Harmonic transition state theory (TST)

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

The rate constant, k, can be interpreted via thermodynamicor statistical mechanical routes. TST is the foundation of theseformulations.

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 has been factored out from the partition function. Prefactor ν ≈ 10 13 s-1 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 N2 dissociation and NH

hydrogenation over Ru

NH*+H*->NH2*+*

N2+2*->2N*

Rate-limiting stepEmmett, Brunauer, JACS 55 1738 (1933)

Experiment, Dahl, Chorkendorff

Logadottir, Nørskov, J. Catal. 220, 273 (2003)

N2 dissociation

NH hydrogenation

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.

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)

MK is used to bridge the gaps !

Basic Assumptions:

Pros & Cons

No adsorbate-adsorbate interactions One rate-limiting (=slow) reaction step All the other reaction steps are in quasi-equilibrium

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

Z A=s!

s−n!n !zAi

n e−nEA /kBT

Chemical potential: μ= dGdn

=−kBT ln zn

A=−kBT ln1−A

A−kBT ln zAEA

Equilibrium constant: K=∏izii

Reaction Scheme and Rate Expression for NH3 synthesis

First, one has to identify all the elementary reaction steps that may be involved. Is there a rate-limiting step?

The overall reaction is N2+ 3H

2 ↔ 2NH

3

The elementary reaction steps:

N2(g) + * ↔ N

2*

N2*

+ * ↔ 2N*

N* + H

* ↔ NH

*

NH* + H

* ↔ NH

2*

NH2*

+ H* ↔ NH

3*

NH3*

↔ NH3*

(g)

H2(g) ↔ 2H

*

Diffusion of adsorbates is fast !!

Equilibrium equations:

N2gas =N

2ads

NH=NH

…

NH=NH A=−kBT ln1−A

A−k BT ln zAEA

Equilibrium equations:

Equations forcoverages as a functionof Θ

*

The rate for the rate-limiting step

r2=k2N 2adsfree−k−2Nads

2

r2 is defined as the number of turnovers per site per second

For k2 and k

-2 we use an Arhenius expression: k=e

−EkBT

The prefactor ν is calculated using Harmonic transition state theory

From the equation: θ* +θ

N2 +θ

N +θ

H +θ

NH +θ

NH2 +θ

NH3 =1

From equilibrium equations we can derive expressions for coverages θ !

θ* =1-θ

N2 -θ

N -θ

H -θ

NH -θ

NH2 -θ

NH3

r2=k2N 2adsfree−k−2Nads

2

Now we know the coverages but want to write the rate in more userfriendly way.

First:

Second at equilibrium

Third

r=2k2K 1 pN2 1−pNH3

2

KG pH23 pN2

free2

Above the rate equation is written in terms which can be expressedwith the help of molecular properties !!!

Micro-kinetic model for ammonia synthesis

S. Dahl et al. Appl. Cat. A 222,19 (2001)

Calculated NH3 TOF as a function of the

potential energy of adsorbed N

S. Dahl et al. Appl. Cat. A 222,19 (2001)