Break

Link between Thermodynamics and Kinetics

1

1

k

kK

Kinetics

Modern Methods in Heterogeneous Catalysis

F.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

]][[

][

k

kK

BA

AB

A* + B

ABA + B k1

k-1

k2

k-2

k3

k-3

32

32

]][[

][

kk

kkK

BA

AB

Unidirectional reaction with identical rates is not an option

Steady State Approximation

Bodenstein’s approximation for consecutive reactions

If k1*>>k1, then

A B Ck1 k1

*

0][

dt

Bd

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

mol

l

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

catalystg

mol

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

ssite

molecules

TOF, TON, Catalysis

TON

Total number of product formed molecules per active site

TON= TOF*catalyst life time

TON = 1 stoichiometric reaction

TON 102 catalytic reaction

TON = 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

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

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 Theory

Collision Theory

Transition State Theory (Eyring)

Arrhenius Theory

BAk1

k-1 1

1

k

kK

2

ln

RT

H

T

K

p

van’t Hoff’s Equation

211 lnln

RT

H

T

k

T

k

211ln

RT

E

T

k

211ln

RT

E

T

k

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

RT

EAk 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

Tk

NA 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)

Kh

kTkn

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”

RT

EAk 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