecting back reaction contributions (simply drop the second exponential in Equation 9), (b) a polarogram with a = aeffectlve and k , = keffectlveincluding back reaction curvature, and (c) the polarogram resulting from the cumulative effects of the two curvature causing phenomena, i.e., a = g ( E ) with the back reaction contribution included. Clearly, the cumulative effect of the two curvature producing factors tend to cancel yielding an essentially linear function. There is one polarographically obtainable criterion which, in conjunction with curvature in the shape analysis plot, gives both necessary and sufficient conditions for verification of Marcus’ prediction. This is the presence of an inflection point shifted in a positive direction from the polarographic half-wave potential. The disparity between the inflection point and E l p is a result of decreases in the potentially dependent transfer coefficient as the negative overpotential increases. This decrease in a forces the current due to the slow electron transfer to slowly approach the mass transport limited value. The net result of this slow approach is that a greater fraction of the polarogram appears beyond the inflection point (Figures 1 and 2). The magnitude of the discrepancy between E1/2 and the inflection point is enhanced at slow rates in much the same way as the magnitude of curvature of shape analyses. The differences between current-voltage curves characterized by C Y e f f e c t l v e and keffectlveand those characterized by a = g ( E ) and k,,,, are also largest at the smallest k,, e.g., in Figure 5 the largest difference occurs a t i/id = 0.75 and is 30 mV; however, the average deviation is only 8 mV. Inability to properly correct for double-layer effects due to imprecise data or inadequacies of the theory could substantially decrease these differences and result in nearly linear shape analysis plots for systems with n = g ( E ) . Thus, absence of curvature in these plots is not sufficient evidence to disprove Marcus’ prediction. In this case, however, trends in aeffeCtlve with k e f f e c t l v e as shown in Table I should be observed. It is noted that as k e f f e c t l v e decreases by one order of magnitude, aeffectlve decreases by one-half. Failure t o observe this trend along with absence of curvature in the shape analysis plots would constitute evidence disproving Marcus’ prediction. Due to the incomplete nature of previous studies (16), sufficient accurate data to substantiate or disprove this behavior is not available. From our comparison of the two shape analysis methods, it appears that both are equally sensitive t o curvatures produced by the potential dependence of the transfer coefficient. However, since Equation 1’7 contains a small intrinsic curvature in the opposite direction to that expected from a = g ( E ) , we feel that at larger constants,
ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 1 , J A N U A R Y 1974
fL
Table I.Trends in Kinetic Parameters for Polarograms Simulating Marcus' Behavior Simulated polarograms Marcus' behavior
a
k,, cm/sec
+ 0.26 ( E 0.50+ 0.23 ( E - EO') 0.50+ 0.21 ( E - EO') 0.50 + 0 . 1 9 ( E - EO')
0.50
EO')
aefrectlve
5 X
0.40
5X
0.33
5 X IO-'
0.27 0.20
5 X lo-'
k s , effective. cm/sec
6.9 X 1.5 X 5.2 X 4.3 X
IO-'
it is desirable to use Equation 18, which is perfectly linear for constant a. At small rate constants either method is suitable. The results of this study suggest a direction of further experimental work on this subject. Systems with heterogeneous rate constants less than 2 x 10-5 cm/sec should be investigated polarographically. Deviation between the inflection point of the polarogram and curvature in the shape analysis plots should be sought; but most importantly the possible existence of the trend given in Table I should be thoroughly investigated.
APPENDIX I a = charge transfer coefficient. Co* = bulk concentration of the oxidized species, mmol/ liter. Do = diffusion coefficient of the oxidized species, cm2/ sec. DK = diffusion coefficient of the reduced species, cm2/sec. E = potential applied to electrode, corrected for double layer effects, volts. EO' = thermodynamic formal potential of the couple
E"'
=
E"
+
= activity coefficient of ith species.
F = faraday. 3G" = molar free energy of activation of charge transfer
RT nF
__ l n ( f o / f K ) , v o l t s
reaction. AGRZ = molar free energy of activation of charge transfer reaction at the thermodynamic formal potential. I = instantaneous kinetically controlled polarographic current, bA. i d = instantaneous diffusion limited current, PA. 2 , er = instantaneous theoretical reversible polarographic current, PA. K = kinetic transition probability. k , -- reverse heterogeneous rate constant, cm/sec. k f =- forward heterogeneous rate constant, cm/sec. k, = standard heterogeneous rate constant, cm/sec. m = inass flow rate of mercury, mg/sec. m , = atomic weight of a single reactant species 0. n = total electrons involved in charge transfer reaction. n, = total electrons from oxidized state to and inclusive of the potential determining step of the reduction. R = gas constant, joule/deg/mole. t = instantaneous time, sec. T = temperature, "K. Zhet = heterogeneous collision frequency, cm/sec. [ = ratio of t h e mean square deviation of the distance between ion a n d electrode to the mean square deviation of the perpendicular distance from the reaction hypersurface. Received for review April 27, 1973. Accepted July 25, 1973. Presented in part at the 165th National Meeting, American Chemical Society, Dallas, Texas, April 1973. Receipt of an NDEA fellowship by D. R. Lewis is gratefully acknowledged. Taken in part from the M.S. thesis of D. R. Lewis submitted to the Graduate School University of Georgia.
Quantitative Determination of the Polymeric Constituents in Compounded Cured Stocks by Curie-Point Pyrolysis-Gas Chromatography Anoop Krishen and Ralph G. Tucker The Goodyear Tire & Rubber Company, Research Division, Akron, Ohio 44316
Quantitative estimation of the polymeric constituents of a compounded and cured stock, without extensive chemical treatment, has been accomplished earlier by pyrolysis-gas chromatography. The increasing complexity of the mixtures now used in various applications, has necessitated the use of more precise analytical techniques. The Curie-point pyrolysis technique, where the sample is heated indirectly by a high frequency current, was used to achieve precise control over a number of the experimental parameters. The higher precision resulting from this technique was combined with the advantages of using a dual gas chromatographic column. Under these conditions, hydrocarbon products ranging from methane to diprene/limonene were quantitatively analyzed. Conditions were established for determining the percentages of
polyisoprene, styrene-butadiene rubber, ethylene-propylene terpolymer rubber, polybutadiene, and chlorobutyl rubber. I t was possible to identify the components of unknown compounded cured stocks by this method. One sample can be analyzed in about an hour and the standard deviation was found to be 1.5%.
Pyrolysis-gas chromatography has been shown to be an excellent technique for quantitative estimation of polymers ( 1 ) . Its acceptance is apparent from the large number of publications reported in the past year (2). The in(1) A . Krishen, Anal. Chem., 44, 494 (1972). (2) C. W . Wadelin and M . C. Morris, Anal. Chem.. 45, 333R (1973).
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 1, J A N U A R Y 1 9 7 4
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