plot of d[F-]jdt os. the function on the right hand side of Equation 13 is given in Figure 2 and is seen t o be a straight line passing through the origin. Considering the wide range of fluoride concentrations employed, the straight line plot obtained can be taken as proof of the hypothesis that FeF2+ is inactive toward I- and that the fluoride complex inhibits the reaction by reducing the concentration of the free ferric ion, which is the active reactant. This is in striking contrast t o the behavior of FeC12+ and FeBr2+ as reported by Sykes (3). The slope of the plot in Figure 2 is found t o be 17.1 M-l. sec-l which can be taken as the value of kl in Equation 12 a t 25 “C and an ionic strength of 1.OM. This is consistent with the values of 86.6 and 808 at 25 “C and ionic strength of 0.09 and 0.0, respectively, obtained by Hershey and Bray (8), in that the rate constant diminishes with an increase in ionic strength, which is the expected primary salt effect for a reaction between oppositely charged ions.
Thus, the kinetic data obtained with the fluoride ionselective electrode for the reaction between Fe3+and iodide ions in the presence of fluoride ions when FeFZ+is the major complex can be reasonably interpreted on the basis that FeF2+ is not attacked by iodide ions and that the concentration of the free ferric ions reacting with the iodide ions is governed by the equilibrium existing between the free ferric ions and the fluoride complex, i.e.,
Fe3+ f I-
+
Fe2+
+
’/?
I?
(19)
RECEIVED for review June 11, 1968. Accepted August 16, 1968. Work supported by Office of Saline Water, U.S. Department of the Interior.
Alternating Current Polarography: An Extension of the General Theory for Systems with Coupled First-Order Homogeneous Chemical Reactions Thomas G. M c C o r d ’ and Donald E. Smith2 Departinent of Chemistry, Northwestern Unioersity, Ecanston, Ill. 60201
A more exact and complete treatment than previously accorded the general theory of the fundamental harmonic ac polarographic response of systems involving first-order homogeneous chemical reactions coupled to a single charge transfer step is presented. The theory is extended to include an explicit solution for the current amplitude within the framework of the expanding plane electrode model. Rate control by diffusion, a single heterogeneous charge transfer step and any number or type of coupled first-order chemical reactions is considered. The solution presented represents a rigorous treatment of the expanding plane boundary value problem by the Matsuda method which involves just two minor limitations on the magnitudes of the relevant rate parameters. It permits one to write the theoretical expression for the fundamental harmonic ac polarographic response associated with a particular mechanistic scheme in this class simply by inspection of appropriate surface concentration expressions which a r e readily available.
theoretical ac polarographic relationships for a particular mechanism can be ascertained by noting coefficients of convolution integrals in readily obtained surface concentration expressions ( I , 3). To date, this work has been based o n the stationary plane electrode model and current amplitude expressions have been formulated only for special circumstances ( I , 3). Only the phase angle expression (I-3), which is not influenced by electrode growth and geometry with the mechanistic class in question ( 3 - 9 , is of adequate rigor and scope for general application to data obtained with the dropping mercury electrode (DME). Because of the importance and ubiquitous nature of electrode reactions characterized by coupled first-order chemical reactions, we have endeavored to alleviate these shortcomings in the general theory. The results of this effort are presented here.
IT has been pointed out that the theory of the fundamental harmonic ac polarographic wave for systems characterized by homogeneous, first-order chemical reactions coupled to a single charge transfer step can be generalized to include any number arid type of first-order chemical steps (1-3). The general solution has been formulated in such a manner that the
The general theory of the ac polarographic wave for the case in question can be approached on two levels. The first involves a derivation beginning with a general formulation of the boundary value problem as described by Ashley and Reilley (2). The second uses general surface concentration expressions formulated in terms of convolution integrals as a point of departure ( I , 3). The latter method presupposes the availability of the necessary surface concentration expressions. The greater elegance of the first approach is unquestionable. Nevertheless, we have selected the second because the interpretation of the resulting “general” equation is less involved
NIH Graduate Fellow; present address, General Electric Corp., Materials and Processes Laboratory, Schenectady, N. Y . , 12305.
* To whom correspondence should be addressed.
(1) H.L.Hung, J. R. Delmastro, and D. E. Smith, J. Electroanal. Chem., 7, 1 (1964). (2) J. W. Ashley, Jr. and C . N. Reilley, ibid., 7,253 (1964). (3) D. E. Smith in “Electroanalytical Chemistry,” A. J. Bard, Ed., Vol. 1, Marcel Dekker, New York, N. Y.,1966, Chapter 1.
THEORETICAL
(4) J. R. Delmastro and D. E. Smith, J . Electround. Chem., 9, 192 (1965). (5) J. R. Delmastro and D. E. Smith, ANAL.CHEM., 38, 169 (1966). VOL. 40, NO. 13, NOVEMBER 1968
1959
and because a table of coefficients necessary to apply it to ten mechanistic schemes, including the most important, is readily available (Table V, Reference 3). Although originally based on the stationary plane model, this table is equally valid for the expanding plane model (see below). I n the event that the surface concentration expressions are unknown for a particular mechanism, they are readily derived in the appropriate form by using the expanding plane electrode model. Such a derivation is outlined in Appendix I. Although we d o not employ the approach of Ashley and Reilley, one should recognize that it will lead t o completely equivalent results in all respects if properly adapted to the expanding plane boundary value problem. Because it is based on the expanding plane model of the D M E (6), which accounts for mercury drop growth, the theoretical development given below is expected t o yield current amplitude expressions which are slightly inexact under some conditions because of neglect of drop curvature ( 4 , 5). However, the error will normally be small and its effect on conclusions based on data analysis can be made negligible if appropriate care is exercised. Widespread successful application of the expanding plane model in dc polarographic studies (6) demonstrates its quantitative merits. Other assumptions or approximations incorporated in the theory are the usual ones employed. They have been enumerated elsewhere (7). Because most of the mathematical procedures employed in the theoretical development under consideration have been described in the literature (3, 7-11), the derivation presented here will be more in the nature of an outline than a step-by-step description. A detailed derivation may be obtained from the authors on request. The surface concentration expressions for the oxidized and reduced forms (species 0 and R, respectively) obtained by solving the expanding plane boundary value problem for any mechanistic scheme involving first-order homogeneous chemical reactions coupled to a single charge transfer step have the general form (notation definitions are given below):
and
d$ 1‘
EXP [-kt(t - u ) ] u ~ ’ ~ ~ ( u ) ~ u nFADUZ(t713 - ~ 7 / 3 ) 1 / 2
appear in place of the stationary plane integrals,
S
0
and
‘ EXP (- kiu)i(t - u)du nFA(Dau)1‘2
respectively. Thus, the significance of the constants W , X d , etc. are the same in the two cases and Table V of Reference 3 is applicable to the present discussion. The method whereby such surface concentration expressions are deduced is discussed in Appendix 1. It should be mentioned that the assumption of equality of certain diffusion coefficients is required t o obtain Equations 1 and 2 (see Appendix 1). The derivation to follow will consider the most common situation where the reduced form is initially absent from the solution (CR* = 0, dc reduction waves). The case of oxidized form initially absent (dc oxidation waves) is readily treated by the same procedures as is the frequently implemented situation where both redox forms are initially present. By substituting Equations 1 and 2 in the absolute rate expression,
inserting the expression for E(t) appropriate t o ac polarography, E(t) = Ed, - A E sin ut
ksf
(6) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, N. Y., 1966, pp 77-83. 35,602 (1963). (7) D. E. Smith, ANAL.CHEM., (8) H. Matsuda and Y. Ayabe, Bid/. Chew. Soc. Jup., 28, 422 (1955). (9) H. Matsuda and Y.Ayabe, ibid., 29, 134 (1956). (IO) H. Matsuda, 2. Elekrroclzeni., 61, 489 (1957). (11) H. Matsuda, ibid., 62, 977 (1958). 1960
ANALYTICAL CHEMISTRY
(4)
setting CR* = 0 and employing step-by-step operations identical to those used by Matsuda (11) (see also Reference 3), one obtains the system of integral equations D1’2Qep(t) -
where Y , w,X d , zm,kd, k, are constants determined by the nature of the chemical reaction scheme. No and N R represent the number of chemical reactions coupled with species 0 and R, respectively. Equations 1 and 2 differ from their stationary plane counterparts ( I ) only in the form of the convolution integrals. The integrals u2/3i(u)du nFADl/2(r7!3 - ~ 7 / 3 ) 1 / 2
i(t - u)du nFA(Dnu)1/2
e--a3 aP(sin
P!
ut)p -
5
(si;;[)’
r=O
where p = 0, 1, 2, 3
...
D = DfDRa
X
f
ki
= hPfRU
gi = w
p = 1 - , The significance of a set of integral equations of this type and their solutions for particular values of p has been explained in detail (3). In particular, for small amplitudes ( A E 6 10/n mV.), the fundamental, harmonic current is given by (3) Z(wf)
=
(-)
nFACo*D,1'2
nFAE Ql(t) RT
Thus, of primary interest is the solution of Equation 5 for
These expressions are very precise except for the first few cycles following application of the alternating potential. Matsuda ( / I ) has derived Equations 17 and 18. Equations 19 and 20 can be obtained by a similar procedure. The validity of these relationships can be rationalized in a much simpler manner than that of Matsuda as shown in Appendix 2. Application of Equations 17-20, solution of the resulting system of two algebraic equations in the unknowns A ( t ) and B(t) (3),and algebraic rearrangement yields
where
As shown previously (3, 7, I I ) , such integral equations describing the small amplitude fundamental harmonic component may be solved by recognizing that Ql (1) will have the form Q i ( t ) = A ( t ) sin w t
+ B ( t ) cos w f
The expression for G,(t) may be simplified with the aid of the integral equation for p = 0 (Equation 5)
(16)
Substitution of Equation 16 in Equation 14 yields a n integral equation which is readily solved with aid of the steady-state relationships
and the relationship
which is accurate for k,t > 10. Equation 30 is rationalized in Appendix 2. Its application is equivalent t o the dc steady state assumption employed by Koutecky (12, 13) and Matsuda and Ayabe ( 9 ) t o solve the dc polarographic problem with coupled first-order chemical reactions. Equations 29 and 30 are applied to Equation 15 to eliminate the integrals in the Go(r) formulation. Following this operation by application of the basic relationship of Equation 13 and alge(12) J. Koutecky, Collect. Czech. Chem. Commirii., 20, 116 (1955). (13) J. Koutecky and J. Koryta, Electrochim Actu, 3, 318 (1961). VOL. 40, NO. 13, NOVEMBER 1968
1961
braic rearrangement yields the expression for the fundamental harmonic alternating current : I(wt) = I,,,,F(t)G(w) sin (ut
+ 4)
(31)
where Ire,.
=
n2F2ACO*(uDO)1’2AE
4RT cosh2
(i)
(32)
The factors which control the various terms in Equation 31 are readily recognized by extension of previous discussions (3, 7 ) . Briefly, the I,,, term represents the amplitude of the purely diffusion-controlled ac polarographic current (chemical reactions nonexistent). The F(t) term is responsive to effects of heterogeneous charge transfer kinetics and/or the kinetic-thermodynamic effects of the homogeneous chemical reaction o n the dc process, while C(u) and @ are responsive to the same effects on the ac process. It is readily shown that the foregoing equations reduce to the expression for the simple quasi-reversible case (3, 11) when all chemical reactions are either kinetically inoperative (ki-+ 0 for all ki’s) or thermodynamically inoperative [Ki >> 1 for all Ki’s - Ki’s are equilibrium constants defined in the usual manner (1, 3, 7 ) ] . For the case where the chemical reactions are thermodynamically inoperative one finds that
(34) Substituting Equations 40 and 41 in Equations 23, 24, 33, 35, 38, and 39 yields V=l+-The +,(t) function is simply related to the dc polarographic current function as seen by combining Equation 36 with the Qo(t) definition given by Equation 7. The explicit form of the &(r) function is obtained by solving Equation 29 by using the approximation of Equation 30. One finds that the +,(t) function is identical to the well known Matsuda-Ayabe +(E) function (8) where
(37)
For the general case under consideration
4‘=-
X k t 1’
Y
(38) (39)
and y is given by Equation 35. A proof of Equations 37-39 is given in Appendix 3. __ The formulation of Equations 31-39 and the appropriate subsidiary relationships (Equations 8-12, 23-28) comprises a closed-form theoretical expression for the fundamental harmonic ac polarographic wave for the broad mechanistic class under consideration. Within the framework of the expanding plane electrode model, the only restrictions on the kinetic parameters are that k i t > 10 for all ki and that certain diffusion coefficients are equal (see Appendix 1). As indicated earlier, these expressions are implemented for a particular mechanistic scheme by inserting appropriate expressions for W , Y, x d , Z,, k d , and km. It might appear that difficulties will arise whenever W = Y = 0 - e.g., as with the well known catalytic system ( 7 ) -because an apparent indeterminancy arises in Equation 33. However, this is not the case. Using the fact that +o(t) = 1 when W = Y = 0 and algebraic rearrangement removes the apparent indeterminant terms from Equation 33. 1962
ANALYTICAL CHEMISTRY
(2u)”2
x
U = l F(t)
=
1
+ (cue-’
(43) -
P)+W
y = l [ =
Xt”2
(44) (45) (46)
Equations 42-46 together with Equations 31, 32, and 34 represent the theoretical formulation for the quasi-reversible case, The same result is obtained by judicious insertion of the condition k -+ 0 into the general formulation. Note that this condition must be introduced into the F(r) expression by using a generally applicable form of this dc responsive function--i.e., introduce k -+ 0 into Equation 15 and follow with appropriate rearrangement. One cannot apply this condition to an F(r) expression which is based on the incompatible assumption that k t > 10 (Equation 33). The predictions of the above theoretical relationships for specific mechanisms of importance have been studied in detail in these laboratories. The results of these studies will be presented elsewhere. NOTATION DEFINITIONS
concentration of species i initial concentration of species i surface concentration of species i diffusion coefficient of species i activity coefficient of species i applied potential in European convention dc component of applied potential amplitude of applied alternating potential reversible dc polarographic half-wave potential (planar diffusion theory) instantaneous total faradaic current instantaneous fundamental harmonic faradaic alternating current phase angle of fundamental harmonic faradaic alternating current relative t o applied alternating potential
F R T
a_ co --
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 Eo = 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 = equilibrium constant for ith chemical reaction (defined so large value favors electroactive form) = dummy variable of integration = distance from electrode surface = Euler Gamma Function = number of chemical reactions coupled with species 0 = number of chemical reactions coupled with species R = =
A n
t W
k, a
ki kd km Ki
11
X
r
No NR
at
Do
azc, ac, -+ 2~ - - + k3Cy - k4C0 ax2
3t ax
(A3)
('413)
w,y, Xdr Z m , kd, k,, k+, k- = constants (combinations of ki's and Kt's dependent on nature of reaction scheme)
With the aid of the independent variable transformations z = Xt2/3
y =
APPENDIX 1
Surface Concentrations. The demonstration that the expanding plane boundary value problem can be manipulated to yield surface concentration expressions in the form of Equations 1 and 2 for mechanisms in the class under consideration requires a simple combination of mathematical techniques which have been described in the literature (1, 2 , 7-9, 14). The general approach will be outlined here by using as an example the mechanism involving a two-step preceding reaction; kl
F3
j12
ka
Z e Y*
0
+n e e R
31,
t713
('415) (A161
and the dependent variable transformations (1)
e
=
c, + c, + co
(A171
(R1)
The manner whereby the term in the differential equation, 2 ~ / 3 t ( a C / a xhas ) , been accommodated, using slight modifications of published techniques (8, 9, I d ) , and its manifestations on the resulting convolution integrals are the most important features of the following derivation outline as they are the points of distinction from the analogous treatments based on the stationary plane model ( I , 2). The expanding plane boundary value problem for Mechanism R1 is defined by the equations (14)
("1
+ kz +2k4k3 - kq - J ) C,}eL-t
(A19)
where J = ((ki
+ kz - k3 - k4)' + 4k2k3)"' (kl + kz + k3 + k4 + J ) (ki + kz + k3 + k4 - J )
(AZO)
k+ = '/z
(A21)
k- =
('4.22)
'12
together with the assumption that D, = D, = Do
('423) the boundary value problem of Equations A1-A14 may be written
(14) J. Koutecky, Collect. Czech. Chem. Commun., 19, 1093 (1954). VOL. 40, NO. 13, NOVEMBER 1968
1963
k+
(A43)
k+z = k-
(A441
k6-i
a_ CR aY
a ZCR
=
Substitution of Equations A37-A44 in the general ac polarographic wave equation given above produces the theoretical expression for the fundamental harmonic ac polarographic response with mechanism R I . The need to assume Do = D y = DZ to effect the foregoing derivation should be noted. In general, the diffusion coefficients of all components coupled by a sequence of homogeneous chemical steps (not electrochemical) must be assumed equal.
DRY az
APPENDIX 2
Do-a*+ az
(
7I‘-)?
EXP
[(F) n FA
3i7
k+] i(y) H+(Y) --
n FA
(‘436) Equations A24-A36 represent a boundary value problem which is formally identical t o the stationary plane case (1) and can be solved without difficulty by the method of Laplace Transformation (15) as shown elsewhere (7, 10, 11). A key step in the reduction of Equations A1-A14 to this more tractable formulation is the derivation of the transformations defined by Equations A17-A22. The general approach to obtaining these transformations is the same as that given for the stationary plane problem (1). Application of the Laplace Transformation method to Equations A24-A36 in the usual manner (7, 10, I ] ) , followed by implementing the inverse of the transforms of Equations A1 5-A19 and algebraic rearrangement yields expressions for Co,=o and CRz=Oidentical in form to those of Equations 1 and 2 with
Steady State Relationships. Equations 17-20 and 30 represent steady state relationships whose validity can be demonstrated by the method of Matsuda (or slight modifications thereof), which has been applied explicitly to Equations 17 and 18 (11). We wish to present here a simpler approach to deducing these relationships which will serve to provide a formal demonstration of Equations 19, 20, and 30 which has been lacking to date. Although lacking the strict mathematical rigor and elegance of the Matsuda approach, the method shown here is less cumbersome and possibly better suited to providing a n intuitive rationale for the relationships in question. Interestingly, the approach shown below is most convincing for the integrals whose steady state values have not been accorded formal demonstration (Equations 19, 20, 30). THEA C INTEGRALS (AC STEADYSTATE). Consider the integral I where
4: 1
u213A(u)sin wudu
=
(t7/3
- u7/3)1/Z
By the variable transformation u’ that I =
&s,
(t
-
=
t
- u, one can prove
u)2i3A(t- u ) sin w ( t - u)du [t713 - ( t - ~ ) 7 / 3 ] 1 i Z
(A461
To express the integral in terms of dimensionless quantities, one introduces the variable transformations
6
=
wt
(‘447)
x
= wu
(A481
which gives
Applying the trigonometric relationship sin (6
- X) = sin 6 cos x
- cos 6 sin x
(A501
one obtains
(15) R. V. Churchill, “Modern Operational Mathematics in Engineering,” McGraw-Hill, New York, N. Y., 1944.
1964
ANALYTICAL CHEMISTRY
The integrands of Equation A51 are periodic functions with amplitudes which are dependent on the variable of integra-
4
Figure 1. Plots of the functions Y(X) and r d x ) us. x
-
0 80
160-
060.
120-
Y(X).
Y;(X) -
040-
080-
A - ~ ( x )cs. x for 6 = 300 B - yl(x) cs. x for 6 = 15 and g, = 1
B
020-
040-
-
-
,
,
I
80
240
I60
2 00
X
tion. The A
(%)
component of the amplitude terms
is a very slowly changing function and can be considered constant for purposes of the ensuing argument. This is equivalent t o stating that the amplitude o€ alternating current density is a slowly changing function of time. Considering the remaining portion of the amplitude function, which we will define as y(x) where
one finds for experimentally realistic 6 values that it varies rapidly with x until x exceeds about 20, beyond which the variation with x is relatively small. This is shown in Figure 1A. F o r a finite integral to arise from the integrands of Equation 51, it is essential that the amplitude function undergo a significant change over each period of the periodic function. Clearly, this will occur only for small values of x so that the value of the integral ceases t o be a function of x--i.e., a steady state is reached-while x is still small compared to 6 which exceeds a value of about 200, even a t the lowest frequencies normally employed (10-20 Hz). Thus, the condition 6
>> x
Equation A53 also implies that A
=
A
(:)
It is well known that (16)
Thus, one obtains the final result which corresponds t o Equation 17. By a n identical procedure, Equation 18 can be verified. The other type of ac integral differs from the first by a n exponential term in the integrand. We define Il
=d?;
J'EXP[-k,(f - ~ ) ] u ~ ' ~ A ( u ) s i ~ u (AGO) (ti/3 - ui/3) 112 37r 0
By using the same operations required to convert Equation A45 t o A51, one obtains I1
3no
expanding the (6 - x ) ~term / ~ in a binomjal series (two terms only) and rearranging gives
Y(X) =
(")
Substituting Equations A56 and A57 in Equation A51, recognizing that the steady state condition allows one to equate the upper integration limit with infinity and retransforming in accord with Equation A47 yields
(A531
applies for values of x where a finite contribution to the integral is realized. Rearranging Equation A52 to
6 00
400
x
(A59
p
3 (76 - 4
4
C'
*
I
The condition of Equation A53 not only justifies the truncated binomial expansion used in Equation A55, but allows one to further simplify Equation A55 to /
1
\
112
(16) C. D. Hodgman, Ed., "Handbook of Chemistry and Physics," 41st ed., Chemical Rubber Publishing Co., Cleveland, Ohio, p 275, 1959. VOL. 40, NO. 13, NOVEMBER 1968
e
1965
In this case the x-dependent portion of the amplitude term in the integrand is the function yl(x) where
is reasonably accurate for x values which contribute most to the steady state magnitude of the integral. Thus, the manipulations of Equations A54-A57 may be employed which yield
Normally, ri(x) not only will cease to vary rapidly with X, but unlike the y(x) function, it also will become vanishingly small long before x becomes comparable to 6--i.e., the integrand will become negligible. This is illustrated in Figure 1B for g = 1. Thus, except for the trivial case where g = 0 [yl(x) = y(x)l, the steady state condition and Equations A53-A57 will be even more precisely obeyed for the integral Il than for I. Applying Equations A56, A57, etc. to Equation A61 yields
('472) The integral of Equation A72 is equal to xl/* (16) so that the final result is Equation 30. APPENDIX 3
Solutiori of the dc Integral Equation. The solution of Equation 29 is related to the dc polarographic current by the expression idc(r) = nFACo* D o 1 i 2 Q o ( t )
('473)
The Qo(t) function is obtained in the following manner. Substituting Equation 30 in Equation 29 and rearranging yields
Introducing the identities (17)
yields Equation 19. Equation 20 is developed by a n identical procedure. THE DC INTEGRAL (DC STEADYSTATE). The integral
Defining the quantities h K ,y and $o(t) (see Equations 35, 36, and 39) and substituting gives
can be shown to be identical to Introducing the variable substitution
by using the variable substitution u' substitutions 6 2 = kit
=
t - u. The additional
('477)
('468) gives the integral equation
The integrand of I z is formally identical to the amplitude component of the integrand of Il for gi = 1. Thus, Figure 1B depicts the characteristics of the I2 integrand. Clearly the integrand becomes vanishingly small and I z reaches a steady state value for X ? > 2 (ca.). If > 10 ( k t > 10) the condition 62 > > XZ ('471) (17) A. Erdelyi, Ed., "Tables of Integral Transforms," Vol. 1, McGraw-Hill, New York, N. Y.1954.
1966
0
ANALYTICAL CHEMISTRY
Equation A78 was solved by Matsuda and Ayabe (8) who showed its solution is given by Equation 37. In addition to representing an important part in the development of the general ac polarographic theory, the foregoing steps also provide a general expression for the dc polarographic wave for the mechanistic class in question. Combining Equations 35-39 and Equation A73 produces the dc wave expression. RECEIVEDfor review June 28, 1968. Accepted August 13, 1968. Work was supported by National Science Foundation Grants GP-5778 and GP-7985.
