Second Harmonic Alternating Current Polarography: A General Theory for Systems with Coupled First-Order Homogeneous Chemical Reactions Thomas G . McCordl and Donald E. Smith2 Department of Chemistry, Northwestern University, Ecanston, Ill. 60201
A general theory for the second harmonic ac polarographic response of systems characterized by firstorder homogeneous chemical reactions coupled to a single heterogeneous charge transfer step is presented. Based on an approach recently outlined in detail for the fundamental harmonic case, the theory accounts for rate control by diffusion, a single heterogeneous charge transfer step, and any number or type of coupled first-order homogeneous chemical reactions, within the framework of the expanding plane model of the dropping mercury electrode. The solution of the boundary value problem is modeled after the method of Matsuda, which invokes just two minor limitations on the magnitudes of the rate parameters. The final master equation allows one to deduce theoretical expressions for the second harmonic ac polarographic wave associated with a particular mechanistic scheme in this class simply by inspection of readily available surface concentration expressions.
where p = 0 , 1, 2, 3
(4)
D
A
of the fundamental harmonic ac polarographic response for systems with first-order homogeneous chemical reactions coupled to a single heterogeneous charge transfer step has been presented recently ( I ) . The theoretical treatment accounts for kinetic contributions of diffusion, the heterogeneous charge transfer step, and any number or type of coupled, first-order homogeneous chemical reactions using a rigorous treatment of the expanding plane boundary value problem. The purpose of the present work is to show that the same ideas can be extended readily t o the case of second harmonic currents, yielding a theoretical expression for the second harmonic wave which is characterized by the same scope and rigor as that developed for the fundamental harmonic current. GENERAL THEORY
=
DoBDRa
f
= fOBfRU
p = 1 - a
Notation definitions are given below. The basic assumptions implicit in the foregoing expressions are the usual ones which have been enumerated elsewhere (2-4). For small amplitudes (AE 5 10/n mV), the second harmonic current, as well as the dc faradaic rectification component, arise from the Qn(t) function (2). Writing Q2(t)as Qdt)
=
Qz(dc)
+ Qz(2wt)
(9)
the second harmonic current Z(2wt) is given by (see Equation 3)
THEORETICAL
For the mechanistic class under consideration with the reduced form initially absent, all current components flowing under ac polarographic conditions are defined, within the framework of the expanding plane model of the dropping mercury electrode, by the system of integral equations ( I ) ; D'"QAr) ksf
-
e-a3Lyp
p!
5
r = ~
r!
X
while the dc faradaic rectification component Z2(dc)is
Because the faradaic rectification component is difficultly measured under ac polarographic conditions (5), the Qs(dc) term is not of interest here. Of primary concern is the solution of Q42wt). Both are obtained by solving Equation 1 for p = 2, i.e.,
D1lzQ2(t) a 2e-"isin2wt ksf 2
~-
NIH Graduate Fellow: present address, General Electric Corp., Materials and Processes Laboratory, Schenectady, N. Y. 12305.
* To whom correspondence should be addressed.
___..
(1) T. G. McCord and D. E. Smith, ANAL. CHEM.,40 (1959)
1968.
-
(2) D. E. Smith in "Electroanalytical Chemistry," A. J. Bard, Ed., Vol. 1, Marcel Dekker, New York, N. Y., 1966, Chapter 1. (3) H. Matsuda, Z . Elektrocliern., 62,977 (1958). (4) D. E. Smith, A N A L . CHEM., 35, 602 (1963). (5) P. Delahay "Advances in Electrochemistry and Electrochemical Engineering," Vol. 1, P. Delahay, Ed., Interscience, New York, N. Y., 1961, pp 279-300. VOL. 40, NO. 13, NOVEMBER 1968
1967
-{+
P = (1
e-j)
F(t)
(a - @Ye3 [1 W Ye3
+
b w 1 + a2!W -
The integrals involving the dc function Qo(r) are evaluated with the aid of the dc integral equation (Equation 1 with p = 0) and the relationship
which is valid for kit > 10 (1). The integrals containing the fundamental harmonic function Ql(t) are evaluated using its solution and the ac steady-state relationships given previously (1). After carrying out these operations and some algebraic rearrangement, Equation 12 reduces to
Introducing the well known trigonometric identities sinzwt
1 2
= - (1
coswtsinwt
- cos2wt)
(28)
1 2
(29)
.
= - sin2wt
and Equation 9 into Equation 14 yields an integral equation which is separable into the two subsidiary integral equations ( 2 ): 1968
ANALYTICAL CHEMISTRY
LW
f+(hi)
The second harmonic term is obtained by solving Equation 31. This is achieved in the same manner as for the analogous fundamental case ( I ) . One recognizes that Q2(2wt) will be of the form
Q2(2wf)= A2(t)sin2wt
+ B2(t)cos2wt
(32)
and substitutes this expression in Equation 31. This step is followed by application of the appropriate ac steady state relationships (Equations 17-20 of Reference I ) , solution of the resulting two algebraic equations for the unknowns A 2 ( t )and B2(t),application of Equation 3 and algebraic rearrangement (see Reference 2 for details of this approach). One obtains the final result
1(2wt)
=
Z(2w) W(w) sin (2wt
+
$2)
(33)
where
Z(2w)
=
n2F2AC,,*(2wD,)1/2AE F(t)Ww) 4RTcosh2
(i)
(34)
(35)
=
[(I
L
+ h22)1/2+ 1
+ hi2
Equations 15-27 and 33-42 comprise a master formulation of the second harmonic ac polarographic response which can be applied to a specific mechanism in the class under consideration simply by inserting the appropriate values of the constants W , Y , xd, Z,, k d , k, which are obtained as shown elsewhere ( I ) . Table V of Reference 2 tabulates these constants for ten different mechanisms involving first order coupled chemical reactions. The significance of the Z(2w) and W(w)components of the second harmonic current amplitude formulation is worth noting. A simple extrapolation of the interpretation of the analogous terms in the theory for the quasi-reversible case ( 6 ) is applicable. The Z(2w) term is identical to the corresponding fundamental harmonic current amplitude term ( I ) except that 2w is found in place of w. I n other words, Z(2w) is a measure of the linear faradaic admittance at the second harmonic frequency which is a significant factor controlling the second harmonic current (6, 7). The magnitude of Z(2w) is determined by the manner whereby the relevant rate processes (diffusion, heterogeneous charge transfer, and coupled chemical reactions) influence the linear faradaic impedance at the frequency 2w. The term W(w) represents a measure of the faradaic nonlinearity at the fundamental harmonic frequency. The effects of the various rate processes o n the faradaic nonlinearity at frequency w determine its magnitude. As in the case of the general fundamental harmonic theory ( I ) , one can readily show that the above formulation reduces to that of the simple quasi-reversible case (6) whenever conditions corresponding to thermodynamically or kinetically inoperative chemical reactions ( I ) are introduced. Although cumbersome, even when applied to a specific mechanism, the foregoing second harmonic theory is readily subjected to study with the aid of modern digital computers. Detailed studies of its predictions have been carried out in these laboratories for several important mechanisms. The calculational results, together with parallel experimental studies, strongly support the validity of the foregoing theory and the utility of second harmonic measurements. A description of these investigations greatly exceeds the scope of the present work and will be given elsewhere. A detailed step-by-step derivation of the above equations may be obtained from the authors on request. NOTATION DEFINITIONS
(37)
e3
E
m=l
Zmf-(h,))
(39)
Ci*
=
Di
=
fi
=
EdC AE El:;
=
i(t) Z(2wt)
=
= =
=
initial concentration of species i diffusion coefficient of species i activity coefficient of species i dc component of applied potential amplitude of applied alternating potential reversible dc polarographic half-wave potential (planar diffusion theory) instantaneous total faradaic current second harmonic faradaic alternating current
(6) T. G. McCord and D. E. Smith, ANAL.CHEM., 40,289 (1968). (7) J. Paynter, P1i.D. Thesis, Columbia University. New York, 1964. VOL. 40, NO. 13, NOVEMBER 1968
1969
dc faradaic rectification current component phase angle of second harmonic faradaic current relative t o applied alternating potential Faraday’s constant ideal gas constant absolute temperature electrode area number of electrons transferred in heterogeneous charge transfer step time angular frequency of applied alternating potential heterogeneous charge transfer rate constant at E a charge transfer coefficient rate constant for ith homogeneous chemical reaction step
rate constant for dth homogeneous chemical reaction coupled to species 0 = rate constant for mth homogeneous chemical reaction coupled to species R = dummy variable of integration = Euler Gamma Function = number of chemical reactions coupled with species 0 = number of chemical reactions coupled with species R =
X m9 =
constants dependent on nature of chemical reaction scheme
RECEIVED for review June 28, 1968. Accepted August 13, 1968. This work was supported by National Science Foundation Grants GP-5778 and GP-7985.
Optically Transparent Thin-Layer Electrodes: Studies of Iron(l1)-( 111) Acetyl aceto nate Ligand Excha nge Reactions in AcetonitriIe W. R. Heineman,’ J. N. Burnett,* a n d Royce W. M u r r a y Department of Chemistry, Uniaersity of North Carolina, Chapel Hill,N . C. 27514 Experiments based on an optically transparent thinlayer electrode cell have been applied to the elucidation of iron acetylacetonate electrochemistry in acetonitrile solvent. The cell allows rapid and simultaneous acquisition of potential-step coulometric and spectral data. Reduction of Fe(acac)2+ complex at 0 volt is followed by ligand exchange between Fe(acac)* electrode product and unreacted Fe(acac),+ to yield Fez+ and Fe(acac)s as ultimate products. Analogous product-reactant interactions are found for reduction of Fe(acac)*+ and oxidation of Fe(acac)?. I n each case, coulometric and spectral data are used in concert to verify the nature of the electrolysis products.
ELECTROCHEMICAL REDUCTION of iron(II1) fris-acetylacetonate [ F e ( a ~ a c )in ~ ]acetonitrile solvent and the effects of supporting electrolyte thereon formed the subject of a recent report ( I ) . Spectrophotometric results presented for Fe(acac)*+ and Fe(acac),+ demonstrated that these complexes form stoichiometrically in 1 :1 and 1 : 2 mixtures, respectively, of iron(II1) and acetylacetonate. Polarogram of the 1 :1 and 1 :2 complexes exhibit cathodic currents less than expected, however, and in addition waves appear for higher coordinated complexes. This polarographic behavior was tentatively interpreted in terms of exchange of acetylacetonate ligands between iron(II1) reactant complex and iron(I1) product complex in the electrode diffusion layer. We now report a study of this effect derived from simultaneous spectral and coulometric observations using an optically transparent thin-layer electrode developed in these laboratories (2). 1 Present address, Hercules Incorporated, Research Center, Wilmington, Del. Present address, Department of Chemistry, Davidson College, Davidson, N. C.
(1) R. W. Murray and L. K. Hiller, Jr., ANAL.CHEW,39, 1221
(1967). (2) R. W. Murray, W. R. Heineman, and G. W. ODom, ibid., p 1666. 1970
ANALYTICAL CHEMISTRY
EXPERIMENTAL
Chemicals. Eastman Spectrograde acetonitrile was used as received. LiC104 (dried in cacuo at 60 “C) and Fe(ClO&. 6 H 2 0 were obtained from G. F. Smith Company. Solutions of the iron(II1) acetylacetonates were prepared by addition of weighed quantities of Li(acac) to a stock (-SmM) Fe(C104)3 solution 0.1M in lithium and tetraethylammonium perchlorates as supporting electrolytes and were used within eight hours of preparation. The stock iron(II1) solution was standardized by conversion to Fe(acac)? and spectrophotometric determination at 430 mp ( E = 3.31 x l o 3 1 mole-1 cm-l) in a 500-micron cell. Li(acac) was prepared according to West and Riley (3). Apparatus and Procedure. The electrochemical apparatus has been described ( I ) . Spectra were recorded on a Cary Model 14 spectrophotometer fitted with a 0-0.1 slidewire. The optically transparent thin-layer electrode (OTTLE) (2) employed a gold minigrid (1000 lpi, 1.8 cmz gross area) as working electrode. Teflon tape spacers (50 micron) center the minigrid in the microscope slide-minigrid “sandwich” to enhance the cell’s optical thickness without appreciable sacrifice of short diffusional paths (4). Adequate spectral sensitivity ( A > 0.04) for the t = 1-3,300 complexes was attained at cell thicknesses ( I ) of 110-120 microns and 5 m M sample concentrations. In a typical experiment, the OTTLE is mounted in the Cary compartment over a reservoir cup containing the Pt gauze auxiliary and aqueous NaCI-SCE reference (cia acetonitrileTEAP salt bridge) electrodes. The portion of the light beam not impinging on the minigrid is masked with black tape. After purging the spectrophotometer compartment (sealed with polyethylene film) with nitrogen, the sample solution is transferred from a degassing bottle by nitrogen pressure to a level in the reservoir solution cup just contacting the lower edge of the OTTLE. The OTTLE is filled after each experi( 3 ) R. West and R. Riley, J . Znorg. Nucl. Chem., 5, 295 (1958). (4) W. R. Heineman, J. N . Burnett, and R. W. Murray, ANAL. CHEM., 40, 1974 (1968).
