Break
 Link between Thermodynamics and Kinetics
k1
K
k 1
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
 Compare catalysts: Activation energy EA
E
Catalyst A
E
Catalyst B
EA
EA
Reactants
Reactants
Products
Reaction coordinate
 Design Parameters for Setup
Products
Reaction coordinate
Microscopic Reversibility
k1
A+B
AB
k-1
k2
k3
k-2
k-3
A* + B
 Equilibrium conditions
[ AB]
k1
K 
[ A][ B ]
k 1
k 2 k3
[ AB]
K 
[ A][ B ]
k  2 k 3
 Unidirectional reaction with identical rates is not an option
Steady State Approximation
A
k1
B
k 1*
C
 Bodenstein’s approximation for consecutive reactions
If k1*>>k1, then
d [ B]
0
dt
 Simplifies Rate Equations
Rate Equations I
 Typical rate equation:
r  k[ A]a [ B]b [C ]c ...
 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
Rate Equations II
 Typical rate equation:
r  k[ A]a [ B]b [C ]c ...
 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
n 1


1  l 
 
min 
mol

 

Catalysis in Solution:
Specific Acid / Base Catalysis
 Proton donor:
H3O+ (solvated protons)
Proton acceptor: OH Rate equation (analogous for base catalysis)
r  k cr
pseudo 1st order
log k  log cH O   const.
3
 Rate constant a linear function of pH
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
r  k cr cHA
2nd order
log k   log K A  const.
H+ + A-
HA
KA 
c H  c A
cHA
General Acid Catalysis
Rates in Heterogeneous Catalysis
 Rate with respect to mass or surface area


mol


 min g catalyst 


mol
 min m2

catalyst surface 

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)
 molecules 
1

s
 site s 


 
TOF, TON, Catalysis
 TON
Total number of product formed molecules per active site
TON= TOF*catalyst life time
 TON = 1
TON  102
TON = 106-107
stoichiometric reaction
catalytic reaction
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
A
B
 Both species are adsorbed, adsorption follows Langmuir
isotherm (see class next week)
K A pA
A 
1  K A pA
k K A p A K B pB
r  k  A B 
1  K A p A  K B pB 2
Eley Rideal Mechanism
B
A
 Only one species is adsorbed, adsorption follows
Langmuir isotherm
k K A p A pB
r  k  A pB 
1  K A pA
How to Derive a Rate Equation I
2 C2H5OH
H+
C2H5-O-C2H5 + H2O
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
 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
C2H6 + H2
2 CH4
 also: ammonia synthesis (reactions involving C-C, N-N
bond breaking)
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
A
k1
k-1
B
k1
K
k 1
H
  ln K 



2

T
RT

p
van’t Hoff’s Equation
  ln k1    ln k 1  H



2

T

T
RT

 

E1
  ln k1 


2
 T  RT
  ln k 1  E1


2
 T  RT
E1  E1  H
Arrhenius Theory
EA
ln k  ln A 
RT
 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
Collision Theory
 According to the simple collision theory, the
preexponential factor is dependent on T1/2
A  N A
8k

T
 with NA Avogadro’s number, σ cross section, μ reduced
mass, k Boltzmann’s constant
Activated Complex Theory
A + BC
A
B C
 Evans/Polanyi, Eyring
 based on statistical thermodynamics
AB + C
Results of Activated Complex Theory
 Rate constant (based on number of moles)
kT 
kn 
K
h
 Function of T
 From the equilibrium constant for the activated complex,
a standard free enthalpy of activation can be calculated
G    RT ln K 
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 k  ln A 
EA
RT
 ln A and EA/RT have the same order of magnitude but
different signs
ln A  mEA  const.
 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