At low stirrer speeds (20-30 rpm), the limiting current is directly proportional to the cube root of the flow rate, to the analytical concentration C, and only slightly dependent on stirrer speed: iL
vy3c
(low speed)
Intermediate stirrer speeds probably represent a transition between laminar and turbulent flow. The current also depends to some extent on the relative geometry of the cell, but is constant as long as the electrode position is fixed. A stirrer pitch of about y2 turn per centimeter gives more efficient mass transport than a pitch of about 3 turns per centimeter. The turbulent tubular electrode may have some important advantages. It is relatively simple to construct and is highly sensitive, with a limit of detection below 0.1 gM. I t
also has a limiting current which is independent of flow rate a t high stirrer speeds. The latter would be an important advantage for making measurements in flowing systems with poorly controlled flow rates. The high mass transport rates attainable with this electrode might prove advantageous for hydrodynamic voltammetric studies of electrochemical kinetics.
ACKNOWLEDGMENT The assistance of R. Schmelzer in the machining of the cell is highly appreciated. RECEIVEDfor review January 28, 1974. Accepted April 29, 1974. This research was supported in part by an Office of Water Resources Research Grant, No. A-053-WIS. Additional support in the form of a du Pont Summer Research Assistantship (G.W.S., 1972) is gratefully acknowledged.
Fundamental and Second Harmonic Alternating Current Polarography of Electrode Processes with Coupled First-Order Catalytic Chemical Reactions: Theory and Experimental Results with the Iron Triethanolamine-Chlorite System Kathryn R. Bullock’ and Donald E. Smith2 Department of Chemistry, Northwestern University, Evansfon, 111. 6020 7
Some results of studies on the fundamental and second harmonic ac polarographic responses with the first-order catalytic mechanism are presented. Some unpublished aspects of the theoretical predictions for the ac polarographic behavior with this mechanism are discussed. Theoretical predictions reveal simple schemes for characterizing the homogeneous and heterogeneous rate parameters from current amplitude measurements in kinetic regimes where a previously-applied simple procedure based on phase angle measurements is inapplicable. Experimental data obtained with a previously unreported catalytic process, which occurs when ferric triethanolamine is reduced in the presence of chlorite ion, are given. These data provide additional support for the fidelity of ac polarographic rate laws which have been derived for the catalytic mechanism. Heterogeneous charge transfer rate parameters for the ferric-ferrous = 0.21 triethanolamine redox couple were found to be cm sec-’ and a = 0.50 in an aqueous electrolyte primarily composed of O.lOMNaCI, O.OSOMNaOH, and 0.1OMtriethanolamine. The second-order rate constant for the homogeneous oxidation of ferrous triethanolamine by chlorite ion in the same electrolyte was calculated to be (8.0 X 1 0 3 ) W 1 sec-’ from polarographic data obtained under pseudo firstorder conditions.
The initial fundamental harmonic ac polarographic theory for the first-order catalytic mechanism,
’
Present address, T h e Gates Rubber Co., 999 S o u t h Broadway, Denver, C o l a 80217. T o w h o m correspondence should be addressed.
was derived by Smith in 1963 on the basis of the stationary plane electrode model ( I ) . Quantitative predictions were limited to the phase angle and the current amplitude’s frequency response profile, because these observables were indicated to be insensitive to the electrode model’s accuracy. The phase angle theory was validated experimentally by studying the polarographic reduction of Ti4+to Ti3+in the presence of chlorate ion (2). In 1968, McCord and Smith published a general theory for the fundamental ( 3 )and second harmonic ( 4 ) ac polarography of systems with firstorder homogeneous chemical reactions coupled to a single heterogeneous charge transfer step. These theories were derived on the basis of the expanding plane electrode model and included the case of the catalytic mechanism. Through this development and Delmastro’s theory ( 5 ) for the dc process with a catalytic mechanism, which encompasses all combinations of heterogeneous and homogeneous rate parameters, h,, a , and h,, a quite general theory for the ac polarographic rate law with the catalytic mechanism is made available in the context of the expanding plane electrode model. Subsequently, Sluyters-Rehbach and Sluyters (6, 7 ) extended the theory to include the effects of reactant adsorption under Nernstian conditions and provided expressions for the overall interfacial admittance by introducing the double layer parameters. Efforts to obtain mathematically simpler rate law formulations also characterized their efforts. Despite these apparently successful efforts in developing (1) D. E. Smith, Anal. Chem., 35, 602 (1963). (2) D. E. Smith, Anal. Chem., 35, 610 (1963). (3) T. G. McCord and D. E. Smith, Anal. Chem., 40, 1959 (1968). (4) T. G. McCord and D. E. Smith, Anal. Chem., 40, 1967 (1968). ( 5 ) J. R . Delmastro, Ph.D. Thesis, Northwestern University, Evanston. 111. 1967. (6) J. H. Sluyters and M. Sluyters-Rehbach. J. Nectroanal. Chem., 23, 457 (1969). (7) M. Sluyters-Rehbach and J. H. Sluyters, J. Elecfroanal. Chem., 26, 237 (1970).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
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accurate rate laws for fundamental and second harmonic ac polarography with the first-order catalytic mechanism, the supporting foundations for applications of ac polarography in this context are somewhat incomplete. For example, theoretical predictions for the second harmonic response have not been surveyed in the literature to provide detailed in. sights into expected response characteristics and recommended measurement guidelines for the experimentalist. Although results of the latter type of endeavor have been published for the fundamental harmonic case ( I , 2, 6, 7), we have found that certain useful data interpretation guidelines have been overlooked. Furthermore, experimental validation of the theoretical predictions has been rather sketchy, since the quantitative theory-experiment comparisons have emphasized the fundamental harmonic phase angle characteristics ( 2 ) .To eliminate some of these unexplored areas, we undertook further investigation of the predictions of the theory, as well as an experimental study. Theoretical predictions were scrutinized for useful data analysis guidelines, with emphasis on non-Nernstian conditions. Experimental studies focused on the catalytic mechanism encountered when ferric triethanolamine is reduced to the ferrous complex in the presence of chlorite ion and basic media. The latter emphasized theory-experiment correlations for the fundamental and second harmonic current amplitude-d.c. potential profiles. Results of these investigations are reported here.
NOMENCLATURE The following notation is used here. A = electrode area. Co* = initial concentration of oxidized form of redox couple. D, = diffusion coefficient of Species i. 1.F = amplitude of applied alternating potential (one half peak-to-peak value). E& = dc component of applied potential. Er1/2 = reversible dc polarographic half-wave potential (planar diffusion theory). Eo = standard redox potential in European convention. F = Faraday's constant. I ( w t ) = instantaneous fundamental harmonic current. Z(2 ut) = instantaneous second harmonic current. h , = pseudo-first-order rate constant for homogeneous catalytic reaction. h , = heterogeneous charge transfer rate constant a t EO. n = number of electrons transferred in heterogeneous charge transfer step. R = ideal gas constant. t = time. T = absolute temperature. (Y = charge transfer coefficient. r = Euler Gamma Function. 9 = phase angle of fundamental harmonic alternating current relative to applied alternating potential. w = angular frequency of applied alternating potential.
b = c0t-l
$ ( t ) = P,
+
c'
(for k,t
p, 1.732h.t"'
2r P, =
V
-
2
2.5)
(q)
r ( q )
Second Harmonic. Similar adaptation of the general theory for second harmonic ac polarography - . . ( 4 ) to the catalytic mechanism yields: I ( 2 w t ) = Z(2w)W(w) s i n ( 2 o t
+
6,)
THEORY Fundamental Harmonic. Specialization of the general theory for the fundamental harmonic ac polarographic response ( 3 ) to the catalytic mechanism and introduction of the Delmastro ( 5 ) solution fdr the dc process yields the theoretical formulation:
I ( w t ) = I,,,F(t)G(w) sin(wt where 1568
+
b)
(1)
V2 and C'2 are obtained by replacing w by 2w in Equations 9 and 10. respectively. Z (20) is obtained by performing this
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
same substitution on the current amplitude term in the fundamental harmonic rate law [I,," F ( t ) G ( w ) ] . Special Cases and Simplifications. The foregoing equations simplify considerably under several special circumstances. Some of those which are important for subsequent discussion will be considered here. N e r n s t i a n Conditions. The Nernstian case is probably the most frequently acknowledged "special case." Sluyters and Sluyters-Rehback (6, 7 ) gave this situation close attention in examining the fundamental harmonic response with the catalytic mechanism. Nernstian conditions in the dc sense require that h , be sufficiently large to validate the expression
(35) If Nernstian conditions also prevail (A,
>> I ) ,
and Equation 33 reduces to the simple expression which was previously given for the combination of Nernstian conditions and h , >> o (9). In the opposite extreme where h, is sufficiently small that the heterogeneous charge transfer step may be considered irreversible, the relationship
Equation 23 leads to the simplifications is applicable (9) and Equation 33 reduces to
and where
(25) The applicability of Equation 24 for Nernstian conditions has been recognized for some time (8). The existence of Nernstian behavior on the ac time scale demands that
which leads to the further simplifications:
P =
1 - e-j 1 + e-j
-
sinh(j/2) cosh(j/2)
Equation 39 bears a close resemblance to the Nernstian form of Equation 33 [ F ( X K )= l)],differing only by the factor of a in the amplitude term and the nature of the dc potential dependences embodied in Equations 39 and 13, respectively. Inserting the conditions given by Equations 29-31 in the second harmonic expression produces the result
I(2Ut) = (28)
L=O
and deletion of the term containing A in the expressions for Vsand V . T h e Case W h e r e h , >> o.The rapidity of many homogeneous redox reactions and the possibility of using large excesses of catalytic agent makes the condition h , >> o readily accessible with most systems without utilizing unduly low frequencies. When h , >> o,one has
(29)
g>> 1
I
I
(40) where
a$(1 1
A,
+
+
e-j) 2 ( a e - j - $)A, ae-j -$ (I + e - j ) ( l +
hK)
As in the fundamental harmonic case, invoking Nernstian conditions reduces the F ( X K ) term to unity, W ( o x ) becomes
n F h E sinhb/2) W(h,) = 4RT cosh(j/2)
n
The application of Equations 30 and 31 to Equations 4, 9, and 10 yields
and one obtains for 1(2 u t ) a simple relationship which has been given previously (9). In the opposite extreme of electrochemical irreversibility where Equation 37 applies, one obtains
I(2ut)irrev = Equation 32, combined with Equations 2, 3, and 5, leads to the relatively simple, frequency-independent expression for the fundamental harmonic response,
and by the difference in the dc potential-dependent parameters, j and J . T h e Case W h e r e h ,
