1967
J . Phys. Chem. 1990, 94, 1967-1973
Unconventional Temperature Dependence of the &Factor for the Oxygen Evolution Reaction at Pt Electrodes In Acid Solution: Significance of the Recovered Activation Energies A. Damjanovic,* A. T. Walsh, Allied-Signal Inc., Morristown, New Jersey 07960
and D. B. Sepa Faculty of Technology and Metallurgy, University of Belgrade, Belgrade, Yugoslavia (Received: May 23, 1989; In Final Form: September 13, 1989)
Oxygen evolution reaction is examined at Pt electrodes in acid solutions at temperatures ranging from 273 to 348 K. Tafel slopes at all temperatures are close to 120 mV, Le., the symmetry factor /3 = &T ( E 1/2 at room temperature). When /3 = PsT, the Arrhenius plots, log i at a given E vs 1/T, are not linear. The common assumption that, from quasi-linear plots and/or at zero Galvani potential difference, and isothermal conditions, the enthalpy of activation at the reversible potential, eR, AH*am=o,are recovered does not hold. It is suggested that AH* at any potential is close to zero when 6 = PsT and that the activation energy is predominantly given by the entropic term, T A 9 , which is potential dependent. The rate equation for the reaction is given in terms of the electron energies of the reacting complex in its ground state, to, and activated state, t . It is shown that, for the case /3 = &T, to becomes less negative as temperature increases, and this results in the increased rates. In turn, the change in to with temperature arises from the entropic term, TASo,in the ground state of the reacting complex. This leads to the linear dependence of AS* at a given potential on temperature. From the plot of E vs T , or log i vs T , both of which are now linear, ASo is determined.
Introduction Current-potential relations for O2 evolution reaction at Pt electrodes are reexamined in acid solutions of different pH and at different temperatures. It is known that these relations are characterized, at room temperature, by the Tafel slope close to 2.3RT/(/3F)for over five decades of current density (cd). The symmetry factor, /3, at room temperature was reported14 to be close to 1/2; Le., the first electron-transfer step is rate determining (rd). However, it appears that very little attention has been paid to analyzing the temperature dependence of the reaction and in particular the dependence of /3 on T . It was reported that the heat of activation at the reversible potential, as determined from the Arrhenius plots, is 1 I s and 19 kcal/mol (cf. ref 4). These data contain elements of uncertainty from an experimental point of view and, even were the experimental side satisfactory, their significance is unclear. Thus, only in the past 15 years it was realized that the rate of 0, evolution critically depends on the thickness of an oxide film (OF) anodically grown over the Pt e l e ~ t r o d e . ~In ~ ~order ~ ~ *to obtain meaningful kinetic data, OF thickness must be kept constant at all potentials and, when the effect of temperature was being studied, at all temperatures as ell.^^^ Even if the thickness was kept constant, which seems to be the case for most of the recent experiments at room temperature, but not necessarily for experiments at different temperatures, the significance of the data obtained at different temperatures is obscure. This is because possible dependence of on T was
not examined with certainty in all experiments. In the older literature, (3 was reported to increase with T,9 but no indication was given whether /3 = PsT, or /3 = PH PsT, and also to be invariant with T$Io Le., /3 = OH #AT),these specific distinctions for a reaction, in general, being made much It was recently shown that, depending on the temperature behavior of @, the experimentally obtained heats of activation from Arrhenius plots for a given electrode potential, A H t E ,may have different ~ignificance.'~Thus, only if /3 = OH = constant, the enthalpy of activation at any potential, including the reversible potential for a given reaction, can be obtained.I6 However, if /3 = PsT, the Arrhenius type of activation energies are commonly reported (cf. ref 12 and 17) to represent the enthalpy of activation at the zero Galvani potential difference at the electrode-solution inor the enthalpy of activation at the reversible terface, potential, AH*R,which are then supposed not to uary with electrode potential. The assumption that or AH*R, is obtained from Arrhenius plots when @ = PsT is, however, not correct,15 as will be shown further below. Another difficulty in the interpretation of the experimental activation energies, regardless of the temperature behavior of @, arises with the controversy regarding whether the experiments are to be conducted, or analyzed, with reference to an electrode kept at the same temperature as the test electrode, Le., under isothermal conditions, or with reference to an electrode kept at a constant temperature, i.e., under nonisothermal conditions (cf. ref 11, 15, and 18).
(1) Hoare, J. P. J . Electrochem. SOC.1965, 112, 602. Pushnograeva, I. I.; Skundin, A. M.; Vasil'ev, Yu. B.; Bagotskii, V. S. Sou. Electrochem. 1970, 6 , 134. Bockris, J. O'M.;Shamshul Huq, A. K. M. Proc. R . Soc. London, Ser. A 1956, 237, 277. Visscher, W.; Devanathan, M. A. V. J. Electroanal. Chem. 1964.8. 127. Damianovic. A.: Dev. A,: Bockris. J. O M . Electrochim. Acta 1966, / I ; 791. Kokbulina,'D. V.; Krasovitskaya, Yu. I.; Kristalik, L. I. Sov. Electrochem. 1970, 7, 1172. (2) Damjanovic, A.; Jovanovic, B. J. Electrochem. SOC.1976, 123, 374. (3) Birss, V. I.; Damjanovic, A. J . Electrochem. SOC.1983, 130, 1694. (4) Hickling, H.; Hill, S. Trans. Faraday SOC.1950, 46, 550. (5) Damjanovic, A. In Modern Aspects of Ehtrochemistry; Bockris, J. O'M., Conway, B. E., Eds.; Plenum: New York, 1969; Vol. 5, p 369. (6) Vetter, K. J.; Schultze, J. W. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 945. (7) Damjanovic, A . Special Publication 455. National Bureau of Standards: Washington, DC, 1975; p 259. (8) Gilroy, D. J. Electroanal. Chem. 1977, 83, 329.
(9) Bowden, F. P. Proc. R . SOC.A 1937, 142 628 (as reported in ref 4). (10) Guggenheim, E. A. Thermodynamics; North Holland: Amsterdam, 1957; p 103. (1 1 ) Agar, J. N. Discuss. Faraday SOC.1947, I , 8. (12) Conway, B. E. In Modern Aspects of Electrochemistry; Bockris, J. O'M., Conway, B. E., Eds.; Plenum: New York, 1985; Vol. 16, p 103. (13) Conway, B. E.; Tessier, D. F.; Wilkinson, D. P. J . Electroanal. Chem. 1986, 249. (14) Conway, B. E.; Wilkinson, D. P.; Tessier, D. F. Ber. Bunsen-Ges. Phys. Chem. 1987, 91,484. (15) Damjanovic, A.; Sepa, D. B., to be published. (16) Damjanovic, A.; Conway, B. E.; Sepa, D. B. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 510. (17) Sepa, D. B.; Vojnovic, M. V.; Vracar, Lj. M.; Damjanovic, A. Elecrrochim. Acta 1986, 31, 91. (18) Conway, B. E. In Modern Aspects of Electrochemisfry;Conway, B. E., White, R. E., Bockris, J. O M . , Eds.; Vol. 16, Plenum: New York, 1985; Vol. 16, p 103.
0022-3654/90/2094-1967$02.50/0
+
0 1990 American Chemical Society
1968 The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
Damjanovic et al.
The aim of this paper is to provide reliable kinetic data for the 0, evolution reaction at Pt electrodes in acid solutions at different temperatures under controlled, although not actually determined, thicknesses of the OF. Since, as recently d i s c ~ s s e d , ~ ~ . ~ ~ J ~ meaningful kinetic data for the analysis of the activation energy should be given with respect to the reference electrode kept at the same temperature, e.g., room temperature, the reference electrode should be kept either at a constant temperature, in which case a correction for the temperature junction potential difference in solution has to be applied to the experimental data, or it should be kept at the same temperature as the test electrode, in which case a correction for the potential change of the reference electrode with temperature has to be applied to the experimental data. Temperature variations of the kinetic data are presented and the significance of the obtained “activation energies” is discussed.
Experimental Procedure and Data Pt wire electrodes (Alfa, m5N, 0.5 mm diameter) “sealed” in a glass tube were used in these experiments. After initial cleaning with an aqueous H 2 0 2 / N H 3mixture, electrodes were further treated by a few anodic/cathodic current pulses (a few seconds duration) in the test solution ending with the cathodic pulse. Solutions were prepared from the purest grade acids available (Fisher Reagent, ACS) and water was passed consecutively through two nanopurification systems (Millipore). Highest grade oxygen (99.999%), or nitrogen (99.999%), was used in these experiments. A two-compartment glass cell was jacketed so that a heating/cooling mixture surrounds both compartments. However, when desired, the reference electrode compartment could be maintained at a constant, usually room temperature (23 “C), irrespective of the temperature in the test compartment. Experiments were done with the test solutions kept at 273, 296 or 298, 323, and 348 K. Before each experiment, a constant current (from lo4 to 10” A) is applied across the initially prereduced electrode (geometric area 0.5 cm2) for a specified time (from lo2 to lo4 s). It is known that an oxide (or hydroxide) film grows initially over a platinum surface at potentials above about 1.0 V vs R H E without any detectable oxygen evolution, and that it follows the Cabrera-Mott formalism of high field assisted migration of ions through anodic film.19-22 It is also known that the oxide film continues to grow according to the same mechanism, although ever so slowly, even when oxygen evolution becomes the major and eventually, for any practical purpose. the only net electrode process.23 The reason for such electrode pretreatment is related to the fact that meaningful linear Tafel relations for oxygen evolution are obtained only when it is ensured that the thickness of the O F does not change significantly during the determination of the Tafel relat i o n ~ . ~ ~Normally, ~ , ~ $ ~subsequent ~ * ~ ~ to the electrode pretreatment, these relations are determined at cd’s lower than the one used in the anodic pretreatment of the electrode. In this case, the electric field within the OF is decreased and the rate of growth of the OF even further reduced. An excursion to higher currents, e.g., for one decade of cd for a short time, does not significantly affect the Tafel relation measurements once the O F film has practically ceased to g r o ~ . * - ~ * ~ ~ A saturated KCI calomel electrode, or a saturated NaCl calomel electrode (for HCIO., solutions), is used as a reference electrode. An electrode is kept either at a constant temperature (nonisothermal experiments), or at the same temperature as the test electrode (isothermal experiments). Finally, in most experiments the initial growth of the O F was done at room temperature. After the Tafel relation at room (19) Ord, J. L.; Ho, F. C. J . Elecfrochem. SOC.1971, 118, 46. (20) Damjanovic. A.; Ward, A. t.; Ulrick, B.; O’Jea, M. J . Elecfrochem. SOC.1975, 122, 47 1. (21) Ward, A. T.; Damjanovic, A,; Gray, E.: O’Jea, M. J . Elecfrochem. SOC.1976, 123. 1599. (22) Damjanovic, A,; Yeh, L. S. R . J . Electrochem. SOC.1979, 126, 555. (23) Birss, V. I.; Damjanovic, A. J . Electrochem. SOC.1983, 130, 1688. (24) Birss, V. I.; Damjanovic, A.: Hudson, P. G . J . Elecfrochem. SOC. 1986, 133, 1621. (25) Birss, V . 1.; Damjanovic, A . J . Electrochem. Soc. 1987, 134, 113.
I
1
I
-3 Log i [i in Amps]
-5
-2
-4
Figure 1. E-log i relations at different temperatures. Solution: pH 1.45 H2S04. For each measurement, a prereduced Pt electrode was anodically pretreated at room temperature with 3 X lo-) A for 1000 s. These are isothermal experiments. Electrode area is 0.5 cm2. Data are not cor-
rected for roughness factor.
I I
?
T I
I
1
9, Oi
mt
NI
-5
I
1
2.9
3.1
1
,
3.3 3.5 Reciprocal Temperature [K”]
3.7
x
10.3
Figure 2. Arrhenius plot. Currents at 1.6 V vs SCE at room temperature are plotted against reciprocal temperature (nonisothermal data). No linearity exists for isothermal data also.
temperature was determined, the test electrode was subjected to a low anodic cd (3 X - 3 X 10-4 A for the time required to decrease temperature in the test electrode compartment to 273 K (in about 5 min). The Tafel line is then determined at that temperature. Subsequently, temperature is raised to 323 K and the Tafel line is determined at that temperature. Finally, the procedure is repeated and the Tafel line at 348 K is determined. To check whether the O F thickness is altered in the process of lowering and raising temperature, the Tafel relation at room temperature is redetermined at the end of the experiments and compared to the initial relation. Alternatively, for measurements of the Tafel lines at each temperature, an electrode is prereduced and anodically pretreated at room temperature before the temperature is brought to the required test temperature. In Figure 1, E-log i relations are shown at 273, 296, 323, and 348 K in pH 1.45 oxygen-saturated H2S04solutions. For each temperature, the electrode was first cathodically reduced and then subjected to a constant anodic current of 3 X IO-) A for 1000 s at room temperature. The current is then lowered to 5 X IO-5 A while the temperature in both the reference and the test electrode compartments is brought to the required temperature. It is evident that the Tafel lines in the temperature interval 273-348 K are strictly parallel. The slopes are all close to 120 mV/decade, Le., @ = @sT 1 / 2 (1) which gives Ps 0.0016 K-I. In Figure 2, currents at 1.6 V are plotted against reciprocal temperature. For this plot, the correction
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990 1969
P-Factor for Oxygen Evolution at Pt Electrodes 1
I
I
I
I
-4
-3
1.7
1.11-
i
1
29
30
31
I
32 33 34 35 Reciprocal Temperature [K ‘1
i i
,
37x103
36
-5
Figure 3. Electrode potentials at IO4 A plotted against reciprocal temperature. Potentials refer to SCE at room temperature (Le., for nonisothermal conditions). No significant change is observed when potentials refer to SCE at the working temperature (Le., for isothermal conditions).
t
.1,8
1
\
-2
Log Current [in Amps]
Figure 5. Tafel lines at 348 K in pH 2.5 H2S04solution. A prereduced Pt electrode was polarized with i, = IO-’ A for 300 and 1000 s. Higher “catalytic” activity at electrodes prepolarized for 1, = 300 s is due to thinner surface oxide films at this electrode.
polarization, t,. The higher activity of the electrode pretreated for 300 s relative to the activity of the electrode pretreated for 1000 s with the same i, (see Figure 5) reflects a thinner OF and hence higher probability for electron tunneling through the OF. In fact it was observed previously that for the same i, at room temperature26 d E / d log t i= 76 mV (3) Present data for i = A and t , = 300 and 1000 s, although obtained at 75 are in accord with the previously reported dependence.
‘8,
1
1
273
1
298
323
,
i
348
Temperature [K]
Figure 4. Electrode potentials at IO4 A plotted against temperature. Potentials refer to SCE at room temperature (Le., for nonisothermal conditions). Different symbols represent one series of experiments under the same experimental conditions (Le., oxide film thickness and pH of solution). Upper line is for much thicker oxide films.
was made to account for the temperature variation of the KCI SCE; Le., currents in Figure 2 are plotted at 1.6 V with respect to the SCE a t room temperature. It is difficult to draw with sufficient reliability a straight line through these nonisothermal experimental points, as is commonly expected for an Arrhenius type of plot. Similarly, plots of electrode potentials against reA), again corciprocal temperature at a constant current ( rected for the temperature coefficient of the SCE, deviate from a straight line (Figure 3) and this too is not expected. In contrast to this, electrode potentials at a given current change convincingly linearly with temperature (Figure 4). The spread of experimental points in this figure does not represent uncertainty in measurements of a single set of Tafel lines at different temperatures. Rather, they represent parallel shift of the Tafel lines due to different experimental conditions, e.g., different pH and O F thickness. The linearity should be judged by following the same symbol in this figure with temperature. Nevertheless, a straight line is drawn through all the points obtained at different experimental conditions so that the slope can be determined with an increased accuracy. It is (dE/dT)i = -0.0047 V K-’ (2) Obviously, an excessive anodic pretreatment (e.g., with A for 3 h) in low pH solutions will shift the Ei-T line toward much higher potentials, as also shown in Figure 4. The effects of the OF thickness on the kinetics of OE are shown in Figure 5 with the Tafel lines obtained at 75 OC. To obtain this dependence, a constant anodic current for electrode prepolarization, i,, was applied to the electrode for different times of
Discussion It has recently been s h ~ w n ’ ~that - ’ ~ the symmetry factor, 0, can be split into a temperature-independent component, OH, and a temperature-dependent component, Le.,
os,
@ = OH
PST (4) where is a constant. PH was formally associated with much stronger variation with electrode potential, E, of the enthalpy of activation, AH*, than of the entropy of activation, AS*,which virtually becomes independent of E when os = 0. Similarly, Ps was associated with much stronger dependence on E of the entropy of activation than of the enthalpy of activation, which virtually becomes independent of E when PH = 0. Thus,
For fl = PH, Tafel slopes increase (decrease for a cathodic process) as T increases, and Tafel lines at different temperatures intersect, in extrapolation, at a common point (if the temperature dependence of the preexponential factor is ignored). This was shown to be the case for oxygen reduction at Pt in acid solutions.I6 For the OE at Pt in acid solutions, Tafel lines at different temperatures are parallel, and consequently, in the temperature interval of the experiments, /3 = &T. In this case, the rate of an anodic reaction, in terms of the Galvani potential difference, A$, is given by2’
and, with /3 = PsT, by (26) Damjanovic, A.; Ward, A. T. In Iniernaiional Review of Science; Physical Chemistry Series Two; Butterworths: London, 1976; Vol. 6 , p 103. (27) Temkin, M. I. Zh. Fiz. Khim. 1948, 22, 1081.
1970
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
Here, AG* is the Gibbs energy of activation at a given electrode potential. AC*A,=o,AH*,+=o,and are respectively Gibbs energy of activation, enthalpy of activation, and entropy of activation, at the electrode potential for which the Galvani potential difference, Ag, across the electrode/solution interface is zero. These energies and entropy at A+ = 0 are expected to be constant for a given metal/solution interface and are commonly assumed to be temperature independent, the assumption that is now being questioned. I I n order to express the rate in terms of a measured electrode potential, E , A@ in the rate equation (8) has to be related to the potential of a reference electrode, Le., A+ = EA+=O,RE + ERE
(9)
Here, EA+=O,RE is the potential, vs the same reference electrode, at which the Galvani potential difference at the reference electrode is zero. [For a reference electrode of the same electrode material, e.g., the reversible hydrogen electrode at Pt in the same solution, E,+,, is nothing else but -A+RE, the negative of the Galvani potential difference at the reference (hydrogen) electrode. In this case the Galvani potential difference at the test electrode at E = 0, AdEDO, is equal to A$RE.] For simplicity, subscripts R E for the reference electrode are omitted in subsequent equations. With eq 9, one obtains
Damjanovic et al. In either case, it is assumed that the potential corresponding to the zero Galvani potential difference with respect to an electrode kept at a constant temperature is the same at all temperatures. In most of the literature data, activation energies are given at a constant overpotential, q, including 7 = 0, rather than at a constant E. This case corresponds to isothermal conditions with the reference point between the reversible potential of the reaction itself for which the activation energy is sought. The Gibbs energy of activation is then given with respect to the energy at the reversible potential, i.e., for an anodic process:
IC' = AC*, = AG*,
+ PFq
which clearly shows that the representations in terms of 7 and A@, which is also valid for isothermal conditions, are equivalent. Subscript R here and in the following stands for "reversible", meaning that the measurements, and analysis, are done with the reference point being the reversible potential of the reaction itself, Le., with internal reference point. Rate equation with the qrepresentation for the case /3 = PsT is given by i =
-AH* (14)
and in the differential form, at any 7, by d In i/K
Thus, for the observed case p = &T, Tafel slopes at any T are given by
What is, however, surprising and difficult to understand is why at all temperatures Tafel slopes are close to the "ideal" value of 120 mV/decade, which is reserved only for the room temperature first electron-transfer step as rd when P = PH = 1/2, and not to any other value. It is also surprising why PS should have a singular value even for different reactions exhibiting the Tafel slopes of 120 mV when P = PsT. It implies that an increment in the activation energy, at a constant T , required to increase the rate by one decade, or by any constant factor, is inversely proportional to T . Furthermore, for the current at a given constant E to increase with temperature, the activation energy decreases as temperature increases. It has, however, commonly been assumed that the "activation energy" does not depend on temperature. Equation I O can now be used to discuss the observed temperature dependence of the OE reaction. It is customary to use Arrhenius plots in analyses of temperature effects on reaction rates. Differentiation of eq 10 with respect to 1 / T a t a constant potential yield
The latter terms in eq 12a and 12b will be zero if the reference electrode is kept at a constant temperature, Le., for nonisothermal conditions. If the reference electrode is kept at the same temperature as the test electrode, Le., for isothermal conditions, for which eq 12a and 12b hold, the change in the potential of the reference electrode with temperature must be taken into account.
AH'R
It is interesting to note that, in the case of = PsT, the enthalpy of activation, as recovered from the Arrhenius plots for the isothermal conditions, is independent of 7. For the nonisothermal conditions, also, the enthalpy term at a constant E , or A+, including A@ = 0, is the same at all electrode potentials (see eq 12a and 12b). This is a direct consequence of the experimental fact that the Tafel lines at different temperatures are parallel. The question, however, can be raised, when = PsT what is actually obtained from Arrhenius plots in isothermal or in nonisothermal conditions. It may be noted, also, that in the representation of the rates in terms of A@,or E, even for isothermal conditions, the recovered activation energies are independent of A+, or E (see eq 12a and 12b). = AH*R. From eq 12 and 15, it seems to follow that This could be possible only if AH* does not vary with electrode potential, as suggested by ConwayI2 and Conway and co-worke r ~when ~ ~/3 =, &T. ~ ~However, if AH* is the same at all potentials, it must imply that AH in any activated state is the same as in the ground state, Le., AH*A+=Q = AHIR = 0
(16)
This relation reemphasizes the importance of the significance of the "energy of activation" as recovered from the Arrhenius plots when p = psT, both in the representation of the rates in terms of E and in terms of q. Such uncertainty in the meaning of the energy of activation from Arrhenius plots does not exist when p = PH = constant. In this case, which will not be discussed further below, the isothermal and nonisothermal conditions yield, respectively,
[
d In ( i / K ) d(l/T)
1.-
AH*A+=o PFE,+=o R R
+-
-~
- --AH'E R
+ -PFE +R - -
PF dEA+o T d(I/n
PF ~ E A + = o
+--F
41/73
(17)
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990 1971
@-Factorfor Oxygen Evolution at Pt Electrodes and
["a;,'/,"]
AH* A
= -E
R
p o
PFE +-=-R +-PFEApo R
AH*E R
(18) Thus, under both conditions, the enthalpy of activation increases with electrode potential, as expected. However, in order to obtain the enthalpy of activation at a given E under isothermal conditions, the correction has to be made for the variation of EA6-O with temperature. This variation is the same as the variation of the reference electrode potential with temperature, providing the potential-vs a reference electrode at a constant potential-at which A 4 = 0 does not change with T. Since for the nonisothermal condition this correction is not required, the nonisothermal condition is preferred when AH* at a given E (not 11) is being determined. In the derivations leading to eq 12 and 14, it is assumed, according to the generally accepted procedure, and without proof or justification, that EA+o, AH*A,,o,and all of which are inaccessible to experimental determinations, as well as AH*R and AS*R,are independent of temperature. Recently, the validity of these assumptions has, however, been questioned,15 as will be discussed below in the context of the present data for the OE at Pt electrodes. I n a recent analysis of kinetics at electrodes,'5~i6~28 the Gibbs energy of activation, AG*(E), at a given electrode potential, E, was expressed in terms of the electron energies in the activated state, t, and the ground state, to, of the reacting complex in the rds, i.e., for an anodic process AG*(E) I AG* = P(t-t,J (19)
--E
I
I
7 \
\ --\
or, in the familiar form, by
-1
i = K ex,[ PFE For the case P = PsT, eq 22 and 23 are modified respectively to
and
-1
i = K exp[ PsFE
Density of electron energy levels
Solution Side
Figure 6. A model for electron transfer in oxygen evolution a t thin (- 10
A) anodic oxide film, OF, a t Pt electrodes. In a simple representation,
electrons for solution with energy t tunnel through the OF if its energy equals the electron energy a t the Fermi level, tF. For more details see ref 15.
\
*
Reactlon Coordliate Products In Rate
Reactants Determlng Step
Figure 7. Graphical representation of Gibbs energies of activation a t constant electrode potentials at different temperatures. The ground-state energy of the reactants changes with temperatre due to the entropy factor in AGO. It is arbitrarily taken that the ground-state energy of the products in a rate-determining step also changes with temperature.
It is clear that the anodic current at a given E vs the reference electrode at a constant temperature can increase only if 6 increases (becomes less negative) with temperature. For a cathodic process, which will not be discussed here, signs in front of P in eq 21-25 change. For cathodic current to increase as temperature increases, 6 must decrease (become more negative) as temperature increases. to for an anodic process becomes less negative as temperature increases because the ground-state energy of the reacting complex, AGO,decreases with temperature. When @ = PsT, AGOdecreases according toi5 6AGo = 6TASo = -PsT 660 (26) and dto = -6to =-dT
(28) Damjanovic, A,; Birss, V. I.; Boudreaux, D. S . J . Electrochem. SOC., submitted for publication. ( 2 9 ) Lohmann, F. 2. Naturforsch. 1967, 220, 843. Gerischer, H. 'Electrocatalysis on Non-metallic Surfaces". National Bureau of Standards Special Publication 455, 1976, p 1.
\
Electrodb
where P is the symmetry factor. Electron transfer occurs via quantum mechanical tunneling process when t in the activated state matches the electron energy, tF, at the Fermi level, FL, in the metal electrode, Le., when t = tF. As E increases by AE, t F decreases by AtF = -AE. This is illustrated in Figure 6. The value of to is usually an unknown quantity and tF can be related to a reference electrode, e.g., to the hydrogen electrode at Pt for which the electron energy at the FL with respect to vacuum, tHE, is k n o ~ n .Then, ~ ~ for ~ ~a ~process occurring at Pt electrodes t F = t = 'HE - eE(HE) (20) Here, E(HE) is the potential at the test electrode with respect to the hydrogen electrode. With eq 20, the Gibbs energy of activation for an anodic process as a function of potential is given by AG*(E) = P(tHE-tO) - @E(HE) (21) The rate equation is then given by
I
6T
AS0
PsT
(27)
Here, ASo is the entropy in the ground state of the reacting complex. It is a negative quantity when P = psT.I5 The change of AGOwith T is illustrated in Figure 7 on the potential energy diagram. In Figure 8, the distribution of the electron energy states
1972
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
Damjanovic et al. This type of the compensation effect between the enthalpy and/or entropy of activation on one side, and on the other is easy to visualize when P = PH. In this case, tois not a function which is otherwise independent of T and any change in of T if E,,=o does not change with T , is compensated by an equivalent change in eE,,=,, i s . , Thus, for the case /3 = PH,AH*at any potential does not depend on the position of the zero of the Galvani potential difference with respect to a reference electrode at constant T (see Figure 8). Finally, when the ground-state energy of a reacting complex and to vary with T , as in the case /3 = &T, electrode potentials at any given current density are expected to vary linearly with temperature. The slopes of the E-T lines should then be the same at all currents. This is indeed observed for the oxygen evolution at Pt electrodes as shown in Figure 4. When P = pH = constant, E-T relations at a constant i are also linear, but the slopes would be different for each current. The linearity in the case /3 = &T is not immediately evident. From eq 27, tois obtained in the form
Solution
Figure 8. An illustration of the distribution of electron states, 6 , in the activated reacting species in solution, N ( t ) , at two temperatures. The ground-state energy of the reacting species changes with temperature and consequently, at the same eF, and hence the same electrode potential, activation energy decreases as T increases. A position of the electron energy level in the electrode at which A@ = 0 is sketched. Note that in the case a = aH = constant, zo is invariant with T.15
and positions of to at two temperatures are illustrated. At a given tF, Le., at a given electrode potential against the reference electrode at the same temperature, the activation energy, AGtE, decreases, as temperature increases, according to 6AG*E = -&T 6to
(28) These equations will be used below in the analyses of Arrhenius plots and energies of activation when 0 = PsT. Meaning of the Activation Energies Recovered from Arrhenius Plots When /3 = PsT. Only when @ is constant, slopes of the log i vs 1 / T plots at a given E against a reference electrode kept at a constant temperature give AH*at that E. When @ = &T, AH* does not vary with the electrode p ~ t e n t i a l l l -and ~~ temperat~re.'~ In fact, AH* may be assumed to be zero at all potentials and temperature. Consequently eq 28 can be written in the form
6AG*E = -T 6 A S * E = PsT S t o = -6TASo
(29)
Since to increases with 7,AS*E must decrease with T, Le., AS* at a constant E is a function of T . The differentiation of eq 10, therefore, does not lead to eq 12, but rather to
which with eq 29 gives, at any E (including dlni d(l/T)
Tso
R
Thus, when /3 = PsT an Arrhenius plot recovers, not AH*,,Po, or AH*R, but rather the entropy of the reacting complex in the ground state. What is more important, in this case, the dependence of log i on 1 / T is not expected to be linear. In the derivative of eq 3 1 , it is assumed that E,,=, is invariant with T . If E,,=o varies with T , an additional change of AS*,,=O with temperature will compensate such a variation. Thus, at E = 0, for instance, the energy of activation, AG*E=o AG*ESo = - T L ~ S *=~ -eE,,=o ,~ + TAS*,,=,= PsT(t-t0) (32) will have the same value irrespective of the position of A 4 = 0 with respect to the reference electrode at constant T. Therefore, at any E , eq 31 holds irrespective of the value of (see Figure 8).
where T R and are room temperature (in kelvin) and the ground-state electron energy a t room temperature, respectively. For a small temperature interval around room temperature, T I T , is close to one (1 f 0.1 for the present experimental conditions) and In TIT,, which is obtained in the integration of eq 27, can be approximated to ( T I T , - I ) , i.e.,
Introducing to into eq 24, it can be seen that, at any constant current, the electrode potential for an anodic process vs a reference electrode at a constant temperature decreases linearly with temperature. From the slopes of the E-T lines at any constant thickness of the OF, Socan be obtained. For the oxygen evolution at Pt it is (see eq 36) So = -PsTRF
dE - = 0.00235 eV K-' dT
(35)
Significance ofthe Arrhenius Plots When P = PsT. Combining eq 24 and 34b, we can write the rate equation in the form
It is now clear that in the case = PsT, no linear In i vs 1 / T relation at a constant E is possible. Any "activation" energy obtained from some quasi-linear In i vs 1/ T relations not only does not represent AH*,,=o or AHtR, as frequently assumed and discussed above, but it cannot represent any constant energy at all. A plot of In i-T at a constant E for nonisothermal conditions should in this case be linear (see Figure 9) and should provide information regarding So,much the same as the plot of E vs T at a constant current does. Arrhenius type of plots, i.e., In i vs 1 / T, may be used only when P = PH = constant in which case AHtEare obtained, but only for the log i vs 1 / T plots at constant E vs a reference electrode a t a constant temperature, Le., for nonisothermal experiments. When P = PsT, or P = PH + PsT, Arrhenius plots should not be used. It seems that activation energies reported in the literature are frequently determined without regard for the behavior of the symmetry factor with respect to temperature. The observed kinetics for the case P = PsT is formally explainedLz-I4in terms of much stronger decrease of the entropy of activation as electrode potential increases (for an anodic process) than of the enthalpy of activation, which, in fact, has to be close to zero at all electrode p0tentia1s.I~ A critical point, however, remains as to why PsT at room temperature is so close to 1/2 and not to any other value. Such a close correspondence of the Tafel
1973
J . Phys. Chem. 1990, 94, 1973-1981 273
I ' -2
298
323
I
1
348 K I
t
/I
(Note that AS* has a negative value.15) Since A€ always corresponds to -AE, and N(e) is the main factor controlling rates, it follows that
/
1 /;
-5
/
0
/
Since, with /3 = &T, it is implied that AS* is the predominant factor controlling rates, AH*being close to zero at all ptentials,I5 the distribution of electron energy levels is governed by the entropy term in the activation energy, Le.,
/
T 6AS* = -p 6 c = pe 6E 1
I
25
50
I
75 "C
Temperature
Figure 9. A plot of log i vs T at a constant E (=1.6 V vs SCE) for nonisothermal conditions. This plot should be compared with the Arrhenius plot in Figure 2. Clearly, in the case p = BsT, a linear dependence is obtained not with the Arrhenius plots but rather with the log i vs T plot. i, = 3 X lo-) A, t , = 1000 s, surface area 0.5 cm2.
slopes of 120 mV to the value 2.3 X 2 R T / F for the room temperature cannot be coincidental but must have a real physicochemical significance. A kind of compensation effect leading to the temperature-independent symmetry factor is suspect. Another look at the whole problem of the apparent symmetry factors, with very carefully collected data over as wide a temperature range as experimentally and for various systems, seems to be - possible, . required.
(38)
It is reasonable to assume that, at any temperature, AS* increases linearly as E increases, and, therefore, p in eq 37 and 38 must be equal to PsT. Then,
ps = --sAs* = -6As* f f ( T ) e6E
6t
(39)
This is, however, only another representation of the definition of ps previously given by Conway12and Conway and c o - ~ o r k e r s . ' ~ J ~
Acknowledgment. A.D. and A.T.W. are grateful to Drs. Lance A. Davis, Robert C. Morris, and Dave Narasimhan for creating an atmosphere in which this work could be done. They are also grateful to Ms. Marcy Daboul for her assistance in technical preparation of the manuscript. Registry No. Pt, 7440-06-4;02,7782-44-7; H2S04,7664-93-9.
A New ab Initio Potential Energy Surface for H on Ru(0001) and I t s Use for Variational Transition State Theory and Semiclassical Tunneling Calculations of the Surface Diffusion of H and D Thanh N. Truong, Donald G . Truhlar,* Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431
James R. Chelikowsky,* Department of Chemical Engineering and Materials Science and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431
and M. Y. Chou* School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (Received: May 26, 1989; In Final Form: August 18, 1989)
We have applied variational transition state theory with a semiclassical tunneling method to calculate the rates of hydrogen and deuterium diffusing on a Ru(0001) surface. The hydrogen-metal interaction is modeled by a pairwise potential which has been fitted to electronic structure calculations based on a local density approximation and a pseudopotential. We present diffusion coefficients and kinetic isotope effects for the temperature range from 80 to 800 K using the fitted potential and also for two modified potentials-one in which the potential along the minimum-energy path is scaled by a constant and one in which the pairwise potential parameters are modified to change the barrier height. The results for all three potentials are compared to experimental rate coefficients determined recently by laser-induced thermal desorption.
1. Introduction
Understanding hydrogen chemisorption and diffusion on transition-metal surfaces is a fundamental issue for a variety of technologically important chemical reactions. For example, the synthesis of hydrocarbons may be catalyzed by passing C 0 2 and H2 over certain transition-metal surfaces.' A key ingredient in ( I ) Storch, H. H.; Golumbic, N.; Anderson, R. B. The Fisher-Tropsch and Related Synthesis; Wiley: New York, 1951.
0022-3654/90/2094-1973$02.50/0
formulating a microscopic picture of this reaction and many others is to understand how H behaves on the transition-metal surface. The attempt to achieve this understanding is central to our effort. Ruthenium is especially interesting from this point of view since it is a catalyst often used for hydrocarbon synthesis. Moreover, ruthenium appears to be well understood in terms of its intrinsic bulk and surface properties.2" The Ru surface has been ex(2) Feibelman, P. J. Phys. Reu. B 1982, 26, 5347.
0 1990 American Chemical Society
