Theory of the Faradaic Impedance Relationship between Faradaic Impedances for Various Small Amplitude Alternating Current Techniques DONALD E. SMITH Department of Chemistry, Northwestern University, Evanston, 111.
b Four experimental methods are possible for combining control of direct current or potential with control of sinusoidal alternating current or potential to study the small amplitude faradaic impedance. The theories have been examined and compared for various types of electrode reaction mechanisms. Differences in the faradaic impedance appear only between techniques involving a different mode of control of the d.c. variable, and are predicted only when the d.c. wave is influenced kinetically b y rate processes in addition to diffusioni.e., charge transfer, chemical reactions, etc. With diffusion-controlled d.c. processes, theory suggests the same faradaic impedance for all methods. The phase angle is predicted to b e relatively insensitive to the experimental approach with phase angle dependent on method of d.c. variable control only in special situations. Existing a x . polarographic theory may b e applied in many cases to other small amplitude a.c. methods which have been suggested, but not examined theoretically. Inspection of the differences in the faradaic impedance observed with different techniques may find application in kinetic and mechanistic diagnosis of electrode processes.
S
techniques have been applied to the study of the small amplitude a.c. response of an electrolytic cell under polarographic conditions. By far the best known and most n-idely applied is a x . polarography ( 3 ) , in which the d.c. and sinusoidal alternating potentials applied to the polarographic cell are controlled and t!ie resulting alternating current is measured as a function of the d.c. potential (alternating potential amplitude held constant). For brevity, a.c. polarography is referred to as Technique 1. A second approach is aiternating current chronopotentionietry (23),in which the direct and sinusoidal alternating currents are controlled (constant d.c. current and constant a x . amplitude) and the alternating potential resulting from these influences is measured as a EVERAL
962
ANALYTICAL CHEMISTRY
function of time (Technique 2). Bauer (1) has recently proposed a method which he entitles (‘ax. polarography with controlled alternating current” (Technique 3), in which the direct potential is controlled as in a.c. polarography, and the alternating current as in a.c. chronopotentiometry. The resulting alternating potential is measured as a function of d.c. potential. A final possibility (Technique 4), not proposed in the literature but considered here for completeness, involves control of the direct current and alternating potential and measurement of the alternating current. These four approaches to the study of the faradaic impedance may be considered analogous to the four possible approaches to faradaic rectification studies (11). I n a x . polarography, rather than measuring alternating current directly, the cell impedance is often determined through use of an impedance bridge, an approach usually referred to as faradaic impedance measurements rather than a.c. polarography-e.g., Delahay’s ( 7 ) discussion. In faradaic impedance measurements, it has been customary to measure the a s . impedance under conditions of zero direct current flow-that is, finite concentrations of oxidized and reduced forms are present in the bulk of the solution and the direct potential is set a t the equilibrium potential of the system. However, nothing fundamental dictates against faradaic impedance measurements with d.c. current flow, so conventional procedures are not considered a reason for considering a.c. polarography and faradaic impedance measurements as separate techniques. For the present discussion, techniques are considered as “different” only if they involve control of different d.c. and/or a.c. variables. KO detailed theoretical or experimental work examining the relationships among these four methods has been reported. The possibility that some of these techniques may prove complementary, that variations in faradaic impedances observed with different small amplitude methods might be employed to advantage in mechanism diagnosis, or that existing theory for a
given technique might be applied to another technique suggests the desirability of comparing the theories. Such a comparison is the primary concern of the present work. The relative advantages of these methods on either a theoretical or practical basis, are not examined. BASIS FOR COMPARISON
OF THEORY
I t has been customary in work on a.c. chronopotentiometry to consider the measured a.c. potential as a function of time (23). Presumably, this practice would be followed also in Technique 4,in which the direct current is controlled, as in a.c. chronopotentiometry. However, the use of time as an experimental variable provides no direct basis of comparison between Techniques 3 and 4 and Techniques 1 and 3. Because the ratio of the mean (or d.c.) surface concentrations is a primary factor in determining the faradaic impedance and because this ratio depends on the direct potential, the latter is a more fundamental variable to be considered in examining the faradaic impedance, regardless of whether the d.c. potential is controlled or not. In addition, the d.c. potential often represents a more convenient variable for expression of theoretical equations. These factors, in addition to the need for a unified basis of comparing theories for all methods, suggest the desirability of expressing theoretical results for Techniques 2 and 4 in terms of d.c. potential rather than time whenever a choice exkts. This practice is followed in theoreticzl relations given below. BASIC ASSUMPTIONS
Inherent in the theoretical relations compared are the assumptions that alternating potential amplitudes (controlled or uncontrolled) are very small (a few millivolts) and that steady state in the a x . diffusion profile is achieved ( 7 ) . These represent the usual approximations in derivation of 8.0. polarographic theory and normal ax. polarographic experimental conditions render such assumptions accurate. For Techniques 2 and 4, the latter assump-
tion places a lower limit on transition times which will depend on ax. frequency. Normally, transition times must be sufficiently long that only stationary electrodes are practical (conventional chronopotentiometry may be accomplished a t the dropping mercury electrode if transition times are much shorter than drop life). For this reason, and for simplicity, we consider only theory for diffusion to a stationary plane. Qualitative conclusions of the theory will not be altered by other electrode geometries. The small amplitude approximation implies neglect of hig'7er harmonics and the d.c. faradaic rectification component. Faradaic rectification influences the d.c. concentration profile to some extent. Neglect of this effect on the f~~ndamentalharmonic component a t small amplitudes represents a minor approximation. The neglect of higher harmonics nced not be considered an approxima tion because, even if significant, they can be eliminated from experimental measurements through the use of tuned amplifiers. Only sinusoidal appl.ed a.c. functions are considered.
amplitudes and an applied current function
i(t) =
id.o.
- LUsin ut
(6)
the solution was shown to be
where T is the transition time (9). Employing the relations obtained from solution of the d.c. wave (9)
one may express Equation 7 in terms of d.c. potential rather than time. The resulting expression is
E(wt) =
($
4RTAI cosh2 n2F2ACo*(WDo)112 sin (ut
-
:)
z-
'
with diffusion to a statior.ary plane and mass transfer uninfluenced by chemical reaction and adsorption, the surface concentrations of the electroactive species are given by Reinmuth (19) (assuming species R initially absent from the solution)
(10)
Equation 10 corresponds to a faradaic impedance given by Equations 4 and THEOkY 5. The difference in sign of the phase angle terms of Equations 2 and 10 arises The Reversible Case. TECHNIQUE because Equation 2 gives the phase 1. The expression for the a.c. polaroangle of current relative to potential graphic wave with a diffusion-conwhile Equation 10 gives the phase angle trolled faradaic impedance has been of potential relative to current. Therederived independently by several fore, the actual phase angle is identical workers (5, 10, 15, 22;. For an applied for both methods, with the alternating potential (see Komenclature section current leading the alternating potential for notation definition;) by 45". The origin of this difference in sign must be kept in mind Fhen comE(t) = Ed.c. - hE sin ut (1) paring phase angle relations for conit may be given in the form trolled alternating current and controlled alternating potential techniques I(wt) = for other types of systems considered nZF2ACo*(~l~,) ll2AE below. -__sin (ut TECHNIQUE 3. To our knowledge, 4 RTcosh2 (2) no theory for this technique has been published. However, it can be shown where that, for the reversible case, the measured alternating potential, E(ut), is given by Equation 10, so that the (3) faradaic impedance corresponds to Equations 4 and 5. Specific derivation Equation 2 corresponds to a faradaic of this result will not be given, but it impedance may be obtained as a limiting case of more complex mechanisms for which 4 R T cosh2 theoretical derivation is presented be(4) low. - n2F2ACO*m:do)ll* TECHNIQUE 4. For a diffusion-controlled process, it can be shown that the which is equivalent to a series RC ciralternating current obeys Equation 2, cuit with so that the faradaic impedance is given by Equations 4 and 5. Again, specific derivation of this result will not be given, as it follows as a limiting caFe for systems whose theory is derived TECHXIQUE 2. Takemori and cobelow. workers (23) have derived equations Quasi-Reversible Case. T o our for the steady-state solution of the a.c. knowledge, a.c. polarography is the chronopotentiometric wave. For small only technique of those under con-
(i)
sideration for which a theoretical treatment has been published considering the quasi-reversible system. The most rigorous treatment is that of Rlatsuda (16). For the present discussion, it is advantageous to obtain theoretical expressions for controlled alternating potential and controlled alternating current methods allowing for any possible mode of control of the d.c. variable. For controlled alternating potential techniques, the desired equations can be obtained by slight modification of llatsuda's equations. However, for completeness, a simplified derivation of the desired equations is outlined. For the electrode process
TECHNIQUES 1 AND 4. To obtain theoretical equations for the measured alternating current with controlled alternating potential techniques and a quasi-reversible system, one performs the following operations. Equations 12 and 13 are substituted in the current potential expression (12 ) .
+ i)
(i)
where (15)
@ = l - C t
The potential function, E ( t ) , is expressed as in Equation 1, where the a x . portion is controlled and the d.c. term may or may not be controlled. The exponential functions are expanded in the form EXP
x
(1 RTffnF
-~( E d . c .
-
- AEsinut - E")
(:l?)a"2e-uj VOL. 36,
[I
NO. 6 ,
mFAE
+ RT
MAY 1964
1
=
sin ut
e
1
963
EXP
x
Substituting Equation 27 in Equation 25 yields
Z ( 4 (17) which is applicable only a t small values of AE. Not considering higher harmonics, the faradaic current, i ( t ) , is expressed as
+
i(t) = id.o.(t)
6 sin w t
+ y cos ut
=
Zrev.F(t) X 2
[ut
1/2
Xsin
+ cot-'
(1
+ x)] 6
E(t) =
-
AI sin wt
&.e.
+ 6 sin wt + y cos wt
and I,,,. is the amplitude term of the reversible a.c. wave-Le., the coefficient of the sine term in Equation 2. Equation 28 corresponds to a faradaic impedance
EXP
[sF ( E ~ .+~ . sin + wt - E " ) ] (%> e-aj X 6
=
y COS
EXP (19) 6
[
where
id.c. ( t )
(22)
DOBDBa
(23)
Qo(t) = nFAC,*D,I/2
D
f
=
(24) Thus, the alternating current function may be written
[g
(Ed.c.
[1 + PnFG sin RT --
('
T)]"' 1/2w (30)
where Z,. is the faradaic impedance with a reversible process given by Equation 4. The equivalent series RC components may be represented by
wt
coswt]
(35)
+ 6 sin ut +
- EO)]
y cos ut
Ut
a/2
RT
1 + - -
(34)
where the a.c. terms are the measured a.c. components, higher harmonics being disregarded. The exponential functions are expanded in the forms
(18)
' = ( T)
(33)
(28)
where
In the resulting equation one performs the indicated integrations involving a x . terms assuming steady state (7, 19, ZO), equates coefficients of sin wt and cos ut, respectively, and solves the resulting equations for 6 and y. One then obtains
id.c.
where the a x . term is the controlled a s . variable. The d.c. variable may or may not be controlled. The potential function is expressed as
+qy]
[1+(1
i(t) =
=
(
x
$)-"2ebj
+ PnRT Fy
cos a t ]
(36)
which apply only to small amplitude a x . potentials. Performing the indicated integrations involving a x . terms, assuming steady state, equating coefficients of the sine and cosine terms, respectively, and solving for 6 and y yield
z/zw
= fo!fEa
Rf =
+ -T--)
R,,,. (1
(31)
F(t)
(37) L
sin [ut
+ cot-'
(1
+
e)]
964
ANALYTICAL CHEMISTRY
(38)
(25)
which corresponds to an alternating potential function given by
This result may be further simplified by applying the integral equation for the d.c. process
from which me obtain the relationship
-6
No assumptions have been made regarding the mode of control or form of the d.c. variables in obtaining Equations 28 through 32. TECHNIQUES 2 AND 3. To obtain theoretical expressions for the alternating potential observed with controlled alternating current techniques and a quasi-reversible system, one proceeds in the following manner. Equations 12 and 13 are substituted in Equation 14, as before. The current function is written as
RTAZ (1
E(wt)
= -
c +)""
(e-*j
+
+ epj)
1
+
_.
n2F2ACo*(2wD0) u2
Applying Equations 26 and 27 yields
sin [ u t - cot-' (1
.+ x)] .\/w
corresponding to Equations 41 and 42 may be obtained by carrying out the derivation in a manner identical to that employed above for the quasi-reversible system. Integrations involving a.c. terms employ relations given in a previous communication (60) (Equations 43, 44, and 45). The results are
G(t)
=
(1
+ e')Y
X
Qo:"T;)::du
(52)
and the d.c. components of the surface concentrations are given by [Coz-o]d.o.
=
wx
co* -
(40)
where E,,,, is the Eimplitude of the alternating potential observed with a reversible process which is given by the coefficient of the sine term in Equation 10. Equation 40 corresponds to the faradaic impedance expressed by Equations 30, 31, and 32. As with Equations 28 through 32, Equation 40 applies to any form of d.c. variable control, provided the steady- state and small amplitude assumptions are valid. One may obtain the reversible a.c. wave equations from these results for the quasi-reversible system by letting kh + m , in which esse &/A + 0 and F ( t ) + 1. Systems with First-Order Chemical Reactions Coupled with Single Electrochemical Ste:?. For reasons which will become apparent, we consider first the general case, developing the theory as completely as we find possible. Two specific mechanisms are also considered : the case of a first-order chemical reaction preceding charge trsnsfer and the caSe of a chemical reaction parallel to charge transfer-Le., the catalytic process. In general, surface concentrations of the electroactive species for systems with any number of firs t-order or pseudofirst-order chemical r:actions coupled with a single charge transfer step may be expressed in the form ( I S ) :
where Substituting Equations 53 and 54 in the Nernst equation and rearranging yield
so that G(t) =
(44)
s=1 {w + Yd + (1
(1
+ e9Y
Yd
+w
(56)
For any type of first-order chemical reaction scheme, when charge transfer kinetics are very rapid and chemical reactions so slow as to be essentially inoperative, one has
+ e')
(45)
(47) (59)
(48)
These expressions correspond to a faradaic impedance given by
Upon application of the Nernst equation, one obtains
1 with equivalent series RC components
(for equal diffusion coefficients, diffusion to a stationary plane and species R initially absent froin the solution) , where K, X,, Y, Z,, k,, and k , are constants determined by the nature of the chemical reaction ticheme. N o and A;R represent the nurr.ber of chemical reactions coupled with species 0 and R, respectively. TECHNIQUES 1 AND 4. Theoretical equations for the alternating current with controlled alternating potential techniques and surface concentrations
and
G(t) For reasons which will become apparent, it is useful to consider two special cases. When chemical reactions and charge transfer are so rapid that they exert only thermodynamic influence on the d.c. wave (applies to reversible chemical reactions not including the catalytic case), Equation 46 reduces to
=
(61) For a system with a single first-order chemical reaction preceding charge transfer VOL 36, NO. 6, MAY 1964
965
+ ne
ki
Yg0 kt
charge transfer step simultaneously operative at a given d.c potential (21). Among the systems which have been considered are
$R
+
G(t> = G(t), = (1 e-3 X {aQ,,(t)D1/z + e j Jot e-kcuQe(t khe(TU)I/’
one hss (ZO)
2,
X,>l = 0
=
Y - 1
xi = 1+R
The expression for G(t)$ can be simplified further by consideration of the integral equation for the d.c. wave with a catalytic process which provides the relation
w=-l +K K where
e - k$uQ,(t = ki/kz
Finally, for p:= 1
k, =
h =
- u)du -
+ kz
and g,fg
+ nle * R1 + %e e RZ
and, upon substitution of Equation 81 in Equation 80, one obtains
+
ki b w
Thus
v = v , = - a+-{&+ (1
h
+ e5)
TECHNIQUES 2 AND 3. Theoretical equations for the controlled alternating current techniques and surface concentrations corresponding to Equations 41 and 42 are obtained also by carrying out the derivation as for the quasi-reversible cme. The final results are
O1
+ nle
ki
$ R1
wt
(7 1)
G(t) = G(t), =
For a system with a catalytic chemical reaction
O+neeR b
(73)
I
one has (20)
w = Y = x,>1= xi = For p
=
21
1 and m
k,
=
Zm>l = 0 (74)
= 1
=
(75)
1
k , = k,
(76) (77)
Thus
v = v c = -a+ x
966
ANALYTICAL CHEMISTRY
- cot-’!)
S
(83)
where V, S, and G(t) are given by Equations 44, 45, and 46. Therefore, the faradaic impedance corresponding to Equation 83 is given by Equations 49, 50, and 51. For a system with a first-order preceding chemical reaction, Equations 70, 71, and 72 apply. For a catalytic wave, Equations 78, 79, and 82 are used. For special cases of very rapid or very slow chemical reactions combined with rapid charge transfer, Equations 56 and 61 apply. These equations for systems with coupled first-order chemical reactions have been derived without specifying the mode of control or form of the d.c. variable, except that steady state in the a x . variable is obtainable under existing experimental conditions. Reilley and Ashley (18) have recently formulated an alternative generalized theoretical approach to systems with coupled first-order chemical reactions applicable to the experimental techniques in question. Our application of their approach to the systems considered above yields the same results. Systems with Multistep Charge Transfer Processes. Calculations have been carried out for a number of mechanisms involving more than one
e k2 Rz
+ nze
$ Ra
(86)
Derivation and detailed discussion of theoretical equations for these mechanisms are beyond the scope of the present work. However, one interesting dserence between this type of system and those considered above is worth indicating. The source of this difference can be illustrated simply by considering the somewhat trivial case of the faradaic impedance when two chemically and electrochemically independent charge transfer reactions occur a t the same d.c. potential with neither system influenced by chemical reaction or adsorption-that is, when the quasireversible reactions 0,
+ nle e R1
(87)
+ nze + Rz
(88)
and
O2
t
(84)
(80) 1
R
0
are contributing simultaneously to the faradaic impedance (unresolved 8.0. polarographic waves). For reasons which will become apparent, only the phase angle relations are considered. The faradaic impedance can be calculated for this system by considering the two electrode reactions as two independent faradaic impedances, 21 and Zz, in parallel, each consisting of a series resistance and capacitance obeying Equations 31 and 32. For such an impedance, the phase angle of the alternating current relative to alternating potential, with either controlled alternating current or potential, is given by
where
XI
=
1
-
wc1
(90)
R1, CI and Rz, C2 represent the series resistance and capacitance comprising impedances Z1 and Z2. Substituting Equations 30, 31, and 32 in Equation 89, with subscripts 1 and 2 designating to which charge transfer process the various parameters correspond, one obtains after some rearrangement
6 - ( + q9[1+(1+_\_)L]+
nlZCo1*D,,11~2Fl(t)cosh2 @ 2 1
m?C,2*D,21/2F2(t) The significance of this result is examined below. While use of the parallel impedance concept is both convenient and correct, Equation 92 (ran be obtained without its introduction. For example, with controlled alternating potential techniques, simple addition of the ax. current expressions for two independent quasi-reversible systenls (Equation 28), followed by minor 1,rigonometric rearrangement effects derivation of Equation 92. For controlled alternating current techniques, the derivation proves more complicated, but still yields Equation 92 (91). Other Types of Systems. We haye not discussed the influence on the faradaic impedance of absorption effects and second or higher order coupled chemical reactions. Very little theoretical work on these types of systems is to be found in the literature for experimental conditions corresponding to net d,c. current flow. Thus, regarding adsorption effects, only some brief observationn are given below. For systems with second or higher order chemical kinetic effects, we have developed recently a method for approximately cslculatillg the faradaic impedance under conditions of net d.c. current flow (21). Because it is beyond the scope of the present discussion, derivation and resulting theoretical equations will be pre6,ented elsewhere. Some results of significance in the present discussion are mentioned below. PREDICTIONS 01: THEORY
To make meaningful comparisons of predicted faradaic impedances between any two techniques, i; is necessary to consider the faradaic impedance a t the same direct potential and frequency. Thus, all statements below regarding differences in the faritdaic impedance imply a comparison ~ n d e rthese conditions. Perhaps one of the most apparent and important general conclusions indicated by the above theoretical considerations iq that for any electrode reaction considered, the theoretical expressions for the famdaic impedance are identical in form for a11 four experiment techniques. Since the form of the direct current has not been specified, this does not suggest an identical faradaic impedance for all
techniques with a specific electrode process. Rather, one concludes that a difference in faradaic impedance may be observed only between techniques involving different forms of the direct current function. The faradaic impedance is predicted to be independent of the mode of control of the a x variable and any differences in faradaic impedances between techniques must be due to differences in the mode of control of the d.c. variable. Controlled d.c. potential techniques (Techniques 1 and 3) will always show identical faradaic impedances with a given redox system (among the systems considered). The same applies to the controlled direct current techniques (Techniques 2 and 4), if the direct current densities are the same. However, the faradaic impedance observed with Techniques 1 and 3 may differ from that observed with Techniques 2 and 4 because of the different mode of control of the d.c. variables. Further examination of the theoretical equations indicates that a difference in direct current is a necessary but not sufficient condition for the observation of a difference in faradaic impedances between techniques. This is apparent from inspection of the equations for the reversible case (Equations 4 and 5 ) , which prove t o be independent of the direct current density. All techniques under consideration are predicted to show the same faradaic impedance with a reversible electrode process. The quasi-reversible system, on the other hand, shows a term dependent on direct current: specifically, the Qo(t) term in F(t) (Equation 29). It is possible then that techniques employing different direct current functions-e.g., a.c. polarography and as. chronopotentiometry-will show a different faradaic impedance with a quasireversible system. However, under conditions corresponding to n reversible d.c. process,
and
F(t) = 1 (94) Thus, the term containing the direct current is negligible. With a quasi-reversible process it is concluded that a difference in faradaic
impedance will be observed only between techniques involving different modes of control of the d.c. variable and only if the d.c. process is not Nernstian (reversible). The above conditions do not indicate a reversible a.c. wave, the condition for this being that &/X
