1538
J. Phys. Chem. 1991, 95, 7538-1542
dependence on the choice of r, and z . Improper choice of these parameters leads to a good quantitative fit only over a limited range of ion masses. Changing ro and z directly affects the calculated value of rmrand through it of to, V(r),and Q,. The sensitivity of the fitting procedure to the choice of n is not as
dramatic as with the other parameters, as its effect is manifested only in the reduced mass and reduced mobility. The effect becomes important only under conditions of low temperatures with highly polarizable drift gases, and affects light ions more strongly than heavy ions.
Chronoamperometry at Channel Electrodes: A General Computational Approach Adrian C. Fisher and Richard C. Compton* Physical Chemistry Laboratory. South Parks Road, Oxford OX1 3QZ, United Kingdom (Received: January 29, 1991)
A conceptually simple and computationally efficient implicit approach to the simulation of current/time transients at the channel electrode is presented. The generality of the method is illustrated with reference to the simple case of a step between potentials mmponding to zero and transport-limited currents for a simple redox reaction (where good agreement with existing analytical theory is found), to ECE and to DISPl processes, and finally to the case of double-potential steps. The latter are confirmed to be a possible means of distinguishing between ECE and DISPl reactions.
Introduction Channel electrodes are now recognized as well-characterized and advantageous flow-through hydrodynamic electrodes both for the study of electrode processes and for electroanalytical purp0ses.l Their merits include the huge range over which mass transport can be varied, operation under chemostatic conditions, and the mechanistically discrimating power conferred by the nonuniformity of the diffusion layer over the electrode surface.2 Hitherto the great majority of channel electrode studies have confined themselves to steady-state conditions, although of course the extra information available from time-dependent work is established.) Accordingly, timedependent theory for channel electrode problems has begun to emerge, albeit rather slowly. Thus we have derived and experimentally verified a theory for the ac voltammetry of an electrochemically reversible redox couple a t the channel and this work has been extended by Currane6 Additionally, Aoki has treated theoretically the problem of chronopotentiometry a t the channel electrode’ while Johnson* has devised a pulsed anodic technique for electroanalytical purposes: a full, review of the area appears in ref 1. The purpose of the paper is to provide a general implicit computational strategy for the calculation of the chronoamperometric responses arising from potential-step experiments at channel electrodes. The method is applicable to a wide range of electrode reaction mechanisms and, by way of example, theoretical results are given for single- and double-potential-step experiments for ECE and DlSPl reactions as well as for a simple electron transfer. The capability for general extension to other electrode processes should be evident. The ability to treat mechanistically complex processes arises from the computational efficiency of the implicit calculations used here, as opposed to the explicit finite-difference treatment developed by Marcoux for channel flow systems? We (1) Unwin, P. R.; Compton, R. G. Compr. Chem. Klner. 1989, 29, 173. (2) Compton, R. G.; Fisher, A. C.; Tyley, 0. P. J . Appl. Electrochem.
1991, 21, 295.
( 3 ) Sluyters-Rehbach, M.; Sluyters, J. H. Compr. Chem. Kiner. 1986, 26, 203. (4) ComPton, R. G.; Sealy, G. R. J . Elecrroanal. Chem. 1983, 145, 35. ( 5 ) Compton, R. G.; Laing, M. E.; Unwin, P. R. J . Elecrroanal. Chem. 1986, 207, 309. (6) Kingsley, E. D.; Curran, D. J . Electroanalysis 1990, 2, 273. (7) Aoki, K.; Matsuda, H. J . Electroanal. Chem. 1978, 90,333. (8) Polta, J . A.; Johnson, D. C. Anal. Chew. 1985, 57, 1373.
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note that for the very simple case of a single potential step in the absence any homogeneous kinetics Aoki et a1.I0 have derived an approximate analytical theory valid under conditions where the LEvQue approximation (i.e., treating the convective flow velocity profile as linear near the electrode surface) is applicable: this will be shown to be in excellent agreement, under the appropriate conditions, with our computations.
Theory We consider first the computation of the current/time transient resulting from a step between a potential at which no current flows and one corresponding to the transport-limited reduction or oxidation of a single solution-phase species, A: A&e+B The convective-diffusion equation describing the distribution of A in time ( 1 ) and space is
where D is the diffusion coefficient of A and the Cartesian coordinates x and y can be understood with reference to Figure 1 which is a schematic diagram of a channel electrode. u, is the solution velocity in the x direction, the components in t h e y and z directions being zero. Provided a sufficiently long lead-in length’ exists upstream of the electrode so as to allow the full development of Poiseuille flow then v, is parabolic 0,
=
.( ( 1-
;)2)
vv = 0,0, = 0
(2)
-
where h is the half-height of the cell (see Figure l ) , y ’ * h y , and vo is the velocity of flow at the center of the channel. The boundary conditions defining the problem specified above are t < 0, y = 0,O < x < x,: [A] = [A]hlk (3) t l O , y = O , O < x < x , : [A]=O all t , y = 2h, all X: a [ ~ ] / a=y o (9) Flanagan, J.
E.; Marcoux, L. J . Phys. Chem. 1974, 78, 718.
(4) (5)
(IO) Aoki, K.; Tokuda, K.; Matsuda, H. J . Elecrrwnal. Chem. 1986.209,
241.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 7539
Chronoamperometry a t Channel Electrodes
-
-2-
x Figure 1. A schematic diagram of a channel electrode which defines the coordinate system used. 0
xe
d;.=
b l c1 0 a Z b c 2 0
U1
4
.. .. .. .. .. .. 0
~j
wall
bj
cj
0
.. .. .. .. ... .. aJ-2
dJ-2 dJ-l
dj =
bj = 2ky
cJ-2 bJ-I
uJ-2 uJ-l
+1 ;
'0j,1
dj = ('ajc } +
bJ-2 OJ-1
'j
k;{'+'~j~l}
+ k; + 1
uj =
l+l
aj4
cj = -2
Figure 2. The finite difference grid.
where x, is the electrode length and [AIbulkis the bulk concentration of A. The problem set by eqs 1-4 may be solved by the use of a twdimensional finite-difference grid which covers the x-y plane. This is shown in Figure 2. Spacings in the two directions are Ax and Ay respectively so that Ax = x , / K Ay = 2 h / J
We may then define concentrations at a point (j, k ) corresponding to x = kAx, y = j A y . We use the symbol aj,&to denote the concentration of A (normalized relative to [AIbulk)at this point. Equations 1 and 2 may be then cast into finitedifference form if we define the parameters 6VfAyArj(2h- j a y ) Aj' = (6) d(2hyhx A" = D A t / ( A y ) 2
(7)
where Vr is the volume flow rate (=4uodh/3) of solution. Ar represents an increment of time. We use the notation raj,kto indicate that normalized concentration of A at time rAr. Equation 1 then becomes
Or,rearranging, 'U,,k
+ Aj('+la,,k-I) =
+
-AY{f+la,-l,k] ( 2 V
+ Af + I){f+'aj,kJ- AY{'+laj+l,k]( 9 )
Application of the boundary condition ( 4 ) gives 'a1.k
+ A~C(f+lU~,k-l)
(2A"
+ + l)('+'al,kI - Ay('+lU2,k1
(10)
while the boundary condition ( 5 ) results in faJ-l.k
+ A ~ - l ( f + l a J - l , k - I ~= -Ay{'+laJ-2,k)
+ (Ay + Aj-1 + I ) ( ' + ' a J - l , k }
where the matrix elements are given by
-2
=
+ k;-1 + 1
Note that the matrix equation ( 1 2 ) shows how the concentrations throughout the cell a t time ( r + 1)At may be calculated if we know those at time rAr, To do this we have to find the set of vectors (u): each k value has its own vector ( u ) ~ The . matrix [TI being of tridiagonal form allows us to use the Thomas alThe boundary condition 3 gorithm1'-I4 to give ( U } ~from supplies the vector {d), from which {ul0is calculated. The (d]k+l = (ulk,so (uIl is calculated from (dIl,and so on until (uJKis obtained. The calculation is then repeated. In this way the concentration profile of A within the flow cell may be calculated as a function of time. The current a t the electrode may thus be evaluated at any instant from
By use of the theory outlined above, single-step chronoamperometric transients were computed (on a Sun Sparcstation) and convergence examined by varying J , K , and At values. For a representative electrode geometry of 2h = 0.04 cm, d = 0.6 cm, x, = 0.4 cm, w = 0.4 cm and for a flow rate of Vf = 0.01 cm3 s-I with D = 1 X 10" cm2 s-l, values of J = 500 and K = 250 gave convergence to four significant figures. Values of Ar in the range 0.02-0.00002 s gave identical results except a t very short times in the transient ( Eoc/s (oxidation). Different mechanisms arise depending on which step is rate-determining. If the second electron transfer takes place heterogeneously so that reaction iv may be neglected, an ECE mechanism is said to operate. Alternatively, if this transfer is homogeneous then step iii is neglected and a DISP mechanism results. The latter is subdivided according to whether step ii ("DISPI") or step iv ("DISP2") is rate-determining. The appropriate transport equations for the ECE mechanism are
k = l k>1
where 6 = [B], c = [C] and the boundary conditions relevant to the evaluation of the effective number of electrons transferred under transport-limited conditions are t < 0,y 0,0 < x < x,: [A] = [A]bulk, [Bl = [ c ] = 0 (24) t L 0,y = 0,o < x < x,: [AI = 0, a [ ~ ] / a y= - a [ ~ ] / a y (25) all t , y = 2h, all X: a [ ~ ] / a y= a [ ~ ] / a y= a[c]/ay = o (26) where we have assumed A, B, and C to have identical diffusion coefficients. On applying the general theory outlined above, three matrix equations akin to eq 12 result, one for each species. The new matrix elements are given in Table 1. Note that the matrix equation for B is dependent on a/,&and that for C on b,,&. Thus the equations are solved, for each column on the grid, in the
Species B
'bj,i - ki44'bj,il 'bj,k + A{('+'bj,k-I]- klAt('bj,kJ bl = 2AY Aj 1
2
+ +
Species C unchanged from Table I sequence A, then B, and finally C. The current is the sum of that due to the reduction/oxidation of both A and C:
We consider next the pure DISPl mechanism. The pertinent transport equations are
The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 7541
Chronoamperometry at Channel Electrodes and the boundary conditions become
< 0 , =~0,O < x < x,: t 1 0,y 0,O < x < x,: t
[A]
[A]hlk, [B] = 0
(30)
[A] = 0, ~ [ B ] / ~=J -d[A]/ay J (31)
all t , y = 2h, all x: d[A];dy = a[B]ay = 0
2
i_ 1
(32)
-& . 5 -1.
I
These expressions are again used to construct matrix equations for A and B. Table I shows the matrix elements. The equations are solved in the sequence A, then B, and the current is evaluated by using eq 20. Double-Potential Steps. We now consider the response of a simple electron-transfer reaction and also ECE and DISPl processes to a double-potential step defined by the sequence of potentials E l , E2,and E3. El (t'< 0 ) corresponds to no current flow: E2 (0 I t ' I t*) corresponds to the transport-limited reduction (oxidation) of A to B and E3 ( t ' > t*) corresponds to a potential at which B is oxidized (reduced) back to A but which is still capable of reducing (oxidizing) C (Le., E3 lies between and Eocls). The chronoamperometric behavior up to the time t ' = t* may be calculated as outlined above. Afterwards the boundary conditions require the following modification. simple electron transfer and DISPl: t ' > t*, y = 0, 0 < x < x,: [Bl = 0, a[BI/ay = d [ A ] / d y
05
10
-2. -3
(33) I
0.5
ECE: 1'1
0, y = 0, 0 < x < x,: [B] = [C] 0, d[B]/dy
tm /
-a[A]/dy
025
(34)
Notice that we are assuming that the standard electrode potentials EoA/B and Eoc are sufficiently separated that the surface concentrations of the intermediates (B and C) can take the value of zero when the potential E3 is applied. For reversible couples this requires (EoA/B- Eoc/sl > 0.24 V. The development of the finite-difference equations for the relevant systems of partial differential equations leads again to matrix equations involving tridiagonal matrices and Table I1 shows the new matrix elements. Note first that the required matrix elements for the case of simple electron transfer can be derived from those for ECE by straightforwardly putting kl = 0 and, second, that the matrix equations can be solved with a knowledge of the concentration distributions prevailing at time t*. Results and Discussion We have already displayed the results for the no kinetics/single-step case in Figure 3 and recognized the excellent agreement with approximate analytical theory but also the improved generality of the numerical approach since that is not constrained to those flow rates and for which the *Que approximation has validity' Figure shows the double-step transient calculated for exactly the conditions pertaining to Figure 3 except for the occurrence of the second potential step at time t' = t*: Immediately following this step thecurrent flow changes direction due to B trapped within the diffusion layer being reconverted into A, the current falling to zero once the concentration of B has been exhausted. Figures 5 and 6 show typical double-potential step transients for, respectively, ECE and DISPl processes both characterized by a rate constant of kl = 1.5 s-l, It can be seen that in the case of the DlSPl process the current switches sign immediately following the second step and then monotonically decreases to zero as with the simple no kinetics case discussed above. This happens since, by definition, the only electrode process is the A/B redox event given by reaction i and this reaction changes direction at time t*. In contrast, the ECE behavior displayed in Figure 5 shows an initial current reversal, followed by a return to current flow in the original part (0 < t' < r*) of the transient and then a decay to zero current. This leads to a characteristic "hump" in the transient.I5 The origin of this behavior lies in the fact that the
050
0'15
l a
125
I/
150
175
200
t'
2 5
Figure 5. A computed double-potential step chronoamperometric transient as in Figure 4 except for an ECE process characterized by a rate constant of kl = 1.5 s-I.
I \
Fiprc'6. A computed double-potential step chronoamperometric transient as in Figure 4 except for a DlSPl process characterized by a rate of &, ,5 ~
ECE process involves two electrode reactions: (i) and (iii). The former involving the A/B couple is reversed at time t*, leading to the initial change in direction of current flow, whereas the latter which concerns the C/S couple is not reversed. Thus the presence of C in the diffusion layer is capable of providing a (transient) current flow in the original direction and since at time t* most C is located further (on the average) from the electrode surface than B this contribution dominates afrer the current reversal due to the conversion of B to A. In this way is generated the Note, however, that current reversal as described by the theory presented here will only occur when E 0 , p < E, < Eoc s (for a reduction) and, with reversible couples, when IE,,/B - koc/sl > 0.24 V so suitable values of E3 for providing mechanistic dis(IS) Compton, R. G.; Mason, D.; Unwin, P. R. J . Chcm. Soc., Faraday Trans. I 1988, 84, 2057.
7542
J . Phys. Chem. 1991, 95, 7542-1545
crimination will not exist for all systems. It is apparent that double-potential step transients can provide qualitative mechanistic insight. Moreover, the precise fitting of theoretical transients to experiment could confirm a distinction between ECE and DISPl and, further, provide a rate constant.
In conclusion, we have given a general approach to the calculation of current transients at the channel electrode and shown it can be applied to a wide range of electrode reaction mechanisms. The basis for the use of channel electrodes in time-dependent experiments for mechanistic studies has thus been provided.
Investigation of the Aggregation and Reactivity of Neat and Concentrated Solutions of Polar Alcohols by Time-Domain Spectral Studies Ngai M. Wong and Russell S . Drago* University of Florida, Department of Chemistry, Gainesville, Florida 3261 1 (Received: September 10, 1990; I n Final Form: April 29, 19911
The dielectric relaxation processes found in aliphatic alcohols were studied by using timedomain reflectometry. The aggregation of the alcohol leads to a three-dimensional dynamic network structure through the coordination of the two lone pairs on the oxygen and the OH proton. The behavior of both the static dielectric constant (Q) and the critical frequency (f, frequency of relaxation process) were studied as a function of the dilution of 1-butanol by CC14. The aggregate size of butanol remains fairly constant at high concentrations but tends to shift the aggregate-size distribution toward the smaller species at the lower concentrations, giving rise to nonlinear behavior in €0 and& Linear behavior in €0 andf, was seen in the binary system of I-propanol and I-hexanol. This mixed-alcohol system maintains a constant aggregate size but varies in the composition of the aggregate. The linear behavior was attributed to the similar H-bond strength of the two alcohols and the difference in the alcohols molecular size. The addition of pyridine to 1-butanol greatly shifts the critical frequency. This was attributed to pyridine complexation, leading to the formation of smaller aggregates. Adduct formation with pyridine hinders the formation of large aggregates by blocking the alcohol's OH proton-coordination site. After terminal OH groups on the exterior of the aggregate are complexed at low pyridine concentrations, reduction in the average size of the aggregate results in order to provide coordination sites for additional pyridine. This was manifested in a nonlinear behavior of the static dielectric constant and an increase in the frequency of relaxation over that expected for linear behavior.
Introduction The study of donor-acceptor interactions in protic solvents are complicated by self-association of the solvent, hydrogen bonding of the solvent to the adduct, and Born-type nonspecific solvation effects.' These effects contribute to the position of the equilibrium and the enthalpy of complexation. Studies have been reported on the enthalpy of complexation of various bases to aliphatic alcohols in carbon tetrachloride* or the gas phase.3 The solution studies are typically done under dilute conditions of alcohol and base with the results extrapolated to infinite dilution using the assumption that the alcohol and base are monomeric species. Similar assumptions are made in studies of the change in the 0-H stretching frequencies of alcohol-base adducts in dilute CCI, ~ o l u t i o n . ~In a recent analysis of the "anomalous basicity of amines"? it was shown that the acidity of hydrated proton species, as well as the relative importance of covalent and electrostatic bonding contributions, changed appreciably with the extent of hydration, Le., the n value of H(H20),,+. Similar changes in the acidity or basicity of (ROH),may be expected as n varies. Thus, gas-phase studies of monomers and solution studies in inert solvents may determine properties of species that do not exist in pure solvents. In this context it was of interest to study the variation of the extent of aggregation of alcohols in the binary systems of ~
~
( 1) LeMer, J. E.; Grunwald, E. Rates and Equilibria of Urganic Reactionr;
Wiley: New York, 1963; p 266. (2) (a) Stephensen, W. K.;Fuchs, R. Can. J . Chcm. 1985,63, 342. (b) Enthalpia not found in ref 2a were mtimated with the E and C equation: Drago, R. S. Cwrd. Chem. Rev. 1980, 33. 251. ( 3 ) Keesee, R. G.; Castleman, A. W., Jr. J. Phys. Chem. Re/. Data 1986,
- ..
I -, T i.n.i i
~
(4) Drago, R. S.; Wong, N. M.; Bilgrien, C.; Vogel, G. C . Inorg. Chem. 1987, 26, 9. (5) Drago, R. S.; Cundari, 7 . R.; Ferris, D. C. J . Org. Chem. 1989, 54, 1042.
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CC1,-butanol and pyridine-butanol, as well as the ternary system of pyridine-butanol-CCl,. Such information is relevant to understanding the chemistry in pure protic solvents and in mixedsolvent systems. Frequency-dependent studies of the dielectric constant show dielectric relaxation corresponding to the making and breaking of hydrogen bonds in alcohol aggregates.6 Three different relaxation times have been found for each alcohol a t 10 normal alcohols from propyl to dodecyl. The two fast relaxations correspond to the rotation of monomeric molecules (17-50 ps) and to rotation of the O H group around the C-O bond (1.7-4 ps). The long relaxation occurs over the range 0.1-2 ns (80-1600 MHz) and corresponds to breaking hydrogen bonds of'terminal ROH groups in an aggregate, concurrent with rotation of the alcohol molecule as it forms a new hydrogen bond in the same aggregate. The relaxation time for this process increases regularly with increasing chain length. The activation energy for this process ranges from 21 to 33 kJ/mol, increasing in a regular fashion with chain length. This is of the order of magnitude expected for breaking a hydrogen bond and rules out rotation mechanisms that require the breaking of two hydrogen bonds. Accordingly, the molecular motion involves terminal groups of three-dimensional structures. Since the hydrogen-bond dissociation energies are not expected to increase with increasing length of the alkyl group, the size of the group involved in the molecular rotation is proposed to contribute to the observed activation enthalpy and entropy. In neat alcohol, one expects a distribution of aggregate sizes and hence a distribution of relaxation times. Since this is not observed, the energetics of the processes described must not differ enough with aggregate size to be resolved. Failure to observe a distribution of relaxation times is taken as evidence to rule out the existence (6) Gar& S.K.; Smyth, C. P. J . Phys. Chem. 1965, 69, 1294.
0 1991 American Chemical Society
