J. Phys. Chem. B 2002, 106, 8681-8688
8681
Hydrogen Evolution Reaction on Electrodes: Influence of Work Function, Dipolar Adsorption, and Desolvation Energies S. Harinipriya and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras 600 036, India ReceiVed: June 25, 2001
The dependence of the standard exchange current density of the hydrogen evolution reaction on electrode surfaces has been investigated using the work function of metals, free energy of bond formation with adsorbed hydrogen atoms, surface potential of the reactant, and adsorption characteristics of solvent dipoles. The explicit influence of the parameters is derived using a postulated transition state and a satisfactory agreement with experimental data is noticed for a large number of sp and d metals.
1. Introduction The study of electron-transfer reactions at electrode/electrolyte interfaces continues to be a fascinating exercise in view of its importance in all electrochemical technological processes. The activated complex theory, which expresses the electron-transfer rate constant (ket) in terms of the free energy of activation (∆Gq), is given by1
ket )
kBT -∆Gq exp h RT
(1)
where kB and h denote the Boltzmann and Planck constants, respectively. Because the free energy of activation can be further analyzed in terms of either statistical thermodynamic versions or quantum mechanical considerations with varying levels of sophistication,2 the above equation plays a crucial role in electrode kinetics. Further, mechanistic analysis of electrochemical reactions can be accomplished using a variety of experimental techniques.1,3 Among different electron-transfer processes, the hydrogen evolution reaction (HER) represented as H+ + e- f 1/2H2 is one of the half-cell reactions pertaining to hydrogen-oxygen fuel cells, electrodeposition of metals in acid media,4a hydrogen embrittlement, etc.4b Thus, the study of HER at different metal surfaces has been a focus of investigation employing work function variations, metal-hydrogen bond energies, and elucidation of rate-determining steps. Further, experimental studies in this context deal with investigation of the potential dependence of current, as well as extraction of exchange current densities and transfer coefficients, using polarization measurements, radiochemical and open circuit potential decay techniques,3a impedance analysis, etc.5 While the transfer coefficient in relation to the symmetry factor is a measure of the asymmetry in the potential energy vs reaction coordinate description, the magnitude of exchange current density (i0) determines the velocity of the reaction at equilibrium potential. Amidst different strategies adopted for the analysis and estimation of i0 for HER, mention may be made of (i) correlation with work function and metal-hydrogen bond energy for sp and d metals derived from experimental data,3b (ii) the nature of coupling between the electrode surface and the reactant (adiabaticity vs nonadiabaticity),2b (iii) the choice of adsorption isotherms for H atoms in relation to the rate-determining step,6 and (iv) formulation
in terms of electronic density of states and distribution functions.7 Despite such multifaceted attempts in elucidating the mechanistic details of H2 evolution reactions using a variety of electrochemical and surface characterization studies,8 the calculation of i0 for different electrodes incorporating diverse system parameters within a thermodynamic framework remains formidable. The primary reason behind this lacuna consists of the widely varying experimental values of exchange current density for different metals (cf. 10-3 A m-2 for Pt to 10-9 A m-2 for Pb) apart from diverse dipolar interactions arising out of the spill over of the electronic density profile pertaining to different metals.9a,9b Within the realm of nonequilibrium statistical thermodynamics, it is customary to introduce the concept of canonical forms for formulating the transition state of the elementary chargetransfer processes. This methodology in conjunction with the principle of microscopic reversibility leads to the classical Butler-Volmer equation9c almost effortlessly. This implies that electrochemical parameters, standard exchange current densities, symmetry factors etc., possess strict statistical thermodynamic interpretations, too.9c Among several electron-transfer processes that are amenable for analysis, HER is especially convenient because it has been studied on a variety of metals and alloys. In this communication, we report (i) the calculation of the standard exchange current density of HER at different metals using a postulated transition state involving the electrode surface, solvent molecules, and adsorbed hydrogen atoms, (ii) the influence of work function, desolvation energies of ions, desorption characteristics of solvent dipoles, etc. on the magnitude of log i0, and (iii) the interpretation of the volcano plot using the formalism developed here. 2. Transition State and Exchange Current Density Within the classical Butler-Volmer description of electrode kinetics, the exchange current density, i0, is given as1,9d
i0 )
βnFEe nFCRket exp A RT
(2)
where β denotes the transfer coefficient or symmetry factor, CR is the bulk concentration of the reactant (in moles), and Ee denotes the equilibrium potential, A being the area of the electrode under consideration. n equals unity in the case of HER.
10.1021/jp012399z CCC: $22.00 © 2002 American Chemical Society Published on Web 07/31/2002
8682 J. Phys. Chem. B, Vol. 106, No. 34, 2002
Harinipriya and Sangaranarayanan
SCHEME 1: Representation of the Postulated Transition State (Depicted as a Large Circle)
The evaluation of ∆Gq for any electron-transfer reactions requires postulating the transition state. In the present context, the transition state is visualized as a loosely held fictitious species and is represented as S‚‚‚M‚‚‚e-‚‚‚H+, where S, M, and e- represent, respectively, the solvent, metal surface, and electrons (Scheme 1). This implies that solvent molecules adsorbed on electrode surfaces along with the desolvated H+ ions dictates the rate of the reaction (hydronium ions are not considered here because the reactant species gets rid of its solvation sheath while getting transported from bulk to the reaction zone).7a The approach of H+ toward the surface leading to the formation of transition state may be schematically represented as shown in Scheme 1. The assumption of the transition state implies that desolvation of H+ ions and electron transfer occur in a synchronous manner. This visualization is somewhat isomorphous with the bimolecular nucleophilic substitution reaction (S2N mechanism) wherein the bond breaking between the leaving group and the carbocation, as well as the bond formation of the carbocation with the nucleophile, occurs simultaneously. Our transition-state assumption is entirely identical with the above S2N reaction mechanism. 2.1. Composition of the Free Energy of Activation. To estimate ket and hence i0, it is essential to analyze ∆Gq in terms of various parameters constituting the transition state. ∆Gq is assumed to be the sum of the Gibbs free-energy change (∆G1) involved in the bond formation of adsorbed hydrogen atoms (Hads) with the metal (M) and the free-energy change of desorption of solvent dipoles from the electrode surface (∆G2),
∆Gq ) ∆G1 + ∆G2
(3)
Recalling that ∆G2 incorporates the energetics of solvent molecules with metals, we note that the coordination characteristics of solvent dipoles in the case of sp and d metals are different. The orientational states of the solvent molecules depend on the nature of the metal, as well as the surface charge density. In the case of sp metals, it has been demonstrated that adsorption of water dipoles occurs via oxygen atoms because of positive surface charge densities.3b,3d On the other hand,
SCHEME 2: Orientation of Water Dipoles Pertaining to (a) d and (b) sp Metals
hydrogen atoms of the water dipoles are involved for d metals because of negative surface charge densities (see Scheme 2). The free energy of desorption of solvent dipoles arises in view of the assumption that the approach of H+ ions toward the electrode is synchronous10a with desorption of solvent dipoles in the visualization of the transition state. In fact, it has recently been stated10b that “water gets desorbed during the reaction because the charge-transfer products (adsorbed hydrogen atoms) knock the water off the surface...”. Thus, the magnitude of ∆G2 consisting of contributions from dipolar interactions with metal surfaces is dependent upon the nature of the metal.10c Consequently, one may write ∆G2 ) ∆Gdesor/a, where ∆Gdesor represents the free-energy change associated with desorption of water molecules and “a” takes into account the nature of coordination of water dipoles with the electrode surface. As described earlier, ∆Gdesor equals ∆GM-H and ∆GM-O, respectively, for d and sp metals. Thus, eq 3 becomes
∆Gq )
∆Gdesor + ∆GM-Hads a
(4)
Because the probability for a water molecule to coordinate with the electrode surface via hydrogen is 1/2, a becomes 2 in eq 4 in the case of d metals, while a equals unity for sp metals (because the coordination is through the oxygen atom of the water dipole). 2.2. Estimation of ∆GM-Hads and i0. The above strategy exploiting the known adsorption behavior of solvent dipoles at different metals enables the computation of ∆Gdesor. However,
Hydrogen Evolution Reaction on Electrodes
J. Phys. Chem. B, Vol. 106, No. 34, 2002 8683 in conjunction with the definition of µM el yields an explicit er 14 that the as follows: It is well-known expression for µint el , is electrochemical potential of a metal, µM el
SCHEME 3: Processes Dictating the Composition of ∆GM-Hads
µM el ) µM + zFχM
(7)
µM el , χM, and µM denote, respectively, the electrochemical potential, surface potential, and the chemical potential of the electrode under consideration.14 Further, µM ) -F(zχM + ΦM), where ΦM denotes the work function. Thus, eq 7 becomes
µM el ) -FΦM
(8)
Consequently, the estimation of ∆GM-Hads requires consideration of the processes represented in Scheme 3, namely, (i) transport of solvated H+ ions from the bulk to the inner Helmholtz plane (IHP) governed by the Galvani potential11 (∆φ), which is composed of the Volta potential and the surface potential of H+ (because the magnitude of the Volta potential is small, the surface potential of H+ ions (χH+) is assumed to be equal to Galvani potential, ∆φ), (ii) removal of the solvation sheath of H+ ions at the IHP in relation to the reaction zone,12 the freeenergy change being ∆GHinter +-s, and (iii) transfer of electrons to the bare H+ ions dictated by the work function12 of the metal (ΦM). On the basis of the above considerations, it follows that inter ∆GM-Hads ) WH+ + ∆GHinter +-s + µel
(5)
WH+ ) zH+FχH+
(6)
where
and zH+ equals unity here. int er are interfacial quantities representing the ∆GHinter +-s and µel desolvation of H+ ions and electrochemical potential of the metal, respectively. It is essential to express these in terms of tabulated parameters so that the calculation of i0 becomes feasible. An approximate procedure to accomplish this is as + at the follows: ∆GHinter +-s is the desolvation energy of H interface dictated by various factors, nature of the electrode, solvent dielectrics, extent of imaging,11 etc. Consequently, an exact calculation of ∆GHinter +-s is a formidable task. Nevertheless, the fact that it is considerably diminished from its bulk value (∆GH+-s) is well-known.7 For example, when solvated ions move from the bulk to the reaction zone, they get rid of the solvation sheath, and at the IHP, the electron transfer is to the unsolvated H+ ions. This feature is incorporated by formulating + ∆GHinter +-s as ∆GH -s/SN, where SN denotes the hydration number of H+ ions. The rationale behind the division by SN consists of the fact that the reactant ions are unsolvated at the reaction zone. er µint represents the electrochemical potential of the metal in el the reaction zone and is dependent upon its work function and surface potential. The transition state visualized in the present context envisages the bond formation of adsorbed hydrogen atoms with electrode surfaces. Hence, it is appropriate to consider the coordination number of the metals as a parameter in determining the extent to which adsorbed hydrogen atoms are involved in the bond formation. Obviously, the probability for a hydrogen atom to adsorb on the metal surface and subsequently form M-Hads will be inversely proportional to the coordination number of the metal (which is customarily inferred from its crystal structure13). The foregoing assumption
er )µint el
FΦM CN
(9)
CN being the coordination number of the metal lattice. Therefore,
∆GM-Hads ) FχH+ +
∆GH+-s FΦM SN CN
(10)
Equation 10 enables the computation of ∆GM-Hads from the surface potential of H+, its desolvation energy, the work function of the metal, and its lattice coordination number. An alternative approach consists in formulating the work function of the metal M in the solvent environment (ΦsM) in terms of ΦM (at the metal-vacuum interface) and other molecular parameters pertaining to ions and solvent dipoles. In effect, eq 10 incorporates the work function of the metal, as well as reactant and solvent characteristics, and is hence amenable for generalization to other electron-transfer schemes with similar mechanistic details. Rearrangement of eq 10 leads to
-
∆GM-Hads FΦM
+
χ H+ ∆GH+-s 1 )+ )X ΦM FSNΦM CN
(11)
where X is a dimensionless constant. The parameter X is influenced by hydration numbers, desolvation energies of ions, and the lattice coordination number of the metal. Hence,
∆GM-Hads ) FχH+ - FXΦM
(12)
The magnitude of X varies from 0.02 to 0.21 (see Table 1). It is interesting to note that a plot of ∆GM-Hads (obtained either from theoretical considerations or from experimental data) vs ΦM for metals may yield a straight line, the intercept providing the surface potential of H+. Using eqs 4 and 12, we obtain
∆Gq )
∆Gdesor + FχH+ - FXΦM a
(13)
On substituting eq 13 in eq 1,
ket )
{
}
kBT ∆Gdesor FχH+ FXΦM exp + h aRT RT RT
(14)
Thus,
i0 )
{
}
FCH+kBT ∆Gdesor FχH+ FXΦM βnFEe exp + Ah aRT RT RT RT (15)
8684 J. Phys. Chem. B, Vol. 106, No. 34, 2002
Harinipriya and Sangaranarayanan
TABLE 1: Estimation of the Exchange Current Density for Hydrogen Evolution Reaction at 298 K Using Eq 17 of the Texta ΦM in eV3b
X
Ag Au Cd Cr Cu Mn Ni Pt Ti Zn Feb Cob Hg Mob Nbb Pdb Rhb
4.30 4.79 4.12 4.40 4.70 3.90 4.73 5.03 4.10 4.30 4.65 4.70 4.50 4.30 4.20 5.01 4.99
0.1170 0.1067 0.0278 0.0367 0.0450 0.0533 0.1291 0.1363 0.1105 0.1170 0.0437 0.0116 0.1229 0.0336 0.1138 0.1358 0.0520
d Metals 0.1 M H2SO4 0.5 M H2SO4 1M H2SO4 0.5 M H2SO4 0.5 M H2SO4 0.05 M H2SO4 0.5 M H2SO4 1 M H2SO4 H2SO4 (pH ) 2.2) 1 M H2SO4 0.5 M H2SO4 0.5 M H2SO4 7.5 M HClO4 0.1 M HCl 0.5 M H2SO4 1 M H2SO4 0.5 M H2SO4
Bi Al Gac Inc Pb Sbc Sn Tl
4.36 4.78 4.25 4.08 4.18 4.56 4.35 4.02
0.1188 0.2133 0.1984 0.1928 0.1132 0.1246 0.1185 0.1907
1M H2SO4 1M H2SO4 1M H2SO4 0.1M H2SO4 1 M HCl 1 M H2SO4 1 M H2SO4 0.1 M HClO4
metals
electrolyte3b
sp Metals
∆Hdesor (kJ mol-1)16
-log iexpt 0 (i0 in A cm-2)3b
-log icalcd 0 (i0 in A cm-2)
215 ( 8.4 292 ( 8.0 372 ( 4.0 280 ( 50 277.80 234 ( 8.0 252.3 ( 14.0 335 ( 20.0 159 ( 21.0 368.02 268 ( 20.0 301.5 ( 12.5 39.84 421 ( 10 473.5 ( 5 256 ( 8.0 361.2 ( 25.0
7.90 6.50 11.60 7.00 7.80 10.90 5.25 3.00 8.30 10.50 5.60 5.30 12.30 7.30 8.40 3.10 3.50
7.70 ( 0.32 6.64 ( 0.30 10.998 ( 0.15 6.86 ( 1.90 7.55 10.45 ( 0.30 4.94 ( 0.53 3.30 ( 0.76 7.86 ( 0.78 9.60 5.01 ( 0.76 5.61 ( 0.63 12.57 7.40 ( 0.41 8.30 ( 0.23 2.85 ( 0.30 3.66 ( 0.95
337.2 ( 12.6 512.1 ( 9.2 353.6 ( 41.8 320.1 ( 41.8 382 ( 12.6 434.3 ( 41.8 531.8 ( 12.6 249.23 ( 8.0
7.80 8.00 8.40 9.50 11.40 5.10 7.80 9.60
8.27 ( 0.96 8.80 ( 0.70 8.43 ( 3.20 9.14 ( 3.20 11.55 ( 0.96 4.63 ( 3.20 8.30 ( 0.96 9.89 ( 0.61
a Area of the electrode is assumed to be 0.1 cm2. b ∆H M-H was calculated as the arithmetic mean of ∆HM-M and ∆HH-H, where M refers to Fe, Co, Mo, Rh, Pd, or Nb as the case may be and ∆HH-H is 435.99 kJ mol-1 and ∆HM-M values for these metals are 100 ( 21, 167 ( 25, 406 ( 20, 285.4 ( 20.9, and 75 511 ( 10 kJ mol-1 (see Weast16). c In view of the large uncertainities of ∆Hdesor for Ga, In, and Sb, the calculated values of log i0 too exhibit a similar trend.
At the equilibrium potential15a pertaining to HER, eq 15 becomes
i0 )
{
}
FCH+kBT ∆Gdesor FχH+ FXΦM exp + Ah aRT RT RT
(16)
Interestingly, eq 16 contains no adjustable parameters at this stage and incorporates all of the essential system parameters. Because it is customary to investigate the dependence of log i0 on various quantities, we rewrite eq 16 as
{
log i0 ) log
}
FχH+ ∆Gdesor FCH+kBT + Ah 2.303aRT 2.303RT FXΦM (17) 2.303RT
χH+ denotes the energy required for the unsolvated H+ ions to pass through the vacuum-solution interface.15b Its value equals 1.47 eV in a vacuum.15c 2.3. Calculation of log i0 at Different Electrodes. Because ∆Gdesor representing the Gibbs free-energy change involved in the desorption of solvent dipoles is dependent upon the nature of the electrode (sp or d metals),3b the applicability of eq 17 may be illustrated separately for these two classes. In the case of sp metals, eq 17 yields
log i0 ) log
{
}
∆GM-OH2 FCH+kBT FχH+ FXΦM + Ah 2.303RT 2.303RT 2.303RT (18)
Further, ∆GM-OH2 ) ∆HM-OH2 - T∆Sdesor, where ∆Sdesor represents the entropy change involved in desorption of water dipoles from the electrode surface. For monovalent and poly-
valent metals, T∆Sdesor has been reported as 12.552 and 125.52 kJ mol-1, respectively, employing the Born charging process.15b ∆HM-OH2 (as ∆HM-O) for different sp metals is available in tabular compilations16 with an approximate error of ∼1-5%, in general. Further, X contains the desolvation energy, ∆GH+-s, and the solvation number SN, which are 1156.0 kJ mol-1 and 5, respectively.15b The coordination numbers of sp and d metals are 12, except for Pb, Al, and Sn, which have 8 on the basis of their crystal structures.13 Consequently, log i0 may be estimated from eq 17. For all sp metals, this procedure is employed leading to the entries of Table 1. Analogously, in the case of d metals, eq 17 leads to
log i0 ) log
{
}
∆GM-H-O-H FCH+kBT FχH+ + Ah 2.303RT 2.303(2RT) FXΦM (19) 2.303RT
The estimates of T∆Sdesor were obtained as mentioned above, while ∆HM-H values are employed from the literature.16 The calculated log i0 values for d metals are also shown in Table 1. Whenever ∆HM-H-O-H (as ∆HM-H) is unavailable, the arithmetic mean of ∆HM-M and ∆HH-H is assumed as is customary.17 For metals such as Os, Ir, Re, W, Ru, and Ta, neither ∆HM-H nor ∆HM-M values are available and hence these were excluded in the calculation of log i0. 3. Discussion The comparison of log i0 evaluated from eq 17 for different metals with experimental data is shown in Table 1. While the general trend is satisfactory, there are instances (cf. Cr, Pt, Ti,
Hydrogen Evolution Reaction on Electrodes Ga, In, Sb, etc.) when large differences are noticed. This is attributed to the uncertainities reported16 in ∆HM-H-O-H and ∆HM-OH2 (as ∆HM-H and ∆HM-O). The parameters occurring in eqs 12 and 17 are themselves not new in the context of electrode kinetic study of HER. For example, the correlation of log i0 with work function or bond formation of M with Hads is well-known both from classical electrocatalytic versions3b,18 and from recent quantum mechanical considerations.2b,7 However, the quantitative manner in which the system parameters enter into the final prescription of log i0 is reported here for the first time. 3.1. Justification of the Postulated Transition State. The transition state depicted in Scheme 1 leading to the standard exchange current density may be rationalized in the following manner. The electrosorption of ions or organic compounds is visualized as a replacement of adsorbed solvent molecules. Further, electron-transfer reactions at metal surfaces are postulated to occur from solVated electrodes. These considerations imply that any description of electrode processes should incorporate the nature of solvent dipolar arrangement at metals. However, a satisfactory microscopic analysis of interfacial solvent molecules at the electrode surface is still lacking,9a,9b thereby warranting an ad hoc thermodynamic approach. Because the differences in the arrangement of solvent dipoles at sp and d metals are well-documented,3b,3d Scheme 2 seems valid enabling the formulation of ∆Gdesor either as ∆GM-OH2 or as ∆GM-H-O-H. However, the estimation of these quantities in a rigorous manner is still an involved excercise. In a fortuitous manner, T∆Sdesor values have been reported by Bockris et al.15b as 12.552 and 125.52 kJ mol-1 for monovalent and polyvalent metals, respectively. Analogously, the enthalpy changes ∆HM-H-O-H and ∆HM-OH2 have been reported in tabular compilations as ∆HM-H and ∆HM-O. Although diverse values pertaining to ∆HM-H and ∆HM-O are available in the literature, we have employed, in general, the estimates of Weast16 in our calculations. The effect of bonding between adsorbed H atoms and metal surfaces is known to influence log i0, and this has been incorporated via various transport processes as shown in Scheme 3, and hence, factors, surface potential of H+, work function ΦM, coordination number of the metal lattice, hydration numbers, etc., naturally arise in the expression for the standard exchange current density. 3.2. Surface Coverage of Adsorbed Hydrogen Atoms and the Nature of the Rate-Determining Step. It is customary to interpret the influence of various metals upon the magnitude of standard exchange current densities using different pathways3b,7,18 by which Hads leads to H2. Thus, the three commonly postulated mechanisms in this context are as follows:
H3O+ + e- f Hads + H2O (Volmer) 2Hads f H2 (Tafel) H3O+ + e- + Hads f H2 + H2O (Heyrovsky) While the above versions are useful in (i) comparing the experimentally observed Tafel slope at different metals and (ii) studying the potential dependence of surface coverage of Hads, a salient observation made by Breiter20 is that the plot of log i0 vs ∆HM-Hads exhibits a maximum regardless of the mechanism being followed. Hence, it follows that there is an underlying commonness of HER, which is oblivious to the precise manner in which Hads yields H2 at least within the limited objective of
J. Phys. Chem. B, Vol. 106, No. 34, 2002 8685 the estimation of log i0 for different metals. When viewed from this perspective, it appears that the methodology proposed herein incorporates major factors contributing to the magnitude of i0 even if detailed considerations of adsorption isotherms pertaining to Hads are ignored. Our formalism is essentially concerned with the transitionstate description whose origin is a consequence of the above Volmer step and takes no direct cognizance of the manner in which subsequent formation of H2 occurs. These refinements would become essential if the objective is to comprehend and control the rate-determining step for detailed analysis of the entire current-potential region under more stringent protocols. 3.3. HER and Percent d Character of Metals. The percent d character of a metal represents the number of unpaired electrons in its d band and is a measure of the energy required to extract an electron from it. It has been noted using experimental data on a large number of metals and alloys18 that log i0 increases with percent d character.18a In our methodology, the percent d character enters via two parameters, ∆Gdesor and ΦM, whose influence in this context is competitive. For example, when the series Pt, Fe, Co, Ni, Au, Ag, Al, Sn, etc. is considered, percent d character decreases while passing from Pt to Sn. However, for this series, ∆Gdesor decreases in view of the enhanced attraction of solvent dipoles with the electrode surface. On the other hand, work functions of the metals are known to increase with percent d character (As the spin-paired electrons in the d band increase, more stable will be the metal lattice and hence greater energy is required to extract an electron from it). Because these two factors, ∆Gdesor and ΦM, have opposing influences on log i0, an increase in log i0 with percent d character is noticed (Note that ΦM and ∆Gdesor in eq 17 are of opposite signs). 3.4. Rationalization of Volcano Plot. The interpretation of the volcano plot (log i0 vs ∆HM-Hads) has been a central issue in the analysis of HER.2b While i0 values for metals Pt, Pd, Ag, Au, Ni, Fe, Bi, Hg, etc. occur on the left-hand side (lhs) of the volcano plot, electrodes Rh, Sb, Mo, Ga, etc. lie on the decreasing portion of the same. Figure 1 depicts the volcano plot obtained using our calculated log i0 values of Table 1 as a function of ∆HM-Hads extracted from the tabular compilation of Weast.16 Although various estimates of ∆HM-H are available,19 we have employed the values reported by Weast,16 because the qualitative trend is not altered. One may wonder here whether eq 10 with appropriate modifications can also be employed in lieu of the tabulated enthalpy of bond formation. Because eq 10 has underlying approximations in view of incorporation of factors, solvaton number (SN), lattice coordination number (CN), etc., we prefer to use the tabulated data.16 The volcano plot may be interpreted using eq 16 in the following manner. The metals on the lhs of the volcano plot are characterized by increasing work functions and decreasing X values (Note that X is an explicit function of 1/ΦM, as well as lattice coordination number, 1/CN). On the other hand, the decreasing portion of the volcano plot includes metals having lower work functions and larger X values indicating that the estimates of XΦM are smaller in magnitude as indicated in Table 2. Further, the nature of arrangement of solvent dipoles on the electrode surface also plays a crucial role. The lhs of the volcano plot contains metals having a ) 2, in view of the orientation of hydrogen atoms of solvent dipoles toward the electrode surface. Because the magnitude of ∆Gdesor decreases from Hg to Pt, an increase in i0 is observed. On the other hand, on the right-hand
8686 J. Phys. Chem. B, Vol. 106, No. 34, 2002
Harinipriya and Sangaranarayanan
Figure 2. Dependence of log i0 on work function of the metals calculated using eq 17. The errors in individual log i0 are not shown for clarity. Figure 1. Volcano plot for HER obtained using eq 17. Points denote the calculated log i0 values, and ∆GM-Hads values are taken from Weast.16 Line is drawn as a guide to the eye.
TABLE 2: Interpretation of the Volcano Plot on the Basis of Eq 16 volcano plot
ΦM
XΦM
∆Gdesor/a
i0
lhs comprising metals increases increases decreases increases Zn, Hg, Pb, Ti, TI, Ag, Au, Co, Cr, Cu, Ni, Fe, Mn, Sn, Pd, Pt, etc. rhs constituting metals decreases decreases increases decreases Pt, Rh, Sb, Mo, Ga, etc.
side (rhs) of the volcano plot comprising metals from Pt to Ga, a equals unity; thus, the magnitude of ∆Gdesor is higher in this case. Because ∆Gdesor/a occurs as a negative term in eq 16, a decrease in i0 is observed. Hence, the volcano plot is a consequence of the interplay between XΦM and ∆Gdesor/a (cf. Table 2). Similar volcano plots are observed in other contexts too.20 As an example, we may note a recent study on the catalytic activity of hydro-desulfurization of transition metal sulfides as a function of the electron density of the metal.20 3.5. Dependence of log i0 on Work Function. As pointed out earlier, the standard exchange current density of HER spans a wide range of values. This has resulted in the investigation of the work function dependence of log i0, and experimental data conclusively suggest a linear correlation of log i0 with ΦM.3b,7,18 In particular, two different straight lines with identical slopes but varying intercepts are clearly seen in the correlation derived from the experimental data (Figure 1 of Trasatti et al.).3b The two different intercepts pertain to metals Hg, Pb, T1, etc. (lower ΦM) and Pt, Pd, Fe, Co, Ni, Cu, Au, Ag, etc. (higher ΦM). After substituting various system parameters in eq 17, we obtain
log i0 ) (5.97 ( 1.09)ΦM + (-33.09 ( 2.56) (correlation coefficient ) 0.998) (20)
and
log i0 ) (5.97 ( 1.09)ΦM + (-34.09 ( 4.86) (correlation coefficient ) 0.998) (21) for the two different series of metals (Figure 2). Interestingly, the constant slope arising from eq 17 is consistent with 6.7 eV-1 obtained from the experimental correlation of Trasatti,3b as well as the recent estimate of 6.8 eV-1 reported by Sakata7 employing the weak coupling approximation within a quantum mechanical formulation. The constancy of the slope irrespective of the nature of the metal seems puzzling at the first sight; however, the analysis propounded herein yields a rational basis for the same. From eq 17, it follows that
log i0 ) k1 + k2ΦM
(22)
where k2 ) nX/(RT). Although the composition of X (cf. eq 12), as well as its evaluation (cf. Table 1), seems to suggest a metal dependence, the constant slope can be interpreted in the following manner. The variation of X within a group of metals having high or low work functions is quite minor. For example, when the work function difference is 0.5 eV in the case of metals having higher work function values, the corresponding change in X is 0.04 (∼10%). Analogous considerations hold good for metals with lower work functions, too. In view of this, the slope of the log i0 vs ΦM plot is effectively constant. The magnitude of the two intercepts reported herein are in gross agreement with the experimentally observed correlation, 36.6 and 39.3.3b Because ∆Gdesor is different for the two classes of metals (cf. section 3.1), the intercepts are not identical. 3.6. Quantum Mechanical Formulations of HER. The standard exchange current density, in general, can also be represented in terms of electronic overlap integral and density of states using the Levich-Dogonadze21,22 formalism. The metal dependence may be introduced via jellium and related models.23 The description of HER using electronic density of states has
Hydrogen Evolution Reaction on Electrodes
J. Phys. Chem. B, Vol. 106, No. 34, 2002 8687
Figure 3. Three-dimensional mesh plot constructed from eq 17 depicting the simultaneous dependence of log i0 on solvent desorption characteristics and work function of the metal (related to ∆HM-Hads via eq 12).
been extensively discussed.2b,7 A recent contribution,7a which invokes the weak coupling between the reactant and electronic density of states, deserves mention in this context in view of its explanation of the volcano plot using a microscopic theory. The major factors, work function, metal-hydrogen bond formation, etc., are introduced within a quantum mechanical perspective. However, our analysis has delineated various processes contributing to HER and incorporated energetic contributions involving desorption of solvent, desolvation of ions, and M-Hads bond formation within a thermodynamic framework. In contrast to the existing treatments, the present version takes into account the nature of adsorption of solvent dipoles on sp and d metals and its influence in altering the exchange current density in relation to rate constants. These will enable the study of dielectric effects in the mechanism of HER. 3.7. Dependence of log i0 on XΦM and ∆Gdesor/a. While the analysis has so far centered around the dependence of log i0 upon work function and free-energy change of M-Hads bond formation separately, it is far more illuminating to depict the variation of log i0 with two system parameters simultaneously. Thus the influence of ∆GM-Hads and ∆Gdesor/a on log i0 is depicted in a three-dimensional mesh plot (Figure 3) using eq 12. Because the dependence of log i0 is linear with respect to ∆Gdesor/a, as well as XΦM, the mesh plot too indicates this feature. (Because X is influenced by the metal albeit weakly, XΦM is treated as a variable for generality.) The constant and linear regimes can be noticed for the chosen set of parameters. Further, the interplay between solvent adsorption at electrode surfaces and work function terms is discernible. It would be naive to anticipate that the transition-state approach outlined above may hold good for other classes of electron-transfer processes at metals. In particular, in the case of ferrous/ferric redox reaction at electrodes,24-26 the outer sphere mechanism is prevalent. However, if adequate corrections are incorporated in formulating energetics of the system, the extension of the present analysis to other electron-transfer processes may become feasible. One may also interpret the estimation of log i0 as the calculation of activation energy barrier consisting of an intrinsic contribution and the reaction enthalpy. It is then possible to employ various types of potential energy surfaces of which the validity has recently been demonstrated for a large number of hydrogen-transfer reactions.27
A limitation of the present method is the ad hoc manner in which interfacial quantities are expressed in terms of the bulk properties by including hydration numbers of the ions, lattice coordination numbers, etc. This limitation coupled with uncertainities in the thermodynamic quantities, as well as noninclusion of explicit double-layer corrections, needs to be overcome for a comprehensive analysis of HER. 4. Summary The standard exchange current density for HER on different metal surfaces has been estimated using phenomenological thermodynamic considerations in conjunction with the transitionstate formalism. The ionic hydration numbers, lattice coordination numbers, and different Gibbs free-energy changes have been explicitly introduced. A satisfactory agreement with the experimental data is noticed for a variety of sp and d metals. The experimentally observed correlation with work function of the substrate and metal-hydrogen bond strength is rationalized. Acknowledgment. It is a pleasure to thank the reviewers for their valuable suggestions. This work was supported by the Council of Scientific and Industrial Research, Government of India. References and Notes (1) See, for example: Bard, A. J.; Faulkner, L. R. Electrochemical methods Fundamentals and Applications, 2nd ed.; John Wiley and Sons: New York, 2001; p 91. (2) (a) The well-known Anderson-Newns Hamiltonian has been extensively investigated in this context. See, for example: Smith, B. B.; Hynes. J. T. J. Chem. Phys. 1993, 99, 6517 and references therein. (b) See, for example, Khan, S. U. M. In Modern aspects of electrochemistry; Bockris, J. O’M., Ed.; Plenum Press: New York, 1997; No. 32, p 71 and references therein. (c) Schmickler, W. Annual reports on the progress of chemistry; The Royal Society of Chemistry: London, 1998; Vol. 95, Section C and references therein. (3) (a) Kelsall, G. H. In Techniques in Electrochemistry, Corrosion and metal finishing - A Handbook; Kuhn, A. T., Ed.; John Wiley and Sons: New York, 1987; Chapter 6. (b) Trasatti, S. J. Electroanal. Chem. 1972, 39, 163. (c) Rutschi, P.; Delahay, P. J. Chem. Phys. 1955, 23, 195. (d) Trassati, S. J. Electroanal. Chem. 1971, 33, 351. (4) (a) Hamann, C. H.; Hamnett, A.; Vielstich, W. Electrochemistry; Wiley-VCH publications: New York, 1998. (b) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry, 2nd ed.; Kluwer Academic/Plenum Publishers: New York, 2000; Vol. 2B, p 1688. (5) See for example, MacDonald, D. D. Transient techniques in electrochemistry; Plenum Press: New York, 1977.
8688 J. Phys. Chem. B, Vol. 106, No. 34, 2002 (6) (a) Srinivasan, S.; Wroblowa, H.; Bockris, J. O’M. AdV. Catal. 1967, 17, 351. (b) Vijh, A. K. J. Electrochem. Soc. 1971, 118, 1963. (c) Kita, H. J. Electrochem. Soc. 1966, 113, 1095. (d) Conway, B. E. Electrochemical Supercapacitors - Scientific Fundamentals and Technological Applications; Kluwer Academic/Plenum Publishers: New York, 1999; Chapter 3. (7) (a) Sakata, T. Bull. Chem. Soc. Jpn. 2000, 73, 299. (b) In the preceding reference, HER is considered as a weak coupling case wherein the overlapping between (the distribution functions corresponding to) the electrode and H+ is considered to be less. This distinction does not arise here insofar as the microscopic origin of the transition state is not considered, and consequently, the present analysis is strictly phenomenological. (8) Abruna, H. D. Electrochemical Interfaces; Cornell University: Ithaca, New York, 1991. (9) (a) Saradha, R.; Sangaranarayanan, M. V. J. Phys. Chem. B 1998, 102, 5099; 1998, 102, 5468. (b) Parsons, R.; Bockris, J. O’M. Trans. Faraday Soc. 1951, 47, 914. (c) See, for example, Keizer, J. Statistical thermodynamics of non-equilibrium processes; Springer-Verlag: Berlin, 1987, p 238. (d) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, U.K., 1994. (10) (a) Carey, F. A.; Sundberg, J. AdVanced Organic Chemistry, 3rd ed., Plenum Press: New York, 1990; Part B. (b) Bockris, J. O’M.; Reddy, A. K. N.; Gamboa-Aldeco, M. Modern Electrochemistry, 2nd ed.; Kluwer Academic/Plenum Publishers: New York, 2000; Vol. 2A, pp 1191 and 1192. (c) It is possible to interpret the dependence of H2 evolution on the nature of the metal surface qualitatively using the hard-soft acid-base principles (cf. Sastri, V. S. Corrosion Inhibitors, Principles and Applications; John Wiley: New York, 1998, p 239). This feasibility is not unanticipated because the density functional formalism provides a definition of hardness (see Pearson, G. R. Acc. Chem. Res. 1990, 23, 1) in terms of energy density, which can be further improvised to include electron-transfer schemes. A quantitative investigation of hardness of metals in influencing i0 of electrontransfer reactions at electrode surfaces has however not yet been accomplished. (11) Schmickler, W. Interfacial Electrochemistry; Oxford University Press: London, 1996; p 256. (12) Bockris, J. O’M.; Reddy, A. K. N. Modern electrochemistry; Plenum Publishing Corporation: NewYork, 1973; Vol. 2, Chapters 2, 7, 8, 10, and 11.
Harinipriya and Sangaranarayanan (13) Mingos, D. M. P. Essentials of inorganic chemistry 1; Oxford Science Publications, Oxford University Press: London, 1995; p 11. (14) CRC Handbook of solid state electrochemistry; Gellings, P. J., Bouwmeester, H. J. M., Eds.; CRC Press: NewYork, 1996; pp 15 and 16. (15) (a) Conway, B. E. Theory and principles of electrode processes; Ronald Press Company: New York, 1965; Vol 1. (b) Bockris, J. O’M., Conway, B. E., Eds. Modern aspects of Electrochemistry; Butterworths Scientific Publications: London, 1956; p 71. (c) Dogonadze, R. R.; Krishtalik, L. I.; Pleskov, Yu. V. Elektrokhimiya 1974, 10, 507. (16) Weast. R. C., Ed. CRC Handbook of Chemistry and Physics, 68th ed.; CRC Press Inc.: Boca Raton, FL, 1987. (17) Pauling, L. The Nature of Chemical Bond, 3rd ed.: Cornell University Press: Ithaca, New York, 1960; p 92. (18) (a) Bockris, J. O’M.; Conway, B. E. J. Chem. Phys. 1957, 26, 532. (b) Calvo, E. J. In Electrode Kinetics: Principles and methodology; Bamford, C. H., Compton, R. G., Eds.; Comprehensive Chemical Kinetics, Vol. 26; Elsevier: Amsterdam, 1986; p 54. (19) Krishtalik, L. I. In AdVances in Electrochemistry and Electrochemical Engineering; Delahay, P., Ed.; Interscience: New York, 1970; Vol. 7. (20) Aray, Y.; Rodriguez, J.; Vega, D.; Rodriguez, N. E.; Arias, Angew. Chem., Int. Ed. 2000, 39, 3810. (21) (a) Levich, V. G. AdV. Electrochem. Electrochem. Eng. 1966, 4, 249. (b) Levich, V. G. In Physical Chemistry; An adVanced treatise; Eyring, H., Ed.; Academic press, New York, 1970; Vol. 9B, Chapter 12. (22) Dogonadze, R. R. In Reactions of molecules at electrodes; Hush, N. S., Ed.; Wiley Interscience: New York, 1971; Chapter 3 and references therein. (23) Saradha, R.; Sangaranarayanan, M. V. Unpublished results. (24) Straus, B. J.; Calhoun, A.; Gregory, A. V. J. Chem. Phys. 1995, 102, 529 and references therein. (25) Rose, A. D.; Benjamin, I. J. Chem. Phys. 1994, 100, 3534 and references therein. (26) Harinipriya, S.; Sangaranarayanan, M. V. J. Chem. Phys. 2001, 115, 6173. (27) Blowers, P.; Mazel, R. AIChE J. 2000, 46, 2041 and references therein.
