Break

Link between Thermodynamics and Kinetics

1

1

kkK

Kinetics

Modern Methods in Heterogeneous CatalysisF.C. Jentoft, November 1, 2002

Outline

Motivation and Strategy Some Important Concepts Rate Equations Mechanisms and Kinetics Temperature Dependence of Rate Constant 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

EA

Catalyst A

Reaction coordinate

Reactants

Products

E

EA

Catalyst B

Reaction coordinate

Design Parameters for Setup

Compare catalysts: Activation energy EA

Equilibrium conditions

Microscopic Reversibility

1

1

]][[][

kkK

BAAB

A* + B

ABA + B k1

k-1

k2

k-2

k3

k-3

32

32

]][[][

kkkkK

BAAB

Unidirectional reaction with identical rates is not an option

Steady State Approximation

Bodenstein’s approximation for consecutive reactionsIf k1*>>k1, then

A B Ck1 k1

*

0][

dtBd

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

...][][][ cba CBAkr

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

...][][][ cba CBAkr

11min

n

moll

Catalysis in Solution:Specific Acid / Base Catalysis

Rate constant a linear function of pH

.loglog3

constckOH

rckr pseudo 1st order

Proton donor: H3O+ (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 = kobs * Cacetone

Catalysis in Solution:General Acid Base Catalysis

Proton donor HA, H2O...

Proton acceptor B, H2O

Rate equation

.loglog constKk A

HAr cckr 2nd order

HA

AHA c

ccK

H+ + A-HA

General Acid Catalysis

Rates in Heterogeneous Catalysis

Rate with respect to mass or surface area

catalystgmol

min

surfacecatalystm

mol2min

Turn Over Frequency

Rate with respect to number of active sites

low site density high 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)

1

s

ssitemolecules

TOF, TON, Catalysis

TONTotal number of product formed molecules per active siteTON= TOF*catalyst life time

TON = 1 stoichiometric reactionTON 102 catalytic reactionTON = 106-107 industrial application

TON origins from enzyme kinetics, definitions vary

Examples for TOFs

Reaction Steps in Heterogeneous Catalysis

Diffusion of reactant to catalyst Adsorption of reactant on catalyst surface Reaction Desorption of products from catalyst surface 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 fluidglobal 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

AA

AAA pK

pK

1

21 BBAA

BBAABA pKpK

pKpKkkr

Eley Rideal Mechanism

Only one species is adsorbed, adsorption follows Langmuir isotherm

A

B

AA

BAABA pK

ppKkpkr

1

How to Derive a Rate Equation I

2 C2H5OH C2H5-O-C2H5 + H2OH+

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

+ H2

Structure Insensitivity

Structure Sensitivity

also: ammonia synthesis (reactions involving C-C, N-N bond breaking)

C2H6 + H2 2 CH4

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 TheoryCollision Theory Transition State Theory (Eyring)

Arrhenius Theory

BAk1

k-1 1

1

kkK 2

lnRTH

TK

p

van’t Hoff’s Equation

211 lnln

RTH

Tk

Tk

211ln

RTE

Tk

211ln

RTE

Tk

HEE 11

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 -EA/R

which allows the calculation of the activation energy A rule of thumb: the rate doubles for 10 K rise in

temperature

RTEAk Alnln

Collision Theory

According to the simple collision theory, the preexponential factor is dependent on T1/2

with NA Avogadro’s number, σ cross section, μ reduced

mass, k Boltzmann’s constant

TkNA A 8

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)

KhkTkn

Function of T From the equilibrium constant for the activated complex,

a standard free enthalpy of activation can be calculated

KRTG ln

Example for Arrhenius Plot

2 different slopes may indicate change in mechanismor change from reaction to diffusion control

Compensation Effect

A “sympathetic variation of the activation energy with the ln of the pre-exponential factor”

RTEAk Alnln

.ln constmEA A

ln A and EA/RT have the same order of magnitude but

different signs

Change in EA 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 EA,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

EA,app is a weighted sum of the EA,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