Harmonic transition state theory (TST)
The rate constant, k, can be interpreted via thermodynamic or statistical mechanical routes. TST is the foundation of these formulations.
References:
R.I. Masel Chemical Kinetics and Catalysis, Wiley I. Chorkendorff & J.W. Niemantsverdriet Concepts of Modern Catalysis and Kinetics, Wiley Kurt W. Kolasinski Surface Science: Foundations of Catalysis and Nanoscience, Wiley etc
Multidimensional potential energy surface Reactants and products are separated by a transition state An activated complex
Main assumptions
Once the transition state is reached the system carries on to produce the products The energy distribution of reactants follow the Maxwell-Boltzmann distribution The whole system does not need to be at equilibrium, but the concentration of the activated complex can be calculated based on equilibrium theory The motion along the reaction coordinated is separable from the other motions of the activated complex Motion is treated classically – NO tunneling !
Equilibrium between reactants and an activated complex
If reactants and products are in equilibrium there are many equivalent ways to write an equilibrium constant
where
The final expression is
The loose vibrational mode that corresponds the motion leading to a reaction 13 -1 has been factored out from the partition function. Prefactor ν ≈ 10 s at room temperature and if the ratio of partition functions ≈ 1. Thermodynamic treatment
Comparison of the rate constant obtained from quantum mechanical calculations and harmonic transition state theory
van Harrevelt, Honkala, Nørskov and Manthe Journal of Chem. Phys. 122, 234702, (2005)
Rates of N dissociation and NH 2 hydrogenation over Ru
NH*+H*->NH2*+* NH hydrogenation
Experiment, Dahl, Chorkendorff
N2+2*->2N*
N2 dissociation Rate-limiting step Emmett, Brunauer, JACS 55 1738 (1933) Logadottir, Nørskov, J. Catal. 220, 273 (2003) Microkinetics
- a reaction mechanism and the molecular properties of reactants and intermediates are used in simulations of the reaction at the macroscopic level. - estimations of the rates of elementary reactions and surface coverage are a consequence of the analysis not a basis ! - no initial assumptions for which steps are kinetically significant or which surface species are more abundant.
MK is used to bridge the gaps !
References: P. Stolze “Microkinetic simulations of catalytic reactions”, Progress in Surface Science, 65, 65, (2000) I. Chorkendorff, J.W. Niemantsverdriet, “Concepts of Modern Catalysis and Kinetics”, Wiley-VCH P. Stolze and J.K. Nørskov Phys. Rev. Lett. 55, 2502 (1985) P. Stolze Physica Scripta 36, 824 (1987)
Basic Assumptions:
No adsorbate-adsorbate interactions One rate-limiting (=slow) reaction step All the other reaction steps are in quasi-equilibrium
Pros & Cons Bridges the pressure gap between UHV experiments/DFT calculations and catalysis studies done at ambient pressure Based on mean-field assumption Fast Can be used to predict properties Kinetic parameters for each elementary reaction step
Statistical mechanics of chemisorption
Partition function for an adsorbed phase is
s! −nE /k T Z = z n e A B A s−n!n! Ai dG z Chemical potential: μ= =−k T ln dn B n
1− A A=−k B T ln −k B T lnz AE A A
Equilibrium constant: K= z i ∏i i
Reaction Scheme and Rate Expression for NH synthesis 3 First, one has to identify all the elementary reaction steps that may be involved. Is there a rate-limiting step? The overall reaction is N + 3H ↔ 2NH 2 2 3 The elementary reaction steps: Equilibrium equations:
N (g) + * ↔ N N gas=N ads 2 2* 2 2 N + * ↔ 2N 2* * N + H ↔ NH NH =NH * * * NH + H ↔ NH * * 2* … NH + H ↔ NH 2* * 3* NH ↔ NH (g) 3* 3* Diffusion of adsorbates is fast !! H (g) ↔ 2H 2 * 1− NH=NH A A=−k B T ln −k B T lnz A E A A
Equilibrium equations:
Equations for coverages as a function of Θ *
The rate for the rate-limiting step
r =k −k 2 2 2 N 2 ads free −2 Nads r is defined as the number of turnovers per site per second 2
− E k T B For k and k we use an Arhenius expression: k = e 2 -2 The prefactor ν is calculated using Harmonic transition state theory
From equilibrium equations we can derive expressions for coverages θ !
From the equation: θ +θ +θ +θ +θ +θ +θ =1 * N2 N H NH NH2 NH3
θ =1-θ -θ -θ -θ -θ -θ * N2 N H NH NH2 NH3 r =k −k 2 2 2 N 2 ads free −2 Nads Now we know the coverages but want to write the rate in more user friendly way.
First:
Second at equilibrium
Third
2 pNH3 2 r=2k2 K 1 pN2 1− 3 free K G pH2 pN2
Above the rate equation is written in terms which can be expressed with the help of molecular properties !!!
Micro-kinetic model for ammonia synthesis
S. Dahl et al. Appl. Cat. A 222,19 (2001) Calculated NH TOF as a function of the 3 potential energy of adsorbed N
S. Dahl et al. Appl. Cat. A 222,19 (2001)