Influence of the Work Function on Electron Transfer Processes at Metals

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Influence of the Work Function on Electron Transfer Processes at Metals: Application to the Hydrogen Evolution Reaction S. Harinipriya and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras - 600036, India Received January 17, 2002. In Final Form: April 24, 2002 The exchange current density and the corresponding free energy of activation for the hydrogen evolution reaction at metal surfaces is estimated using the work function of the substrate, desolvation energy of the reactant, solvation numbers, and so forth. The computed values are in satisfactory agreement with experimental data for a large number of metals.

1. Introduction The analysis of electron-transfer reactions at the electrode/electrolyte interface is a topic of much current interest on account of its implications in various electrochemical technologies.1 Recent theoretical investigations have focused attention on (i) the role of dielectric polarization,2,3 (ii) the magnitude of solvent reorganization energies,4,5 and (iii) the nature of coupling of the reactant with electrode surfaces.6-8 In contrast to homogeneous redox reactions, electron-transfer processes at metals warrant a detailed description of the orientational states of solvent dipoles and electronic properties of electrodes. Consequently, phenomenological thermodynamic approaches provide insights into factors that govern the reaction rate and serve as a first step in delineating the influence of constituents comprising the electrochemical system. Further, the ferric/ferrous reaction at different metals has been investigated using diverse molecular dynamics simulations in conjunction with the AndersonNewns Hamiltonian4,6 on account of its outer sphere9,10 nature as well as implications in alkaline storage batteries.11 It follows from the above that an adequate description of electron-transfer processes at electrodes requires inclusion of the metal characteristics as well as interfacial solvent organization. The hydrogen evolution reaction (HER) represented as

H3O+ + e S (1/2)H2 + H2O

(1)

occurring in acidic media is of considerable significance in hydrogen embrittlement, fuel cells, electrodeposition * To whom correspondence should be addressed. E-mail: [email protected]; mvs @chem.iitm.ac.in. Fax: 91-044 2352545; 2350509. (1) See for example: Conway, B. E. Electrochemical Supercapacitors; Kluwer Academic/Plenum Publishers: New York, 1999; Chapter 3. (2) Song, X.; Marcus. R. A. J. Chem. Phys. 1993, 99, 7768. (3) Ulstrup, J. Charge transfer in condensed media; SpringerVerlag: Berlin, 1979. (4) Straus, B. J.; Calhoun, A.; Voth, G. A. J. Chem. Phys. 1995, 102, 529 and references therein. (5) Harting, C.; Koper, M. T. M. J. Chem. Phys. 2001, 115, 8540 and references therein. (6) Rose, A. D.; Benjamin, I. J. Chem. Phys. 1994, 100, 3545. (7) Goasvi, S.; Marcus, R. A. J. Phys. Chem. B 2000, 104, 2067. (8) Robert, J. F.; Loughman, P.; Keyes, T. E. J. Am. Chem. Soc. 2000, 122, 11948. (9) Marcus, R. A. J. Chem. Phys. 1993, 98, 5604 and references therein. (10) Schmickler, W. Chem. Phys. Lett. 1995, 237, 152. (11) Hamman, C. H.; Hamnett, A.; Vielstich, W. Electrochemistry; Wiley-VCH: New York, 1998; Chapter 9.

reactions, and so forth.11-13 Since a wide variety of metals and alloys have been employed in studying HER,1,14 it serves as a prototype for analyzing the influence of substrates upon electrode kinetic parameters. Although HER has been interpreted using (i) the nature of ratedetermining steps,12-16 (ii) bond formation between the electrode and adsorbed hydrogen atoms,12,18 (iii) the work function of metals,12-16,19,20 and (iv) electronic overlap between the electrode density of states and the reactant species,21,22 the explicit parametric dependence of the exchange current density (i0) on the reactant characteristics and properties of the substrates is till now unavailable. Further, such a formalism becomes amenable for generalization, thereby providing a complementary version to detailed microscopic theories. The essential feature of the electrode kinetic processes lies in the use of the current (i) versus potential (E) relation23 and the subsequent extraction of parameters such as exchange current density (i0) and transfer coefficient (β). The explicit i-E dependence formulated by Butler24a and Erdey-Gruz and Volmer24b is customarily referred to as the Butler-Volmer equation. While the exchange current density is a measure of the reaction rate under equilibrium, the transfer coefficient β describes the symmetry of the potential energy barrier as a function of the applied electric field.13 The transfer coefficient β being approximately 0.5 for most electron-transfer processes, the detailed analysis of i0 has been an essential focus in this context. Thus several correlations for the variation of the exchange current density with (a) work function, (b) metal-hydrogen bond strength, and (c) percentage d-character have been sought during the past few decades, using the experimental data. While these methodologies emphasize the role of adsorption isotherms, (12) Trasatti, S. J. Electroanal. Chem. 1972, 39, 163. (13) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Kluwer Academic/Plenum: New York, 2000; Vol. 2B, p 1191. (14) Conway, B. E.; Bockris, J. O’M. J. Chem. Phys. 1957, 26, 532. (15) Bockris, J. O’M.; Parsons, R. Trans. Faraday Soc. 1951, 47, 914. (16) Parsons, R. Trans. Faraday Soc. 1958, 54, 1053. (17) Parsons, R. Surf. Sci. 1964, 2, 418. (18) Krishtalik, L. I. In Advances in Electrochemistry and Electrochemical Engineering; Delahay, P., Ed.; Interscience: New York, 1970; Vol. 7. (19) Bockris, J. O’M. Trans. Faraday Soc. 1947, 43, 417. (20) Kita, H. J. Electrochem. Soc. 1966, 113, 1095. (21) Sakata, T. Bull. Chem. Soc. Jpn. 1996, 69, 2435. (22) Sakata, T. Bull. Chem. Soc. Jpn. 2000, 73, 297. (23) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Kluwer Academic/Plenum: New York, 2000; Vol. 2B, p 1052. (24) (a) Butler, J. A. Trans. Faraday Soc. 1924, 19, 729. (b) ErdeyGruz, T.; Volmer, M. Z. Phys. Chem. 1930, 150, 203.

10.1021/la025548t CCC: $22.00 © 2002 American Chemical Society Published on Web 06/19/2002

Influence of Work Function on Electron Transfer

Langmuir, Vol. 18, No. 14, 2002 5573

the nature of rate-determining steps (cf. Tafel, Heyrovsky and Volmer mechanisms),25 and so forth, the evaluation of i0 using properties associated with the constituents of an electrochemical system (electrode, ionic reactant, and dipolar solvent) has remained elusive. This limitation is particularly crucial for HER which plays a central role in electrocatalysis pertaining to batteries and fuel cells,11,13 supercapacitors,1 and so forth. Consequently, strategies which are tailor-made to evaluate electrode kinetic parameters from control variables and system characteristics become valuable. The exchange current density can also be interpreted using the perspective offered by nonequilibrium statistical thermodynamics.26 Hence, any thermodynamic-inspired approach to electron-transfer reactions at electrode surfaces will be useful in analyzing the kinetics of electrode processes27,28 using Onsager’s fluxforce formalism29 and thereby estimating entropy production for these processes. In this communication, we illustrate (i) the validity of an explicit expression for the exchange current density for HER incorporating the work function of electrodes and desolvation energy of the reactant, (ii) the linear dependence of the free energy of activation on the work function of the metal, (iii) the rationalization of the volcano plot (deduced from the experimental data) between log i0 and M-Hads bond formation energies (EM-H), and (iv) the variation of the solvent reorganization energy with the nature of the metal. Our essential input in this analysis consists of identifying various processes constituting the phenomena and representing the interfacial parameters in terms of the corresponding bulk quantities in an approximate manner. The applicability of the present formalism to the ferric/ferrous reaction has been recently demonstrated.30 2. Exchange Current Density and Free Energy of Activation Within the classical transition state theory of electrontransfer processes at electrodes, the frequency of electron transfer, B k (in s-1) is given by31-33

B k)

{

}

kBT ∆Gq exp h RT

(2)

where ∆G q is the free energy of activation (eq 88a of Eyring et al.33); kB and h represent the Boltzmann constant and the Planck constant, respectively. T denotes the absolute temperature, R being the universal gas constant. The corresponding velocity b v may be represented as

b v)B k CR′ exp{-βnFη/RT}

(3)

where CR′ is the concentration of the reactant per unit area of the electrode in mol cm-2 (eq 92 of Eyring et al.33). n denotes the number of electrons, while η is the overpotential. Representing CR′ as CR/A, where CR denotes the bulk concentration of the reactant in moles, A being (25) Delahay, P. Double Layer and Electrode Kinetics; Interscience Publishers: New York, 1966; Chapter 4. (26) See for example: Keizer, J. Statistical thermodynamics of nonequilibrium processes; Springer-Verlag: New York, 1987; p 238. (27) Aldrin Denny, R.; Sangaranarayanan, M. V. J. Phys. Chem. 1998, B102, 2131. (28) Aldrin Denny, R.; Sangaranarayanan, M. V. J. Phys. Chem. 1998, B102, 2138. (29) See for example: Prigogine, I. Thermodynamics of irreversible processes, 3rd ed.; Interscience: New York, 1967; Chapter 5. (30) Harinipriya, S.; Sangaranarayanan, M. V. J. Chem. Phys. 2001, 115, 6173. (31) See for example: Bard, A. J.; Faulkner, L. R. Electrochemical methods, 2nd ed.; John Wiley and Sons: New York, 2001; p 91.

the area of the electrode, we may write the equilibrium velocity of the reaction b ve as

b v e ) (kBTCR/Ah) exp{-∆Gqeq/RT} exp{-βnFEe/RT} (4) and since

be i0 ) nFv

(5)

i0 may be written as

i0 ) (nFkBTCR/Ah) exp{- ∆Gqeq/RT} exp{-βnFEe/RT} (6) when the overpotential is zero. The above equation is identical with eq 101 of Eyring et al.33 where the reactant concentration is expressed in moles. Further, the above representation assumes that the transmission coefficient k is unity.34 In the case of HER, Ee equals zero. To estimate i0, it is imperative to consider various types of coupling such as metal-solvent, electrode-reactant, and reactant-solvent with the help of a schematic representation of the processes depicted in Scheme 1. Hydronium ions are not considered here since the reactant gets rid of its solvation sheath while getting transported from bulk to the reaction zone. In our visualization, the unsolvated H+ ions undergo reduction at the interface. This is consistent with the recent postulate of Bockris et al.13 according to which electron-transfer processes at electrode surfaces involve unsolvated reactants. It follows from Scheme 1 that the free energy of activation consisting of various work terms along with the contribution from the electrode surface may be represented as30

∆Gqeq ) ∆Get +

(wr + wp) 2

(7)

where ∆Get is the Gibbs free energy change involved in the transfer of electrons from the metal to the reactant. The above equation is reminiscent of the customary interpretation of the free energy of activation arising from the classical Marcus theory35,36 which consists of work terms and solvent influences. wr is composed of the work done in bringing the reactant from (a) bulk to the outer Helmholtz plane (OHP) which involves the desolvation energy ∆Gint er and (b) the OHP to the reaction zone dictated by the surface potential χr. Analogous considerations apply for wp. Such a partitioning of the free energy of activation and subsequent formulation of the standard exchange current density have been demonstrated to yield satisfactory agreement in the case of the ferric/ferrous reaction at different metals.30 Here we extend the results derived therein for the hydrogen evolution reaction represented by eq 1. Although the mechanistic analysis of the ferric/ferrous reaction and the hydrogen evolution reaction show significant differences regarding electrodereactant coupling, there is also an isomorphism existing at a phenomenological level of description (cf. section 4). (32) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Kluwer Academic/Plenum: New York, 2000; Vol. 2A, eqs 7.5 and 7.12. (33) Glasstone, S.; Laidler, K. J.; Eyring, H. The theory of rate processes; McGraw-Hill, New York, 1941; pp 584-588. (34) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press, ELBS: Oxford, 1994; p 940. (35) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (36) Marcus, R. A. J. Chem. Phys. 1965, 43, 679.

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Harinipriya and Sangaranarayanan

Scheme 1. Schematic Representation of HER Showing Solvated H+ Ions Reaching the Solvated Electrode Surface Subsequent to Desolvation

Scheme 2. Representation of Work Function Changes in Solvent Medium

and i0 are as follows:30

∆Gqeq )

and

(

(

)(

)

∆Gr-s zrFχes ∆Gp-s zpFχes + + 2SNr 2 2SNp 2 nFξΦM (8)

) (

∆Gp-s nFCRkBT ∆Gr-s zrχes + i0 ) exp Ah 2RTSNr 2RT 2RTSNp

)

βnFEe zpχes nFξΦM + (9) 2RT RT RT

For example, the reactant characteristics are taken into account via desolvation energy vis a vis wr while the electrode influence is incorporated using the work function (which can be expressed in terms of metallic electron density with the help of jellium models37), and these seem sufficient to estimate the required parameter of interest, viz., exchange current density. An essential input in our methodology is the representation of the work function of the solvated electrode (ΦMs) in terms of the work function of the metal (ΦM) and the solvation free energy of electrons in solution (ΦS) (cf. Scheme 2) whose value is estimated to be30 0.83 eV. ΦMs is defined as ΦMs ) ΦM - ΦS. Since ΦM for Pt ranges from ∼5.03 to 5.64 eV depending upon the experimental technique employed, ΦMs may range from 4.2 to 4.81 eV. This quantity may be considered to be analogous to the work function of the standard hydrogen electrode (SHE) having a value of 4.5 ( 0.2 eV, reported by Trasatti.38 Further, ΦMs ) ξΦM, where ξ is a dimensionless constant30 independent of the nature of the metal which equals ca. 0.17 and ΦM is the work function of the metal. For a general electron-transfer scheme represented as reactants + ne T products, the expressions for ∆Gqeq (37) Smith, J. R. Phys. Rev. 1969, 181, 522. (38) Trasatti, S. In Advances in Electrochemistry and Electrochemical Engineering; Gerisher, H., Tobias, C. W., Eds.; John Wiley and Sons: New York, 1976; Vol. 10, p 298.

(cf. eqs 14 and 16 of ref 30). In eqs 8 and 9, ∆Gr-s and ∆Gp-s represent the desolvation energy of the reactant and product; zr and zp denote the charge of the reactant and product, χes being the surface potential of electrons in solution while SNr and SNp denote the solvation numbers of the reactant and product, respectively. The corresponding equations pertaining to HER follow from the above since the product is gaseous hydrogen and consequently wp is zero, since the formation of H-H bonds and the breaking of M-H bonds occur simultaneously and the gaseous product (H2) does not enter the solution, rather escaping through the solution/vacuum interface. Thus wp is assumed to be zero and the terms involving the products do not arise. With this proviso and zr being unity from the stoichiometry of eq 1, we obtain

∆Geq(HER)q )

∆GH+-S zH+Fχse - nFξΦM + 2SNH+ 2

(10)

and

i0(HER) )

(

) (

)

zH+Fχse FξΦM ∆GH+-s FCH+ kBT + exp (11) Ah 2SNH+RT 2RT RT

from eqs 8 and 9, respectively. Under electronic equilibrium,38 the surface potential of electrons in solution equals that of electrons in metal (χes


Langmuir, Vol. 18, No. 14, 2002 5575

) χeM). While an exact value of χeM is unavailable even within stabilized jellium models,39 we employ an approximate value of χes reported as ca. -0.4 eV which holds good when surface anisotropic effects are ignored.38 The calculation of ΦM using eq 11 requires parameters such as ∆GH+-s (11.98 eV),40 the solvation number of H+ (SN ) 5),40 and the work function (ΦM) pertaining to various metals.12 If the fundamental constants kB, h, and F are substituted appropriately and CH+ is represented (in moles) while the area of the electrode is in cm2 at a temperature of 298 K, we obtain

∆Gqeq ≈ 1 - ξΦM

(12)

and

( )

i0 ) 9.9 × 10-7

CH+ exp{-38.9 + 6.6ΦM} (13) A

leading to ∆Gqeq in eV and i0 in A cm-2. 3. Discussion Equations 12 and 13 constitute the central results of the analysis. Since ξ has a value of 0.17 irrespective of the nature of metals as mentioned earlier,30 ∆Gqeq decreases with ΦM linearly. While the tabular compilation of ∆Gqeq pertaining to HER is till now not available, thus precluding a quantitative verification of eq 12 for different metals, ∆Gqeq for HER at an indium electrode is deduced as 0.31 eV from polarization studies.41,42 Since the work function of indium12 is 4.08 eV, eq 12 yields ∆Gqeq as 0.30 eV, in agreement with the experimental data. Interestingly, the dependence of ∆Gqeq upon ΦM of the type predicted by eq 12 has been observed from the experimental data for the ferric/ferrous reaction at various metals.43,44 Thus it may be inferred that the free energy of activation at zero overpotential varies linearly with the work function for electron-transfer processes occurring at metals. Since the exchange current densities for a variety of sp and d metals have been deduced12 from experimental studies in the case of HER, it becomes possible to verify eq 13. A comparison between log i0 calculated using eq 13 with the experimental data is shown in Table 1 (all the calculations reported here were performed using MATLAB version 5.3). While the general agreement with the experimental data is satisfactory, a few aspects regarding the validity of the parameters employed in the calculation of i0 need to be emphasized here: (a) Reactant concentration: The double-layer corrections arising via the Frumkin ψ effect,31 viz., Cs ) Cb exp{-zeφ2/kBT} (where Cs and Cb denote the surface and the bulk concentration of the reactant, respectively, while φ2 is the Gouy-Chapman potential), may play a crucial role when low concentrations of the electrolyte are employed. In Table 1, the electrolytes pertaining to HER at Cd, Cu, Mn, Hg, and Pb employ a concentration of 0.001, 0.05, 0.005, 0.01, and 0.01 M, respectively, and hence i0 estimates in these cases may be inaccurate. In all the cases considered in Table 1, CH+ is converted into moles (39) Perdew, J. P.; Tran, H. Q.; Smith, E. D. Phys. Rev. 1990, B42, 11627. (40) Bockris, J. O’M.; Conway, B. E. Modern Aspects of Electrochemistry; Academic Press: New York, 1954; Vol. 1, Chapter 2. (41) Butler, J. N.; Dienst, M. J. Electrochem. Soc. 1965, 112, 226. (42) Hush, N. S. Trans. Faraday Soc. 1961, 57, 557. (43) Bockris, J. O’M.; Mannan, R. J.; Damjanovic, A. J. Chem. Phys. 1968, 48, 1898. (44) Galizzioli, D.; Trasatti, S. J. Electroanal. Chem. 1973, 44, 367.

from the molar concentrations of the electrolyte, and these are reported in Table 1. (b) Work function of the metal: Different ΦM values45 have been reported for several metals. In view of this, we have provided i0 arising from each of these in Table 1. In general, the work functions reported by Trasatti12,38 yield excellent agreement with the experimental data. (c) Area of the electrodes: The area of the electrodes in eq 13 has been extracted from the experimental data; whenever the same has not been reported, A was assumed to be 0.001 cm2. While this assumption may quantitatively alter the agreement, the qualitative trend predicted by eqs 12 and 13 is still valid. (d) Solvation number of H+: While the solvation number of H+ varies40 between 1 and 5, depending upon the experimental methods (compressibility, ultrasonic measurements, etc.), we have employed SN for H+ as 5. However, the order of the magnitude of i0 is not affected significantly, if other values of solvation numbers are used. In fact, the incorporation of solvation numbers may directly reflect the extent of dehydration of H+. 3.1. Dependence of the Work Function and Free Energy of Activation on the Percentage of d Character. The exchange current density of HER is amenable for interpretation in terms of % d character of metals,14 and hence it is instructive to enquire whether the present methodology yields some new insights into this behavior. The % d character represents the number of unpaired electrons in the d band of a metal and denotes the ease with which it donates electrons.14 In the study of HER for Pt, Pd, Ta, and Ti, it has been shown that log i0 increases with % d character. Since an increase in % d character is associated with an increase in the work function, eq 13 is consistent with the experimental correlation and analogously, eq 12 describes the decrease of ∆Gqeq with an increase in % d character. 3.2. Rationalization of the Volcano Plot. The variation of log i0 with EM-H (the enthalpy of M-H bond formation) provides a volcano plot and is customarily interpreted using the interaction of adsorbed reactant species with the metal surface. A plot of log i0 estimated using eq 13 versus EM-H from tabular compilations18,46 yields a volcano plot, as anticipated (cf. Figure 1); however, new insights emerge regarding the origin of the volcano plot, in view of the explicit expression for ∆Gqeq and i0. Although eq 13 does not contain EM-H per se, empirical correlations of EM-H with work functions have been proposed for sp and d metals on the basis of experimental data,12 viz., EM-H ≈ 120-11ΦM for d metals and EM-H ≈ 26ΦM - 54 for sp metals where the numerical factors yield18 EM-H in kcal mol-1. Substitution of the above equations in eq 13 leads to

[ ]

CH+ - 0.26EM-H A

(14)

[ ]

(15)

log i0 ) 8.53 + log and

log i0 ) -29.9 + log

CH+ - 0.11EM-H A

for d and sp metals, respectively. The above equations imply that when EM-H is small (from Hg, Zn, Cd, Pd, Au, Ag, Ta, Ti, etc. till Pt) log i0 increases and reaches a (45) CRC Handbook of Chemistry and Physics, 80th ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 2000; pp 12-126. (46) Although diverse estimates of EM-H are available, it is customary to employ the values reported by Krishtalik (ref 18 above) in this context.

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Table 1. Estimation of Exchange Current Density for HER Using Equation 13 of the Text metals Zn (ref 53a) Ag (ref 53b) Au (ref 53b) Cd Cr Cu Mn Ni (ref 53b) Fe Pt Ti Co Hg (ref 53a) Mo Nb Pd Rh Os Ir Re Ru Ta (ref 53b) W (ref 53b) Bi Al (ref 53a) Ga In (ref 41) Pb (ref 53a) Sb Sn Tl a

ΦM (eV) 3.63a 4.30b 4.90a 4.30b 4.78b 4.08a 4.12b 4.40b 4.50a 4.70b 3.90b 4.10a 4.73b 4.65b 5.03b 5.64a 4.10b 4.33a 4.70b 5.00a 3.90a 4.48a 4.50b 4.30b 4.60a 4.20b 4.00a 5.01b 5.22a 4.98a 4.99b 5.25a 4.83b 5.23a 5.93a 4.97b 5.30a 4.72a 4.95b 5.50a 4.71a 4.80b 5.00a 4.22b 4.25a 4.55b 4.10a 4.34a 4.36b 4.78b 4.25b 4.32a 4.08b 4.09a 4.29a 4.18b 4.25a 4.55a 4.56b 4.15a 4.35b 4.42a 3.84a 4.02b

CH+ (mol)

log i0exptal (ref 12) (i0 in A cm-2)

-log i0 cald (i0 in A cm-2)

µeM (eV) (ref 38)

0.1 M H2SO4

0.2

10.50

-3.97

0.1

1 M H2SO4 0.5 M H2SO4 0.001 M H2SO4

2 1 0.002

7.90 6.50 11.60

-3.97 -4.37 -3.82

0.001 0.001 0.001

0.5 M H2SO4

1

-4.05

0.001

0.05 M H2SO4 0.005 M H2SO4

0.1 0.01

12.19 10.27 8.55 7.27 6.20 10.90 10.79 7.29 7.00 7.29 10.72 10.15 5.34 5.57 4.50 2.73 7.85 7.19 5.43 4.57 12.42 10.76 10.70 8.57 7.71 7.86 8.43 4.66 4.06 4.62 4.60 3.85 5.18 4.12 2.00 4.65 3.71 5.49 4.84 3.26 5.70 5.44 4.87 8.50 8.42 6.56 7.85 7.16 7.10 8.20 8.42 8.22 10.2 10.18 9.60 11.92 11.72 5.98 5.95 7.70 7.13 6.93 9.89 9.38

-4.30 -3.64

0.001 0.001

-4.33 -4.26 -4.58

0.001 0.001 0.001

-3.80

0.001

-4.26

0.001

-4.14

0.1

-3.97

0.001

-3.89

0.001

-4.56

0.001

-4.54

0.001

-4.41

0.001

-4.53

0.001

-4.51

0.001

-4.38

0.001

-3.90

0.001

-4.18 -4.02

0.001 0.001

-4.37 -3.93

0.1 0.001

-3.79

0.1

-3.87

0.1

-4.19

0.001

-4.01

0.001

-3.74

0.001

electrolyte employed (ref 12)

7.00 7.80 10.90

5 M H2SO4 5 M H2SO4 5 M H2SO4

10 10 10

5.25 5.60 3.00

1 M H2SO4

2

8.30

5 M H2SO4

10

5.30

0.01 M H2SO4

0.02

12.30

0.1 M HCl

0.1

7.30

0.5 M H2SO4

1

8.40

7.5 M HClO4

7.50

3.10

5 M H2SO4 7.5 M HClO4 5 M H2SO4

10 7.50 10

3.50 4.10 3.60

7.5 M HClO4

7.50

3.00

5 M HCl

5

4.20

0.1 M H2SO4

0.2

8.50

1 M H2SO4 1 M H2SO4

2 2

6.40 7.80

0.5 M H2SO4 0.1 M H2SO4

1 0.2

8.00 8.40

0.5 M H2SO4

1

9.50

0.01 M HCl

0.01

11.40

7.5 M HClO4

7.50

5.10

1 M H2SO4

2

7.80

0.1 M HClO4

0.1

9.60

area of the metal (cm2)

Weast (ref 45). b Trasatti (refs 12 and 38).

maximum for Pt. On the other hand, for larger EM-H (metals such as Pt, Rh, Ir, and W) log i0 decreases as can be seen from eqs 14 and 15 thus yielding the decreasing portion of the volcano plot. The interpretation of the volcano plot also arises from eq 12 which yields the free energy of activation ∆Gqeq as (1-0.17ΦM). In the series Hg, Cd, In, Ti, Au, Ag, and Pt, the work function increases

from Hg to Pt and hence ∆Gqeq decreases, whereas for metals such as Pt, Rh, Sb, Mo, Ga, and so forth, ΦM decreases from Pt to Ga and consequently, an increase in ∆Gqeq is observed. We reiterate that the present analysis has led to the explicit dependence of exchange current density upon EM-H via eqs 14 and 15. One may wonder here why the origin and nature of rate-determining steps


Langmuir, Vol. 18, No. 14, 2002 5577

mercury12 as 4.5 eV) thus denoting a satisfactory agreement. The physical origin of the dependence of λel upon ΦM may be attributed to the spillover of the electronic density profile of the metal into the interfacial region containing the solvent dipoles.51,52 3.4. A Critique of Equations 8 and 9. The rationale behind the satisfactory validity of eqs 12 and 13 is prima facie unclear, especially in view of the fact that the exchange current density of HER varies from 10-3 to 10-13 A cm-2 and is postulated to arise from differences in M-H bond energies. On the other hand, the experimental data for i0 in the case of the Fe3+/Fe2+ reaction ranges from 10-3 to 10-6 A cm-2. This would imply that it is too naive to expect a general formalism valid for both reactions. However, since (a) the exchange current density refers to the equilibrium potential which has a constant value for a redox reaction and (b) the transfer coefficient β is =1/2 in general, an unanticipated simplification arises enabling the work function as the only substrate-dependent parameter, in influencing the electrode kinetics at zero overpotential. However, quantitative differences do exist between the two reactions regarding the interdependence of i0, ∆G q, and ΦM. Figure 2 depicts the behavior of the above parameters for HER and Fe3+/Fe2+ wherein the work functions range from 4 to 5 eV. In the case of the ferric/ ferrous reaction, eqs 8 and 9 yield Figure 1. Volcano plot between log i0 (i0 in A cm-2 ) calculated using eq 13 vs tabulated EM-H values (ref 18). The line is drawn as a guide to the eye.

have been not encountered in the above interpretation of the volcano plot in contrast to the existing approaches which employ detailed considerations of adsorption isotherms.12-16 This is attributed to the incorporation of the nature of metals via the work function while the hitherto-available methodologies emphasize the surface coverage of adsorbed hydrogen atoms. The present formalism thus provides a clear delineation of the role of the metal surface in electron-transfer reactions. Similar volcano plots are observed in other contexts too, and a recent study on the catalytic activity of hydro- desulfurization of transition metal sulfides as a function of metal-sulfur electron density deserves mention in this context.47 3.3. Solvent Reorganization Energy and Work Function. Since the free energy of activation at zero overpotential10 is related to the solvent reorganization energy λel for electron-transfer reactions at electrodes as λel ≈ 4∆G q, eq 12 yields λel as

q ∆Geq(Fe 3+/Fe2+) ≈ 1.69 - 0.17ΦM

(17)

and

(ln i0)Fe3+/Fe2+ ≈ 11.29 - 16.91∆Gqeq + 2.88ΦM

(18)

respectively.30 In the above equations, ∆Gqeq and ΦM are in eV while i0 is in A cm-2. From Figure 2a,b, it may be noted that for metals with work functions between 4 and 5 eV, (a) ∆Gqeq for HER varies from 0.14 to 0.34 eV in contrast to that for Fe3+/Fe2+ (0.8 to 1.1 eV) and (b) log i0 inter alia shows a larger variation in the case of HER. These quantitative differences arise since the solvation characteristics of the reactant are different in both cases and wp is nonzero for the ferric/ferrous reaction. 4. Perspectives and Summary

where λel is in eV. A microscopic calculation of λel would however require consideration of image effects and consequent changes in the dielectric polarization.48,49 While values of λel for HER at different electrodes have not yet been reported, a recent experimental study employing kinetic isotope effects50 using H2O and D2O leads to λel as =1 eV at a mercury surface. In comparison, eq 16 provides λel as 0.93 eV (using the work function of

The foregoing analysis has provided an explicit parametric dependence of (a) the free energy of activation, (b) the standard exchange current density, and (c) the solvent reorganization energy upon the work function of metals. The computed standard exchange current densities for various metals are in good agreement with experimental data deduced using polarization studies and impedance data. The electronic structure of metals is incorporated via their work function in the kinetics of heterogeneous electron-transfer reactions. The role of single-crystal surfaces and rationalization of data deduced using them in this context is now feasible. It is instructive albeit retrospectively to analyze the basis behind the general validity of eq 9. There are several factors that have led to this agreement. (i) The first factor

(47) Aray, Y.; Rodriguez, J.; Vega, D.; Nouel Rodriguez-Arias, E. Angew. Chem., Int. Ed. 2000, 39, 3810. (48) Curtiss, L. A.; Halley, J. W.; Hautmann, J.; Hung, N. C.; Nagy, Z.; Ree, Y. J.; Yonco, R. M. J. Electrochem. Soc. 1991, 138, 2033. (49) Schmickler, W. Interfacial Electrochemistry; Oxford University Press: New York, 1996; p 15. (50) Krishtalik, L. I. Electrochim. Acta 2001, 46, 2949.

(51) Saradha, R.; Sangaranarayanan, M. V. J. Phys. Chem. 1998, B102, 5099. (52) Saradha, R.; Sangaranarayanan, M. V. J. Phys. Chem. 1998, B102, 5468. (53) Areas of the electrodes are from: (a) Schuldiner, S. J. Electrochem. Soc. 1959, 106, 891. (b) Hillson, P. J. Trans. Faraday Soc. 1952, 48, 462.

λel ) 4 - 0.68ΦM

(16)

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Figure 2. Three-dimensional mesh plot depicting the simultaneous dependence of log i0 on the work function (ΦM) and free energy of activation (∆Gqeq). The ranges of values chosen for the mesh plot are (a) HER, 3:0.0268:5 and 0.2:0.01784:1.5, and (b) ferric/ferrous reaction, 3:0.067:5 and 1.2:0.01:1.5 for the work function and free energy of activation, respectively.

is the concept of electronic equilibrium38 according to which χeM ) χes at zero overpotential. Although an exact estimate of χeM has not been available till now, this limitation does not lead to significant errors in the order of the magnitude of i0. (ii) The second factor is the decoupling procedure which yields the work function of the solvated electrode (ΦMS) in terms of the work function at the metal/vacuum interface (ΦM) with the help of a numerical constant ξ, viz., ΦMS ) ξΦM (cf. eq 9 of ref 30). This simple prescription finally leads to an explicit thermodynamic calculation of i0. In general, ΦMS shows an involved dependence on work function and dipolar polarization and hence the above equation for ΦMS is possibly a zeroth order approximation valid only at equilibrium potential. (iii) The third factor is the representation of interfacial parameters in terms of bulk quantities. Since the work term in bringing the ionic reactant to the reaction zone at the electrode/ electrolyte interface is not easy to obtain, it seems useful to invoke the processes depicted in Scheme 1 and hence rewrite the interfacial Gibbs free energy of solvation using the corresponding quantity in the bulk (∆Gr-s) and solvation number. Although this division of ∆Gr-s by SN is ad hoc, it is well-known that the solvation energetics at electrode/electrolyte interfaces is diminished from the bulk value on account of imaging, discreteness of charge effects, and so forth.48 Interestingly, the coupling of the

electrode-solvent is represented via ΦMS, the work function of the solvated electrode, while that due to the reactant-solvent is expressed using ∆Gr-s, after incorporating empirical correction factors due to the presence of the interface. Other system parameters such as solvation numbers, work functions, and so forth also have diverse reported values; hence, the agreement of the calculated i0 values with the experimental data might possibly have been due to cancellation of errors in individual terms, and this requires further investigation. However, the fact that a metal-dependent tabulated parameter (viz., work function) can be explicitly introduced for calculating the exchange current density in a phenomenological manner is a favorable outcome of the preliminary analysis carried out here. For a detailed study of the entire currentpotential domain spanning nonzero overpotentials, it may be essential to incorporate finer details regarding solvent polarization, electrostatic potentials, and diverse interactions of the reactant species with the electrode surface. Acknowledgment. It is a pleasure to thank the reviewers for their valuable suggestions. This work was supported by Department of Science and Technology, Government of India. LA025548T