AI terna ting Current Polarography of Electrode Processes with Coupled Homogeneous ChemicaI Reactions 1.
Theory for Systems with First-Order Preceding, Following, and Catalytic Chemical Reactions DONALD E.
SMITH
Department o f Chemistry, Northwestern University, Evanston, 111.
b The theory of a s . polarography with coupled homogeneous chemical reactions is discussed. It is shown that derivation of expressions for the phase angle between applied alternating potential and fundamental harmonic faradaic alternating current may b e accomplished with little difficulty or approximation. The frequency and potential dependence of phase angles for systems with preceding, following, and catalytic chemical reactions are derived considering both charge transfer and chemical reaction kinetics. The results are discussed in detail, including rationalization of equations in terms of well established concepts of the faradaic impedance.
H
of diffusional mass transfer brought about by rapidly changing sinusoidal potentials render a.c. polarography useful for kinetic studies of very rapid electrochemical and coupled chemical reactions. The application of a.c. polarography to the measurement of rapid charge transfer kinetics is vie11 established for electrochemical redox reactions kinetically controlled by diffusion and charge transfer. Sufficient data have been compiled in the past decade t o justify acceptance of early theoretical work of Randles (27), Ershler (8), Gerischer (IO),and Grahame (12) and later, more general, treatments of Kambara (15) and Matsuda ( 2 3 ) . Randles and Somerton (29) found a number of systems obeying the theoretical faradaic impedance-frequency relationship. Randles obtained agreement between kinetic parameters obtained by a.c. and d.c. polarographic studies of the same system (28). Other work also indicated agreement between theory and experiment (I, 2, 14). Recently, workers have found experimental phase anglepotential behavior of several systems to be that predicted theoretically (33, 34) 38). Systems not obeying theoretical predictions can be interpreted as IGH RATES
602
ANALYTICAL CHEMISTRY
involving rate determining steps in addition to diffusion and charge transfer [cf., e.g., (6) for references]. Matsuda (%$) and Matsuda and Delahay (26) have extended theory to account for specific effects of the electrical double layer. The development of a.c. polarography for quantitative studies of systems in which coupled chemical reactions exert kinetic influence has been less rapid. The only case discussed in the literature has been the system with a preceding chemical reaction, initially treated theoretically by Gerischer (IO) and later, in more detail, by Matsuda. Delahay, and Kleinerman (26) and Delahay ( 5 , 6 ) . X communication by Satyanarayana, Reddy, and Doss (32),presenting equations without derivation, is the only other contribution knon n to this author. h’one of these works discuss the potential dependence of the faradaic impedance, although a method of calculating potential dependence, applicable for fast charge transfer processes ( k h > cm. second-l), is pointed out by Delahay (6). To calculate charge transfer coefficients and distinguish between certain chemical reaction schemes, knowledge of the potential dependence is required. Thus, it appeared profitable t o examine theoretical aspects of a.c. polarography for such systems with purposes of obtaining a theory useful for various reaction paths, giving the potential and frequency dependence of a x . parameters and applicable to any values of kinetic and thermodynamic parameters. This paper is concerned with such a theory for firstorder and pseudo-first-order chemical reactions. The excellmt theoretical work of Koutecky and conorkers [cf., e g . , ( 3 ) , (20) for references] on d.c. polarographic kinetic currents, and Matsuda (B),on a x . polarography, serve as models on which the present work is based. I t should be mentioned that alternating current polarography and faradaic
impedance measurements refer to two different experimental methods of studying the faradaic impedance (5,6). Concerning theory, this difference is irrelevant. The term, a x . polarography, is employed in this paper only because it is the experimental approach used in these laboratories. THEORY
rlssumptions made in the following derivations are: (a) Fick’s law is applicable to each species independently; (b) the theory of absolute reaction rates as applied to electrode reaction kinetics (11) is applicable; (e) charge transfer involves one rate determining step; (d) each reacting species is soluble either in the solution or electrode phase; (e) diffusion and chemical reaction are the sole means of mass transfer; (f) chemical equilibrium esibts prior to electrolysis (at t = 0 ) ; (g) adsorption effects are negligible; (h) electrode geometry and convection due to drop growth may be neglected; (i) specific double layer influences are negligible. The first six assumptions, common to most modern electrochemical theory, are discussed in detail in numerous sources [cf., e.g., (5, 6, 30)]. Other assumptions are made for convenience and simplicity rather than from necessity. Double layer corrections employing the ideas of Frumkin (9) can be conveniently incorporated in equations to follow. Only when very rapid processes necessitate very high frequencies are more sophisticated theoretical approaches required (24, 25). That electrode geometry and mercury drop growth are of negligible significance regarding alternating current components iq indicated by the work of Gerischer (IO), Koutecky ( I 9 ) , and Matsuda (23). For small amplitudes of alternating potential, Natsuda’s equations predict drop growth influencing amplitudes of faradaic alternating currents significantly only when charge transfer kinetics are sufficiently .low that the d.c. polarographic n-ave does not correspond to reversible behavior. However, no influence of drop growth on phase angle is predicted for
any magnitude of heterogeneous rate constant. Possible exceptions, discussed below, do not apply to systems presently under con3ideration. If convection due to drop growth is included in the treatment to follow, equations for phase angles are unaltered from those presented below (36). Less general approximations or assumptions will be indicated in the course of derivation. A list of notation definitions is given in Appendix l. System with a Preceding Chemical Reaction. Consider the electrode reaction ki
Y e0 kz
+ ne + R
(1)
where reactants prejent in large excess are neglected for siinplicity, their concentrations being iiicorporated in the rate and equilibrium constants. Specific adaptation of equations presented here for more complex :*eaction schemese.g., dissociation of a metallic complex followed by reduction to metal-can be accomplished mithoL t difficulty (6, 26). For this process, the system of partial differential equations, initial and boundary conditions are
ac,
-= 0 at
azc,
~
t > O , x = O ; -a$ =-- i ( t ) ax nFADo
- b6 ax
i(t)ekt =-=-
KnFADo
H(t) KnFADo
~
(17a)
From the theory of absolute reaction rates (11)one has
where E ( & ,the instantaneous value of the potential, is given by
Employing the method of Laplace Transformation (4, the solutions t o Equations 12, 13, and 14 in transform space are (applying conditions 15a through 16c inclusive)
- AEsinwf
E(t) =
(17b)
(29)
Substituting Equations 26, 27, and 29 in 28 and employing step-by-step operations identical to those used by Matsuda (dS), one obtains the system of integral equations
(18)
where
B
=
C,
=
B,=oesp [ - x ( $ ) " ~ ]
c ~exp~ [-x-(;)'"]~
(19)
(20)
where the definition of the Laplace transformation is ,f(s) = =
0, any
+ C,
C,
2;
=
C: + C: = Co (6a)
Sum
f(t)esp( -st)df
where p = 0, 1, 2 , 3 . . Q(t)
(21)
Differentiating Equations 18, 19, and 20 with respect to x and equating with the Laplace transforms of Equations 17a, 17b, and 17c, respectively, yields
=
. . . ...
I
2
2")
$:(
Qp(t) nFACo*Dol/Z- p = o j =
nF - (Ea.c.RT
AE)'
E1/2Y
f
=
(32)
(33)
P = l - a
D
(31)
(34)
DOPDP
(35)
= joqfrp
[reversible EWZ] (36) t
> 0, x
=
0 ; D-,'
ac
: :
dX
- D ac, -=
ax
Taking inverse transforms of Equations 22, 23, and 24, using the convolution and substituting Equatheorem (4, tions 9 and 10, one obtains for the surface concentrations of Co, C, and C,
Introducing the transformations $ =
c, -E
Co
(9)
=
ki
+ kp
l i d ) = nFACo*Do"'Ql(f)
(37)
(10)
where Ql(t) is the solution to the integral equation for p = 1
(11)
Qi(t)
where k
Equation 30 represents the complete description of all current components flowing through a polarographic cell under the influence of an applied sinusoidal alternating potential of any amplitude. Of interest in this work is the small amplitude ( A E 2 8 mv.) behavior of the fundamental harmonic current given by
and, assuming D, = Do,a new set of equations is obtained in terms of the variables il. and e (3, 20).
j:r2Fo(f)einot -
s,'e-'"&l(t
- u)dlL (TU)l'2
(12) VOL. 35, NO. 6, M A Y 1963
603
solution to the small amplitude ax. polarographic wave in terms of as yet unevaluated Fo(t).. Detailed discussion of this result is given below. System with a Following Chemical Reaction. For the reaction scheme
The significance of F,(t) and its evaluation will be discussed below. The nature of Equation 38 is such that its solution contains only fundamental harmonic alternating current components. Thus, &(t) d l have the form Q(f)
=
A(t)coswt
+ B(t)sinwf
n-here the chemical reaction is represented in a simplified form, as in the above case, the system of differential equations, initial and boundary conditions is
(40)
Substituting Equation 40 in Equation 38 and making use of the trigonometric identities sinw(t - u) = sinwtcoswzc coswtsinwu (41)
cosw(t -
ZL)
=
CoswtcoswZL
+
sinwtsinwu
(42)
yields an expression integrable by neglecting the slight time dependence of A(t) and B(t),applying the steady-
17-here
K,ki
(54)
k.2
t
x; Co
=
Co*
(5a)
state approximation
C,
=
0
(55b)
(6)and employing the identities (?,IS')
c,
=
0
(5%)
CO*
(56a)
0
(56b)
0
(56,)
= 0 , any
t > 0, x
-
m ; Co
c,
c,
-+
-f
-.+
System with a Catalytic Reaction. For t h e reaction scheme
Equating coefficients of sinwt and c o s ~ t , solving for A(t) and B(t) and using the identity asinwt
+ bcoswt
Treating this system of equations in a manner identical to that employed above, one obtains for the surface concentrations of 0, R and Y (assuming
D,= D,) 1
=
(1 ~~
= wt
+ cot-'
?)
+ K)nFAD,l
F?
Z
(67)
where 2 is present in large excess and the chemical oxidation is irreversible, the system of differential equations, initial and boundary condition- for this system is
(46)
h
yields for Q l ( t ) Qi(t)
2
L+ nek , ARI +
0
(58)
=
( + cot-' );
(li~+~?)lrl sin w t
(47)
-
where
1
+ k")' 2 + k]"2
+K
2(w2
+ kZ)
+ (72)
1 __
(2w)'
2
(e +;
(49)
Substituting Equations 47, 48, and 49 in Equation 31 gives the complete
604
ANALYTICAL CHEMISTRY
where k
=
ki+ k2
Substituting Equations 59 and 60 in Equation 28 and proceeding as in the previous case, one obtains a system of integral equations
Solving for the surface concentrations, one obtains ( a w m i n g Do = D,) Go=-, = C'
1 - nFAD1IZ _-
Substituting Equations .73 and 74 in Equation 25 and, again, proceeding as before, a system of iitegral equations is obtained
Obtaining the solution for QI(t) gives
Ha(t)
= ae-al
- (cte-a, - /3eB') t eZ(?o(t-
U)CZ?L
(79)
Because no reversible ( atalytic reaction has been reported, only the irreversible process is considered here. For the reversible case, the only change required in final results is replacement of k IC, by k , (1 where K = ' (kb = h.6 rate of reverv reaction).
+ $),
DISCUSSION
The solutions for the a x . polarographic waves, Equations 47, 63, and 76, are given in terms of functions F,(tj, GO(!) and Ilo(t), respectively, n.hich haye iiot heen evttliiated. IZnon-1edge of these functions requires bhe complete curreiit,-l,oteiitial solutioii to the d.c. polarographii: \Tare for the systems under consid.ration [id.e. = n F A C o * D o 1 ~ 2 Q o ( obtainable t)], by employing the general theory of d.c. polarographic kinetic curre:its of Koutecky and coworkers (3, 20). However, these solutions are quite complicat'ed, their evaluation representing the most difficult part of the a.c. 1)-obleni requiring greater approximation and more cumbersome niatheniat,ical exlxession than elsexhere in the soluti'm. T h a t phase angles between applied alternat'ing potential and fundamental harmonic faradaic alternating current are independent of functions such as Fn(t), Go(t), etc., is most important, representing a major simplification of the theory under consideration. Esani:nation of this work and the derivation of Matsuda (23)
indicates that such functions, dependent on the characteristics of the d.c. process, will appear in solutions t o ax. polarographic problems for any electrode reaction scheme. T h a t phase angles Rill be independent of d.c. characteristics is likely true for all firstorder or pseudo-first-order chemical reaction schemes coupled with a single electron transfer step, but u-ill probably not be true for Some processes involving higher order chemical reaction? and certain systems \\ ith adsorption phenomena, Phase angles for systems n i t h t n o or inore simultaneous charge transfer steps can be shown dependent on d.c. characteristics n hen kh < cm. second-' for a t least one of the steps (35). With these possible exceptions in mind, some considerations can be made regarding calculation of phase angles for systems 17 ith first-order chemical reaction%. Difficulty in obtaining solutions for phase angles will r e d alniost solely on the complexity of the problem of deriving integro-differential equations for wrface concentrations-e.g , Equations 25. 26, and 27. Subsequent niathematical steps remain the jame for most systems representing no barrier to solution and differing from one reaction achenie to another only in algebraic coniplexity. Because eupression of surface concentrations often represents little difficulty, even for rather complex
cot4
phase angles often may be accomplished with greater accuracy than alternating current amplitudes or d.c. polarograijhic currents for the same systems. Perhaps more important in this regard is the probable insensitivity of phase angles to depletion effects, which appcar to have significant, influence 011 d.c, liolarographic currents (21) and, if so, would also influence a.c. amplitudes through the fuiictioii Fait), etc. 'That phase angles would be relatively iinaffccted by depletion is indicated by the independence of phase angles to initial concmtrations of electroactiw species for most' systems. Exceptions mentioned above apply here. For cases under consideratioii, depletion would iiot influence phase angles uiiles =isbuml)t'ion if) is rendered invalid or if concentrations of species present in large cxcess are subject to depletion. Both possibilities are unlikelx for t y i c a l reaction rates and conceiibration?. Because of these considerations and that phase angles are conTreniently determined exlierimrntally, further discussion will be limited to phase angle rclatioiis given above. Rearranged phase angle rclations, n-ith appropriate liniit'ing cases, for preceding, following, and catalJtic reactioris arp. resl)ectively, ki
Y e0 k2
+ ne * R
=
system5 (niechanims in1 olving tn o or more chemical and/or electrochemical .teps), phase angle eymssions will be an illuscorrespondingly accessibl~. ;1$ tration, the theoretically predicted phase angle behavior for a system inT olving simultaneous control by preceding and follon inq reactions is given in Appendis 2 and briefly discussed helon . The best mathcniaticzl description for tliffuqion to the dropping mercury electrode (22) iq difficult and cumber-omr to aiiply, a$ ne11 a5 slightly inexact. Deril ation of Fait). Go(t), etc.- Le., deri\ ation of the d.c. ma1 e-often requires application of t h r approximate reaction laver concept and neglect of drop geometry ( 3 20). Thus, insensitil ity of small amplitude a e . polarograiihic phase angles to factors such as drop gron t h electrode geometry, Fo(t),
where :
etc., indicates that, \\-ith present mathematical tools, theoretical expression of
where g and are given by Equations 81 and 82. The limits for K m,
g=-
+ kz
kl
w
lim cot+ K-m
=
+
lini cot+
=
1
lim cot+
=
1
0-0
(Ow)' --
?
(83)
x
(S-4)
w - r o
lim cot+ = 1
( 2 ~ ) " ~ 1+ + __ ___el
A
L W7 -Zm O
(i&
+
el)
(85)
+ ne S 12 ke Y r 122
0
-
VOL. 35, NO. 6 , M A Y 1963
605
r-
I
I
k , = 500
,
I
'00'-0080
I
0.000
-0040
I
+O 040
I
+0080
(E-E;,~)-+
Figure 2. Potential dependence of cot4 with preceding and following reactions with varying k l
W ' l Z---+
Figure 1. Frequency dependence o f cot@ with preceding or following reactions a t k c .= E; with varying k l and K = 0.100 cm. second-', a: = 0.500, D = 1.00 n = 1 , T = 25OC. , K = 1.00, kl = 50.0 second-' -__ K = 1.00, kl = 500 second-' K = 1.00, k l = 5000 second-' K = 0.1 00, k l = 50 second -. - . - . - K = 0,0100,k l = 5 0 5econd-I
kh
. .. . . . .. . . ---_____ -
(87) 0
T
=
+ ne
R
+2
kc
1
x (88)
where g = k , , ' W , A is given by Equation 82 and the limit for g+ 0 is the same as in Equation 83. The low frequency limit is: lini cot+ = w - 0
m
(89)
g--
To the author's knowledge, Equations 86 and 88 have not been given previously in any form. The phase angle relation for a preceding reaction presented by Gerischer (10) represents a special case of Equation 80 for E d . o . = E1,2r.At this potential Equation 80 is identical to Gerischer's result. It is also in agreement with equations presented by Matsuda, Delahay, and Kleinerman (26) (rearranging their equations for the simple reactionschenie, Y + 0, and restricting Equation 80 to the case K cm. second-Li.e., "fast" charge transfer. No such restriction is placed on the present work because the Nernst Equation is not applied. It is interesting to note that the two methods of calculation yield identical phase angle expressions. Results of the more general method employed here differ from the approximate calculation assuming Xernstian d.c. concentrations only in the alternating current amplitude. The difference is incorporated in the function Fa(t). Thus, phase angle expressions calculated from previous work are more general than was implied. Although not previously indicated, Equation 86 is obtainable from the theory of faradaic impedance (6) by applying the chemical reaction terms to species R rather than 0. Examples of predicted variations of cot$ with frequency and potential for preceding and following reactions arc given in Figures 1, 2, and 3. I t is evident that one can detect but cannot distinguish between preceding and following chemical reactions based on the frequency dependence of
[email protected] is quantitatively identical a t Edo = E112r and qualitatively similar a t other potentials. However, the distinction is immediately apparent from the potential dependence (Figures 2 and 3).
The frequency and potential dependence of cot4 with a catalytic reduction are given in Figures 4 and 5, showing a markedly different behavior from the other cases. With catalytic reductions, theory predicts that cot$ approaches 0) a t low frequencies. infinity (6 Also, while the potential dependence is enhanced (bcot+/bEd becomes larger a t most potentials) by the presence of the chemical reaction, the potential of peak cot$ remains uninfluenced. Without contributions from charge transfer kinetics cot@would be potential independent. The predicted d.c. polarographic behavior for cases considered in Figures 1 through 5 may be obtained by consulting the work of Koutecky (1618).
Intuitive rationalization of the predicted behavior of phase angles may be accomplished best by consideration of effects of chemical reactions in terms of the following points [cf., e.g., (6),for references]. a) The faradaic impedance may be considered a series resistance and capacitance n-hose values are frequency dependent. b) For a series circuit, cot$ is the ratio of the resistive and capacitive impedances (resistive over capacitive). c) The faradaic series resistance, r., is given by the sum of three terms; the charge transfer resistance, T , ~ , the mass transfer resistance for 0, TO, and the mass transfer resistance for R, rr. d) The faradaic series capacity, c8, is derived from mass transfer only and is the sum of mass transfer capacities of 0 and R. e) LIass transfer resistance and capacitance may be influenced by diffusion, chemical reaction, and adsorption.
I
I
5'mt \ -0.080
-0.040
I
I
I
I
I
I I I
-
W'IZ
/
Figure 4. Frequency dependence of cot+ with catalytic reaction a t Ed.c. = E; with varying kc
/ /
/
= 0.100 cm. second-', n= I , T = Z ~ ~ C . kh
0.000
+0.040
+0.08
(E-E:,~
Figure 3. Potential dependence of cot+ with preceding and following reactions, with varying K w i = 20.0 second-$, k1 == 50 second-', k h = 0.100 cm. second-', a = 0 . 5 0 0 , D = l.0OX 1 0 - 5 c m . 2 s e c o n d - 1 , n = l , T = 2 5 ' C . E = Ed.0.
-_-
preceding reaction reaction
- following
The mass transfer terms for 0 and R are inversely proportional to the mean (d.c.) surface concentra;ions of 0 and R , respectively. IXffusion components are . resistive proportional to l / ~ l ' ~The first-order chemical reaction component is proportional to
m-hile the capacitive chemical reaction component is proportional to
f ) Rch 2 C c h so t h a t cot@n-ill show positive deviations when influenced by a chemical reaction. This exemplifies the principle that, the ess efficient the a.c. process, the smaller the phase angle (31J.
The frequency dependence of the resistive and capacitive components of the faradaic impedance with preceding reactions has been discussed by Delahay, et al. ( 5 , 6, 26). Specific application of their arguments to phase angles for such systems is considered here for completeness.
It is evident that, a t sufficiently high frequencies ( W >> k ) , kinetic influence of any type of chemical reaction will disappear because electrolytically induced periodic concentration changes of 0 and R are too rapid t o be influenced by sloner chemical reactions. While the alternating current amplitude still may respond to the presence of chemical reactions at high frequencies (the current being proportional to the equililrium concentration a t t = 0 of 0 and/or R ) , the phase angle, independent of initial concentrations of electroactive Ypecies, is determined by charge transfer and diffusion kinetics (Equation 83). -4t lower frequencies ( W IC), chemical reaction beconies important and cot+ ihows positive deviations from linearity (Figure 1). At very lorv frequencies (w + 0), chemical and electrocheniical equilibria tend to prevail. For the latter situation, with preceding or following reactions, the process becomes diffusion controlled (cot+ = 1). With such reactions, the case m here chemical equilibrium prevails but electrochemical equilibrium does not is considered in Equations 85 and 87. While the chemical reaction exerts no kinetic influence N
a = 0.500,D = 1.00 X 10-5cm.2second-1,
under these conditions, i t manifests itself thermodynamicallj- as a shift in the ha11 wave potential, resulting in these equations differing from t h a t usually given for charge transl'w and diffusion control (Equation 63) by a factor accounting for the shift in the wave. Finally, a hen K becomes large, influence of preceding or follou ing reactions disappears at all frequencies. At low frequencies, a different situation exists M ith catalytic processe.. When k >> w, virtually all I: generated electrolytically is reoxidized chemically. Rapid chemical reo\idation of E to 0 results in a narrov, biit iteep concentration gradient delegating diffucion impedances t o a negligible role. The resistive and capacitir-e diffusion mass transfer components become ncgligible as does the chemical reaction capacitive component (Item e above) Thus ca0 as w -+ 0. The resistir e component due to chemical reaction remains and the component dur to charge transfer niay or may not he significant. I n any event, the faradaic impedance brcomes iwrely resistive a. w + 0 Thcb most familiar cape in 11 hich the faradaic impedance becomes purely resiptive because diffusion impedances of both 0 and R are rendered insignificant is the high frequency limit n ith diffusion and charge transfer hinetic control A4nother case where chemical reaction delegates diffusion to a negligible role yielding a resistive faradaic iniprdance iq mentioned in Appendix 2. Rationali~ationof the d c. potential dependence of cotq requires con-ideration of the influence of mean surface concentrations on mass transfer impedance terms (Item e above). \Then R is the predominant species (negative the potentials with respect to El mass transfer impedance terms for 0 are most significant and vice versa. K i t h VOL 35, NO. 6, MAY 1963
607
a preceding reaction, only mass transfer of 0 is influenced by chemical reaction. Thus, chemical reaction influence will be largest at negative potentials when T~ and co are most significant (Figures 2 and 3). With following reactions, the same argument shows chemical reaction influence greatest a t positive potentials. Catalytic reactions influence mass transfer terms of both 0 and R in an equivalent manner because the rate of chemical removal of R equals the rate of chemical formation of 0. The effect on the concentration gradients of both 0 and E will be identical. It follon-s that catalytic reactions will introduce no asymmetric potential dependence. Potential dependence of cot+ with catalytic reactions is derived from charge transfer only, as pointed out above. Such behavior is also predicted for simultaneous control by preceding and following reactions when these reactions have equal rate and equilibrium constants (Appendix 2). Regarding the potential dependence with catalytic reactions, i t was pointed out' above that potential dependence due to charge transfer is enhanced by the presence of a catalytic reaction. This occurs because greater demands are made on charge transfer by rapid chemical reosidation and diffusion than by diffusion alone. The utility of a.c. polarography relative to d.c. polarography in determining chemical and electrochemical kinetic parameters is worthy of mention. The large variety of experimental problems encountered from system to system precludes a complete discussion of this topic in the present n-ork. However, a single exemplary cace will be considered. For a system involving a preceding chemical reaction in which the species 0, R and 1- have been characterized and the diffusion coefficients and reversible half-wave poteritial are known from separate experiments, a.c. polarographic evaluation of k,, khr K and CY can be acconi1)liwhed in the following manner: Phase angle measurements at frequencies sufficiently high to eliminate chemical reaction kinetic influence will yield values of k h and a. dlternating current amplitudes under the same conditions may be used to evaluate K by employing Matsuda's equation (23) where Co* is the equilibrium concentration of cl)ecies 0. The final parameter, kl, i? then deduced from phase angle data a t Ioiver frequencies. Given the same initial information and completely general theoretical espressions, d.c. polarographic evaluation of K , k l , kh and CY (assuming chemical and electrochemical kinetics influence d.c. results) would be extremely difficult, if not impossible. I n many typical situations a.c. polarographic determination of these param608
ANALYTICAL CHEMISTRY
_-------. -_---- - - - - _ _---__ __---2,0 - ___-----_ _ _------z====== _ - - - - - _ ----___ ---.-= _ _ _ _-------
4.0
-
- / - - -
-
> w and ka 4 ka >> w a ) kl ( g and p approach m ). b) w # 0, i.e., (2w)”?//x f 0
+
(-15)
For this system, if K , m d K j are very small, diffusion of 0 and R will be negligible and Equation A5 predicts cot+ will be very large (lim c c t 4 = a). That Kf*Kp
-+
(1959). \----,-
0
is, the faradaic impedance becomes purely resistive in analogy Kith the catalytic rase at low frequencies. A C K N O W L E D O MENT
Valuable discii>sioni on parts of the subject matter with 11’. H. Reinmuth are appreciated. LITERATURE CITED
(1) Aten, A. C., Hoijtinlr, G. J., in ‘.Advances in Polarography,” I. 9. Longmuir, ed., Vol. 2 , pp. 777-85, Pergamon, S e w York, 1960. (2) Bauer, H. H., Elvirig, P. J., ANAL.
CHEM.30, 334 (1958). (3) Brdicka, R., Hanus, V., Kouteckq;! J., in “Progress in Polarography, P. Zuman, ed., with collaboration of I. 11. Kolthoff, F-ol. 1, Chapt. 7, Interscience, h-en7York, 1962. (4) Churchill, R. V., “Modern Operational Mathematics in Engineering,” McGraw-Hill, New Ylirk, 1944.
(14) Joshi, K. M., Mehl, W ~ ,Parsons, R., in “Transactions of the Symposium on Electrode Processes, Philadelphia, May 1959,” E. Yeager, ed., Chapt. 14, W-iley, New York, 1961. (15) Kambara, T., Z . Physik. Cheni. h7.F.5,52 (1955). (16) Koutecky, J., “Proceedings of the 1st International Congress of Polarography in Prague, 1951,” Vol. 1, pp. 826-38, Prirodovedecke 1-ydavatelsti, Prague, 1952. ( 17) Koutecky, J., CoZlection C‘zechosloz’. Chem. Communs. 18, 311 (1953). (18) Ib
