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