H Tunneling Effects on Sequential Dissociation of Methane over Ni

Article pubs.acs.org/JPCC

H Tunneling Effects on Sequential Dissociation of Methane over Ni(111) and the Overall Rate of Methane Reforming Ernst D. German,*,† Olga Nekhamkina,† Oleg Temkin,‡ and Moshe Sheintuch† †

Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel M.V. Lomonosov Moscow State University of Fine Chemical Technology, Vernadsky Pr., 86, Moscow 117571, Russian Federation

‡

ABSTRACT: The rate constants and the activation energies for the C−H bond dissociation of CH3, CH2, and CH species adsorbed on the surface of Ni(111) are calculated in the framework of a previously developed methodology. It considers the dissociations as a result of quantum transitions between discrete vibrational energy levels of an initial C−H bond in methane and energy continuum of the dissociated final state while the system reaches the transition state along other classical degrees of freedom. These constants along with other ones calculated in our previous papers are used for calculating the overall rate using a microkinetic model of methane steam reforming (MSR) incorporating methane and water dissociation and CO formation either by C + O or by CH + O → CHO → CO + H. The analysis shows that the latter steps cannot be ignored. On this metal, where the rate of water dissociation is much larger than that of methane, the rate can be expressed explicitly. The dependencies of the MSR rate on partial pressures of methane, water, hydrogen, and carbon monoxide are studied and are compared with experiment. The isotope effect (IE) for overall rate of MSR is calculated using the rate constants of all steps including disruption of C−H, O−H, and H−H bonds. For certain applications, like MSR in a membrane reactor, rates are analyzed to show that hydrogen separation may suppress the forward reaction.

1. INTRODUCTION The adsorption and sequential dissociation of methane, on transition metal surfaces, are important mechanistic steps in many catalytic reactions including steam or dry reforming (SR, DR). The reverse steps are of importance in methanation and gas-to-liquid processes. Experimental1−9 and theoretical9−49 studies of the dissociation kinetics, the thermodynamics, as well as of the structure characteristics of reactants and products of CH5‑α (α = 1−4) species, have been made; a classical behavior of a H atom in dissociation process was assumed in most referred works. The disrupted C−H bonds, however, have a very high vibration frequencies (∼3000 cm−1), and the CH vibrations should be considered as having a quantum character50 even at temperatures of commercial SR reactors. This fact suggests that sequential steps during methane dissociation should be described as tunnel processes, and their probability may be calculated using the theory of quantum H transitions, in similarity to O−H and H−H dissociations that were previously considered by us.51,52 The first step of methane dissociation has been studied in terms of this approach in our recent publication,53 and the main ideas of this approach were described in detail in previous works.51−54 In the present work, we apply the same technique to study kinetics of catalytic C−H dissociation in CH5‑α species (α = 2, 3, and 4) on a nickel surface, which is most popular catalyst for methane steam reforming (MSR) CH4(g) + H 2O(g) → CO(g) + 3H 2(g)

and products on this metal are used as input parameters. Performing these calculations is the first target of our work. The kinetic data obtained in this work, combined with those calculated earlier by the same methodology, form the main part of the set of rate constants required for microkinetic modeling of methane steam reforming (MSR). These data (describing methane, water, hydrogen, and OH dissociation) are complemented with the rate constants of molecular adsorption− desorption of CO, methane, of water, and of H2, which are estimated using either a mobile or an immobile model of the transition state (TS) and published adsorption energies. The corresponding microkinetic model is then applied to predict surface coverages and to the overall rate, and determine most abundant reaction intermediates. The study of the MSR microkinetics models is the second aim of our paper. We show that using rate constants of individual steps, which account for the H-tunneling effects, gives an adequate description of the overall MSR reaction kinetics. The MSR kinetics considers CO produced either by C + O reaction or by CO + H → COH followed by the dissociation of COH. The latter step was studied, for example, in refs 21, 40, and 41. These two popular chemical mechanisms, differing in two steps, are compared to show that the choice of steps is crucial. Various chemical mechanisms and microkinetic models have been suggested to describe MSR, differing in number of surface intermediates and reaction steps.21,40,41,55−57 However, despite

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and we calculate the corresponding activation energies and the pre-exponential factors for these dissociation steps. Literature values of the physical characteristics of the adsorbed reactants © 2015 American Chemical Society

Received: December 26, 2014 Revised: March 24, 2015 Published: March 31, 2015 9260

DOI: 10.1021/jp5128964 J. Phys. Chem. C 2015, 119, 9260−9273

Article

The Journal of Physical Chemistry C

Figure 1. Pictorial view of methane fragments adsorbed on metal surface.

the potential energy in the direction z perpendicular to the surface (Figure 3a) is considered to describe a vibrational interaction of a reactant quasi-molecule and the surface; (iv) Breaking a C−H bond occurs by the H quantum transitions along r coordinate under an adiabatic potential barrier, of which height is dependent on distance z, on the orientation φ (of A−H relative to the surface) as well as on the parameter Er which is a characteristic of the oscillator. The assumption (i) simplifies the PESs of reactions 2 (for α = 2 and 3) considerably, reducing the full set of internal coordinates {Qk} of both CH3 and CH2 species to a single coordinate r describing the A−H bond length (A is the carbon atom C for the dissociation of a C−H fragment (α = 4), see Figures 1 and 2). Following the methodology of our previous

all the reported experimental and theoretical studies, the detailed path for conversion of methane to syngas remains controversial and often contradictory. On a Ni(111) surface where, as we show, the rate of water dissociation is much larger than that of methane, the rate can be expressed explicitly. Consequently, the rate is linear with methane concentration and is inhibited by large water pressures. Large hydrogen pressure reduces the oxygen coverage and accelerates the rate within a certain range. Hydrogen separation in a membrane reactor may diminish the overall rate, as we show below. In the framework of these models, we interpret the experimentally observed isotope effect (IE) by calculating the overall MSR rates for the normal and deuterated methane and water molecules. This is the third aim of this work. The paper outline is as follows: Section 2 reviews theoretical background of the approach applied for calculations of the rate constants of hydrogen dissociation reactions. Results of our calculations of numerical values of the H-dissociation rate constants are discussed in Section 3, and the total set of the rate constant using for construction of the microkinetic model is considered in Section 4. Numerical analysis of the microkinetic models is performed in Section 5, which is used for construction of the approximate solution. Isotope effect is studied in this section. The application of the microkinetic models for the membrane reactor design is considered in Section 6. The paper ends with a Conclusions section.

Figure 2. Formal model of reactants and products describing adsorbed species dissociation; x and y are coordinates on the metal surface, z is the perpendicular coordinate.

2. THEORETICAL BACKGROUNDS OF STUDYING H TRANSFER REACTIONS ON A SURFACE In this section, we present only a summary of the main ideas of our approach which has been used for calculations of the rate constants of the A−H dissociation steps (A = O, C, H), because the theory has been described in detail in previous publications.51−54,58 According to the theory, the surface dissociation of CH5‑α (α = 2, 3, and 4) species CH5 − α → CH4 − α + H (2)

studies51,53 in addition to the r coordinate, we introduce polar orientation of the A−H bond (angle φ relative to the surface normal), the distance of the mass center from the surface (z), and the coordinate that describes oscillations of the metal atoms, i.e., the number of degrees of freedom required to describe the dissociation is reduced to four, including one quantum. Thus, the barrier for H-tunneling is not static; its form depends on three classical coordinates including the coordinate describing symmetric metal atom vibrations and coordinates characterizing the variation of the barrier for H tunneling while a pseudo twoatom molecule approaches a surface and rotates. The physical mechanism of a quantum transition is as follows. In the initial state the H atom (in CH5‑α) is localized in the potential well ui(r) (Figure 2 and the left curve in Figure 3), the proton states are characterized by a set of discrete quantum vibration levels {εni }. The proton occupies the vibrational levels in the well according to the Gibbs distribution. In the final state, the system (the dissociated hydrogen atom and CH4‑α fragment) may take any level of energy {εf} in an energy continuum spectrum of the decay potential uf(r). Therefore, one may describe the AH dissociation considering quantum transitions of an H atom from the n-th vibrational level εni in the ui potential well to any level εf in the potential uf, with subsequent averaging of the transition probability over the initial energy distribution and integration over the continuum of the final levels. For this

is described as a result of quantum transitions between discrete vibrational energy levels of an initial C−H bond of adsorbed CH5‑α species and an energy continuum of the dissociated final state while the system reaches the transition state along other classical degrees of freedom. The following assumptions are made: (i) Disruption of a C−H bond in CH3 or CH2 species is considered as dissociation of a quasi-diatomic A−H molecule in which A is a hypothetical atom of equivalent mass of which internal structure is ignored; the assumption is based on the recognition that for most of orientations of the adsorbed reactants, one of the H atoms will be “closest” to the surface and “different” than the remaining three; (ii) The thermal motion of the catalyst metal lattice can be reduced to the vibrations of a single harmonic oscillator whose level populations are thermally distributed at temperature T; (iii) The metal surface is smooth, i.e., the dependence of the potential energy surface (PES) on the catalyst surface coordinates x and y is neglected (Figure 2), and 9261


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The notations of z, φ, and εni are clear from the Figures 2 and 3: a cap is the symbol of the transition state, ΔI is the surface reaction energy of the corresponding reaction step, ζ =Δj/kBT is the dimensionless integration variable, and q is the oscillator coordinate.53 The subscript in parentheses at k identifies the proton vibrational level from which tunneling occurs: zero corresponds to the ground level, unity corresponds to the first excited level etc. The pre-exponential factors include characteristic constants and an electron−proton transmission coefficient which depends on H-tunnel factor. The latter is calculated considering tunneling H atom through the variable adiabatic potential barrier along the r(C−H) coordinate of the broken bond (Figure 3a) in quasi-classical approximation; as it was mentioned above, the form of this barrier is modulated by metal atom vibrations and depends on distance and orientation A−H molecule relative to metal surface. As has been shown in our methane dissociation reaction study, only two terms in eq 4 are important for calculation of the rate constant and the activation energy, k(0) and k(1). This conclusion holds in our estimations for other CHα‑5 species considered here. The relative contribution of these terms to the rate constant depends on the metal. The effective activation energy characterizing temperature dependence of the total k is determined by calculation of the rate constant at two temperatures within a narrow temperature interval using the Arrhenius law. The surface reaction energies of the C−H dissociation steps are calculated referring to Figure 4 as

Figure 3. Potential energy curves: (a) along z coordinate; vi(z) is the potential energy describing the interaction of a nondissociated AH species with a surface; vf(z) is the potential energy describing the total interaction of dissociated fragments A and H with the surface. (b) along the r coordinate; ui(r) is the potential energy of C−H bond in an adsorbed methane molecule, uf(r) is the repulsion energy of dissociated hydrogen atom and CH5‑α fragment; uad is the adiabatic curve; Δj is the transition energy along the r coordinate; arrows show positive (up) and negative (down) energy values.

ΔI(CH5 − α) = D(CH5 − α) + Eads(CH4 − α) + Eads(H) − Eads(CH5 − α)

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quantum transition, it is necessary, according to the Franck− Condon principle, that the conservation of energy is obeyed (i.e., see Figure 3) εf = εin − Δj(n , εf )

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where Δj determines the mutual arrangement of the potential curves ui and uf and is identified with transition energy along the r coordinate. The initial and final energy states of the considered system include also vibrational energies of k classical metal lattice oscillators (phonon “bath”), so that the H atom transfer is assisted by transitions from a set of these initial oscillatory levels to a set of the final oscillatory levels. The physical picture presented is the basis for the analytical expression of the rate constant. In the framework of this approach the probability W (i.e., the rate constant k) is calculated using nonadiabatic perturbation theory. It is important that we consider electrons of the C−H bond to be broken to behave adiabatically, i.e., the barrier for proton tunneling is of the adiabatic form (Figure 3). This is because the motion of the electrons is rapid relative to nuclear motions (Born−Oppenheimer approximation). Due to hydrogen tunneling, the total electron−proton matrix element may be small, i.e., the total nonadiabaticity may be observed for this system even in the case of a rather large electronic coupling. The final expression for each specific step-rate constant k is written as the sum53 k = k(0) + k(1) + k(2) + ...

Figure 4. Thermodynamic cycle used for calculation of the energy of the CH5‑α species dissociation, CH5‑α(ads) → CH4‑α(ads) + H(ads), α = 2, 3, and 4.

where D(CH5‑α) and Eads(A) are the dissociation energy of the C−H bond of a gas phase CH5‑α and the adsorption energy of the A, respectively. It follows from eq 7 that energy of the total dissociation of a methane molecule over nickel surface, (CH4(ads) →C(ads) + 4H(ads), is equal to (in kcal/mol) 4

ΔItot =

α=1

= 400 + Eads(C) + 4Eads(H)

∫ an̂ (ζ)exp[−Ĥ n(ζ)/kBT ]dζ

Ĥ n(ζ ) = f {(εin − εi0), ΔI , z(̂ ζ ), q(̂ ζ ), ϕ(̂ ζ )}

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where the sum of the gas phase dissociation energies was estimated to be equal to 397 kcal/mol using D values from ref 59 (105, 110, 101, and 81 kcal/mol for CH4, CH3, CH2 and CH, respectively) and the adsorption energy of a methane molecule, Eads (CH4)= −3 kcal/mol.60,61 Applying the above theory for calculation of the kinetic characteristics of the reaction steps (2) requires information about the adsorption parameters of corresponding intermediates. Most of these parameters are listed in Table 1 being compiled from published works. We provide these data with some comments in addition to those which are clear from Figures 2 and 3.

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where k(n) =

∑ D(CH5 − α) − Eads(CH4) + Eads(C) + 4Eads(H)

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The Journal of Physical Chemistry C Table 1. Adsorption Characteristics of CH5‑α Species (α = 1−5) and H Atom on Ni(111) Surface species

−Eadsa

xb

ωc

Ωrd

Die

φ0if

refs

CH4 CH3 CH2 CH C H

3 42 76 136 138 66

3.38 1.86 1.45 1.2 1.0 1.15

360 360 580 590 550 1100

3000 2940 2960 3040

105 107 97 85

51g 65 54 0

this study 53 11, 14, 40, 46 6, 11, 14, 15, 40, 44 11, 40 53

a

In kcal/mol. bDistance surface - C in Å. cStretching vibration frequency in perpendicular to the surface direction. dStretching frequency of the asymmetric C−H vibration (cm−1) in adsorbed species. eDissociation energy of C−H bond in adsorbed species (in kcal/mol) estimated according to eq 9. fThe initial orientation of a model species A−H relative to the surface normal (see Figure 2). gIn ref 53, axis z was directed to the surface from atom C.

Table 2. Calculated (Columns 4−6) and Compared (Columns 7−8) Kinetic Characteristics of H-Dissociation Steps on (111) Nickel Surface α

CH5‑α species

ΔIa

Eaa

lgAb

(kα or S0)b,c

1

2

3

4

5

6

Eaa

(kα or S0)b,c

7 −9

1 2 3

CH4 CH3 CH2

0 10 −25

17.4 21.6 ∼0

11.6 12.3 12.6

1.9 × 10 8.72 × 106 3.65 × 1012

12.6 ± 1.2 ; 17.7 ± 2.5 16.1;29,61 15.840 8.1;29 6.2;40 7.663

4

CH

17

26.7

12.8

1.44 × 106

31.6;21 30.7;29 32.240 32;63 30.464

d

8 d

−8 d

(1.2 × 10 ) ; (2.1 × 10−9)d; (5 × 10−8)d (1.70 × 109);29,61 (2.02 × 109)40 (1.71 × 1011);29 (5.22 × 1011)40 (2.28 × 1011)63 (2.24 × 106);21 (3.77 × 105)29 (1.59 × 105);40 (1.78 × 105)63 (4.49 × 105)64

kcal/mol. bs−1. cInitial sticking coefficient S0 for methane (α = 1, T = 500 K) and rate constants kα for other reaction species (α = 2, 3, and 4, T = 873 K), which were estimated using the published Ea listed in the neighbor column on the left and classical pre-exponential factor kBT/h. d Experimental data cited from ref 53. a

The frequency ω in the Table describes stretching vibrations of a quasi-atom A, relative to the metal surface, in perpendicular direction. The corresponding values were taken based on DFT calculations of C-metal stretching frequency in refs 11, 14, 27, 35. The frequency of retarded rotation of A−H relative to its equilibrium value φ0i was identified with the frequency of scissor vibrations found for an adsorbed CH2 species (∼1300 cm−1 35). The initial A−H bond length, r0i, which is identified with the C−H bond to be disrupted may be approximately considered to be constant for all species and taken to be equal to 1.1 Å.14,46 The A−H stretching vibration frequencies Ωads r modeling the corresponding vibrations in the adsorbed CH3, CH2, and CH species were assigned values of 2940, 2960, and 3040 cm−1, respectively.1,7,13−16,35 The Ωads r , r0i and the dissociation energy of the A−H molecule in the adsorbed state, Di, affect the value of the transmission coefficient53 of the rate constant. The Di is estimated using an approximate equation62 gas Di = D Ωads r /Ωr

are differences of several kcal/mol between our and published results, variation of the Ea in the series of CH3, CH2, and CH species show the same qualitative picture (Figure 5; the Ea

Figure 5. Variation of the Ea in the dissociation series of CH4, CH3, CH2, and CH species: 1 - the present work; 2 - ref 40; 3 (squares) - ref 29; 4 (x) - ref 53

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where D is the corresponding A−H dissociation energy in gas phase (mentioned above), and Ωgas r is the A−H stretching frequency of isolated species. The adsorption energies Di, estimated by eq 9, are equal to 107, 97, and 85 kcal/mol for the AHCH3, CH2, and CH.

for CH4 is adopted from previous work53 and is added for completeness of comparison.) Using the results obtained in the present work and those in ref 53, we construct the energy profile of the complete dissociation of a methane molecule to C and 4H on Ni(111) (Figure 6a): For completeness, energy profiles characterizing some other steps, computed with published data, are also plotted in Figure 6b−d. The overall ΔE ∼ 54.6 kcal/mol for reaction 1 is in a good agreement with experimental value.59 The methane dissociation over nickel to C and 4H (Figure 6a) is a weakly exothermic process (ΔE = −1 kcal/mol), in agreement with calculation of ΔItot using eq 8. The presented energy profile show a very deep potential well corresponding to formation of CH species, which is separated from the final C + 4H state by a barrier of 27 kcal/mol.

3. ESTIMATES OF THE CH5‑α DISSOCIATION CONSTANTS (α = 2, 3, AND 4) ON NI(111) In this section, we present the numerical estimates for the rate constants of the surface dissociation of CH3, CH2, and CH species on the most preferred adsorption sites using eqs 4−6. The activation energies and pre-exponentials are estimated as was described above. These activation energies are compared (in Table 2) with published values. One can see that although there 9263


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corresponding values (column 8) estimated using the published activation energies Eas (column 7) in combination with the classical pre-exponential factor kBT/h (the entropy effect is neglected). The sticking coefficient (column 6 row 1) for methane dissociation is compared with the corresponding experimental values listed in column 8. The greatest difference between ours and the published theoretical estimations is observed for the CH3 dissociation step. Unfortunately, we cannot discriminate between the proposed and the DFT-TST approaches for estimation of the individual reaction rates. Using the proposed approach to estimate the overall MSR reaction rate was justified by comparison with available published data (see discussion in section 5). According to our calculations, dissociations of CH3, CH2, and CH species are characterized by a considerable kinetic isotope effect (KIE), which is about 2 to 4 at reaction temperatures (∼700−900 K). This is much lower than KIE the calculated previously for methane surface dissociation53 at T = 500 K, which is of a nonadiabaticity character. Thus, many steps of CH4 dissociation affect the observable isotope effect of the MSR. More details about the IE are given in section 5.3. The rate constants of the sequential dissociation of methane calculated in this section along with other kinetic information are used for numerical analysis of the microkinetic models of reaction 1 in the next section.

4. INDIVIDUAL STEP-RATE CONSTANTS AND MSR MICROKINETIC MODELS We employ a microkinetic model that accounts for successive dissociation of CH4 and of H2O followed by reaction between products: overall, the model accounts for 12 adsorbed species: CH4*, CH3*, CH2*, CH*, OH*; H*, H2O*, O*, CO*, H2*, C*, and CHO*, and for vacant sites (14 elementary reaction steps). This model, which accounts for both the C*+O* and CH*+O* → CHO* → CO+H reactions, is referred as Model I; the involved reactions and the corresponding thermodynamic parameters are listed in Tables 3, 4; the reaction numbers denoted here are preserved in the text. Other studies have considered additional steps like hydroxyl disproportionation (2OH → H2O + O) or other reaction steps (CH2 + O → CH2O21,40): to the best of our estimate, these additional steps will not affect the overall rates. 4.1. Rate Constants for Microkinetic Models. Below we list the sources of rate constants of the forward (kif) and back (kib) reactions for steps i = 1−14, the corresponding values at the reference temperature (873 K) are listed in Table 3. The gas phase partial pressures are assumed constant. The methane dissociation rate constants (k3f, k4f, and k5f) are calculated in the present work (Table 3). The constant k2f was calculated earlier53 using 4-D PES along with the presentation of the methodology. All these calculations accounted for the tunneling effects. The rate constants of hydrogen and water dissociation (k6b, k9f, and k10f) were previously calculated,52−54,58 while accounting for the tunneling effects, and are recalculated in the present work for the reference temperature (873 K). The rate constants of back reactions k2b, k3b, k4b, and k5b are calculated using the pre-exponential factors, the activation energy, and the surface reaction enthalpy for the corresponding forward ones. The rate constants k6f, k9b, and k10b are calculated similarly using the information in refs 51, 52, 58.

Figure 6. Energy profiles describing: (a) methane dissociation, (b) water dissociation, (c) carbon oxidation, and (d) CH oxidation, (CH+O)ads → (CO)g +Hads; all over a Ni(111) surface.

The activation barrier for dissociation of CH2 species is negligible. This is explained by the fact that this reaction is strongly exothermic in contrast to the dissociation reactions of CH3 and CH species which are strongly endothermic. This implies that MSR and DR kinetics will be insensitive to this activation barrier, but methanation or Fischer−Tropsch kinetics will be highly sensitive. The pre-exponential factors for these reactions are lower (due to quantum effects) than the corresponding classical value at the same temperature (lgAcl ∼13 at T ∼ 900 K). Additionally, we compare the rate constants characterizing methane dissociation steps and calculated in terms of our approach (Table 2, column 6, rows 2 to 4) with the 9264


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The Journal of Physical Chemistry C Table 3. Microkinetic Model and Rate Constants Calculated at the Reference Temperature T = 873 Ka no.

reaction step

kn,f

kn,b

Kn,eq

ΔGn, kcal/mol

ΔHn, kcal/mol

refs

1 2 3 4 5 6 7 8 9 10 11 12 13 14

* + CH4(g) ↔ CH4* * + CH4* ↔ CH3* + H* * + CH3* ↔ CH2* + H* * + CH2* ↔ CH* + H* * + CH* ↔ C* + H* H* + H* ↔ H2* + * H2* ↔ H2(g) + * * + H2O(g) ↔ H2O* * + H2O* ↔ OH* + H* * + *OH ↔ O* + H* C* + O* ↔ CO* + * CO* ↔ CO(g) + * CH*+ O* ↔ CHO* + * * + CHO* ↔ CO* + H*

1.39 + 08 2.23 × 107 8.72 × 106 3.65 × 1012 1.44 × 106 4.22 × 106 8.12 × 1012 1.31 × 108 4.04 × 108 4.22 × 107 8.85 4.0 × 106 1.78 × 104 3.64 × 1012

3.23 × 1012 2.23 × 107 2.71 × 109 2.01 × 106 2.55 × 1010 6.10 × 1011 3.93 × 108 3.21 × 1010 9.94 × 104 9.01 × 107 2.18 × 10−8 1.05 × 107 1.79 × 106 4.78 × 105

4.30 × 10−5 1 3.22 × 10−3 1.82 × 106 5.65 × 10−5 2.08 × 10−5 2.07 × 104 4.08 × 10−3 4.05 × 103 4.68 × 10−1 4.06 × 108 3.81 × 10−2 1.0 × 10−2 7.62 × 106

17.4 0 9.96 −25.0 17.0 20.6 −17.2 17.5 −14.4 1.32 −34.4 1.7 8.0 −27.5

−3 0 10 −25 17 20.6 1.34 −11 −14.4 1.4 −34.4 34.6 8.0 −27.5

60, 61, this study 53 this study this study this study 52, this study 52, this study 58, this study 58, this study 51, this study b b b b

a

k1f, k7b, k8f, and k12b have dimension of 1/(bar.s site); other constants have dimension 1/s site. bEstimated values using data of refs 40, 41 and corrected accounting for H tunneling.

appropriate than the mobile one to describe the corresponding CO adsorption and desorption rates k12f and k12b. This assumption is also supported by experimental data in ref 65 (see Appendix A for details). However, the assumptions regarding the TS will not affect the results in this case: we argue below, in analyzing the approximate rate expression, that the rate is sensitive to the equilibrium adsorption coefficients but not to adsorption or desorption rates. The energetic characteristics of CO dissociation and of C + O association as well as of reactions CH + O → CHO and CHO→ CO + H and of CO adsorption−desorption over nickel surface were studied by many authors by DFT method (see, for example, refs 21, 29, 40, 41, 66−68) This information was used to estimate the rate constants of steps 11−14. From these sources, the rate constants of steps 11−13 were calculated using standard expression with the preexponential kBT/h. The rate constants k14f and k14b describing C−H dissociation in adsorbed CHO and the corresponding back reaction should be explained. We assume step 14 in the forward direction to be nonactivated10 because this is a strongly exothermic nonadiabatic H transfer process (ΔH14 = −27.5 kcal/mol), similarly to step 4 (ΔH4 = −25 kcal/mol); therefore, the back direction activation energy should be equal to the reaction enthalpy.

Table 4. Reaction Rates of Individual Steps for Microkinetic Model Ia r =rf − rb

no.

no.

r = rf − rb

1

r1 = k1PCH4θ* − k1bθCH4

8

r8 = k 8PH2Oθ* − k 8bθH2O

2

r2 = k 2θCH4θ* − k 2bθCH3θH

9

r9 = k 9θH2Oθ* − k 9bθOHθH

3

r3 = k 3θCH3θ* − k 3bθCH2θH

10

r10 = k10θOHθ* − k10bθOθH

4

r4 = k4θCH2θ* − k4bθCHθH

11

r11 = k11θCθO − k11bθCOθ*

5

r5 = k5θCHθ* − k5bθCθH

12

r12 = k12θCO − k12bPCOθ*

13

r13 = k13θCHθO − k13bθCHOθ*

14

r14 = k14θCOHθ* − k14bθCOθH

6

r6 =

7

r7 = k 7θH2 − k 7bPH2θ*

k6θH2

− k6bθH2θ*

θk and θ* denote the coverage concentration and the vacant site fraction, respectively, and ri is the reaction rate of step i. a

The adsorption rate constants of methane (k1f), hydrogen (k7b) and of water (k8f) are assumed to be nonactivated (i.e., sticking coefficient is unity), and the corresponding desorption rates (k1b, k7f, k8b) are calculated assuming a mobile model of the TS (see Appendix A), which refers to the fact that the adsorption of these molecules over Ni(111) is characterized by a rather weak interaction with the surface and by small diffusion barriers. In contrast to these molecules, the CO adsorption energy over Ni(111) and over many others metals is much stronger.40 Therefore, it is reasonable to assume that the immobile model of the TS is more

5. ANALYSIS AND DISCUSSION OF STEADY-STATE RESULTS In this section we present direct numerical simulations (5.1) using microkinetic model I formulated in the previous section,

Table 5. Governing Ordinary Differential Equations for the Model Ia no.

equation

no.

equation

1

̇ = r1 − r2 θCH4

8

θȮ = r10 − r11 − r13

2

̇ = r2 − r3 θCH3

9

̇ = r11 − r12 + r14 θCO

3

̇ = r3 − r4 θCH2

10

̇ = r6 − r7 θH2

4

̇ = r4 − r5 − r13 θCH

11

θĊ = r5 − r11

5

̇ = r9 − r10 θOH

12

̇ θCHO = r13 − r14

6

θḢ = r2 + r3 + r4 + r5 − 2r6 + r9 + r10 + r14

13

θ*̇ = − r1 − r2 − r3 − r4 − r5 + r6 + r7 − r8 − r9 − r10 + r11 + r12 + r13 − r14

7

̇ θH2O = r8 − r9

Dots in equations denote time derivatives. Algebraic relation Σθk + θ* = 1 was used to verify the accuracy of the simulations. The reaction rates ri following mass action kinetics are given in Table 4. a

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The Journal of Physical Chemistry C the appropriate governing equations are summarized in Table 5. In section 5.2 we capitalize on the observation that on Ni(111) the rate of water dissociation is much larger than that of methane and the overall rate can be expressed explicitly, and predict well the rate computed by the detailed model in a wide range of partial pressures of methane, steam and hydrogen. The approximate model was used, in turn, as an effective tool to predict (i) the effects of the governing parameters on the overall reaction rate (section 5.2), (ii) the isotope effect of the overall rate (5.3) 5.1. Direct Solution of Detailed Model I. Typical dependencies of the turnover frequency (TOF, s−1), of the surface coverages (θi) of the main species and of the fraction of the vacant sites (θ*) on one of the partial pressures (either PCH4, or PH2O, or PH2, or PCO) are plotted in Figures 7−10 using the

Figure 9. Effect of the partial hydrogen pressure (PH2) on the MSR reaction rate (r or TOF(1/s)) and on the main component coverages (θi) on a Ni(111) surface with fixed partial pressures of methane (PCH4 = 0.2 bar), water (PH2O = 0.25 bar) and CO (PCO = 0.2 bar). Notations as in Figure 7.

Figure 7. Effect of partial methane pressure(PCH4) on the reaction rate(r or TOF(s−1)) and on the main surface coverages (θi) for MSR on a Ni(111) surface with fixed partial pressures of water (PH2O = 0.25 bar), hydrogen (PH2 = 0.4 bar) and CO (PCO = 0.2 bar). Solid lines and symbols mark the approximate solution (10)−(13) and the exact solution (Table 5), respectively. T = 873 K. Figure 10. Effect of the partial CO pressure (PCO) on the MSR reaction rate ((r) or TOF(1/s)) and the main component coverages (θi) on a Ni(111) surface with fixed partial pressures of methane (PCH4 = 0.2 bar), water (PH2O = 0.25 bar) and hydrogen (PH2 = 0.4 bar). Notations as in Figure 7

(ii) At steady-state conditions the reaction steps 1, 4−10, 12, and 14 are close to equilibrium, while the rate of step 11 (C + O→CO) is null, indicating that steps 13 and 14 are preferred (Table 6). Model II (without CH + O→CHO) leads to much smaller rates than those of full Model I. Thus, CO formation proceeds mainly through steps 13 and 14 (Table 3) rather than that by the direct carbon oxidation, in accordance with results of other works.21,40,66 The rate linear with PCH4 (Figure 7), is in agreement with experiments of previous works.56,69−71 The surface coverage concentrations are practically independent on this parameter, at least, while PCH4 2 bar. All coverages vary monotonically with PH2O. (iv) The rate passes through a maximum with respect to PH2 (Figure 9). The rate at PH2 = 0 is small. With increasing PH2,

Figure 8. Effect of the partial water pressure (PH2O) on the MSR reaction rate (r or TOF(s−1)) and the main component of coverages (θi) on a Ni(111) surface with fixed partial pressures of methane (PCH4 = 0.2 bar), hydrogen (PH2 = 0.4 bar) and CO (PCO = 0.2 bar). Notations as in Figure 7.

kinetic constants listed in Table 3. The inspection of the obtained results shows that (i) The most abundant surface intermediates are O*, OH*, H*, and CO* and the vacant sites (*) within a wide range of partial pressures of PCH4, PH2O, PH2, and PCO (Figures 7−10). 9266


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The Journal of Physical Chemistry C Table 6. Typical Values of the Forward (rf) and Back (rb) Reaction Rates at Steady-State Solution for MSR Model Ia

a

no.

rf

rb

r =r f − rb

no.

rf

rb

r =rf − rb

1 2 3 4 5 6 7

4.3588 × 106 4.7183 0.9392 192.4052 82.0996 2.9031 × 105 2.4647 × 107

4.3588 × 106 4.0181 0.2390 191.7050 82.0996 2.9031 × 105 2.4647 × 107

0.7002 0.7002 0.7002 0.7002 1.1486 × 10−8 2.1007 2.1007

8 9 10 11 12 13 14

5.1349 × 106 1.7557 × 104 2.5600 × 106 1.1768 × 10−8 3.2926 × 105 0.7011 1.0321 × 104

5.1349 × 106 1.7556 × 104 2.5600 × 106 2.8136 × 10−10 3.2926 × 105 8.9882 × 10−4 1.0320 × 104

0.7002 0.7002 0.7002 1.1486 × 10−8 0.7002 0.7002 0.7002

Pressures in bar, rates in s−1; PCH4 = 0.2 bar; PCH4 = 0.25 bar; PH2 = 0.4 bar; PCO = 0.2 bar, T = 873 K

The vacant site fraction (θ*) can be estimated via the main specie coverages normalized with θ* as

as more hydrogen adsorbs on the surface, oxygen decreases monotonically, while θOH and θ* exhibit local maxima which are shifted with respect to the reaction rate maximum toward the lower and the larger PH2 values, respectively. Note that decreasing hydrogen down to stoichiometric level will lead to very low rates. This will have special implications for membrane reactors as we show below. (v) The rate essentially decreases with increasing PCO (Figure 10), due to formation of θCO, which inhibits the reaction. We compare numerical values of the overall rate (TOF) on Ni(111), calculated at typical operating conditions (PCH4 = 20, PH2O = 25 kPa) with the experimental work of Wei and Iglesia69 for supported nickel catalyst under the same conditions. The overall rate for clean Ni(111) surface at these pressures and at PH2 = 0.4 and PCO = 0.2 kPa is equal to about 0.7 s−1 and is in reasonable agreement with experimental result (∼4 s−1), taking into account a difference in the structure of experimental and the model catalysts (see, for example, Figure 7). 5.2. Approximate Model for Steady-State Solutions. Assuming that methane dissociation steps (2, 3) and CO formation steps (13, 14) are the rate-determining steps, while the other steps, i.e., adsorption and water dissociation, steps 1, 4−10, and 12 are in equilibrium and that step 11 is absent (item (ii) above), we derived an approximated algebraic model (Appendix B) showing that all coverages vary linearly with the free site coverage (θ*), whereas the total reaction (i.e., turn over frequency (TOF) or rate r) varies like (θ*)2. Particularly, the main surface coverages and the rate follow: θH =

θ* K 6K̃ 7

, θOH =

θ* ≅ [1 + θH̅ + θO̅ + θOH ̅ + θCO ̅ ]−1 , θi̅ = θi/θ*

The approximate solutions obtained with the proposed model practically coincide (solid lines in Figures 7−10) with the direct simulations within the whole range of parameters considered in the paper. Thus, we address the model as a well validated tool and employ it to elucidate the tendencies observed above as well as to extend the analysis for more complicated regimes. Analysis of the approximate solution (10−13) shows that the reaction rate varies linearly with PCH4 (via K̃ 1). With increasing PCO, the normalized coverage θ̅CO increases (10) leading to decreasing θ* and r as well. Partial pressures PH2O, PH2 affect both the coefficient kef (11) and θ*, and their effect cannot be predicted in a simple way which agrees with items (iii)−(v) above. Obviously, the approximate rate is dependent on equilibrium adsorption coefficients and is independent of adsorption or desorption steps; this will not hold for very low CO concentration. In addition to items (i)−(v) above, we observe that under steady-state conditions the forward reaction rate constant k13f affects the total rate via the coefficient kef, while the main species coverages are independent of reaction 13 rate constants. These observations justify (the approximate solution practically coincide with the exact solution) the preferential CO formation route through steps 13 and 14. Note, lastly, that the derivation of the approximation assumes that the back reaction of step 13 is negligible under conditions studied. Thus, this approximation is valid only far from equilibrium. 5.3. Isotope Effects. Steps 2−7, 9, 10, and 14 include disruption of the A−H bond (where A is C, O, or H) or formation of the new A−H bond. An important mechanistic characteristic of an H transfer reaction is its isotope effect which may take a kinetic form, expressed as a ratio of rate constant of a reaction involving normal molecules to that while H atom is replaced with its isotope (KIE), or may take an equilibrium form, expressed as a ratio of the equilibrium constants of the reaction involving normal and isotope substituted molecules (EIE).73 In previous works51−54,58 the KIEs were considered for several elementary H/D transfer steps on metal surfaces. These steps are a part of the total microkinetics MSR model shown in Table 3. In this section, we investigate (at T = 873 K) isotope effect (IE) for the overall MSR reaction over nickel surface using microkinetic model I. For this (many steps) reaction, the problem is much more complicated than that for elementary H-transfer because the MSR reaction rate depends on both the step rates and on equilibrium constants (eq 11), and therefore, it has no simple interpretation. Analysis of the steady-state solutions of microkinetic model I for reactions with deuterated molecules (rate constants for D transfer were estimated using technique described above and in previous publications and they are listed in Table C1, of

K 6K̃ 7 K̃ 8K 9θ*,

θO = K 6K̃ 7K̃ 8K 9K10θ*, θCO =

1 θ* ̃ K12

(13)

(10)

r = kef (θ*)2 , kef = K1̃ K 2 ⎞−1 ⎛ 1 1 1 ⎟ ⎜ × + + 5 ⎜ k K K̃ k 2b ⎟ k13f K3K4 K 6K̃ 7 K̃ 8K 9K10 ⎠ ⎝ 3f 6 7 (11)

where K i = ki f /ki b

are the equilibrium constants and the auxiliary dimensionless values K̃ i are defined as ̃ = K12/PCO K1̃ = K1PCH4 , K̃ 7 = K 7/PH2 , K̃ 8 = K8PH2O , K12 (12) 9267


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was justified by our calculations53 based on the tunneling model. (ii) The experimental data on the overall isotope effect over supported nickel catalyst (IE = 1.6−1.7)69 are within the range predicted by the present analysis for an effective characteristic of the total reaction (IE < 4.5, Figure 11). (iii) The activation energy (Ea) of CH4 → CH3 + H reaction calculated accounting for tunneling effects agrees with the experiment (see Table 2, row 1, columns 4 and 7). (iv) The comparison with experimental sticking probability measurements is favorable53 (Table 2, row 1, columns 6 and 8).

Appendix C) shows that the observations made above are valid. Therefore, the approximate model (10−13) may be used to calculate the rate (rD) of MSR reaction between CD4 and D2O and to determine the IE as the ratio rH/ rD, which at fixed methane partial pressure is equal to IE = rH/rD ≅ (kef,H/kef,D)(θH*/θD*)2

(14)

where kef,H and kef,D correspond to coefficient kef (eq 11) estimated with the kinetic rate constants for H and D, respectively. Evidently, the IE depends on the rate constants of the many elementary steps as well as on the equilibrium constants, and on fractions of free sites on a surface in reaction systems involved normal and deuterated molecules. The effect of partial pressures of methane, of water, and of hydrogen on IE was studied separately. IE was found to be practically independent of methane partial pressure in range of values from 0.2 to 4 bar (IE is 3.42 to 3.38); however, it increases strongly while the partial pressure of H2O increases (Figure 11a).

6. IMPLICATIONS FOR REACTOR DESIGN The design of MSR bed, in general, and a membrane reactor, in particular, is affected by various considerations: (i) The steam to carbon ratio is usually above the stoichiometric ratio to avoid coking and deactivation; now we realize that that may lead to inhibition by oxygen adsorption on the Ni catalyst. (ii) Hydrogen separation by a membrane will lead to substoichiometric levels, and with the absence of experimental kinetic data, first-principles analysis maybe the only source for estimations. (iii) Strong CO adsorption inhibits the rate significantly; however, MSR is usually followed by Water Gas Shift (WGS) reaction, converting CO to CO2 and limiting this inhibition CO + H 2O → CO2 + H 2 (15) for an overall reaction of CH4 + 2H 2O → CO2 + 4H 2

(16)

A detailed analysis is beyond the scope of this study, and we just point out several effects. For the reaction in a fixed bed with a complete WGS conversion of CO to CO2, we obtain a stoichiometric composition of methane, steam, H2 and CO2; for the latter, we assume negligible adsorption (actually its concentration is equilibrium limited). Now to study the effect of steam/carbon on the rate we follow the rate along the reactor (actually, as a function of conversion, c): The component consumption follows the stoichiometric relations, i.e.

Figure 11. Dependence of IE on water (PH2O, a) and on hydrogen (PH2, b) partial pressures with fixed partial pressures of methane (PCH4 = 0.2 bar) and CO (PCO = 0.2 bar); partial pressures of hydrogen and water are fixed in panel a (PH2 = 0.4 bar) and panel b(PH2O = 0.25 bar), respectively. Simulations with approximate solution (10)−(13).

With increasing hydrogen pressure, the IE essentially decreases (Figure 11b). The isotope effect exceeds unity with PH2 ≤ 1.6 (as in the case of the elementary reaction of hydrogen transfer). At higher hydrogen pressure, the IE falls below unity. This tendency can be explained by analysis of the rate constant kef (11). With large PH2 the reaction rate coefficient kef ∼ (k7b)5/2 and accounting for (k7b)D/(k7b)H < 1, the IE exhibits the inverse behavior. The inverse isotope effect was recently measured in ref 72. At low hydrogen pressures, quantum effects play the main role, particularly, for the step 3 (Table 3). For this reaction step, KIE is equal to ∼4.4 compared with the overall reaction ratio kefH/kefD equal to 3.85 (at PCH4 = 0.2, PCO = 0.2, PH2O = 0.25 bar). Note that the experimental data for supported nickel catalyst by Wei and Iglesia69 show the H/D ratio (rH/rD ∼1.6−1.7) within the range predicted by the present analysis for an effective characteristic of the total reaction (Figure 11a,b). Summarizing the obtained results, we list below the arguments to justify the proposed microkinetic model, based on assumptions of tunnel effect in some H transfer steps: (i) The essential kinetic isotope effect (KIE) for CH4 → CH3 + H reaction [ ∼ 9 over Ni(111) surface at 475 K, ∼20 on Ru(111) at T = 600 K, 3 to 4 on Ir(111) surface]

FCH4 = FCH4 0(1 − c), FH2O = FH2O0 − 2cFCH4 0 , FH2 = FH2 0 + 4cFCH4 0 , FCO2 = FCO2 0 + cFCH4 0

(17)

3

where Fi is the mole density (g·mol/cm ). The total mole density (F∑ = ∑ FI) follows F∑ = F∑0 + 2cFCH4 0 = F∑0(1 + 2cyCH 0 )

(18)

4

where yi = Fi/F∑ is the mole fraction of i-th component. In a practical situation, the total pressure (Ptot) is fixed, and the stoichiometric relations above should be translated to partial pressure as Pk = ykPtot

(19)

For example, PH2 = yH Ptotal , yH = yH 0 (1 + 4cyCH 0 )/(1 + 2cyCH 0 ) 2

2

2

4

4

Typical profiles of the reaction rate and the coverages as functions of conversion (c) show (Figure 12) that the rate is very small at zero conversion (c = 0, i.e., at the feed of the reactor), since the surface is poisoned by O and OH. As conversion 9268



■

CONCLUSIONS The kinetics characteristics of three successive steps of the commercially important methane dissociation over Ni(111) surface were calculated using the methodology developed in our previous work53 based on chemically justified assumptions as well as on commonly employed approximations. The arguments supporting our approach are as follows: (i) measured high kinetic isotope effects for the first step of methane dissociation over different metals are in agreement with our previous calculations; (ii) the observed isotope effect for the overall MSR reaction is within the range of values predicted by the present analysis for an effective characteristic of the total reaction; (iii) the activation energy and the sticking probability for step 2 calculated taking into account the tunneling effect agree53 with experiment. The step rate constants estimated in the present work along with other ones obtained in our previous publications are used for calculating the overall rate based a microkinetic model of methane steam reforming (MSR). The rates predicted by the detailed model can be approximated well by an explicit expression derived by assuming that O is the MARI. Our study shows that the CH + O → CHO → CO + H steps cannot be ignored. The MSR rate is linear with methane partial pressure in agreement with published experiments56,69−72 and monotonically decreases with PCO, while the dependence on the partial pressures of water and hydrogen passes through a maximum. The isotope effect (IE) for the MSR is calculated using the rate constants of all steps including disruption of C−H, O−H and H−H bonds: the IE is almost independent of methane and CO partial pressures, but increases while PH2O increases and declines with increasing of PH2 tending to limiting values. Low IE at high hydrogen pressures is explained by the classical mass effect, and high IE at low hydrogen pressures is due hydrogen tunneling. To the best of our knowledge, it is the first attempt to include all H dissociation steps for estimations of isotope effect of the MSR.

Figure 12. Effect of consistent change of the partial pressures of methane, water and hydrogen following the stoichiometric relation(17) on the reaction rate((r) or TOF(1/s)) and on the main component coverages(θi) Ptot = 6 bar, PCH40 = 2 bar, PH2O0 = 4 bar, PH20 = PCO0 = 0, T = 873 K. Simulations with the approximate solution (10)−(13).

increases, hydrogen is formed, θO declines, and the rate increases. The OH coverage passes through a maximum with c. This implies that the rate at the feed section should be accelerated, either by packing this zone with another catalyst, on which methane dissociation is faster (e.g., Pt), or by feeding with a S/C = 1 and adding more steam downstream. We can address special implications of the obtained model for membrane reactors. Hydrogen separation by a selective membrane leads to substoichiometric levels in the retentate channel. To mimic this process, several simulations were conducted, keeping the stoichiometric relations (17) for FCH4, FH2O, and FCO2 while assuming that FH2 is decreased with respect to the stoichiometric value like FH2 = FH2 0 + 4c(1 − β)FCH4 0 , F∑ = F∑0{1 + cyCH 0 [4(1 − β) − 2]} 4

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■

(20)

where the factor β ≤ 1 shows the hydrogen fraction removed by the separation. Several reaction rate profiles calculated with various β show (Figure 13) that with increasing β the initial rate (low c) is smaller (O is MASI) and the maximum of the reaction rate is shifted toward higher conversion range. Thus, the problem at the feed zone persists and should be solved as described above.

APPENDIX A. MOBILE AND IMMOBILE MODELS FOR ADSORPTION AND DESORPTION While most studies focus on predicting the energies and activation barriers of the various steps, the uncertainty associated with the pre-exponential factor (PEF) is in many cases at least as large as that associated with the energy. We explain below our approach in choosing the PEF and the uncertainty associated with it. Neglecting the activation energy, the rate constant of adsorption over a metal surface of a species with mass M may be written as65,74 kads =

A ·106 f# , [1/(s bar site)] 2πMkBT fg

(A1)

where A is a site area on the surface in cm2, 106 is the conversion factor, and f# and fg are the partition functions for the transition state and gas phase. We do not detail yet explicit forms of these partition functions, but we note that for the mobile adsorption layers model (which is used for estimations of the adsorption rate constants of H2, H2O and CH4, see main text), the ratio f#/fg is assumed to be equal to unity, and for the immobile adsorption layers model (which is used for describing CO adsorption), this ratio called by the sticking coefficient S0 is lower than unity.

Figure 13. Effect of the hydrogen level reduction with respect to the stoichiometric relation (17) by factor β (20) on the reaction rate ((r) or TOF(1/s)). The initial parameters as in Figure 12. Simulations with the approximate solution (10)−(13). 9269


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The Journal of Physical Chemistry C The rate constant of desorption has the form65,74 kdes =

kBT f# exp[−Edes /kBT ][1/(s site)] h fa

are at quasi-equilibrium, (ii) step 11 is absent, (iii) the rate of the back reaction 13 is negligible. The assumption on equilibrium applied to the reaction rates 1, 6−10, and 12 allows us to present the coverages θCH4, θH, θH2, θH2O, θO, θOH and θCO, as linear functions θ*:

(A2)

where Edes is the activation energy for desorption, f# is the same as that in eq A1 with the assumption that adsorption and desorption states are in equilibrium, i.e., it corresponds to the same TS, and fa is the partition function for a molecule in the adsorbed state. The ratio f#/fa. in this equation is taken to be equal to unity in the mobile model. Then, combining eqs A1 and A2, we calculate the equilibrium constant Keq for desorption Keq =

(k T /h)exp[−Edes /kBT ] fg kdes , [bar] = B fa kads A ·106 / 2πMkBT

(B1)

θH2 = K̃ 7−1θ*

(B2)

θH2θ*

θH =

K6

=

θ* K 6K̃ 7

(B3)

θH2O = K̃ 8θ* (A3) θOH =

Applying eq A3 allows us to perform estimation of the corresponding rate constants adsorption and desorption without consideration the TS partition function if one of ratios of the partition functions f# /fg or f#/fa (in eqs A1 or A2) is known, for example, from experimental or other data. Particularly, the experimental data cited in ref 65 show that the value of the ratio f# /fa for the CO desorption to gas phase from a nickel surface is equal to 100. It yields the rate constant CO desorption in immobile model, kdes = 4 × 106 s−1 (taking into account Edes = 34.6 kcal/mol (Tables 3)). Then, using both the kdes and Keq we can calculate the kdes. Performing this manipulation, we only need to estimate the partition functions of CO in a gas phase and in an adsorbed state, which is more reliable than calculation of the f#. According to The Theory of Rate Process,74 the partition function fg for CO includes two translation components, two rotational components, and one vibrational component. The last one is equal unity, so that fg may be written as fg = A(2πMkBT )/h2 ·(8π 2IkBT )/h2

θCH4 = K1̃ θ*

K 9θH2Oθ* θH

(B4) =

K 9K̃ 8(θ*)2 K 6K̃ 7 = θ*

K 6K̃ 7 K̃ 8K 9θ* (B5)

θO =

K10 K 6K̃ 7 K̃ 8K 9(θ*)2 K10θOHθ* = = K 6K̃ 7K̃ 8K 9K10θ* θH θ* / K 6K̃ 7 (B6)

θCO =

1 θ* ̃ K12

(B7)

where Ki are the equilibrium constants and the auxiliary dimensionless values K̃ i are defined by eq 12. The assumption on equilibrium applied to the reaction rates 4 and 5 allow to present the coverages θCH2 and θC as linear functions of θCH: θCH2 =

θCHθH 1 = θCH K 4θ * K4 K 6K̃ 7

(B8)

θC = K5

θCHθ* = K5 K 6K̃ 7 θCH θH

(B9)

(A4)

The partition function fa for a CO molecule adsorbed on the nickel surface has the contribution corresponding two frustrated translation motions and that of two vibrations C−O and metal− CO. The contribution due to C−O vibrations is equal unity, and that due to metal-CO vibration is easy estimated using the perpendicular frequency ω⊥. The frustrated translations may be considered as vibrations74 with an effective frequency ωeff. So, we write the fa as

Coverage θCH, in turn, can be expressed via the reaction rate r and θ* using approximate relation r ≈ k13f θCHθO

yielding, accounting for (B6) r θCH = ̃ k13f K 6K 7K̃ 8K 9K10θ*

fa = {1/(1 − exp[−ℏωeff /kBT ])2 }{1/(1 − exp[−ℏω⊥/kBT ])} (A5)

(B10)

Now the rate dependent steps 2 and 3

Lastly, on the basis of data described in refs 76,77, we take ω⊥ = 400 cm−1 (for hollow adsorption) and ωeff = 75 cm−1 and estimate Keq = ∼0.38 bar and kads = 1.05 × 107 s−1 (Table 3), which leads to TOF = 0.705 c−1. One should note that the variation of the ωeff in range of from 50 to 100 cm−1 leads to the decrease of the kads from 2.33 × 107 to 6.3 × 106 s−1 bar−1; however, the rate of the MSR in this interval increases from 0.58 to 0.75 s−1 (at PCH4 = 0.2, PH2O = 0.25, PH2 = 0.4, and PCO = 0.2 bar−1). The variation of the ω⊥ in range of ±10% does not change the TOF as compared with the value at ω⊥ =400 cm−1.

r = k 2f θCH4θ* − k 2bθCH3θH

(B11)

r = k 3f θCH3θ* − k 3bθCH2θH

(B12)

accounting for (B1), (B3) and (B8) form a system with respect to r, θCH3 and θ* and allow to derive the following relations: r = kef (θ*)2 , ⎞−1 ⎛ 1 1 1 ⎟ ⎜ kef = K1̃ K 2 + + 5 ⎟ ⎜ k K K̃ k ̃ ̃ 2b k13,f K3K4 K 6K 7 K8K 9K10 ⎠ ⎝ 3f 6 7

■

(B13)

APPENDIX B. APPROXIMATE STEADY-STATE SOLUTION OF MSR MODEL I Approximate solution is derived using the assumptions listed in section 5.2, i.e., (i) the elementary reaction steps 1, 4−10, and 12

θCH3 = 9270

⎡ k ⎤ K 6K̃ 7 ⎢K1K 2 − ef ⎥θ* k 2b ⎦ ⎣

(B14) DOI: 10.1021/jp5128964 J. Phys. Chem. C 2015, 119, 9260−9273

Article

The Journal of Physical Chemistry C Coverage θCHO can be determined using reaction step 14 which can be assumed either the rate dependent, or quasiequilibrium. In a general case, we obtain

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−1 r14 = r = k14f (θCOθH − K14 θCHOθ*)

θCHO =

k14f θCOθH − r k14bθ*

(B15)

The reduced dependence takes a form: θCHO =

K14 ̃ K 6K̃ 7 K12

θ* (B15′)

The vacant site fraction (θ*) can be estimated by the equality θ* +Σθi =1 yielding θ* =

■

θ 1 , θi̅ = i 1 + Σθi̅ θ*

(B16)

APPENDIX C

Table C1. Rate Constants for Deuterated Species Estimated at the Present Work and Used for Numerical Calculations Kinetic Isotope Effect in Microkinetic Model of the MSR; T = 873 Ka no.

reaction step

knf(D)

knb(D)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

* + CD4(g) ↔ CD4* * + CD4* ↔ CD3* + D* * + CD3* ↔ CD2* + D* * + CD2* ↔ CD* + D* * + CD* ↔ C* + D* D*+D* ↔ D2* + * D2* ↔ D2(g) + * * + D2O(g) ↔ D2O* * + D2O* ↔ OD* + D* * + OD* ↔ O* + D* C*+ O* ↔CO* + * CO* ↔CO(g) + * CD* + O* ↔ CDO* + * * + CDO* ↔CO* + D*

1.24 × 108 7.33 × 106 1.97 × 106 2.04 × 1012 3.22 × 105 1.2 × 106 8.12 × 1012 1.24 × 108 2.01 × 108 1.11 × 107 8.85 4.0 × 106 1.78 × 104 8.09 × 1011

3.23 × 1012 7.33 × 106 6.51 × 108 1.12 × 106 5.81 × 109 1.73 × 1011 2.78 × 108 3.21 × 1010 5.14 × 104 2.41 × 107 2.10 × 10−8 1.05 × 107 3.17 × 105 1.06 × 105

a

k1f, k7b, k8f, and k12b have a dimension of 1/(bar.s site); other constants have a dimension 1/s site; rate constants k13f and k13b, k11f and k11b, k12f and k12b are the same as for nondeuterated intermediates

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has receiving funding from the European Union Seventh Framework Program under grant agreement no. 279075 (acronym COMETHY) FCH-JU-2010-1 and from the ICORE program of the Israeli Science Foundation. We thank Prof. E. A. Katzman (Moscow State University of Fine Chemical Technology) for useful discussion of some problems of this work.

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H Tunneling Effects on Sequential Dissociation of Methane over Ni

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