Supporting Information for Hydrogen Chemisorption Isotherms on Pt Particles at Catalytic Temperatures: Langmuir and Two-Dimensional Gas Models Revisited
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Supporting Information for Hydrogen Chemisorption Isotherms on Pt Particles at Catalytic Temperatures: Langmuir and Two-Dimensional Gas Models Revisited Mónica García-Diéguez,1 David D. Hibbitts,1,2,* and Enrique Iglesia1,3,* 1 Department of Chemical and Biomolecular Engineering, University of California, Berkeley, CA, 94720 2 Department of Chemical Engineering, University of Florida, Gainesville, FL, 32601 3 Division of Chemical Sciences, E.O. Lawrence Berkeley National Laboratory, Berkeley, CA, 94720 * to whom correspondence should be addressed: [email protected], [email protected] S1 Table of Contents Section S1. Details of Standard Statistical Mechanics Treatments S3 Section S2. Derivation of Thermodynamic Treatments S5 Section S3. Adsorption Isotherms S7 Figure S1. S7 Section S4. Temperature Dependence of the Heat of Adsorption and Entropy Loss S8 Section S5. Local Langmuir Isotherm for Non-Uniform Surfaces: QH Approach S12 Table S1. S13 Figure S2. S14 Figure S3. S15 Figure S4. S15 Figure S5. S16 Figure S6. S16 Section S5. Local Langmuir Isotherm for Non-Uniform Surfaces: ΔG Approach S17 Figure S7. S18 Table S2. S19 Table S3. S19 Figure S8. S20 Figure S9. S20 Figure S10. S21 Figure S11. S21 Figure S12. S22 Section S6. Diffusion Barrier—Frustrated Translations S23 Figure S13. S23 Section S7. Detailed DFT Calculation Results S24 Table S4. S24 Table S5. S25 References (Supporting Information). S25 S2 Section S1. Details of Standard Statistical Mechanics Treatments The enthalpy of a given state can be written as the sum of the DFT-derived energy (E0), zero-point vibrational enthalpy (ZPVE) and vibrational, translational and rotational enthalpy (Hvib, Htrans and Hrot): H= E0 + ZPVE + Hvib + H trans + H rot (S1) similarly, the free energy of a state can be written as: G= E0 + ZPVE + Gvib + G trans + G rot (S2) and entropy can be determined for a state with a known H and G at a given T: HG− S = (S3) T For calculations which include a periodic Ir(111) surface (including adsorbed species and transition states on that surface), there are no translational or rotational degrees of freedom and DFT-derived vibrational frequencies can be used to determine the ZPVE, Hvib and Gvib shown in Eqns. S4-6. 푍푃푉퐸 = ∑(½휈ℎ) (S4) −휈ℎ 휈 ℎ푒 푘푇 퐻 = ∑ ( ) (S5) 푣푏 −휈ℎ 1−푒 푘푇 1 퐺 = ∑ (−푘푇 ln ) (S6) 푣푏 −휈ℎ 1−푒 푘푇 Gas-phase molecules have translational and rotational degrees of freedom; thus Htrans, Hrot, Gtrans and Grot must also be computed: 5 퐻푡푟푎푛푠 = ⁄2 푘푇 (S7) 퐻푟표푡,푙푛푒푎푟 = 푘푇 (S8) 3 퐻푟표푡,푛표푛푙푛푒푎푟 = ⁄2 푘푇 (S9) S3 2휋푀푘푇 3⁄2 퐺 = −푘푇 ln [( ) 푉] (S10) 푡푟푎푛푠 ℎ2 1⁄2 휋1⁄2 푇3 퐺푟표푡 = −푘푇 ln [ ( ) ] (S11) 휎 휃푥휃푦휃푧 ℎ2 휃 = 2 (S12) 8휋 퐼푘 where Ii is the moment of inertia about axes x, y or z and σ is the symmetry number of the molecule, 2 for H2. S4 Section S2. Derivation of Thermodynamic Treatments The surface area (퐴surf) of the two-dimensional adsorbed phase is constant (analogous to a constant volume in a three-dimensional system), and therefore the enthalpy is identical to the internal energy (H = U) and the Gibb’s free energy is identical to the Helmholtz free energy (G = A). 퐺 = 퐴 = −푘퐵푇 ln 푄 (S13) Entropy can be obtained from the Helmholtz free energy via a Maxwell relation: 휕퐴 푆 = − ) (S14) 휕푇 푁,퐴surf For identical non-interacting classical particles, we can describe the single-particle partition function (q) in a two- dimensional potential energy surface as: −퐸(푥,푦) 1 ( ) 푘퐵푇 푞 = 2 ∬ 푒 푑푥푑푦 (S15) Λ푡ℎ The integral above is done over the entire metal surface (퐴surf) and 퐸(푥, 푦) ≥ 0, therefore the integral has a maximum value of 퐴surf. Let’s define (to make things simpler), the area from the PES per surface metal atom as 훼푃퐸푆: −퐸(푥,푦) −퐸(푥,푦) 1 ( ) ( ) 푘 푇 푘푇 훼푃퐸푆 = ∬ 푒 퐵 푑푥푑푦 = ∬ 푒 푑푥푑푦 (S16) 푁Msurf 1×1 Note: If the potential energy surface is flat, then 훼푃퐸푆 is the area of a 1×1 Pt (111) surface. The total canonical partition function is: 푞푁 푄 = (S17) 푁! and we can substitute Eq. S15 into that, and apply Stirling’s approximation, to get: 푁 푁 푁 푁 푁 훼푃퐸푆푁Msurf푒 훼푃퐸푆푒 푄 = 2푁 푁 = 2푁 푁 (S18) Λ푡ℎ 푁 Λ푡ℎ 휃 where N is the number of H* adsorbates and θ is the ratio of H* to surface-metal atoms. Substituting the expression for Q into Eq. S17. 푥푦 푥푦 훼푃퐸푆 퐺H*,avg,PES = 퐴H*,avg,PES = −푅푇 [ln ( 2 ) + 1] (S19) Λ푡ℎ휃 Before defining the entropy, I will put the definition of α back in. The average (or integral) molar translational free energy of a species with a given potential energy surface, 퐸(푥, 푦), is: −퐸(푥,푦) 1 ( ) 푥푦 푥푦 푘푇 퐺H*,avg,PES = 퐴H*,avg,PES = −푅푇 [푙푛 ( 2 ∬ 푒 푑푥푑푦) + 1] (S20) Λ푡ℎ휃 1×1 where we will bound the integral within a 1 × 1 surface. Differentiation gives: 푥푦 휕퐴H*,avg,PES 푆푥푦 = − ) (S21) H*,avg,PES 휕푇 퐴,푁 −퐸(푥,푦) 퐸(푥, 푦) ( ) −퐸(푥,푦) 푘푇 1 ( ) ∬ 푒 푑푥푑푦 푥푦 푘푇 1×1 푘푇 푆H*,avg,PES = 푅 [푙푛 ( 2 ∬ 푒 푑푥푑푦) + −퐸(푥,푦) + 2] (S22) Λ 휃 ( ) 푡ℎ 1×1 푘푇 ∬1×1 푒 푑푥푑푦 The other term we will define is a Boltzmann-averaged energy (훽푃퐸푆): S5 −퐸(푥,푦) 퐸(푥, 푦) ( ) 푘푇 훽푃퐸푆 = ∬ 푒 푑푥푑푦 (S23) 1×1 푘푇 Computing the differential free energy and entropies from these average properties gives (and substituting Eqs. S16 and 23 for α and β): 푥푦 훼푃퐸푆 퐺H*,PES = −푅푇푙푛 ( 2 ) (S24) Λ푡ℎ휃 푥푦 훼푃퐸푆 훽푃퐸푆 푆H*,PES = 푅 [푙푛 ( 2 ) + + 1] (S25) Λ푡ℎ휃 훼푃퐸푆 These equations are used as the starting point for determining the two-dimensional translational free energy and entropy for adsorbed H* as further described in the main text. The free energy of a 2-D system described by a PES is given by Eq. S24. For a flat PES (ideal gas), the ‘area’ of the PES (훼푃퐸푆) is the area of each metal atom (훼0): 푥푦 훼0 퐺H*,ideal = −푅푇푙푛 ( 2 ) (S26) Λ푡ℎ휃 In an excluded area model, with an otherwise-flat PES, this area is decreased by the area of a H* atom, when such an atom is present. As such, 훼0 is replaced by (훼0 − 휃푏): 푥푦 훼0 − 휃푏 퐺H*,co-area = −푅푇푙푛 ( 2 ) (S27) Λ푡ℎ휃 Taking the derivative of this free energy term at constant T and N (number of H*) gives the entropy: 푥푦 휕퐺H*,co-area 푆푥푦 = − ) (S28) H*,co-area 휕푇 퐴,푁 gives an expression for the two-dimensional translational entropy of an excluded-area gas model: 푥푦 1 훼0 − 휃푏 푏 푆H*,co-area = 푅 [푙푛 ( 2 ∙ ) + 1 − ] (S29) Λ푡ℎ 휃 훼0 − 휃푏 S6 Section S3. Adsorption Isotherms 1.2 (a) (b) (c) 1.0 H 0.8 θ 0.6 0.4 H* coverage, 0.2 0.0 0.001 0.001 0.001 P (kPa) PH (kPa) PH (kPa) H2 2 2 Figure S1. Adsorption isotherms for H2 dissociative adsorption on Pt (1.6 % wt. Pt/Al2O3 9.1 nm (a) and 3.0 nm (b) mean cluster size; 1 % wt. Pt/SiO2 1.6 nm mean cluster size (c)) at 673 K (), 623 K (), 573 K (▲), 548 (+), 523 K (), 498 K (×) and 473 K (). S7 Section S4. Temperature Dependence of the Heat of Adsorption and Entropy Loss S2.1. Change in enthalpy: The temperature dependence of the isosteric heat of adsorption as a consequence of the change in enthalpy upon H2 dissociative adsorption is given by the difference in heat capacities ( − Cp ) between the gas phase (Cg) and the adsorbate ( Ca ): QH Hg Ha = − 2 = (C − 2Ca ) = −Cp (S30) T T T g p H n where, Hg and Ha are the molar gas-phase enthalpy and the differential adsorbed phase enthalpy, respectively. The isosteric heat of adsorption ( (Q ) ) is related to the differential heat of adsorption (QH) H H by: (Q ) = Q + RT . H H H Local (S31a) and mobile (S31b) dissociative H2 adsorption can be described by the following expressions: 1 2 1 2 2 K P = H H2 K P = H (S31a) H 1+ K 1 2P 1 2 H H2 1− H H2 H 2 = K 1 2P 1 2 or K P = H exp H (S31b) H H H2 H H2 1− H 1− H where P is the hydrogen pressure at equilibrium and KH is the equilibrium constant given in both cases H2 by: E KH = K0 exp (S32) RT in which E is the potential minimum energy of an adsorbed molecule and K0 can be estimated by statistical mechanics as: 2 (qa (T)) K0 (T) = (S33) Qgas (T) qa and Qgas are the molecular partition functions of the adsorbed (H*) and gas (H2) phases, respectively. Substituting Equations (S31a-b), (S32) and (S33) into Equation (4) gives: 2 ln K0 (T) (QH ) = E − RT (S34) H (T) H The partition function for H2 in the gas phase is given by the product of the partition functions for the different degrees of freedom (vibrational (qg,v), translational (qg,t) and rotational (qg,r)): Qgas (T) = qg,vqg,tqg,r (S35) S8 1 exp 2g,v qg,v = (S36a) 1 exp −1 g,v T hg with g,v = and g,v = g,v (2mT)3 2 q = V (S36b) g,t h3 82IT q = = (S36c) g,r h2 g,r τg,v is the vibrational temperature, νg is the vibrational frequency of H2 in the gas phase, h and κ are the Planck’s and Boltzman’s contants, respectively. m is the mass of H2, V is the volume, I is the moment if inertia, σ is the symmetry factor (2 for H2) and τg,v is the rotational temperature.