both the chromatographic column and the thin-layer electrode between two glass plates. Such a system could easily separate and detect submilligram quantities of material. In addition, a flow-through cell containing two thin workingelectrode bands in succession could be constructed as a sensitive analog to the ring-disk electrode. Product produced at the first would be electroanalytically determined at the second. Since a wide variety of metal surfaces can be thermally deposited, it would be possible to use such surfaces to generate metal cations for a coulometric titration (or for other purposes) within a twin-electrode cell. In addition, the second electrode in a twin-electrode configuration could be incorporated as a reference electrode-e.g., Ag/AgCl. The determination of the oxygen content of both aqueous and nonaqueous solvents utilizing this cell configuration should have immediate practical applications. The independence of thin-layer coulometry from diffusion control (and thus diffusion coefficients) adds the element of temperature independence to such analyses. Combinations of minigrid and thin-metal-film electrodes for use in bulk solution electrochemistry have come to mind. One basic configuration would be a glass plate incorporating the metal film as one working electrode ( W , ) and a minigrid ( W Z )situated (using epoxy and Teflon spacers) in the neighborhood of 100 microns from the film. Using this design, the advantages of thin-layer steady-state methods (18-22) could be brought to bulk solution electrochemistry. A steady-state concentration profile (and thus current) could be established between the grid and metal film for a species present in the bulk solution. The magnitude of the steady-
state current would be directly related to this bulk solution concentration. The electrochemical generation of free radicals for electron spin resonance studies has been reviewed by Adams (26) and Poole (27). The incorporation of a TFT’LE into a cell for this purpose would have several advantages. The exhaustive nature of thin-layer electrolysis would preclude the line broadening attributed to parent-radical exchange. Shortlived species could be rapidly regenerated to a steady-state concentration level in such a design without the necessity of the inefficient flow systems now used. The thin geometry would mitigate the dielectric loss associated with the supporting medium. ACKNOWLEDGMENT The authors thank E. N. Mitchell, J. Zunes, and P. Smejtek of the University of North Carolina Physics Department for technical advice and loan of the vacuum deposition apparatus. Helpful discussions with R. W. Murray and W. R. Heineman are acknowledged.
RECEIVED for review December 1, 1967. Accepted March 20, 1968. Research supported by the Directorate of Chemical Sciences, Air Force Office of Scientific Research, Grant AF-AFOSR-584-66 and by the Advanced Research Projects Agency. (26) R. N. Adams, J . Electroanal. Chem., 8, 151 (1964). (27) C. D. Poole, “Experimental Techniques in Electron Spin Resonance,” Interscience Publishers, Wiley, New York, 1967, p 620.
Pulse Response of an Electrode Reaction Keith B. Oldham Science Center of the North American RockweN Corporation, Thousand O a k s , Calif. 91360
DERIVATION IN TERMS OF CONCENTRATION PROFILE
General expressions are derived for the faradaic current which flows following the imposition of a constant potential on a planar electrode at which an electrochemical reaction can occur. A less exact treatment of the dropping electrode case is included. No assumption is made about the conditions prior to the pulse. The response is related both to the concentration profiles at the instant of pulse application and to the electrical conditions existing at that instant. The results obtained apply to reversible or irreversible electrode reactions. The implications of the theory to the analysis of nonuniform solutions are discussed and a new method is proposed for the determination of kinetic parameters.
may occur. At the instant t = 7 the potential pulse is imposed. The third and final period embraces all times thereafter, but prime concern will be for current behavior in the period r < t < r T where T is small in comparison with r . The initial absence of an electrode reaction may arise from a variety of causes, the simplest being the electrical isolation
SEVERAL ELECTROANALYTICAL methods-square-wave polarography (I), pulse polarography (2,3)-employ the concept of a potential-pulse, as do a number of techniques-potentialstep method (4,voltage-step method (5),pulse-polarographic method (6, 7)-for the measurement of electrode kinetic parameters. By a potential-pulse is meant the sudden imposition of a constant potential to an electrode. The response to this pulse, that is the current passing at times soon after pulse application, reflects the conditions existing at the instant of application. This article seeks to establish the response for any arbitrary condition of concentration or of electrical parameters immediately prior to the pulse.
(1) G. C. Barker, R. L. Faircloth, and A. W. Gardner, Report C/R 1786, Atomic Energy Research Establishment, Harwell, England, 1955. (2) G. C. Barker and A. W. Gardner, Z . Anal. Chem., 173, 78 (1960). (3) E. P. Parry and R. A. Osteryoung, ANAL.CHEM.,37, 1634 (1965). (4) . . H. Gerischer and W. Vielstich, 2.Physik. Chem., (Frankfurt am Main) 3, 16 (1955). (.5.) W. Vielstich and P. Delahav, _ .J . Amer. Chem. Soc.,. 79,. 1874 (1957). (6) . _J. H. Christie. E. P. Parry, .. and R. A. Osteryoung, . - Electrochim. Acta, 11, 1525 (1966). (7) K. B. Oldham and E. P. Parry, ANAL.CHEM.,40,65 (1968).
1024
ANALYTICAL CHEMISTRY
Three distinct periods of time will be considered. Initially, for t < 0, there is no electrode reaction whatsoever. A period 0 < t q r then ensues in which the electrode reaction Ox(so1n)
+ ne-(M)
Rd(so1n)
(1)
+
of the electrode. Irrespective of the cause, the effect of the absence of reaction during t < 0 is to produce uniform solute concentrations throughout the semi-infinite system. For the electroactive species this uniformity is expressed by the equations C(r
2
0, 0) =
?
(2)
and C’(*r
2
01
0,
=
F
(3)
where C(r, t ) and C’(r, t ) will denote the concentrations of Ox and Rd at a distance r from the electrode surface at time t ; C and are bulk concentrations. The alternative negative sign attaching to r in Equation 3 is included to embrace the possibility that Rd is an amalgam-forming metal and that M is Hg. During the period 0 < t < T no electrode reaction other than Reaction 1 occurs but no stipulation is made concerning whether this reaction occurs throughout the whole (or indeed any) of the period, whether it occurs reversibly or irreversibly, and whether it occurs galvanostatically, potentiostatically, or in neither of these simple modes. It is stipulated, however, that any transport of Ox and Rd which occurs is by virtue of linear diffusion. The following equations accordingly apply by virtue of Fick’s and Faraday’s laws:
-
c
a2C D - (r 2 0, t ) ar2
a
D ’ - ( f r 2 0, t ) ar2
(4)
2 0, t )
(5)
aC nFD - (0, t ) ar
(6)
aC’ FnFD‘(0, t ) ar
(7)
Z(t) =
Z(t) =
aC’ at
= -( h r
required for a completely reversible reaction. The other extreme condition,
reduces Equation 11 to
ac
D - (0, t ) ar
5 (r 2 0, t ) at
=
be satisfied. The definitions of K(t), ii(r) and of a third function J(t) are included in Appendix C : the magnitude of each of three depends on E(t). The physical significance of K(t) is that this is the surface Ox concentration which corresponds to thermodynamic equilibrium at potential E(t); it therefore represents the surface concentration that would ultimately be achieved by Ox if the potential was maintained-though not necessarily in an experimentally significant time period. The significance attaching to A(t) is that of a parameter expressing the relative efficacies of electrode kinetics and diffusion as a means of changing the surface concentrations of Ox and Rd. Note that Equation 11 is a completely general boundary condition. It reduces in the limit A -+ a, to the Nernst relation
=
nF k,C(O, t ) exp {-a - [E(t)- E,]} RT
which is the boundary condition appropriate to a totally irreversible electrode reaction. To learn the response to the pulse, we need to solve Differential Equation 4 subject to Boundary Condition 11 and to a prescribed initial condition. To maintain complete generality we choose Expansion 9 as the initial condition, together with the restriction
c(m,r)
(see Appendix C for a complete list of symbol definitions). Appendix A presents a simple theorem showing how there is a direct one-to-one relationship between the magnitudes of C(r, t ) and C’(r, t). Applying the theorem: if the surface concentration of Ox is C(0, t ) , that of Rd must be:
(14)
=
C
(15 )
which is a consequence of the original ( t < 0) concentration uniformity. Equivalently, Differential Equation 5 could be solved subject to Conditions 10, 11, and to ~ ’ ( w t,) =
??
(16)
Appendix B shows how this set of equations may be made to yield the following expression for the current at a time T after imposition of the pulse:
Similarly, if the concentration profile of Ox immediately prior to the pulse is written as the MacLaurin series C(r, T - ) = C(0,
7)
+ r aC ; (0,
7-
)
+
The functions ej(Z) are defined in Appendix C, but their properties for small arguments are more readily appreciated from Figure 1. Equation 17 is exact under all circumstances. However, under convenient conditions, it may be curtailed. Notice from Figure 1 that for small values of the A d ? product @,(A&) approaches zero asymptotically for j 2 2. Hence if Thz T ) = C(0, f > T ) - K(t > 7)
A(f
> 7) ar
(11)
(18) in which the first term will be recognized as the current density Z(7-) immediately before pulse imposition. VOL. 40, NO. 7, JUNE 1968
1025
\
I
1
Tm 0 Figure 2. Plots of Z(t) us. time and us. &for in potential at t = 7
Z-
Figure 1. The functions edz), e,(z),M z ) , e& for z values up to 1.4
e&), e&),
and
The trend in e&) established for j’ equal to 2 through 5 is continued for j 2 6
Conversely, if TA2 >> 1 alternative approximations are possible as apparent in Appendix C, leading to the result
The curves marked a, b, and c refer to ,8 >
Q(T
+ T) -
T+T Q(7)
I(?) dt
=
=
._
(8) J. H. Christie, G. Lauer, R. A. Osteryoung, and F. C. Anson, ANAL.CHEM., 35,1979 (1963).
1026
0
ANALYTICAL CHEMISTRY
3, and B < 4
@,41
(21)
may be derived. This shows that the response after time T depends mainly on what the concentration was at a distance 4 d D T / a from the electrode at the instant of application. Equation 21 could be used to map the concentration profile at t = 7 from observations of the subsequent current density. A more general approach to this problem is discussed in the next section. PROFILE SYNTHESIS Equation 17 shows that the current density at times t > 7 depends on all the MacLaurin coefficients a’C(0, ~-)>ldfi. It is in principle possible to determine all these coefficients by a suitable analysis of the current-time dependence and, hence, using Equations 9 and 10 to reconstruct the concentration profiles C(r,r-) and C’(r, 7-) which existed at the instant of pulse application. Using Equation 17, it may be shown that the fist few MacLaurin coefficients are given by
ANALYTICAL IMPLICATIONS Since the work of Cottrell, it has been known that the current (or charge) passed by an electrode immersed in a solution uniformly concentrated in a reducible solute, following the imposition of a very negative potential, is proportional to the concentration of the oxidant. The foregoing theory permits an answer to the question of to what extent the current following a similar pulse in a nonuniform solution reflects the surface concentration of the oxidant. A very negative potential results in a large A value: Equation 19 is therefore appropriate. Likewise a very negative potential implies that K is negligible. Inserting this condi-
=
tion into Equation 19 and making approximations which introduce serious errors only if ajC(0, 7-)/a# is important whenj > 4, the relation
c (4
These equations show that the current following pulse application is initially determined highly by C(0, 7 - ) and by aC(0, -r-)/ar, but that with time control passes increasingly to the higher derivatives a’C(0, T-)/arJ’. As Equation 18 shows, the faradaic current density jump Z(T+) - Z(T-) is finite. However, dZ(.r+)/dt may be shown to be infinite, which is an experimental inconvenience. For this and other reasons, the measurement of pulse response in terms of the change in charge density Q(7fT) - Q ( 7 ) rather than the current density has been advocated (8). The simplicity of Equation C9 (see Appendix C) permits a ready integration of Equation 17 to
3, ,8
a step-charge
and that the general relation for j
2 1 is
These expressions are exact and in principle enable the entire concentration profile to be reconstructed. In practice, it is likely that the first few MacLaurin coefficients only will be determinable with adequate precision.
late azC(O, s - ) / a r 2 and a3C(0, 7 - ) / a r 3 , the following procedures are adopted :
KINETIC IMPLICATIONS
Equation 23 suggests a method of measuring the kinetics of fast electrode reactions. Rearranging this equation and invoking the relation I(.-) = nFDaC(0, .-)/&, the expression and
is obtained where AZis the change in current density at t = T . By choosing conditions during 0 < t < T such that I(.-) differs from Z(T+) only fractionally, a large A value may be measured from a trace of Z(t) without a large d Z ( T + ) / d d x Notice that no absolute measurement of current is needed and that, because the change in potential at pulse imposition will not generally be large, interference from nonfaradaic current will be small. Note also that Equation 26 is valid irrespective of the history of the reaction during 0 < t < 7. From an experimental standpoint, the simplest history involves potentiostatting the electrode at E(.-) during the entire interval 0 < t < 7: this would give rise to a current density versus time relationship akin to that shown in Figure 2. The ease with which d Z ( . + ) / d d r can be measured from experimental data will depend largely on the magnitude of d * Z ( v + ) / [ d d F If this second derivative has a zero, or other known value, the determination of d Z ( T + ) / d d r is greatly facilitated: this point will be taken up later in the article. RESPONSE IN TERMS OF ELECTRICAL PARAMETERS
Any of the equations numbered 17 through 20 relate the behavior of the current (or charge) density following the pulse to the concentration profile existing at the instant of pulse application. Another interesting problem, which arises for example in pulse polarography (9), is to relate the pulse response to the electrical parameters pertaining immediately before the pulse. In pulse polarography the electrode potential before the pulse has a constant value, so that the only electrical parameters are Z ( T - ) , dZ(.r-)/dt, dzZ(.-)/dt2, etc. In the general case, however, the parameters E(.-), dE(T-)/dt, dZE(7-)/dt2, etc. must also be considered. In general, the MacLaurin coefficients dlC(0, .-)/a# will become less important as j increases. Moreover, especially for small values of T, ej (l/% diminishes in magnitude as j increases. Together, these factors will ensure a rapid convergence of the series in Equation 17, particularly if 7 >> T. We shall henceforth assume, rather arbitrarily, that the terms in Equation 17 corresponding to 4 2 j 2 m ,can be ignored. To this extent the ensuing derivation is approximate. The relation
-v’ii _ ac (0, t ) = C(0, A(t) ar
t)
- K(t) = n F dm) h(t)
a3C a 2C D 7 (0, t ) = - (0, t ) = ar
atar
1 dZ - (t) nFD dt
~
(29)
Inserting from these last three equations into Equation 17 and neglecting terms corresponding t o j 2 4, the following expression may be obtained after much algebra but without further approximation.
Z(7
+ Y) =
Z(7-)
+ T d-dtZ
(7-1
+
+
+
dJ dA (7-) [(T-) J(T-) ])A? 2 (7-1 - A(.+) dt N7-1 This rather lengthy expression becomes curtailed if the electrode reaction is reversible, if the electrode is totally irreversible, if the electrode is held at a constant potential immediately prior to t = 7, or if the current is maintained constant immediately prior to the pulse. Reversible Case. Reversibility is characterized by a large value of k,, and hence of A(.-) and A(.+). Letting these terms approach infinity in Equation 30 and noting that eo(A.\/?>then approaches [h(~+)dZl-l, we find
Z(7
+ T) - Z(7-)
{
=
dZ T - (7-) dt
+ K ( T - ) dA
- (7-)] A(r-) dt
(31)
The second right-hand term is analogous to the Cottrell expression: a similar term is the basis of square-wave polarography ( I ) . The first and third terms respectively represent the effects on the pulse response of a charging current or potential at the time of pulse application. Irreversible Case. For a totally irreversible electrode reaction, the oxidation of Rd can be discounted at all potentials at which the reduction of Ox proceeds to a significant extent. This means that J(t) in Equation 30 may be set equal to zero in this case; the equation
(27)
which will be recognized as a generalization of Boundary Condition 11, permits C(0, 7 - ) and aC(0, T-)/ar to be related to the electrical parameters Z(t), A(t), and J(t). To similarly re~~
~~
(9) K. B. Oldham and E. P. Parry, unpublished data, 1968. VOL 40, NO. 7, JUNE 1968
1027
then remains. Again, the first and third right-hand terms will generally be small perturbations arising from nonuniform conditions of current and/or potential at the instant of pulse application. If these perturbations are negligible,
Z(7
+ T) - Z(7-)
=
and the pulse response is directly proportional to the current density extant prior to the pulse. Equation 33 forms the basis of pulse polarographic theory for irreversible electrode processes (9). Prepotentiostatted Case. If the potential is constant at t = 7 - , then dA(T-)/dt and d1(7-))ldt are both zero. Equation 30 then reduces to
is greater than or less than */2. Several of the terms composing b, particularly 7 , are freely available to the experimenter and therefore it will be possible to iteratively adjust b to arrive at any of the circumstances indicated as a, 6, or c in Figure 2. Not only will this aid in delineating an accurate A value, but it will also provide a powerful check on the validity of the method. APPLICATION TO DROPPING ELECTRODE
If the electrode is an expanding sphere, the equations developed above, based as they are on linear diffusion to a planar electrode of constant area, are inapplicable. Nevertheless, if 7 represents a time comparable to (but of course less than) the natural drop life of a dropping mercury electrode and T is small in comparison thereto, Equations 30 through 35 will not be greatly in error. In this section we seek to amend these equations to reduce the error. In the time interval 7 < t < t T, the electrode area increases by a factor p from A(7) to A(7 r). For a dropping electrode this increase occurs gradually throughout the interval. Here, however, we shall replace the gradual area increase by one of similar magnitude, but occurring instantaneously. First we will consider the effect of such an increase occurring at t = v , Then its effect at time t = T r will be deduced. Finally, we shall assume that the effect of a gradual area increase is the mean of the two instantaneous effects. Such an approximation will be especially accurate if the T/r ratio is small. The effect of an increase in area from A(T =) to A(T-), where T = and 7- are adjacent instants, may be determined by recognizing that a sudden area increase will maintain constant the volume enclosed by the electrode surface and any surface of constant Ox concentration. Symbolically:
+
Note that for Equation 34 to be valid it is not necessary for E(t) to be constant during the whole internal 0 < t < 7 , but only that dE(T-)/dt be zero. Pregalvanostatted Case. Here dZ ( ~ - ) / d tmay be set to zero, whereby Equation 30 contracts to
Z(7 + T ) - I(.-)
=
(35)
+
+
It follows that the surface concentration is unaffected by the sudden expansion, but that
FURTHER KINETIC CONSIDERATIONS
As stated above, a simple experimental situation is one in which the potential is maintained at a constant value E(.-) during the interval 0 < r < 7 and at a slightly different value E(T+) during t > T . Equation 34 applies in these circumstances and from it
Accordingly, Equation 17 must be amended to
(36)
may be derived. Now, if Reaction 1 is fast and 7 is of the order of seconds, the current at times somewhat less than 7 will follow a Cottrell t-lI2 law. Hence, to a good approximaT Equation tion, dZ(7-)/dt may be replaced by - I ( T - ) ) / ~ and 36 by d2Z [ d d r l'
(T+)
= 2A'AZ
-) - AZ(7 TA(T-)
and Equation 30 to
(37)
The sign of the second derivative is seen, to depend on whether the quantity
A = } 1028
ANALYTICAL CHEMISTRY
+ $(.=)- dt
Z(7=)
+
J ( 7 ' )
A(r=)
The last two equations permit a calculation of to what extent the electrode surface expansion has resulted in an increase in current density. The current itself is incremented by an additional factor p resulting directly from the increased area. The derivation for the case in which the sudden expansion T- and occurs ,between the consecutive instants t = 7 t =7 T+ is simpler. Arguing as before
(Ea, t ) = C(a, t ) where a is any arbitrary length. Equations such as 8 and 10 then follow straightforwardly.
+
+
aC P(0,7+T-) ar
i
~
+
r-)+
?E!
li
sL(r, s) - C(r,
7-)
A(7->
(44)
dr
(0, s) = L(0, s )
-
K
S
and L( Q), s) =
qs
where all symbols are fully defined in Appendix C. Equations 9 and B1 define a second-order differential equation, the solution to which may be shown by the variation of parameters method (10) to be
-
Letting r and considering Equation B3, it is evident that 3 0. The other arbitrary function, Pl(s), in Equation B4 may be resolved by combining with Equation B2. After much algebra and the reinsertion of the Pl(s) expression into Equation B4, the result 9(s)
J(7-)}]
dt
=
(43)
+
2 wu)e2(Ad% Li(7f)A(7-)
The Laplace transformations of Equations 4, 11, and 15 yield, respectively: d2L D 2-dr (r, s)
and, since Z ( t ) is proportional to X ( 0 , I)/&, the current density following the expansion is just p times that immediately bebefore. Equations 17 and 30 require modifying only by the replacement of Z(r T) by Z(T T+) and multiplication of their right-hand sides by I.(. Assuming that the effect of a gradual area increase is the arithmetic mean of the effects of sudden increases at t = 7 and at t = 7+T, we find that amended Equation 30 now reads
+
APPENDIX B
Q)
Simplifying assumptions may be made to this equation to produce analogues of Equations 31, 32, 34, and 35; this will not be done here, however. Equation 44 forms the basis of a theoretical treatment of derivative pulse polarography (9). APPENDIX A
The definitions -
r
w
=
* r i $
is obtained. This equation may now be inverse Laplace transformed to yield a straightforward, though lengthy, general expression for C(r, t > T), the concentration profile at any instant after the pulse. For the present purpose, however, it suffices to inverse transform the r-derivative of Equation B5 at r = 0. The latter is obtained as the compact expression
enable Equations 3, 5, and 7 to be recast as
-
a2 aZ: D C(7,t ) = - ( r , t ) ai
at
dL - (0, s) dr
=
(‘44)
These last three relationships should be compared with Equations 2, 4, and 6 : the complete equivalence of the two sets will be apparent. Thus with 7 and t as independent variables, 2 obeys exactly the same laws as does C in terms of r and t. Therefore
Upon inversion aC(0, t > 7)/ar is obtained, which on multiplication by nFD gives Equation 17 for the current density at all times following pulse imposition. (10) I. S.Sokolnikoff and E. S. Sokolnikoff, “Higher Mathematics for Engineers and Physicists,” McGraw Hill, New York, 1941, p 318. VOL. 40,
NO. 7, JUNE 1968
1029
APPENDIX C
t
m)
Notation =
A(t) C(r, t )
electrode area at time t (cm2)
= concentration of Ox at a distance r from
the electrode surface at time t(mo1e cm-9 C’(fr, t ) = concentration of Rd at a distance r from the electrode (measured into or away from the solution phase) at time t (mole cm-3) - C; C’ = bulk concentrations of Ox and Rd (mole, cm-3) E(?, r ) = concentration defined by equation (A2) (mole cm-3) D; D’ = diffusion coefficients of Ox and Rd (cm2 sec-l) EO) = potential of the electrode at time t , with respect to any convenient reference (volt) Eh = a potential defined in Equation C1 and equal, in the case of a reversible reaction, to the polarographic half-wave potential (volt) Es = the standard potential of the Ox/Rd couple with respect to the chosen reference (volt) F = Faraday’s constant, 9.65 X l o 4 coulomb equivI(t) = the faradaic current density at time t (ampere cathodic current being considered positive I(T-); = current densities immediately before and immediately after pulse imposition (amI(T+) pere c m 2 ) dI = value of drjdt when t = T (ampere -- (7) dt sec-I) J(t); K(t)
= the charge density
= the gas constant, 8.31 joule deg-l mole-’ =
2; z
arbitrary variables an arbitrary length (cm) = an electron = running indices = standard heterogeneous rate constant (cm sec-l) = number of faradays to reduce one mole of Ox (equiv mole-’) = distance measured normally from the electrode surface (cm), being positive for a point in the solution phase. = distance variable defined in Equation A1 (cm)
i; j ks n r
r
1030
T
T
[t
- 71
potential-dependent function defined by Equation C3 below (sec-’i2) = abbreviation for A(T+) or A(t 2 7)(sec-1/2) = dimensionless cathodic transfer coefficient = dimensionless parameter defined in Equation 38 = time at which the pulse is applied (sec) = a sequence of three consecutive times negligibly separated. The pulse is applied between instants t = 7 - and t = T+. In the theory dealing with dropping electrodes, a hypothetical expansion is considered to occur between instants t=T= and r = T (sec). 7 + T and T+T+ are similar = a factor by which area increases, p =
A CY
P 7-
7-;
T=;
T+
I.L
A(T
a5c(0,T - ) ar5
f T)/A(T)
= the j f h partial derivative of the Ox con-
centration with respect to distance, measured at the electrode surface immediately prior to the imposition of the pulse (mole cm-3-9
Definitions
Eh = E,
K(r)
RT D’ + 2nF - In D
[ + {1
[E(t) - Eh]>]-’
exp
[
+
dg]
I(r)dt (coulomb cm-2)
Rd T e-
e,(Z)
sd
R
a
AE; AI
= functions of E(t) defined in Equations C4
and C2 below (ampere c r r 2 and mole cm-3, respectively) K = an abbreviation for K(T+) or K(r 2 7) (mole ~ m - ~ ) L(r, s) = the Laplace transform of C(r,t) with respect to [ t - T ] (mole sec cm-3) M = the electronically-conducting substance of which the electrode is composed ox = an electroreducible solute Pl(s); Pz(s) = arbitrary functions of s (mole sec cm-3)
QW
dummy variable of Laplace transformation with respect to [ t - 71 (sec-I) = time from onset of experiment (sec) = gamma function (11) of z = abbreviations for [E(T+) - E(T-)] and for [ I ( T + ) - I ( T - ) ] (volt; ampere c r r 2 ) = the set of functions discussed below = time measured from onset of pulse (sec); =
S
the soluble reduction product of Ox
= absolute temperature (deg) = =
ANALYTICAL CHEMISTRY
=nFks[[c{$}
=/ 2
+ 1 .-n
The Fuctions e,(z) These are defined by (11) M. Abrarnowitz and I. A. Stegun, Eds., “Handbook of Mathematical Functions,” National Bureau of Standards, U. S. Government Printing Office, Washington, D. C . , 1964, p 253.
e,(z>
(-1'
(--& -___
r(1
+ ;)
Li = 0 , 1 , 2
. . .I
(C5)
Z-a e-l(z) = 2 6 are useful. The relation d
The first member of the series may be defined alternative1 by
~
4 2 2>
e,(z)
= exp(zz)erfc(z) = - l m e x p ( z 2 - Z z ) dZ
(C6)
(C9)
is satisfied for j 2 2 while d
G r ~
Figurp 1 shows the behavior of the functions for small argumento. For large arguments the recursion formulae
e,@>= 0,-Z(Z)
4 2
9
e&) = -
d
__ 4 2
'1
1
e&) = e, - -Z
(C10)
G
A Fortran program has been constructed enabling e,(z) to be calculated to an accuracy of 1 part in 7000 for any value
Electrochemistry of Palladium(ll) Ion in Ammonia and Pyridine Media Pulse Polarographic Determination of Palladium E. P . Parry and K. B. Oldharn Science CenterlAerospace and Systems Group, North American Rockwell Corp., Thousand Oaks, Calg. 91360 The reduction of palladous ion from both pykidine- and ammonia-containing solutions has been studied. In both media the reduction is totally irreversible, with no further complications from adsorption or homogeneous chemical kinetics indicated. Because the palladium is not reoxidized at potentials less positive than mercury dissolution, stripping (from mercury) analysis cannot be used. The effects of pH, concentrations of ligand and supporting electrolyte, and potential interferences are reported. The pulse polarographic determination of as little as 50 ng of palladium with few interferences can be made. The effect of the derivative pulse polarographic parameter of pulse amplitude on peak height and peak half width is discussed.
WILLIS (1) first described the polarographic reduction of palladium(I1) ion in various supporting electrolytes. From a cursory study he reported that palladous ion is reduced irreversibly from an ammoniacal buffer (1M ammonia, 1M ammonium chloride), but is reduced reversibly in a solution 1M in KC1 and l M i n pyridine. However, he reported that in the latter medium the half wave potential shifted with change in concentration of palladous ion, an observation not consistent with a reversible reduction of a metal ion to an amalgam. Later, Douglas and Magee (2) presented a method for the determination of palladium and rhodium using oscillographic polarography. In this work they used a pyridine medium for the palladium determination. Although no data are given, they reported that this reduction is reversible. In a resurvey of a number of other supporting electrolytes, they also concluded, from unreported data, that the reduction of palladous (1) J. B. Willis, J . Am. Chem. Soc., 67, 547 (1945). (2) W. H. Douglas and R. J, Magee, J. Electroanal. Chem., 5, 171 (1963).
ion in ammoniacal medium is reversible. In a later publication (.?), based on their previous conclusion that palladous ion is reduced reversibly in pyridine media, Magee and Douglas used the polarographic technique to determine the stability constants of the palladium(I1)-pyridine complexes. There are no reports in the literature which describe the effects of pH, supporting electrolyte concentration, ligand concentration, and presence of potentially interfering ions on the palladous ion reduction. Nor is there any discussion of the nature of the electrode reaction in a complete enough manner to permit optimum analytical application. This paper discusses the electrode reactions in both pyridine and ammonia in some detail and defines conditions which permit the pulse polarographic determination of as little as 50 ng of palladium with few interferences. EXPERIMENTAL
A pulse polarographic instrument designed in this laboratory was used ( 4 ) , modified to permit its use for chronopotentiometry (5). For display of chronopotentiometric curves a Type 564 Tektronix storage oscilloscope and Tektronix C-12 camera were used. The Kemula type hanging mercury drop electrode was obtained from Brinkmann Instruments (Westbury, N. Y.). The dropping mercury electrode had m and t values on open circuit of 2.06 mg per second and 3.8 seconds, respectively. Experiments were carried out at a room temperatiire of 23 + 2 OC. For the coulometric work a Jaissle potentiostat was used ~
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(3) R. J. Magee and W. H. Douglas, J . Electroanal. Chem., 6, 261 (1963). (4) E. P. Parry and R. A. Osteryoung, ANAL.CHEM., 37, 1634 (1965). ( 5 ) G. Lauer, H. Schlein, and R. A. Osteryoung, ibid., 35, 1789 (1963). VOL. 40, NO. 7, JUNE 1968
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