Modeling Diffusion Effects for a Stepwise Two ... - ACS Publications

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J. Phys. Chem. C 2009, 113, 16042–16050

Modeling Diffusion Effects for a Stepwise Two-Electron Reduction Process at a Microelectrode: Study of the Reduction of para-Quaterphenyl in Tetrahydrofuran and Inference of Fast Comproportionation of the Dianion with the Neutral Parent Molecule Stephen R. Belding, Ronan Baron, Edmund J. F. Dickinson, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, UniVersity of Oxford, South Parks Road, Oxford, OX1 3QZ United Kingdom ReceiVed: July 4, 2009; ReVised Manuscript ReceiVed: July 27, 2009

This paper presents a study of voltammetry at microdisc electrodes for a stepwise two electron reduction (A + 2e- h B + e- h C) where the relevant formal potentials of the A/B and B/C couples are such that two distinct voltammetric waves are seen. Simulations are used to explore quantitatively the effect of comproportionation (A + C f 2B) on the observed cyclic voltammetry and potential-step chronoamperometry. It is found that, in the limit of electrochemical reversibility, the presence of comproportionation can be discerned using cyclic voltammetry only at high scan-rates and when the diffusion coefficients of the species (DA, DB, and DC) are significantly different, such that DB/DA > 1.5 and DC/DA > 1.5 or DB/DA < 0.75 and DC/DA < 0.75. The theory is used to provide clear evidence for diffusionally controlled comproportionation in the stepwise two electron reduction of para-quaterphenyl in the solvent THF. 1. Introduction In many organic electrochemical processes two distinct oxidation or reduction waves are observed:1 Q A + e- h B(Ef,A/B )

(1)

Q B + e- h C(Ef,B/C )

(2)

In the case of consecutive reductions with electrode kinetics fast relative to mass transport, this implies EQf,A/B > EQf,B/C. Under these conditions, the forward (comproportionation) reaction

A + C h 2B

(3)

is thermodynamically downhill. The question arises as to the extent, if at all, to which the presence of comproportionation can be discerned by voltammetric studies alone. Insightful pioneering work by Save´ant and Andrieux2 showed that, for conditions of planar diffusion, voltammetry is blind to any homogeneous chemistry in situations where both electron transfers are reversible and the diffusion coefficients of A, B and C are equal. Otherwise, it is, at least in principle, possible to infer kinetic information about the comproportionation from the voltammetric signals derived from the A/B and B/C reduction steps. Usually, the diffusion coefficients of A, B, and C are comparable, especially if the three species are structurally similar.3-8 Recent research has voltammetrically identified three distinct areas in which significant inequality of diffusion coefficients occurs. The first is when room temperature ionic liquids (RTILs) are used as solvents1,9-14 in which different levels of charge in A, B, and C lead to major differences in the * Corresponding author. Fax: +44 (0) 1865 275410. Tel.: +44 (0) 1865 275413. E-mail: [email protected].

Figure 1. Structure of para-quaterphenyl (PQ).

three diffusion coefficients. For example, the reduction of O2 to O2- (superoxide) in the ionic liquid [N6,2,2,2][NTf2]9,15 shows the ratio of diffusion coefficients to be approximately 25 with the O2- ion diffusing much more slowly. Also, Evans has shown that neutral N,N,N,N-tetramethyl-para-phenylenediamine and its radical cation and dication have significantly different diffusion coefficients in the ionic liquid [C2 mim][NTf2].16 The second area involves the redox chemistry of metal ions in the presence of chelating ligands. Since the chelate molecules are large, coordination leads to a marked reduction in the diffusion coefficient of the associated metal ion.17,18 The equilibrium constant for chelation varies with the nature of the coordination sites on the ligand as well as the charge on the metal ion. Therefore, the presence of the chelate can lead to a large discrepancy in diffusion coefficients for ions in a redox process. The third area is that of voltammetry in solvents of very low polarity such as tetrahydrofuran (THF).19,20 If A is neutral, species B and C are anions and dianions, respectively, and are likely to be involved in ion pairing with cations contained in the supporting electrolyte.19 In both types of medium considered above, RTILs and nonpolar solvents, the preferred voltammetric method is one involving microelectrodes: the tiny currents minimize ohmic distortion in the voltammogram. Accordingly, in this paper we address the EE mechanism at microdisc electrodes with particular emphasis on the influence of any comproportionation kinetics on the voltammetric response. Experiments are reported for the reduction of para-quaterphenyl (PQ; Figure 1) in THF. Two voltammetric peaks are seen due to the formation of the mono and dianions (PQ- and PQ2- respectively). Comparison between simulated and experimental data as a function of voltage scan-rate allows one to infer the presence of comproportionation:

10.1021/jp906323n CCC: $40.75  2009 American Chemical Society Published on Web 08/19/2009

Voltammetry at Microdisc Electrodes

J. Phys. Chem. C, Vol. 113, No. 36, 2009 16043

PQ + PQ2- h 2PQ-

(4)

and that this occurs at a fast rate on the voltammetric time scale.

We consider an EE reaction at a microdisc electrode, as defined by eqs 1 and 2, where we additionally allow for comproportionation (eq 3). The heterogeneous rate constants 0 0 for the first two steps are kA/B and kB/C , respectively. The homogeneous rate constant for comproportionation is kcomp and for disproportionation is kdisp. The equilibrium constant for eq 3 is therefore Keqm ) kcomp/kdisp. Specifically, we consider two problems of cyclic voltammetry and potential step chronoamperometry. The mass transport equations defining diffusion and reaction, in which both diffusional and kinetic terms are present, of each species are as follows:

(

)

∂cA ∂2cA ∂2cA 1 ∂cA + + kdispcB2 - kcompcAcC ) DA + ∂t r ∂r ∂r2 ∂z2 (5)

∂t

( )

∂cC DC ∂z

2. Theory

∂cB

DB

(

) DB

(

∂2cB 2

∂r

+

)

∂2cB 1 ∂cB + - 2(kdispcB2 - kcompcAcC) 2 r ∂r ∂z

2

∂cB ∂z

( )

) -DA

0

(

∂cA ∂z

- DC

0

( ) ∂cC ∂z

0

)

Q F(E - Ef,B/C ) + -RB/C RT Q F(E - Ef,B/C ) 0 kB/CcC,0exp (1 - RB/C) RT

0 -kB/C cB,0exp

(

)

The remaining boundary conditions are

cA ) c*A ∂cA 0, r > re, z ) 0 )0 ∂z ∂cA 0, all r, z f ∞ )0 ∂z ∂cA 0, r f ∞, all z )0 ∂r ∂cA 0, r ) 0, all z )0 ∂r

cB ) 0 ∂cB )0 ∂z ∂cB )0 ∂z ∂cB )0 ∂r ∂cB )0 ∂r

t ) 0, all r, all z t> t> t> t>

cC ) 0 ∂cC )0 ∂z ∂cC )0 ∂z ∂cC )0 ∂r ∂cC )0 ∂r

2.2. Chronoamperometry. In potential step chronoamperometry, the potential, E, is stepped from a value of zero current to one corresponding to the transport controlled current of A forming either B or C. When the applied potential is reducing enough to form only B, when t > 0, z ) 0, and r e re, the boundary condition is

(6)

( )

cA ) 0DB

)

2

)

0

( )

∂cC ∂ cC ∂ cC 1 ∂cC + + kdispcB2 - kcompcAcC ) DC + 2 ∂t r ∂r ∂r ∂z2 (7) where all nontrivial symbols are defined in Table 1. 2.1. Cyclic Voltammetry. In the cyclic voltammetry experiment, the applied potential, E, is swept from an initial value, Ei, to a more reducing potential, Ef, and then back to the initial value. The value of E is therefore calculated at any time on the forward sweep using eq 8 and on the reverse sweep using eq 9, where V is the voltage scan-rate and t is the time

Eforward ) Ei - Vt

(8)

∂cB ∂z

0

( ) ∂cA ∂z

) -DA

0

cC ) 0

When the applied potential is reducing enough to form both B and C, when t > 0, z ) 0 and r e re, the boundary condition is:

cA ) 0 cB ) 0 DB

( ) ∂cB ∂z

0

) -DC

( ) ∂cC ∂z

0

In each case, the remaining boundary conditions are identical to those given above for cyclic voltammetry. 2.3. Computational Details. The problem is generalized by means of a coordinate transform using a conventional set of normalized parameters (defined in Table 2) in order to reduce TABLE 1: Dimensional Parameters

Ereverse ) 2Ef - Ei + Vt

(9)

When t > 0, z ) 0, and r e re, the boundary conditions are the Butler-Volmer conditions describing species A and C and conservation of mass for species B:

DA

( ) ∂cA ∂z

0

(

)

Q F(E - Ef,A/B ) RT Q F(E - Ef,A/B ) 0 kA/B cB,0exp (1 - RA/B) RT

0 ) kA/B cA,0exp -RA/B

(

)

dimensional parameter RX/Y cX,0 c*X DX E Q Ef,X/Y 0 kX/Y kcomp kdisp i j V

definition transfer coefficient for X + e- h Y/unitless concentration of species X at the electrode surface/mol m-3 concentration of species X in bulk solution/mol m-3 diffusion coefficient of species X/m2 s-1 applied potential/V formal reduction potential for X + e- h Y/V electrochemical rate constant for X + e- h Y/m s-1 rate constant for comproportionation/dm3 mol-1 s-1 rate constant for comproportionation/dm3 mol-1 s-1 current/A flux across the electrode/mol m2 s-1 scan-rate/V s-1

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Belding et al.

∂(2cA + cB) ) DA∇2(2cA + cB) ∂t

TABLE 2: Normalised Parameters normalized parameter

definition

J K0 Kcomp Kdisp θ R σ τ x x0 Z

i/(Fc*ADAre) k0re/DA kcompc*Are2/DA kdispc*Are2/DA F(E - EfQ)/RT r/re re2FV/DART DAt/re2 cX/c*X cX,0/c*X z/re

In the electrochemically reversible limit, the chemical species on the electrode surface obey the Nernst equation. In addition, when DA ) DB ) DC, the relationship cA,0 + cB,0 + cC,0 ) c*A holds, such that when t > 0, z ) 0, and r e re, the boundary condition is

(

2

2+ )

)

1+

∂a ∂a 1 ∂a ∂a + ) + 2 + Kdispb2 - Kcompac 2 ∂τ R ∂R ∂R ∂Z

(10)

DB ∂2b ∂b 1 ∂b ∂2b + - 2(Kdispb2 - Kcompac) ) + 2 ∂τ DA ∂R2 R ∂R ∂Z (11)

(

cB,0 cA,0

cB,0 cC,0 + cA,0 cA,0

c*A

( ) (

)

DC ∂2c ∂c 1 ∂c ∂2c + + Kdispb2 - Kcompac ) + ∂τ DA ∂R2 R ∂R ∂Z2 (12) The boundary conditions given in the previous sections are readily formulated in terms of normalized variables using the definitions given in Table 2. The problem is then discretized using the alternating direct implicit method21,22 (ADI) and solved numerically using the iterative Newton-Raphson scheme.23 The discretized spatial mesh is analogous to that reported by Gavaghan24 and converged to within 0.1%. Each temporal increment is optimized during runtime such that convergence to within 0.01% is obtained. All programs were written in C++ and compiled using a Borland compiler. The simulations were run on a desktop PC with a processor speed of ∼3 GHz. Approximately 20 min of CPU time were required to simulate a single voltammogram or chronoamperogram.

(17)

)

(

)

(18)

The remaining boundary conditions are

(2cA + cB) ) 2c*A ∂(2cA + cB) DA )0 ∂z ∂(2cA + cB) DA )0 ∂r ∂(2cA + cB) DA )0 ∂z ∂(2cA + cB) DA )0 ∂r

t ) 0, all r, all z t > 0, r > re, z ) 0 t > 0, r f ∞, all z t > 0, all r, z f ∞ t > 0, r ) 0, all z

The flux through a microdisc is given by

j ) 2πFDAc*Are

∫0r

e

(( )

( ))

DB ∂cB ∂cA r+ r dr ∂z 0 DA ∂z 0

2

3. Voltammetric “Blindness” to Comproportionation2,25 This section presents a generalized form of Save´ant’s argument,2 originally applied to planar diffusion, extended here to cover the case of microdisc voltammetry. The current drawn at the electrode is the sum of the reductions represented by eqs 13 and 14

A + 2e- h C

(13)

B + e- h C

(14)

We therefore consider the linear combination (2cA + cB) formed by adding eqs 5 and 6. It follows that, when DA ) DB ) DC, the homogeneous reaction terms in eqs 5 and 6 cancel and a diffusion-only mass transport equation applies for (2cA + c B)

(16)

Q F(E - Ef,A/B ) RT ) c* A Q Q Q F(E - Ef,A/B ) ) F(2E - Ef,A/B - Ef,B/C 1 + exp + exp RT RT

2 + exp -

)

(

2cA,0 + cB,0 c* cA,0 + cB,0 + cC,0 A

2cA,0 + cB,0 )

the number of independent variables. The normalized diffusion equations become 2

(15)

(19)

Since DA ) DB ) DC, this expression reduces to

j ) 2πFDAc*Are

∫0r

e

((

))

∂(2cA + cB) r dr ∂z 0

(20)

The problem, comprising the diffusion equation (eq 15) and the relevant boundary conditions, is expressed exclusively in terms of the linear combination (2cA + cB) and hence from eq 20 the current possesses no dependence on the rate constants kdisp and kcomp. The voltammetric response is independent of the existence and kinetics of the comproportionation reaction in solution provided DA ) DB ) DC. If this latter equality does not hold, then the measured currents reflect kdisp and kcomp when, at least in principle, are amenable to voltammetric measurements. 3.1. Theoretical Results and Discussion. In this section we evaluate the results of the simulation of the EE process in which,



Figure 2. Ratio of τ1/2 for Kcomp ) 106 and Kcomp ) 0 as a function of DB/DA and DC/DA.

for the purpose of illustration, we assume RA/B ) RB/C ) 0.5, a Q normalized separation of the A/B and B/C waves of ([F(Ef,A/B Q - Ef,B/C)]/RT) ) 40 (corresponding to ∼1 V at 298 K) and 0 0 KA/B ) KB/C ) 103. This corresponds to two stepwise electrochemically reversible reductions in which comproportionation is thermodynamically downhill. Two values of Kcomp are considered: 0 and 106 (the latter corresponds to the limit of diffusion control). 3.1.1. Chronoamperometry. Two experimentally useful potential steps may be considered. First, the potential can be Q such that A is reduced stepped sufficiently negative of Ef,A/B infinitely rapidly but the product, B, does not undergo reduction Q to C. Second, the potential can be stepped negative of Ef,B/C such that both A and B are reduced infinitely rapidly to form C. In the former case, only species A and B are present in solution at all times and comproportionation cannot occur: the transient is independent of Kcomp and can be used to calculate c*A and DA using the Shoup and Szabo equation.26 In the latter case, comproportionation can occur and the transient is dependent on Kcomp. The shape of the chronoamperometric transient for the conversion of A to C is quantified by defining a diffusional half-life τ1/2 as the time taken to achieve a flux equal to twice the steady-state value. Figure 2 shows shows the ratio of τ1/2(Kcomp ) 106) to τ1/2(Kcomp ) 0) as a function of DB/DA and DC/DA. The surface may be explained by consideration of the equation for the normalized flux

J ) 2π

∫01

(( 2

)

DB ∂b ∂a R+ R dR ∂Z 0 DA ∂Z 0

)

( )

(21)

Over short time scales, the comproportionation reaction decreases the concentration of B close to the electrode and increases the concentration of A. Hence it increases (∂b/∂Z)0 and reduces (∂a/∂Z)0. From eq 21 it follows that, as DB/DA increases, the effect of comproportionation will be more significant in enhancing the observed flux. In addition, eq 21 implies no direct dependence on DC/DA. Simulation studies over a range of DB/DA and DC/DA are found to exhibit a common steady-state limiting flux equal to that for a two electron reduction.26 Over long time scales, B diffuses from the electrode surface into bulk solution and (∂b/∂Z)0 f 0. The residual flux in eq 21 is then independent of DB/DA and double that expected for the one electron process A + e- h B, corresponding to overall conversion of two equivalents of A. 3.1.2. Cyclic Voltammetry. We next consider cyclic voltammetry which leads to two peaks corresponding to the A/B

Figure 3. Cyclic voltammogram (-) for which DB/DA ) DC/DA ) 1, Kcomp ) 0, and σ ) 10. Peaks are labeled as shown. Baselines A, B, and C used to defined relative peak heights are also shown (- -).

and B/C couples. The voltammetric peaks are numbered as shown in Figure 3. For the purposes of discussion, we measure the voltammetric currents relative to difference baselines. Baseline A corresponds to J ) 0. Baselines B and C correspond to the steady-state limiting fluxes derived respectively from one and two electron transfer calculated using the equation for the steady-state current at a microdisc electrode27

iss ) 4nFc*ADAre

(22)

Jss ) 4n

(23)

In dimensionless form

The voltammogram is characterized by measuring two different peak heights. First, the absolute peak height Jabs is defined relative to Baseline A. Second, the peak height, Jrel, is defined relative to the steady-state current flowing before the respective electron transfer as indicated by the arrows in Figure 3. By immediate extension of the discussion in Section 3.1.1, the corresponding steady-state voltammograms are independent of DB/DA and DC/DA. Voltammograms for which respectively Kcomp ) 0 and Kcomp ) 106 are compared using the ratios of Jrel and Jabs for each peak. In each case, the normalized scan-rate (σ) is set equal to 10 corresponding to ∼ 10 V s-1 when DA ) 10-5 cm s-1 and re ) 10 µm at 298 K. Peak 1 arises from a single redox process

A + e- h B

(24)

Only species A and B are present in solution and comproportionation cannot occur; therefore, peak 1 is independent of Kcomp. Figure 4a shows the ratio of the peak 2 fluxes for Kcomp ) 0 vs Kcomp ) 106 relative to baseline B. Peak 2 arises from two redox processes

A + e- h B

(25)

B + e- h C

(26)

and the dimensionless flux through the microdisc relative to baseline B is

16046


J2rel ) 2π

∫01

(( 2

Belding et al.

)

DB ∂b ∂a R+ R dR - 4 ∂Z 0 DA ∂Z 0

)

( )

(27)

where “4” corresponds to the normalized steady-state current for one electron reduction as defined in eq 23. As introduced above, the comproportionation reaction increases the flux of B to the electrode (∂b/∂Z)0, and hence, as DB/DA increases, the effect of comproportionation increases the size of peak 2 more significantly. Figure 5a shows the ratio of peak 3 fluxes for Kcomp ) 0 vs Kcomp ) 106 relative to baseline C. Peak 3 arises from two redox processes

A + e- h B

(28)

C h B + e-

(29)

and the flux through the microdisc relative to baseline C is given by

J3rel ) 8 + 2π

∫01

(

)

DC ∂c DB ∂b R+2 R dR DA ∂Z 0 DA ∂Z 0

( )

( )

(30)

where “8” corresponds to the normalized steady-state current for two electron reduction as defined in eq 23. Again, comproportionation increases Peak 3 more significantly when DB/DA is larger. When DC/DA is small, the comproportionation reaction decreases the concentration of C close to the electrode leading to a reduction in the flux of C, (∂c/∂Z)0. Therefore, comproportionation reduces the size of peak 3 more significantly as DC/DA becomes larger. When DC/DA is very large, C diffuses into bulk solution and (∂c/∂Z)0 f 0. In this case, the enhancing effect of (∂b/∂Z)0 is dominant and comproportionation increases peak 3 more significantly. Figure 6a shows the ratio of peak 4 fluxes for Kcomp ) 0 vs Kcomp ) 106 relative to baseline B. Peak 4 arises from two redox processes

B h A + e-

(31)

C h B + e-

(32)

Prior to the voltage scan reaching peak 4 after peak 3, some B and C formed as a result of peaks 1 and 2 will have diffused from the electrode surface into bulk solution. The comproportionation reaction converts C into B, and so increases the size of peak 4 as the bulk concentration of B available for oxidation is enhanced. This effect is most significant when DC is large and DB is small. Therefore, as DC/DA increases and DB/DA decreases, comproportionation increases peak 4 more significantly. The effect of scan-rate was also investigated. The case of peak 2 is considered as peak 1 is devoid of useful homogeneous kinetic information (explained above). Figure 4 shows the flux ratio to be relatively insensitive to DC/DA and so this ratio was held constant at unity. Figure 7a plots the ratio of peak 2 fluxes for Kcomp ) 0 vs Kcomp ) 106, relative to baseline B. At high normalized scan-rates (corresponding to the “macroelectrode” limit re f ∞ and V f ∞) as DB/DA increases, comproportionation increases peak 2 more significantly. At low normalized scan-rates (corresponding to the “microelectrode” limit re f 0 and V f 0) the fluxes for peak 2 converge toward a steadystate corresponding to overall conversion of two equivalents of A. This behavior is directly analogous to that explained in section 3.1.1 for chronoamperometry at long times. The surface shown in Figure 7a testifies the essential need for experimentalists to conduct cyclic voltammetry over a wide range of scanrates. In an experimental situation, the presence of diffusionally controlled comproportionation can only be discerned when disparity in the cyclic voltammetry is greater than experimental error. Figures 4b, 5b, 6b, and 7b show the ratio of absolute fluxes (Jabs) for each peak (relative to baseline A). Assuming experimental error, perhaps optimistically, in each flux measurement to be less than 5% with a corresponding error of 10% in the flux ratio. It can be concluded from these figures that the presence of diffusionally controlled comproportionation with an electrochemically reversible EE reaction can be only discerned using cyclic voltammetry at high scan rates (macroelectrode voltammetry) when DB/DA and DC/DA > 1.5 or when DB/DA and DC/DA < 0.75, if the two voltammetric peaks are Q Q - Ef,B/C )]/RT) ) 40). As the two well separated ([(F(Ef,A/B voltammetric peaks becomes closer together, there is less time during which the concentration profiles can be altered by the comproportionation reaction in solution and larger values of DB/

Figure 4. Ratio of fluxes for Kcomp ) 106 and Kcomp ) 0 for peak 2 as a function of DB/DA and DC/DA. (a) Ratio of fluxes relative to baseline B in Figure 3. (b) Ratio of absolute fluxes.



Figure 5. Ratio of fluxes for Kcomp ) 106 and Kcomp ) 0 for peak 3 as a function of DB/DA and DC/DA. (a) Ratio of fluxes relative to baseline C in Figure 3. (b) Ratio of absolute fluxes.

Figure 6. Ratio of fluxes for Kcomp ) 106 and Kcomp ) 0 for peak 4 as a function of DB/DA and DC/DA. (a) Ratio of fluxes relative to baseline B in Figure 3. (b) Ratio of absolute fluxes.

Figure 7. Ratio of fluxes for Kcomp ) 106 and Kcomp ) 0 for peak 2 as a function of DB/DA and σ (DC/DA ) 1). (a) Ratio of fluxes relative to baseline B in Figure 3. (b) Ratio of absolute fluxes.

DA and DC/DA are required to voltammetrically distinguish the comproportionation reaction in solution. 4. Experimental Procedure 4.1. Reagents and Materials. p-Quaterphenyl (99.5 wt %; PQ) was supplied from Alfa Aesar and was used without further purification. Ferrocene (purum, 98 wt %), ferrocenium hexafluorophosphate, and tetra-n-butylammonium hexafluorophosphate (TBAF, 98 wt %) were obtained from Aldrich. Tetra-nbutylammonium perchlorate (TBAP, electrochemical grade) was

obtained from Alfa Aesar (Heysham, U.K.) and was purified by recrystallization from diethyl ether. Acetonitrile (HPLC Gradient grade), acetone (99.82 wt %), and diethyl ether (Analytical reagent grade, 99.97 wt %) were obtained from Fisher Scientific, Loughborough, U.K. Tetrahydrofuran (THF, anhydrous) was purchased from Rathburn Chemicals Ltd., U.K.; it was kept under a nitrogen atmosphere and filtrated before use through two columns of activated alumina (Alcoa, grade DD-2). Argon (Pureshield) was obtained from BOC Gases, Guildford, U.K.

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Figure 8. Voltammetric fits: experiment (-); simulation (kcomp > 106 dm3 mol-1 s-1) (O); simulation (kcomp ) 0 dm3 mol-1 s-1) (•). (a) 1, (b) 10, (c) 40, and (d) 80 V s-1.

4.2. Instrumental. The setup, protocol and procedures used for electrochemical studies in super dry THF have been detailed in a recent publication.20 Electrochemical measurements were performed using a PGSTAT20 potentiostat (Eco-Chemie, Utrecht, Netherlands). The electrochemical cell used was an airtight standard three-electrode cell. The working microdisc electrode was a 10 µm diameter Pt electrode (Cypress Systems Inc., Kansas, USA). The auxiliary electrode was a 0.5 mm diameter platinum wire (Goodfellow Cambridge Ltd., Cambridge, UK). A ferrocene/ferrocenium hexafluorophosphate (Fc/FcPF6) reference electrode has been developed for use in THF.28-30 It consists of a platinum wire partially immersed in a solution of 4 mM equimolar ferrocene/ferrocenium hexafluorophosphate (Fc/FcPF6) and 0.3 M tetra n-butylammonium hexafluorophosphate (TBAF) in THF. The reference electrode is housed in a distinct compartment, separated from the cell solution compartment by a porous frit. The working microelectrode was polished on a clean polishing pad (Kemet, U.K.) using 1.0 and 0.3 µm aqueous alumina slurries (Beuhler, Lake Buff, IL, U.S.A.), and subsequently rinsed in deionised and doubly filtered water. The radius of the disk microelectrode was electrochemically calibrated before each experiment by recording a voltammogram corresponding to the ferrocene oxidation at 0.01 V s-1 in a 3 mM ferrocene and 0.1 M TBAF solution in acetonitrile. The diffusion coefficients assumed for ferrocene in acetonitrile is 2.3 × 105 cm2 s-1 at 298 K.31 4.3. Procedure. All electrochemical experiments were performed in a Faraday cage. The solutions were degassed with argon for 1-3 min to remove oxygen traces and an inert argon atmosphere was maintained throughout all measurements using inflatable plastic balloons filled with argon. For each voltammogram, the background signal was recorded by performing the voltammetry in the absence of analyte. The background voltammogram was subtracted from the voltammogram recorded in the presence of the analyte. A similar background correction was performed for the chronoamperograms.

5. Experimental Results and Discussion Cyclic voltammograms were recorded for a 1.13 mM solution of PQ in THF containing TBAP as supporting electrolyte. Experiments were conducted with a Pt microelectrode of radius 5.46 µm. Voltage scan-rates in the range 1-80 V s-1 were studied. A Fc/FcPF6 reference electrode was used. Typical cyclic voltammograms are shown in Figure 8. Two distinct waves can be seen corresponding to the formation of PQ- and PQ2- respectively. Note that as the scanrate increases, the voltammograms changes from almost steady-state with no back peaks to full peak shaped voltammetry with prominent back peaks. That is, the diffusion changes from almost fully convergent to almost fully planar over the range of scan-rates studied. Potential step transients were recorded by stepping the potential from -2.5 V at which no current flows to a value of -2.75 or -3.15 V (vs Fc/FcPF6) corresponding to the diffusionally controlled reduction of PQ to form PQ- or PQ2-, respectively. Typical transients are shown in Figure 9; these were simulated using the parameters in Table 3. Cyclic voltammograms were fitted, to the model described above, across a range of scan rates (1-80 V s-1). Four representative fits for the limits of zero comproportionation and diffusionally controlled reaction are shown in Figure 8. Notice that the mechanism of the homogeneous chemical reaction provides a significantly better fit in all cases. Potential-step transients were fitted for a step to just past peak 1 (Figure 9a) and to beyond peak 2 (Figure 9b). The parameters used for the simulations are again given in Table 3. Note that exactly the same parameters are used to fit the cyclic voltammograms and the potential-step transients. The high level of concordance suggests confidence in the fits. The diffusion coefficient of PQ is consistent with value estimated from the empirical Wilke-Chang expression.32 DPQ is greater than DPQ- and DPQ2- because PQ- and PQ2- are negatively charged and are ion paired to ions in the background electrolyte. PQ2- is doubly charged and is expected to be ion paired with two ions of supporting



Figure 9. Chronoamperometric fits: experiment (-) and simulation (O). (a) Potential step just past peak 1. (b) Potential step beyond peak 2.

TABLE 3: Fitted Data for p-Quaterphenyl parameter

experimental value

RPQ/PQRPQ-/PQ2cPQ Q Ef,PQ/PQ Q Ef,PQ /PQ20 kPQ/PQ 0 kPQ /PQ2DPQ DPQDPQ2kcomp

0.5 unitless 0.5 unitless 1.13 ( 0.03 mM -2.68 V vs Fc/FcPF6 -2.88 V vs Fc/FcPF6 0.35 ( 0.015 cm s-1 0.07 cm s-1 1.14 × 10-5 cm2 s-1 5.1 × 10-6 cm2 s-1 3 × 10-6 cm2 s-1 >106 dm3 mol-1 s-1

electrolyte. In contrast, DPQ is singly charged and will coordinate to only one.20,33 This theory is consistent with the observation that DPQ- is greater than DPQ2-. 0 - is five times greater than k0 The rate constant kPQ/PQ PQ /PQ2-. There is evidence in the literature34 that the geometry of PQ changes with charge as shown in Figure 10. In molecules PQ and PQ-, each ring is out of plane by approximately 40 degrees.34 When electron transfer occurs, forming species PQ2-, the rings move into plane and the bond lengths shorten. This behavior is consistent with Marcus theory35 which states that electron transfer is more rapid in redox couples where the change in molecular geometry is smaller. The voltammetric data in comparison with the simulations shows the existence of comproportionation. In order to estimate the value value of the rate constant kcomp, simulations were carried out for various values of this parameter; values greater than 106 dm3 mol-1 s-1 were voltammetrically indistinguishable from the best fit value and used to simulate the voltammograms -

in Figure 9. Accordingly we conclude that the rate constant for the reaction

PQ + PQ2- f 2PQ-

(33)

is at least 106 dm3 mol-1 s-1. Given the difference in diffusion coefficients and formal potentials, this conclusion is consistent with the simulated data presented in previous sections. 6. Conclusion In this paper we have used simulation studies to show that, in the limit of electrochemical reversibility, the presence of comproportionation, in which A reacts homogeneously with C (A + C h 2B), can be discerned only at high scan-rates when the diffusion coefficients of the species (DA, DB, and DC) are significantly different such that DB/DA > 1.5 and DC/DA > 1.5 or DB/DA < 0.75 and DC/DA < 0.75. The theory is used to provide clear evidence for fast comproportionation in the stepwise two electron reduction of para-quaterphenyl in the solvent THF. Note that the theory given above and the associated discussion assume Q Q and Ef,B/C . In a large separation in the formal potentials Ef,A/B the event that these potentials are closer together we can conclude that even greater differences in the ratios DB/DA and DC/DA may be required to insightfully probe comproportionation kinetics. Acknowledgment. R.B. thanks the EPSRC for support. E.J.F.D. thanks St John’s College, Oxford, for support via a Graduate Scholarship. References and Notes

Figure 10. Schematic representation of the mechanism showing the thought change in molecular geometry.

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