J. Phys. Chem. B 2004, 108, 13047-13051
13047
Marcus Theory for Outer-Sphere Heterogeneous Electron Transfer: Predicting Electron-Transfer Rates for Quinones Neil V. Rees, Antony D. Clegg, Oleksiy V. Klymenko, Barry A. Coles, and Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: May 26, 2004
Steady-state voltammetry is used to measure the heterogeneous electron-transfer rates for the reduction of quinones to determine the dependence of k0 on molecular size, according to Marcus theory. This dependence is then used to predict the electron-transfer rate constants of related quinones, and the predictions are compared to experimental measurements.
κel ) κ0el exp[-B(r′ - σ)]
Introduction In this article, the third in a series of quantitative investigations into Marcus theory for outer-sphere electron transfer1-6 using high-precision measurements of outer-sphere heterogeneous electron-transfer rate constants, we consider the electroreduction of quinones with a view toward predicting the rate constants of quinines of biological interest. Our previous publications have confirmed Marcus theory and demonstrated that there is a relationship between molecular size and the standard electrochemical rate constant for outer-sphere heterogeneous electron transfer and, in particular, that the hydrodynamic radius is a suitable measure of molecular size.7,8 The basic results derived from Marcus’ work for the outersphere electron-transfer process suppose the formation of a precursor complex between the reactant molecule and the electrode surface,9 leading to the following expression for the standard electrochemical rate constant
( )
k0 ) κelKpυn exp
-∆G‡ RT
(1)
where Kp is the equilibrium constant for the precursor complex formation, ∆G‡ is the free energy of activation for the electron transfer, υn is the frequency of crossing the free energy barrier, and κel is the probability of electron tunneling in the transition state.6,9,10 It has been shown that, if the reaction free energy is zero and the “weak overlap” limit is assumed, that is, the electronic coupling is small, then the nuclear frequency can be expressed as10
υn ) τL-1
( ) ∆G‡ 4πRT
1/2
(2)
where τL is defined as τL ) τD∞/s; τD is the experimental Debye relaxation time; and ∞ and s are the high-frequency and static dielectric permittivities, respectively.9 To account for any nonadiabaticity in the electron transfer, which is often observed in outer-sphere reactions,9 the electronic transmission probability, κel, can be stated as * To whom all correspondence should be addressed. Tel.: +44 (0)1865 275 413. Fax: +44 (0)1865 275 410. E-mail: richard.compton@ chemistry.ox.ac.uk.
(3)
where r′ is the molecule-electrode separation, σ is the distance of closest approach of the molecule and electrode, and B is a constant.9 If the free energy of activation is solely due to the outer-sphere reorganization energy
λo )
(
)(
)
NAe2 1 1 1 1 8π0 r 2d op s
(4)
where e is the electronic charge; r is the radius of the molecule; and d is the distance from the reactant to the metal surface,11 which is usually set to infinity following Hale,12 then eq 1 can be expressed as
k0 ) Q
(4πψ ) (1r) 1/2
1/2
[(
exp - Br +
ψ r
)]
(5)
where
Q ) Kpκel-1τL-1 exp[-B(δ - σ)] ψ)
(
)
NAe2 1 1 32π0RT op s
and r′ ) r + δ. We have recently developed a methodology for measuring k0 for fast electron transfers with a view toward testing Marcus theory expressions such as eq 5, using the high-speed channel electrode (HSChE)13-15 to make the necessary precision measurements of electron-transfer rates.7,8 In choosing the hydrodynamic radius as a measure of molecular radius, rather than crystallographic or computed values from the mean spherical or ellipsoidal approximations,10,16 it is assumed that the electron transfer is orientation-independent. In the absence of specific orientational requirements or adsorption, the hydrodynamic radius should be related to the true effective radius by some factor and therefore should show the correct trend for our interpretation. The hydrodynamic radius is conveniently measurable from experimental voltammetry, as it can be simply calculated from the Stokes-Einstein equation17
10.1021/jp040382l CCC: $27.50 © 2004 American Chemical Society Published on Web 08/04/2004
13048 J. Phys. Chem. B, Vol. 108, No. 34, 2004
Rees et al. vitamin K, which plays vital roles in the biochemistry of both green plants and animal/human systems.20-23 In green plants, vitamin K1 has been identified as occurring as an electron acceptor in photosystem II. In human systems, it plays a vital role in the γ-carboxylation of glutamic acid to form γ-carboxyglutamic acid, a precursor of prothrombin and several coagulation proenzymes (factors II, VII, IX, and X). Experimental Section
Figure 1. Structures of (a) coenzyme Q0, (b) coenzyme Q2, and (c) vitamin K1.
r)
kT PπηD
(6)
in which η is the viscosity, D is the diffusion coefficient, and P is either 4 or 6 depending whether the “stick” or “slip” limit is assumed for eq 6.17 Furthermore, eq 5 can be linearized, and also rendered dimensionless for convenience by making the substitutions y ) r/ψ, β ) ψB, K ) k0/k′ and q ) Q/k′, where k′ ) 1 cm s-1. It then becomes
( )
1 q ln(Kxy) + ) -βy + ln y 1xπ
(7)
In this report, we present experimental results for the reduction of quinones in acetonitrile for comparison with eq 7 and subsequent investigation into the practical use of eq 7 for predicting the electron-transfer rates for some quinines of biological interest: coenzymes Q0 and Q2 and vitamin K1 (see Figure 1). The coenzymes Qn are of widespread importance in biological systems, with generally low aqueous solubility. Although coenzymes Q0-Q5 are not naturally occurring, Q10 (ubiquinone-10) is an essential electron and proton carrier within the lipid phase of membranes and has diverse applications for important functions in all cellular membranes.18,19 Of the naturally occurring naphthoquinones, the most important is
Reagents. The chemical reagents were obtained from commercial sources and used as received without further purification. These were 1,4-benzoquinone (BQ), 2,5-dichloro-1,4-benzoquinone (DCQ), 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ), tetrachloro-1,4-benzoquinone (TCQ), vitamin K1 (all Aldrich, 98%), 2-chloro-1,4-benzoquinone (CBQ; Aldrich, 95%), tetrabromo-1,4-benzoquinone (TBQ; Aldrich 90%), anthraquinone (AQ; BDH, GPR grade), 2,6-dimethoxy-1,4-benzoquinone (DMQ; Lancaster, 98%), 3,5-di-tert-butyl-1,2benzoquinone (DTB; Lancaster, >98%), naphthoquinone and 1-aminoanthraquinone (NQ and AAQ, respectively; Aldrich, 97%), coenzymes Q0 (Lancaster, 97%) and Q2 (Sigma, >90%), and tetrabutylammonium perchlorate (TBAP; Fluka, Puriss >99%). The solvent MeCN (Fisher Scientific, >99.99%) was stored over molecular sieves (Linde 5Å, Aldrich) for several hours prior to use and thoroughly degassed with argon (Pureshield Argon, BOC Gases Ltd., U.K.) before experimentation. All solutions contained 0.10 M TBAP as the supporting electrolyte, and all experiments were conducted at a temperature of 293 ( 2 K. Instrumentation and Electrodes. The high-speed channel electrode (HSChE) and pressurized apparatus have been described previously13,14,24 (see Figure 2). High flow rates are achieved by pressurizing a chamber containing the solution and electrode assembly to 1.5 atm. The solution passes through the flow cell (width, d ) 0.200 cm; height, 2h ) 126 µm) and out through one of three capillaries of varying internal bore size to exit at atmospheric pressure. This produces volume flow rates between 0.10 and 3.2 cm3 s-1 (corresponding to linear flow velocities close to the electrode of 0.7-20 m s-1), and the Reynolds number, Re, given by25
Figure 2. Schematic diagram of the channel cell and microband electrode showing the geometrical parameters used.
Marcus Theory for Outer-Sphere Electron Transfer
Re )
3Vf 2hdV
J. Phys. Chem. B, Vol. 108, No. 34, 2004 13049
(8)
can attain maximum values of 9000 under well-defined laminar conditions,13,14 as the channel flow cell has been designed to ensure that these Reynolds numbers are present for less than 2 mm before the electrode whereas a “lead-in” length of ca. 4 mm is needed for the development of turbulent flow.14 Voltammograms are measured by means of a built-in potentiostat at a scan rate of 400 mV s-1 with a platinum microband electrode of length (xe) 40.5 µm. The microband electrode was fabricated by fusing platinum (99.95%, Johnson Matthey plc, London, U.K.) into soda glass according to a literature method,7,14 and the working surface ground and polished to a mirror finish. The microdisk electrode used for steady-state measurements had a working radius (rd) of 12.1 µm. The dimensions of both electrodes were confirmed by electrochemical calibration.26 The electrodes were cleaned with ultrapure water, polished using 0.25-µm alumina slurry on soft lapping pads, and finally rinsed in ultrapure water and dried carefully before use. The counter electrode was a smooth, bright, platinum mesh, and a silver wire (99.95%, Johnson Matthey plc, London, U.K.) was used as a quasi-reference electrode. Analysis of Hydrodynamic Voltammetry. The methodology used to analyze a steady-state voltammogram recorded at the HSChE has been reported.15 First, the current response is normalized by division by the respective limiting current, Ilim, and the middle 60% of the wave is plotted against the potential. The data are then compared by computer to a calculated voltammogram of (I/Ilim) vs E for a selected range of R, k0, and E0f values based on a published analytical solution for quasireversible electron transfer in a channel electrode13,15,27,28
i ) 1 - 2u + 2u2 ln(1 + u-1) irev
(9)
where
u)
0.6783DB2/3(3Vf/4dxeh2)1/3 k0[exp[(1 - R)θ] + (DA/DB)2/3 exp(-Rθ)]
(10)
Figure 3. Levich plot for 1.17 mM TBQ in MeCN/0.1 M TBAP (gradient ) 7.04 × 10-6 A cm-1 s1/3, R2 ) 0.989). Inset shows a typical steady-state linear-sweep voltammogram (Vf ) 1.15 cm3 s-1).
and E0f .61 Contour plots of MSAD as a function of R and k0 and as a function of k0 and E0f were produced, showing the existence of a single minimum in every case. Results and Discussion Reduction of Quinones in Acetonitrile. First, a solution containing 1.17 mM TBQ and 0.10 M TBAP in MeCN was introduced into the pressure chamber of the HSChE apparatus fitted with the 40.5-µm Pt microband electrode, and the solution was thoroughly purged with argon. A linear sweep voltammogram was then recorded at an arbitrary flow rate (1.14 cm3 s-1) that yielded an effectively steady-state response, enabling a limiting current (Ilim) to be measured for the first reduction wave as shown in Figure 3. This was repeated for a range of volume flow rates (Vf) from 0.15 to 3.10 cm3 s-1. Figure 3 also shows a “Levich plot” of measured limiting currents against Vf1/3 according to the Levich equation29 for channel flow cells
( )
Ilim ) 0.925nFw[A]bulk(xeD)2/3
and
irev )
0.925nFw[A]bulk(xeDA)2/3(h2d)-1/3Vf1/3 1 + (DA/DB)
2/3
exp(-θ)
(11)
Vf is the volume flow rate, [A]bulk is the bulk concentration of the electroactive species, Di is the diffusion coefficient of species i, and the geometrical parameters are as given in Figure 2. The quantity irev is the current that would flow if the electrode kinetics were reversible. For each simulated voltammogram, a mean scaled absolute deviation (MSAD) given by
MSAD(R,k0,E0f )
)
N
|Iexp(Ek) - Ith(R,k0,Ek - E0f )|
k)1
Iexp(Ek)
∑
(12)
was calculated as the sum of the differences between each simulated point (Ith) and each experimental point (Iexp). Here, Ek, k ) 1, ..., N, represents the potentials of the experimental data points under analysis, and N is typically above 20. Each simulated voltammogram was therefore assigned its own MSAD value and this enabled a minimum to be found (by the direct search method) corresponding to the optimum values of R, k0,
Vf 2
hd
1/3
(13)
where n, F, Vf, and [A]bulk have their usual meanings and the other geometrical terms are shown in Figure 2. The gradient of this plot was used to obtain a value for the diffusion coefficient of TBQ in MeCN of (1.30 ( 0.18) × 10-5 cm2 s-1, which compares with a literature value of 1.90 × 10-5 cm2 s-1 at 298 K.30 These data were then input into the computer program described above, and optimum values of k0, R, and Ef 0 were found simultaneously for each voltammogram and the mean values taken. Figure 4 shows the 3-dimensional surface plots for the pairs k0, Ef 0 and k0, R as they vary independently with MSAD. The plots show single minima, demonstrating that there is only one set of optimized values. The same procedure was repeated for the reductions of DMQ (1.38 mM), CBQ (1.85 mM), DCQ (1.25 mM), TCQ (1.18 mM), DTB (1.27 mM), DDQ (1.22 mM), BQ (1.03 mM), NQ (1.10 mM), AQ (2.40 mM), and AAQ (1.66 mM). In all cases, excellent Levich plots were obtained (with R2 g 0.985), which yielded values for the respective diffusion coefficients reported in Table 1. The analysis confirmed that these compounds exhibit a simple one-electron oxidation, and the kinetic parameters derived are listed in Table 1 (mean ( standard deviation).
13050 J. Phys. Chem. B, Vol. 108, No. 34, 2004
Rees et al.
Figure 5. Best fit plot of [ln(Kxy + 1/y] vs y for the quinones in MeCN with gradient ) -80.083 and intercept ) 14.949 (R2 ) 0.851). Concentrations are as given in the text, and the compounds are identified as follows: (a) BQ, (b) TCQ, (c) CBQ, (d) NQ, (e) DDQ, (f) AQ, (g) DTB, (h) DCQ, (i) TBQ, (j) DMQ, and (k) AAQ.
TABLE 2: Estimated and Measured Diffusion Coefficients for the Three Biological Quinonesa DWC (×10-5 cm2 s-1) k0,WC (cm s-1) Dexp (×10-5 cm2 s-1) k0,int (cm s-1) k0,exp (cm s-1)
Figure 4. Contour plots for TBQ in MeCN, with Vf ) 2.66 cm3 s-1, showing (a) k0 vs E0f and (b) k0 vs R. Numbers shown on contours are the MSAD values.
TABLE 1: Diffusion Coefficients and Kinetics Parameters Obtained for the Oxidation of Substituted Quinones in 0.1 M TBAP/MeCN D compd (×105 cm2 s-1) AQ BQ NQ AAQ CBQ DCQ DDQ TCQ TBQ DMQ DTB Q0 Q2 K1
1.40 ( 0.17 1.97 ( 0.24 1.52 ( 0.18 1.20 ( 0.14 1.27 ( 0.19 1.58 ( 0.15 1.14 ( 0.17 1.65 ( 0.20 1.25 ( 0.15 1.24 ( 0.15 1.33 ( 0.16 1.51 ( 0.18 1.04 ( 0.13 0.90 ( 0.11
k0 (cm s-1)
R
E0f (V vs Ag)
0.097 ( 0.02 0.13 ( 0.02 0.10 ( 0.02 0.095 ( 0.028 0.12 ( 0.03 0.12 ( 0.03 0.26 ( 0.09 0.20 ( 0.03 0.30 ( 0.08 0.062 ( 0.01 0.073 ( 0.02 0.11 ( 0.02 0.029 ( 0.007 0.032 ( 0.003
0.59 ( 0.02 0.55 ( 0.04 0.47 ( 0.03 0.53 ( 0.01 0.67 ( 0.04 0.56 ( 0.04 0.52 ( 0.04 0.55 ( 0.04 0.49 ( 0.02 0.62 ( 0.02 0.74 ( 0.02 0.55 ( 0.03 0.52 ( 0.04 0.70 ( 0.02
-1.004 ( 0.004 -0.626 ( 0.005 -0.668 ( 0.009 -1.023 ( 0.008 -0.459 ( 0.017 -0.293 ( 0.006 +0.204 ( 0.006 -0.137 ( 0.003 -0.114 ( 0.007 -0.751 ( 0.009 -0.720 ( 0.007 -0.750 ( 0.005 -0.606 ( 0.014 -0.942 ( 0.007
Interpretation of Results. These results can be interpreted by means of eq 7, and a graph of this function is shown in Figure 5, yielding a linear plot with a gradient of 80.083 and R2 ) 0.851. From this plot, values of Q and B can be extracted using the known value for ψ of 37.3 Å. This gives B ) 2.15 Å-1 and Q ) 1.35 × 107 cm s-1, which compare reasonably with other values for B of 1-2,9 1.37,7 and 2.31 Å-1.8 Prediction of Electron-Transfer Rates. The linear plot in Figure 5 can now be used to predict the heterogeneous rate constant, k0, for a quinone molecule with a known hydrodynamic
Q0
Q2
K1
1.67 0.15 1.51 0.16 0.11 ( 0.02
1.10 5.7 × 10-2 1.04 3.8 × 10-2 (2.9 ( 1.0) × 10-2
0.86 7.0 × 10-3 0.90 1.2 × 10-2 (2.7 ( 0.3) × 10-2
a DWC is the Wilke-Chang estimate for the diffusion coefficient, k0,WC is the rate constant inferred from Figure 5 based on DWC, Dexp is the experimentally measured diffusion coefficient, k0,int is the rate constant inferred from Dexp, and k0,exp is the experimentally measured rate constant.
radius provided that the internal reorganization energy (λi) does not make a significant contribution to the total reorganization energy (i.e., λo . λi). The hydrodynamic radius can be obtained from the diffusion coefficient, which is measured directly or can indeed be estimated using, for example, the Wilke-Chang method. Three test cases were selected for estimation of k0, and then subsequent experimental investigations were performed to measure the actual k0. These molecules were the coenzymes Q0 and Q2 and vitamin K1. As a first step, diffusion coefficients were estimated for these molecules using the Wilke-Chang expression, DWC
DWC ≈
(7.4 × 10-8)Txφms ησ0.6
(14)
where T is the absolute temperature, φ is the solvent affinity factor (unity for aprotic solvents), ms is the molecular mass of the solvent, η is the solvent viscosity, and σ is the molecular volume. The vales for the estimated diffusion coefficients heterogeneous rate constants are reported in Table 2. Next, experiments were conducted on these compounds using the HSChE method for Q0 and steady-state microdisk linear-sweep voltammetry for Q2 and K1. Microdisk voltammetry was selected for the latter because of the small masses of compounds available at commercial prices. Values for the diffusion coefficients Dexp could then be found from the limiting currents using the relation
Ilim ) 4nF[A]bulkDrd
(15)
Marcus Theory for Outer-Sphere Electron Transfer
J. Phys. Chem. B, Vol. 108, No. 34, 2004 13051 Acknowledgment. We thank the Clarendon Fund for partial funding for O.V.K. and both the EPSRC for a studentship and Avecia Ltd. for CASE support for N.V.R. References and Notes
Figure 6. As for Figure 5 but with the following data added: (v) Q0, (w) Q2, (x) K1.
and are given in Table 2 along with the interpolated value for the heterogeneous rate constant, k0,exp. Then, the microdisk voltammograms were analyzed using the Mirkin and Bard method 31 to find the experimental results for k0, included in Table 2 and displayed in Figure 6. As can be seen, the estimated values for k0 compare well with the experimentally determined values. The function in eq 7 is particularly sensitive to k0 for values of D < 1 × 10-5 cm2 s-1, thus apparently producing relatively large errors between estimated and actual rate constants for slow diffusion coefficients. Even in the absence of data on diffusion coefficients in the solvent of interest, the Wilke-Chang estimation for the diffusion coefficient can be used to obtain a reasonable value for k0. Conclusions Hydrodynamic and stationary electrodes have been used to measure the heterogeneous electron-transfer rates for the reduction of quinones under steady-state conditions to determine the dependence of k0 on the hydrodynamic radius according to Marcus theory. This dependence has then been shown to be useful in predicting the rate constants of related quinones, which, for reasons such as low solubility or electrode passivation, can be experimentally difficult to directly determine. This can be done with or without knowledge of the experimental value of the diffusion coefficient.
(1) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (2) Marcus, R. A. J. Chem. Phys. 1956, 24, 979. (3) Marcus, R. A. J. Chem. Phys. 1957, 26, 867. (4) Marcus, R. A. J. Phys. Chem. 1963, 67, 853. (5) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (6) Marcus, R. A. Int. J. Chem. Kinet. 1981, 13, 865. (7) Clegg, A. D.; Rees, N. V.; Klymenko, O. V.; Coles, B. A.; Compton, R. G. J. Am. Chem. Soc. 2004, 126, 6185. (8) Clegg, A. D.; Rees, N. V.; Klymenko, O. V.; Coles, B. A.; Compton, R. G. ChemPhysChem 2004, in press. (9) Weaver, M. J. In ComprehensiVe Chemical Kinetics; Compton, R. G., Ed.; Elsevier: 1987; Vol. 27, p 1. (10) Opallo, M. J. Chem. Soc., Faraday Trans. 1 1986, 82, 339. (11) Calef, D. F.; Wolynes, P. G. J. Phys. Chem. 1983, 87, 3387. (12) Hale, J. M. In Reactions of Molecules at Electrodes; Hush, N. S., Ed.; Wiley: London, 1971. (13) Rees, N. V.; Alden, J. A.; Dryfe, R. A. W.; Coles, B. A.; Compton, R. G. J. Phys. Chem. 1995, 99, 14813. (14) Rees, N. V.; Dryfe, R. A. W.; Cooper, J. A.; Coles, B. A.; Compton, R. G.; Davies, S. G.; McCarthy, T. D. J. Phys. Chem. 1995, 99, 7096. (15) Rees, N. V.; Klymenko, O. V.; Coles, B. A.; Compton, R. G. J. Electroanal. Chem. 2002, 534, 151. (16) Grampp, G.; Jaenicke, W. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 325. (17) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry 1: Ionics, 2nd ed.; Plenum Press: New York, 1998; Vol. 1. (18) Morton, R. A. In Biochemistry of Quinones; Morton, R. A., Ed.; Academic Press: London, 1965; p 1. (19) Crane, F. L.; Navas, P. Mol. Aspects Med. 1997, 18, 1. (20) Hart, J. P.; Shearer, M. J.; McCarthy, P. T.; Rahim, S. Analyst 1984, 109, 477. (21) Hart, J. P.; Wring, S. A.; Morgan, I. C. Analyst 1989, 114, 933. (22) Ksenzhek, O. S.; Petrova, S. A.; Kolodyazhny, M. V.; Oleinik, S. V. Bioelectrochem. Bioenerg. 1977, 4, 335. (23) Olson, R. E. Annu. ReV. Nutr. 1984, 4, 281. (24) Coles, B. A.; Dryfe, R. A. W.; Rees, N. V.; Compton, R. G.; Davies, S. G.; McCarthy, T. D. J. Electroanal. Chem. 1996, 411, 121. (25) Brett, C. M. A.; Oliveira Brett, A. M. C. F. In ComprehensiVe Chemical Kinetics; Bamford, C. H., Compton, R. G., Eds.; Elsevier: Amsterdam, 1986; Vol. 26, p 355. (26) Brookes, B. A.; Lawrence, N. S.; Compton, R. G. J. Phys. Chem. B 2000, 104, 11258. (27) Blaedel, W. J.; Klatt, L. N. Anal. Chem. 1966, 38, 879. (28) Klatt, L. N.; Blaedel, W. J. Anal. Chem. 1967, 39, 1065. (29) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (30) Wang, R. L.; Tam, K. Y.; Compton, R. G. J. Electroanal. Chem. 1997, 434, 105. (31) Mirkin, M. V.; Bard, A. J. Anal. Chem. 1992, 64, 2293.
