# Break  Link between Thermodynamics and Kinetics.

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Break  Link between Thermodynamics and Kinetics

Kinetics Modern Methods in Heterogeneous Catalysis F.C. Jentoft, November 1, 2002

Outline 1.Motivation and Strategy 2.Some Important Concepts 3.Rate Equations 4.Mechanisms and Kinetics 5.Temperature Dependence of Rate Constant 6.Compensation Effect

What Kinetics Will (Not) Deliver…  Reaction rates  Rate equation / reaction order  Rate constant  Apparent activation energies  Will not deliver a mechanism…..  But any mechanism we think of should be consistent with the kinetic data….

Motivation Reactants Products E EAEA Catalyst A Reaction coordinate Reactants Products E EAEA Catalyst B Reaction coordinate  Design Parameters for Setup  Compare catalysts: Activation energy E A

 Equilibrium conditions Microscopic Reversibility A* + B ABA + B k1k1 k -1 k2k2 k -2 k3k3 k -3  Unidirectional reaction with identical rates is not an option

Steady State Approximation  Bodenstein’s approximation for consecutive reactions If k 1 *>>k 1, then ABCABC k1k1 k1*k1*  Simplifies Rate Equations

Rate Equations I  With a,b,c, the individual reaction order with respect to a particular reactant and the total reaction order n the sum of the exponents  With r the reaction rate in units of mol/l per time  Typical rate equation:

Rate Equations II  Typical rate equation:  With k the rate constant in units of min -1 for a first order reaction, for higher orders in inverse units of concentration in different powers

Catalysis in Solution: Specific Acid / Base Catalysis  Rate constant a linear function of pH pseudo 1 st order  Proton donor: H 3 O + (solvated protons) Proton acceptor:OH -  Rate equation (analogous for base catalysis)

Specific Acid Catalysis  Dependence of the observed rate constant for oximation of acetone on pH at 25°C. The rate equation is r = k obs * C acetone

Catalysis in Solution: General Acid Base Catalysis  Proton donorHA, H 2 O... Proton acceptorB, H 2 O  Rate equation 2 nd order H + + A - HA

General Acid Catalysis

Rates in Heterogeneous Catalysis  Rate with respect to mass or surface area

Turn Over Frequency  Rate with respect to number of active sites low site densityhigh site density  Turnover frequency is the number of molecules formed per active site per second (in a stage of saturation with reactant, i.e. a zero order reaction with respect to the reactant)

TOF, TON, Catalysis  TON Total number of product formed molecules per active site TON= TOF*catalyst life time  TON = 1stoichiometric reaction TON  10 2 catalytic reaction TON = 10 6 -10 7 industrial application  TON origins from enzyme kinetics, definitions vary

Examples for TOFs

Reaction Steps in Heterogeneous Catalysis 1.Diffusion of reactant to catalyst 2.Adsorption of reactant on catalyst surface 3.Reaction 4.Desorption of products from catalyst surface 5.Diffusion of products away from catalyst We want to know the reaction kinetics. Diffusion should thus not be a rate limiting step.

Interfacial Gradient Effects  Mass transfer bulk of fluid to surface  Case 1: reaction at surface instantaneous global rate controlled through mass transfer “diffusion control”, favored at high T  Case 2: reactant concentration at surface same as in bulk fluid global rate controlled through reaction rate “reaction controlling”, favored at low T and high turbulence

Intraparticle Gradient Effects  Mass transfer within the pores of a catalyst  Vary particle size!

Langmuir Hinshelwood Mechanism  Both species are adsorbed, adsorption follows Langmuir isotherm (see class next week) A B

Eley Rideal Mechanism  Only one species is adsorbed, adsorption follows Langmuir isotherm A B

How to Derive a Rate Equation I 2 C 2 H 5 OH C 2 H 5 -O-C 2 H 5 + H 2 O H+H+

How to Derive a Rate Equation II

How to Derive a Rate Equation III

Structure Insensitivity  rate per exposed metal surface area is NOT a function of the metal particle size  active site 1-2 atoms  Example: the hydrogenation of cyclohexene + H 2

Structure Insensitivity

Structure Sensitivity  also: ammonia synthesis (reactions involving C-C, N-N bond breaking) C 2 H 6 + H 2 2 CH 4  rate per exposed metal surface area is a function of the metal particle size / the exposed facet plane  active site an ensemble of atoms  Example: the hydrogenolysis of ethane

Structure Sensitivity

Temperature Dependence of Rate Constant  Once a rate equation has been established, a rate constant can be calculated  The rate constant is temperature dependent  There are three different ways to derive this relation: Arrhenius Theory Collision Theory Transition State Theory (Eyring)

Arrhenius Theory BA k1k1 k -1 van’t Hoff’s Equation

Arrhenius Theory  With E the apparent activation energy in kJ mol -1 A the frequency factor  Plot of ln k vs. 1/T gives a slope of -E A /R which allows the calculation of the activation energy  A rule of thumb: the rate doubles for 10 K rise in temperature

Collision Theory  According to the simple collision theory, the preexponential factor is dependent on T 1/2  with N A Avogadro’s number, σ cross section, μ reduced mass, k Boltzmann’s constant

A + BC A B C AB + C Activated Complex Theory  Evans/Polanyi, Eyring  based on statistical thermodynamics

Results of Activated Complex Theory  Rate constant (based on number of moles)  Function of T  From the equilibrium constant for the activated complex, a standard free enthalpy of activation can be calculated

Example for Arrhenius Plot 2 different slopes may indicate change in mechanism or change from reaction to diffusion control

Compensation Effect  A “sympathetic variation of the activation energy with the ln of the pre-exponential factor”  ln A and E A /RT have the same order of magnitude but different signs  Change in E A may b compensated by change in A

Compensation Effect  Observed for the same reaction on a family of catalysts

Compensation Effect  Observed for similar reactions on the same catalyst

Compensation Effect: Explanations  “Apparent” activation energy E A,app derived from measured rate and rate equation  With increasing temperature, the “true” reaction rate will increase  With increasing temperature the coverage decreases (exothermic adsorption), leading to a smaller measured rate  E A,app is a weighted sum of the E A,true and the enthalpy of adsorption

Literature  Gabor A. Somorjai, Introduction to Surface Chemistry and Catalysis, John Wiley, New York, 1994  Bruce C. Gates, Catalytic Chemistry, John Wiley, New York, 1992  G Ertl, H. Knözinger, J. Weitkamp, Handbook of Heterogeneous Catalysis, Wiley-VCH, Weinheim 1997  G. Wedler, Physikalische Chemie, Verlag Chemie Weinheim  G.F. Froment, K.B. Bischoff, Chemical Reactor Analysis and Design, Wiley 1990  Compensation effect: G.C. Bond, Catal. Today 1993, J. Catal. 1996