1406
HARVEY B. HERMAN AND HENRYN. BLOUNT
I
NC /
-0
A
B
C
D some systems to drive the reaction to D. Observed esr spectra may be due to complex B or C, depending on their relative stabilities. The existence of such complexes would explain several phenomena. First, since the paramagnetic intermediate has no C-N or C-0 bond there should not be any strong interactions between the unpaired electron of NOzand the protonsof most olefins. Second, it explains why the release of energy in formation of B or C can lead to the disappearance of esr signals if the reaction proceeds to D. Third, it accounts for the mode of addition of NO2 and ON0 across the double
bond. Finally, it is in agreement with Khan’s16 suggestion, which is itself supported by considerable experimental evidence. Interaction between the unpaired electrons and protons in solvents which possess both olefinic and aromatic character (Table I ) probably results from a greater stability of the NOZ-solvent complex. This stability is presumably achieved through resonance since it does not appear to be strongly related to ncelectron densities. However, the inability of benzene and some of its derivatives to complex NOz suggests that a minimum r-electron density of about 0.73 (as in furan) may be required.’O The sequence of esr spectra in Figure 3 shows that frozen olefinic solvents are inert towards NOz which can only complex with Nz04, as it does in solvents with no unsaturated character.2 On warming, a competition for NOz between Nz04 and solvent leads gradually to the formation of the NOaolefin complex which persists in the liquid state. Acknowledgments. The authors wish to thank Dr. David Christman for the a,@-dideuteriostyrene,which he was kind enough to synthesize. This research was carried out in part under the auspices of the U. S. Atomic Energy Commission. (16) C. A. Coulson and A. Streitwieser, Jr., “Dictionary of ?r-Electron Calculations,” W. H. Freeman & Co., San Francisco, Calif., 1965.
Chronocoulometric Measurement of Chemical Reaction Rates. The ECE Mechanism at Plane and Spherical Electrodes by Harvey B. Herman and Henry N. Blount Department of Chemistry. The University of Georgia, Athens, Georgia
30601
(Received October 4, 1 9 6 8 )
Chronocoulometry has been applied t o the measurement of the kinetics of coupled homogeneous chemical reactions of the ECE type at spherical electrodes. Theoretical expressions have been derived and tested with the p-nitrophenol and the p-nitrosophenol systems. Direct comparisons of chronocoulometric and chronoamperometric results obtained simultaneously have been made and the relative merits of the two techniques have been discussed. These comparisons offer support t o the chronocoulometric technique as a useful tool for the measurement of kinetic parameters of coupled chemical reactions.
Introduction The technique of chronocoulometry or “integral chronoamperometry”1 has been applied to a number of electrochemical problems in the past several years. Adsorption and electrical double-layer studies have been made employing both the single potential step2-B and the double potential step methods’s8 as well as the The Journal of Physical Chemistry
charge step chronocoulometric techniq~e.~JOHelbig“ has pursued some analytical applications of chronocou(1) P. Delahay, G. Charlot, and H . A. Laitinen, Anal. Chem., 3 2 , 103A (1960). (2) J. J. Magenheimer and J. E. Boggio, Anal. Chem., 39, 326 (1967). (3) F. C. Anson and D. A. Payne, J . Electroanal. Chem., 13, 36 (1967). (4) R. W. Murray and D. J. Gross, Anal. Chem., 3 8 , 392 (1966).
1407
CHRONOCOULOMETRIC MEASUREMENT OF CHEMICAL REACTION RATEs lometry and Christie, et a1.,12Jahave applied the double potential step chronocoulometric technique to the study of homogeneous catalytic reactions and homogeneous succeeding chemical reactions following the chargetransfer process. Because chronocoulometry seemed to provide certain advantages such as first-order nonfaradaic charging corrections14over other electrochemical techniques, this present work was undertaken to examine the relative merits of the chronocoulometric technique as a tool for the study of coupled chemical reactions other than those already treated,12Ja particularly the ECE mechanism at plane and spherical electrodes.
Theory The ECE mechanism in which a homogeneous chemical reaction is coupled between two charge transfers has been well characterized by a variety of electrochemical techniques (see ref 15 and references therein). This mechanistic scheme may be represented by kf
A
+ nle*B$C
+ n2e@D
(1)
kb
where C is reducible at the potential of the A reduction. The assumptions made in this treatment are that all species are soluble and that diffusion is the sole means of mass transport. Three different cases are treated here. Case I : Plane Electrode, Irreversible Chemical Reaction. Alberts and Shain16 have shown that for the ECE mechanism, the chronoamperometric current is given by
Two approaches were used to evaluate this expression in order to generate a “working curve,” with both approaches giving rise to the same result. The first involved expansion of the exponential in eq 7 and subsequent term by term integration. In the second approach, a substitution ( X = (Ict) 1/2) was made which allowed eq 7 to be written in the form
=
+
1 - { n z ~ l / ~ / [ z ( n nz) l ( k t ) 1 / 2 ] ] erf[
(kt)l/z]
(8)
Hasting’s approximation” of the error function in eq 8 allowed easy computation of the normalized charge parameter as a function of kt. As can be seen from the uppermost curve in Figure 1, the value of the normalized charge parameter varies smoothly between limits corresponding to electrolysis involving nl electrons at small values of kt and (nl n2) electrons at large values of kt. The region of choice for experimental measurement of rate constants lies between 0.60 and 0.85 and corresponds to a relative error of ca. 5% for an absolute error of 0.01 in the measured parameter. Case 11 : Plane Electrode, Reversible Chemical Reaction. Alberts and Shain16 derived an expression for the chronoamperometric current observed at a plane electrode under reversible homogeneous chemical reaction conditions in the ECE scheme. This expression is of the form
+
+
i/FAD1I2C0~ = [nl n2(l - e - k t ) ] / ( r t ) 1 / 2 ( 2 ) where k is the rate constant for the irreversible chemical reaction (kr in eq 1). At very long times of measurement, Le., Ict -+ 00, eq 2 reduces to i m / F A D 1 / 2 C= o ~[nl n z ] / ( ~ t ) l / ~ (3) such that the normalized current expression appears as it1/2/(it1/2)= 1 - [e-ktnz/ (nl Q) ] (4) Time integration of the expressions for the chronoamperometric currents yields the chronocoulometric charge relationships, namely Q / F A D I W o=~ [2 (nl nz)t l / z / d ~ z ]
+
+
+
[
- n,/r1/2 and
[
e-kt/t1/2 dt]
(5)
+
Qm/FAD112CQ~ = 2(nl nz)t1‘z/r1/2 (6) with the normalized charge parameter having the form
(Q/W/ (Q/tl/z)>m =1
1‘
- {nz/C2(n1 + n~)t‘/~]]
0
e-kt/tl/z dt
(7)
x
[perf [
( f z 2 t ) lI2]
- erf [K ( r f z 2 t )l/z]} perf
+
where k = k f kb, z2 = k/(K2 - l ) , K = k b / k f , the upper signs and erf (arg) are used for K > 1 , the lower (5) B. Case and F. C . Anson, J. Phys. Chem., 71, 402 (1967). (6) 0. W.O’Dom and R . W. Murray J . EZectrounaL Chem. Interfacial Electrochem., 16, 327 (1968). (7) F. C. Anson, Anal. Chem., 38, 54 (1966).
(8) F.0. Anson, J. H. Christie, and R . A. Osteryoung, J . ElectroanaZ. Chem., 13, 343 (1967). (9) F. 0. Anson, Anal. Chem.. 38, 1924 (1966). (10) F. C. Anson, J . Phys. Chem., 7 2 , 727 (1968). (11) H. Helbig, Chem. Tech. (Berlin), 19, 496 (1967);Chem. Abstr. 68, 35542 (1968). (12) J. H. Christie, J . Electroanal. Chem., 13, 79 (1967). (13) P.J. Lingane and J. H. Christie, tbdd., 13, 227 (1967). (14) F. 0 . Anson, Anal. Chem.. 36, 932 (1964). (15) H. N. Blount and H. B. Herman, J . Phys. Chem., 7 2 , 3006 (1968);H. B. Herman and A. J. Bard, J . Electrochem. Soc., 115, 1028 (1968). (16) G.S. Alberts and I. Shain, Anal. Chem., 35, 1859 (1963). (17) M. Abramowitz and I. A. Stegan, Ed., “Handbook of Mathematical Functions,” National Bureau of Standards Applied Mathematics Series, No. 55,TJ. 9. Government Printing Omce, Washington, D. C., 1965,p 299. Volume 75, Number 6 May 2869
1408
HARVEY B. HERMAN AND HENRYN. BLOUNT I
’
1
parameter
?=
l‘
erf K ( fz2) 1/2r1/2ea21 [( fz2t)1/2] dt perf
LOG k t
Figure 1. Chronocoulometric working curves for the ECE mechanism with nl = n2 at plane electrodes. From top to bottom: irreversible kinetics; reversible, K = 0.1; reversible, K = 0.95; reversible, K = 2; reversible, K = 5; reversible, K = 10; reversible, K = 20.
which with change of variable and eq 6 becomes
signs and perf(arg) are used for K < 1, and the “positive error function,” perf, is defined by perf (a) = 2/?r1l2/ a e+r2dy
(10)
0
In the limit of very large values of kt, eq 9 can be expanded for the error function terms18 and rearranged to the form given by eq 3 such that the normalized current parameter has the form
erf perf Time integration of eq 9 and 3 gives rise to
The normalized charge parameter is shown in Figure 1 as a function of kt for various values of the equilibrium constant K . As is evident from that figure, chronocoulometry at plane electrodes provides a useful means for the study of ECE schemes with coupled reversible homogeneous kinetics. Case 111: Spherical Electrode, Irreversible Chemical Reaction. The results of Alberts and Shaid6show that the current observed during the potentiostatic reduction of an ECE system at a spherical electrode is given by
i/ (nl
+ n2)FAD’/’COA = [1/ ( nt ) + [D1/2/r~] - f p r k t /(d)lI2J + pD1I2/ro ll2]
X
((1 - [D/ro2qP]) e-@$ erfc [ Dt/r02) (
112]
+ D/rt+a - (D112k1/2/ro+2)erf [(kt)’12]
and eq 6 which provides the normalized charge The Journal of Physical ChernistTl,
(15)
(18) See, for example, P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, Inc., New York, N. Y.. 1954, p 74.
1409
CHRONOCOVLOMETRIC MEASUREMENT OF CHEMICAL REACTION RATES
+
where p = nz/(nl 4, d2 = k - (D/ro2), 10 is the modified Bessel function, the upper sign applies if 42 > 0, and the lower one applies if q52 < 0. The limiting value of eq 15 at very large values of kt is given by i,,,/(nj
+ %)FAD1I2C0~= [ ( 1 / r 1 l 2 ) + P1/2]/t1/2
sented by
(16)
where @ = Dt/ro2. Hence the normalized current parameter may be expressed as
2pl12e-* perf (A$) erf
+-X
[’
expC(1
I1
rkt
ape-*
c(*$)1/21 - kt
- ( ~ P / W ) & / ~ Y O ( dQ]} &/~)
-
where $ = kt Dt/ro2,the upper signs and perf(arg) are used if 1c, > 0, and the lower signs and erf(arg) are used if $ < 0. The time integrals of eq 15 and 16 are the corre-sponding chronocoulometric charge relationships and are given by eq 18 and 20, respectively
where all terms have their usual or already defined significance. Figure 2 shows the dependence of the normalized charge parameter in eq 21 on the term kt for various values of D/rOek ( i e . , p / k t ) . l @ The feasibility of using the dimensionless parameter & / i t as a measure of coupled kinetic parameters was investigated. For the case of irreversible kinetics at plane electrodes, the ratio of eq 5 to 2 may be expressed as
&/ (n1 + n2)FAD’/2CO* = ( (2/7r’/2) + p)t l l 2
- ( p / W 2 ) erf [( k t ) l I 2 ] + pD1l2/r0
X
[C
(1
- (Dt/r&i2t)) ro2/D]
/@
e-((kc/@)-l)Eerfc (f*/2) d[
The limiting values of this expression at very large and very small values of kt are both numerically equal to 2.
0
09
8
-
E 0.83
3
-5
0.70.6-
-
05 1
.
-240
I
1
-1.0
0.0
1.0
1
2.0
LOG kt
where
1e(’-(28/kt))E/z10([/2) d l E
E1(f) =
0
+ na)FAD1W0.4 = ( (2/7r”2) +
(19)
(20) Thus the normalized charge parameter may be repre-
Figure 2. Chronocoulometric working curves for the ECE mechanism at spherical electrodes, irreversible kinetics. From top to bottom the value of D/r+% is 0 (plane case), 0.01, 0.05, 0.1, 0.2, and 0.6.
pl/z)tl/a
(19) 8. W. Feldberg, in “Electroanalytical Chemistry,” Vol. 111, A . J. Bard, Ed., Marcel Dekker, Inc., New York, N. Y., 1969, in press.
Volume 75,Number 6 M a y 1960
1410
HARVEYB. HERMANAND HENRYN. BLOUNT
At intermediate values of kt, eq 22 behaves as is shown in Figure 3. For the case of irreversible kinetics a t spherical electrodes, the parameter &/it may be obtained from the ratio of eq 18 to 15. The value of this ratio varies with kt for various values of D/$k as is shown in Figure 4.
Experimental Section The potentiostat used in this work employed one of the basic designs of Schwarz and ShainZ0and was based on operational amplifiers manufactured by G. A. Phillbrick Researches, Dedham, Mass. The particular configuration chosen incorporated modular stabilization, a grounded signal source of the type described by Underkofler and ShaiqZ1and a current follower in the working electrode branch of the circuit such that the potentiostat could be coupled with a grounded integrater for chronocoulometric work. The potentiostat employed a Philbrick SK2-V octal plug-in for control coupled with an SK2-B booster amplifier. An SK2-V amplifier was also used in the current follower application because of the stability of this plug-in. The integrater was chopper stabilized using Philbrick K2-P/K2-W amplifiers. The signal generator was based on a K2-W octal plug-in. The entire configuration is similar to that reported by Anson and Pnyne. Measurements were made using a hanging mercury drop electrode (radius 4.8 X cm) formed by a Metrohm extruder, an isolated platinum foil counter electrode, and a saturated calomel reference electrode equipped with a Lugin capillary. As the potential of the working electrode was stepped from a value less cathodic than the reduction potential of A (ie., either p-nitrosophenol or p-nitrophenol) into the diffusion plateau of the A reduction wave, chronoamperometric and chronocoulometric data were obtained simultaneously. This was accomplished by I
1
'
- 1.0
I
I
0.0
1.0
2.0
I
3.0
LOG k t
Figure 4. &/it working curves for the ECE mechanism at spherical electrodes with n1 = m. From top to bottom, D/r& = 0, 0.00001, O.OOOl,O.OOl, 0.01, 0.06, 0.1, 0.2, and 0.9.
sampling the current signal a t the output of the current follower and by sampling the charge signal a t the output of the integrator and then displaying both signals on the screen of a Tektronix Model 564 storage oscilloscope equipped with a Type 3872 dual trace amplifier. This display was recorded by means of a Tektronix C-27 oscilloscope camera and the data were taken from the resulting photograph. All measurements were made a t 25.00 f 0.05" using a jacketed cell in conjunction with a thermostated reservoir. All solutions were 1.0 mM with respect to A in 20 vol % aqueous ethanol of pH 4.94. The pH was maintained at this value by means of an equimolar acetate-acetic acid buffer system having a total ionic strength of 1.0 M . Solvents were purified by distillation. The p-nitrosophenol was obtained as the sodium salt. This was converted to the acid form and purified according to Alberts and Shain.16 The p-nitrophenol was obtained as the indicator and used without further purification. The buffer system was prepared from fractionally crystallized and distilled acetic acid which was neutralized to the desired pH with carbonate-free sodium hydroxide prepared from the reagent grade chemical. Resulting acetic acid and sodium acetate concentrations were both 0.5 M . Adjustment of the ionic strength to 1.0 M was made with reagent grade potassium nitrate. The determination of the pH of this solution has already been described.I6 Prior to each set of experiments, the solid phenol was preweighed and dissolved in a measured volume of deaerated buffer solution. Chronoamperometric and chronocoulometric working curves were obtained using an IBM System 360/65 computer. Integrations were performed by portions of the IBM Scientific Subroutine Package.22 Results
LOG kt
Figure 3. Working curves for &/it measurements a t plane electrodes as described in the text. Upper curve is for nl = na = 2; lower curve is for 121 = 4, na = 2. The Journal of Physical Chemistry
(20) W.M.Schwarz and I. Shain, Anal. Chem., J I , 1770 (1983). (21) W. L. Underkofler and I. Shain, (bid., 35, 1778 (1963). (22) IBM Application Program H20-0206-2,International Business Machines Gorp., White Plains, N. Y., 1967.
1411
CHRONOCOUMMETRIC MEASUREMENT OF CHEMICAL REACTION RATES
obtained through the use of Simpson's rule and Gaussian quadrature techniques were indistinguishable.
Table I: Limiting Current and Charge Parameters* System
Results and Discussion
Parameter
The reductions of both p-nitrophenol (PN02P) and p-nitrosophenol (PNP) have been shown to follow the ECE scheme (see ref 23 and references therein) and were therefore chosen as model systems for this study. The similar reduction schemes of PNP and PNOzP may be represented by 0
and I)
(W)O, A sec1/2 Q (nonfaradaic), C (Q/t1/2)a, C sec-ll2 (Q/t1/2)ma,a C s e d a (itl/z)ma~oA se@ (it1/2)m5rdA secllz
7
PNOiP
PNP
7.95 ( ~ 0 . 0 6 x ) 10-6 1.57 (&o.oi) x 10-5 4.60 (10.5) X 10-7 6 . 8 (f0.7) X 10-7 1.56 (10.04)X 3.03 (rt0.06) X 10-6 4.54 (fO.06) X 3.11 (10.04)X 1.59 (fO.O1) X 2.36 (rtO.01) X 1.17 (10.08)X 2.05 (fO.12) X 10-6
First term only in eq 25 and 27. * Evaluated from ( Q / t 1 / 2 ) ) o as described in text. OEvaluated from ( i W ) o as described in text. Evaluated from limiting-slope techniques of Alberts and Shain.16 @ T O = 4.8 X cm; 0 = 1 X 10-~mol/cma;pH 4.92 in 20 vol % aqueous ethanol at 25.0'.
time product for the first charge transfer only. The first term in eq 25 for (iP) is then obtained by multiplying (it1/2)oby a factor of (nl nz)/nl. The spherical term (the second term in eq 25) must then be calculated from values of D , coA,rO, F , and n (where n = nl+ n2) for each time of measurement, t, and this result added to the first term such that the sum of these two would give rise to the value of (it1/2)m at the time of measurement. This is tantamount to determining the value of (it1'2)mas described in eq 25 where n = nl n2 and t is the time of measurement. In the case of chronocoulometry, all measurements must be corrected for nonfaradaic charging. This was done according to Anson.I4 After nonfaradaic corrections, a treatment very similar to the one used for the determination of the limiting value of it1'2was employed in the determination of the limiting value of Q/t112, Time integration of eq 25 and 26 gives rise to
+
respectively. Determination of Limiting Current and Charge Parameters. Examination of the current-time relationship for chronoamperomety a t spherical electrodesz4 which is given by it1l2 =
+
( T L F A D ~ / ~ C O A / T ~(nFADCOAtl/z/ro) /~) (25)
shows that indeed the it1l2product is time dependent. This presents something of a problem in the determination of the (iW)co term necessary for calculation of the normalized current parameter W 2 /(itt12) used in the determination of kinetic parameters. Alberts and Shain16 chose to determine experimentally the value of (itlIz)mfrom the limiting slope of a plot of current us. the inverse square root of the time for long-time measurements. The results obtained in that study exhibited a time-dependent trend in the observed value of the rate constant for the coupled chemical reaction. Another approach to the determination of the limiting current parameter has been employed in this present study. If in the course of derivation of eq 2 the exponential resulting from the inverse Laplace transformation is expanded for small values of kt, then one obtains an expression of the form
it'"
+
= (nlFktD1/2c0A/d1z) (k n 2 F A D 1 / 2 C o ~ / ~t 1 / 2 )
(26) where no spherical contribution to the total current is assumed (Le., the last term in eq 25 is negligibly small). Then by plotting the current parayeter it1I2 as a function of time for short-time measurements only, a straight line results, the intercept of which is the first term in both eq 25 and 26. This term is in fact (it1/2)htBo corresponding to the current square root of the
+
+
Q/t112= ( 2 n F AD 1 / 2 C o ~ / d / 2 )(nFADCoA/ro1/2) t112 (27)
and
+
Q/tllz = (2nlFAD 1 / 2 C o ~ / ~ 1 / 2(2) k n 2 F A D 1 / 2 C o ~ / 3t~ 1 / 2 ) (28) respectively, with the same restrictions applying to eq 26 and 28. For very short-time measurements, the corrected charge parameter, Qo [&(cor) = Q(obsd) - &(nom faradaic)], was plotted as a function of time and a linear dependence was observed with an intercept corresponding to the first term in eq 28. h i s value of the intercept (Q/t1I2)owas converted to the first term in eq 27 for (Q/t1/2)mby multiplication by the factor (nl nz)/nl. Correction for spherical effects was made by numerical evaluation of the last term in eq 27 and addition of it to the previously determined first term, the sum providing a measure of (Q/t1/2)m at the
+
(23) H.B. Herman and A. J. Bard, J . P h y s . Chem., 70, 396 (1966). (24) P. Delahay, ref 18, p 61.
Volume 75,Number 6 May 1969
1412
HARVEY B. HERMAN AND HENRYN. BLOUNT
Table 11: Kinetic Results of Simultaneous Chronoamperometric and Chronocoulometric Experiments0 A
-lOW/2,
Time, sec
Mean
sec1/L--. errorb
Std
-
sec-1-
-k,
Mean
Std
-lO4Q/W,
errorb
Mean
0 Sec-LlkStd error)
-k,
Mean
sec-1--. Std errorb
A. PNP System, D = 6.52 X 10-6 om8 sec-1 0.50 1.oo 1.50 2.00 2.50 3.00 4.00 6.00 8.00
0 * 101 0.114 0.119 0.129 -_ 0.132 0.141 0.148 0.161 0.171
0.001 0.002 0.002 0.001 0.003 0.002 0.001 0.001 0.001
0.51 0.46 0.35 0.36 0.31 0.33 0.30 0.26 0.26
0.02 0.03 0.03 0.02 0.03 0.02 0.02 0.02 0.01
0.169 0.178 0.186 0.191 0.195 0.206 0.217 0.234 0.244
Weighted mean 0.36, std dev 0.09
0.178 0.191 0.203 0.212 0.218 0.226 0.233 0.246 0.257
0.001 0.001 0.002 0.002 0.002 0.002 0.002 0 * 002 0.003
0.36 0.37 0.37 0.35 0.32 0.34 0.29 0.25 0.25
=
Experimental conditions are the same as in Table I.
0.04 0.02 0.03 0.03 0.03 0.03 0.03 0.02 0.05
0.321 0.337 0.333 0.347 0.348 0.358 0.372 0.389 0.401
+
(2nFAD1/2CoA/d/2) [I
+ (d/2D1/2t1/2/2ro) 3 (30)
respectively. In the above cases, the added corrective term does not involve concentration. It has been found in the present study that this latter approach gives rise to more self-consistent results. Equations 26 and 28 predict that the values of ( i W ) k t = o and (&//1/2)k1,0 should differ by a factor of 2. Experimentally this was observed to be the case. This is taken as evidence that at the times of measurement greater than ca. 20 msec the nonfaradaic portion of the observed current is negligibly small. Results of the determination of limiting values of chronoamperometric and chronocoulometric parameters for both P N P and PNOzP are shown in Table I. For the experimental conditions of this study, the calculated values of D/ro2k were 0.0071 for the P N P system and 0.0063 for the PN02P system. Working curves having D/ro2kvalues of these magnitudes were therefore used in the determinations of kinetic paramThe Journal of Physical Chemistrv
0.38 0.41 0.25 0.36 0.27 0.34 0.35 0.33 0.29
0.02 0.07 0.07 0.05 0.04 0.04 0.04 0.04 0.03
C. Eisenhart, Science, 160, 1201 (1968).
it1l2= (nFAD1/2Co~/?y1/2) [1 (.rr112DW/z/ro)] (29) and =
0.001 0.003 0.004 0.003 0.003 0.004 0.003 0.004 0.004
Weighted mean 0.33, std dev 0.05
time of measurement. An alternative method for the calculation of limiting current and charge parameters at the time of measurement which circumvents exact knowledge of concentration is to write eq 25 and 27 in the forms
Q/t”2
0.04 0.04 0.07 0.03 0.04 0.03 0.02 0.02 0.02
5.81 X 10-8 cmz/sec
Weighted mean 0.32, std dev 0.05 (I
0.39 0.36 0.34 0.32 0.29 0.34 0.32 0.33 0.30
Weighted mean 0.33, std dev 0.03
B. PNOaP System, D 0.50 1.00 1.50 2.00 2.50 3.00 4.00 6.00 8.00
0.001 0.002 0.004 0.002 0.004 0.003 0.002 0.002 0.003
eters. The kinetic results of simultaneous chronoamperometric and chronocoulometric experiments covering a broad time range are shown in Table 11. These results are in good agreement with other published values of the rate constant for the dehydration of p-hydroxyphenylhydroxylamine,16the initial reduction product of both P N P and PN02P. On the basis of Student’s t test,26the mean values of chronoamperometric and chronocoulometric results are not statistically different at the 90% level of significance for both the PNP and the PN02P systems. Figure 5 shows &/it working curves for both PNP and PN02P systems. The experimental points shown in this figure are for an assumed rate constant of 0.34 sec-l, this value being based on the average of the chronoamperometric and chronocoulometric results. The use of &/it measurements is attractive in that the limiting values of it112 and Q/t1l2 do not have to be known and also in that certain experimental fluctuations during the course of a single experiment tend to some extent to be self-canceling. However, the working curves show areas of large relative error in kinetic parameter determinations as well as regions of nonsingle-valued behavior, two undesirable characteristics for a kinetic method. Comparison of the ohronoamperometric and chrono(26) A. M.
Neville and J. B. Kennedy, “Basic Statistical Methods
for Engineers and Scientists,” International Textbook Co., Scranton, Pa.,1964,p 148.
1413
CHRONOCOULOMETRIC MEASUREMENT OF CHEMICAL REACTION RATES 1.0,
- 20 ,LOG k t
Figure 6. &/it working curves for the ECE mechanism at spherical electrodes. Upper curve: n1 = 4, n2 = 2, D/$k = 0.0063, Lower curve: n1 = nz = 2, D/@k = 0.0071. Experimental points are for an assumed rate constant of 0.34 sec-’ (see text): 0, PNOzP system; 0,PNP system.
coulometric working curves for the same value of D/$k (Figure 6) shows that for the same value of the rate constant, k, a longer chronocoulometric time of measurement will give rise to a more accurate determination of that rate constant. This fact permits the study of somewhat faster coupled chemical reactions of the ECE type at spherical electrodes with the chronocoulometric technique than is possible with chronoamperometry. On the other hand, however, such chronocoulometric measurements will inherently give rise to greater relative errors in the kinetic constants because of the less steeply rising chronocoulometric working curve. The integral signal is advantageous in that the noise which is characteristic of high-sensitivity, long-time chronoamperometric measurements is not present in the corresponding chronocoulometric result. Moreover, the integral output is less severely affected by convective diffusion which occurs at long times of measurement. The chronocoulometric technique is more ideally suited to kinetic analysis of a system of known mechanism than to diagnosis of mechanism. The latter task is more amenable to treatment by electrochemical techniques such as cyclic voltammetry or cyclic chronopotent iometry. Even in kinetic analysis of systems of known mechanisms, a reversal technique such as current reversal chronopotentiometry is more desirable than a “oneway” technique such as first-transition chronopotentiometry, chronoamperometry, or chronocoulometry in that exact electrode area, diffusion coefficients (with certain assumptions), and other parameters (depending on the technique) do not have to be known because of the use of a ratio measurement. Therefore,
I
I
I
4
-1.0
0.0 LOG kt
1.0
2.0
Figure 6. Chronoamperometric (A) and chronocoulometric (B) working curves for the ECE mechanism at a spherical electrode; D/@k = 0.0071; nl = Q; irreversible kinetics.
it is unnecessary to evaluate limiting parameters at short or long times. These above-mentioned one-way techniques are still necessary because the use of a reversal technique is not always possible. The proximity of another redox couple may render the reverse measurement impossible or irreversibility of the couple in question may cause the reverse electrolysis step to occur at more extreme potentials than solvent electrolysis or electrode dissolution. For example, in an earlier paper16 it was found at low pH that a reverse transition time for the PNP was imperceptible because of the interference of the mercury oxidation wave. The study reported here was prompted by that observation. Without question, there is a trend present in the data shown in Table 11. While this trend is not as pronounced as that in the chronoamperometric results of Alberts and Shain,16 it is nonetheless present. Perhaps the chronoamperometricand chronocoulometric working curves are more sensitive to perturbation by the slow hydrolysis of p-benzoquinoneimine16 than are those for current reversal chronopotentiometry. Also the perturbation of the PNP and PN02P ECE systems by an ECC scheme would qualitatively correct this trend.26 While the above possibilities have not been investigated in the chronocoulometric treatment described in this work, they are offered in addition to those advanced by Alberts and Shain16 as means of rationalizing the observed trend in the kinetic parameters.
Acknowledgment. This work was supported in part by grants from the Petroleum Research Fund (No. 488-G2) and the National Science Foundation (No. GP-6596). (26) M. D. Hawley (1966).
and S. W. Feldberg, J. Phys. Chem.,
70, 3459
Volume Y8, Number 6 May 1969
