Catalytic Kinetics: Cycles and the LHHW Formalism kinetics Physical adsorption Chemisorption From...

© Michael T. Klein Reactions 1

Catalytic Kinetics: Cycles and the LHHW Formalism

"A Catalyst is by definition a substance that increases the rateof approach to equilibrium of a chemical reaction without beingsubstantially consumed in the reaction; a catalyst works byforming chemical bonds to one or more reactants and therebyfacilitating their conversion. A catalyst does not significantlyaffect the reaction equilibrium."1

"The definition of a catalyst rests on the idea of reaction rateand therefore the subject of reaction kinetics is central,providing the quantitative framework."1

Gates, B.C. Catalytic Chemistry, Wiley (1990).1

Common theme: Catalysis ≡ Kinetics


Reaction Chemistry: The Practice of Catalysis

Reaction: A → B + C

Thermodynamics: ∆HoR = ∆Ho

f B + ∆Hof C - ∆Ho

f A

Role of Catalyst: Increase the rate

Implications: Smaller vesselsBetter selectivities (relative rates)


Reaction Coordinate Diagram Motivates and Explains Catalysis

EA

A poor catalyst

Homogeneous

A good catalyst

B + C

ξ

∆HR = EB + EC - EA is independent of catalyst


Kinetics of Heterogeneous Catalytic Reactions

External Transport Adsorption/Desorption Internal Transport Surface Reaction


Series and Parallel Events in Catalytic Reactions

1

2

34

5

6

7Steps 1,3,4,5,7 in series Steps 2,3,4,5,6 in parallel

Al

Bl

Bs

A

BA

A B

s

ss

ss


SITES AND ACTIVE CENTERS

"|" USED TO INDICATE A CATALYTIC SITE

Single Site Dual Site

A R S A R S

Al Rl Sl Al1=Rl1 Rl2 Sl2

Surface Intermediates, Active Centers

=


Adsorption

Adsorbent

Adsorbate

Temperature range

Heat of adsorption

Rate and activation energy

Coverage

Reversibility

Importance

All solids

All gases below critical point

Low T

Low, ~ ∆Hliq

Very rapid, low E

Multilayer

Highly reversible

For determination of surfacearea and pore size

Some solids

Some chemically reactivegases

Generally high temperature

High order, about enthalpyof reaction

Nonactivated, low E;activated, high E

Monolayer and less

Often irreversible

For determination of surfaceconcentration, rates ofadsorption and desorption,estimates of active centerarea, and elucidation ofsurface-reaction kinetics

Physical adsorption Chemisorption

From Carberry, J. J. , Chemical and Catalytic Reaction Engineering, McGraw Hill


Adsorption Isotherms

Langmuir Model:Homogeneous solid surface Energetics independent of coverage Adsorbed species don't interact

Mass Action Steps:

A + l = Al

ra = kaCACl rd = kdCAl


Adsorption Isotherms

The Equilibrium Isotherm:ra = rd kaCACl = kdCAl

or, withCt = constant = Cl + CAlΘA = CAl /C t = KACA/(1 + KACA)

1.0

01

order

1st order0

0ΘA

CA CA/ SAT


Extensions of Basic Idea

1. Multicomponent Systems

2. Dissociative Adsorption of "A"

ΘA = (K APA)1/2 /(1 + (K APA)1/2 + ΣKjPj)

Θi = C il/C t = KiCi/(1 + ΣKjCj)


Heterogeneous Rate Laws

Premises

Homogeneous Principles Valid

Surface Concentrations Relevant

Three Different Classes of (Equal) Rates


Heterogeneous Rate Laws

1. Adsorption A + l = Al

2. Surface Reaction Al = Rl

3. Desorption Rl = R + l

K = Adsorption Equilibrium Constant (used by convention)

ra = kA(C ACl-C Al/KA)

rsr = ksr(C Al-C Rl/Ksr)

rd = kR(C Rl/KR-C RCl)

Site Balance: C t = C l + C A + C B + …

i


Steady State Solution

This General Rate Expression Reduced for VariousRate Controlling Steps

Which May:Not ExistChange with TChange with P


Rate Determining Step

1. The Usual Procedure 2. Requires that all other steps be in virtual equilibrium

Two Common Cases:

1. Surface Reaction Controls 2. Adsorption/Desorption Controls


Surface Reaction Controlling

A R

r = k sr(C Al - CRl/Ksr)

Θi = C il/C t = KiCi/(1 + Σ KjCj)

r = ksrCtKA(C A-C R/K)/(1 + KACA + KRCR)

K = KsrKA/KR


Additional Examples

Bimolecular Reactions A + B R + S

r = k srCtKAKB(C ACB-C RCS/K)/(1 + ΣKJCJ)2

rinit1st

0

C A

-1


Additional Examples

Two-Site Mechanisms

A l1 + B l2 R l1 + S l2

r = k sr C Al1 C Bl2 =

k sr C t K A K B C A C B /(1 + Σ K i1 C i )(1 + Σ K i2 C i2 )


Additional Examples

Rideal Mechanism

Al + B (g) Rl + S (g)

r = k srCtKACACB/(1 + ΣKiCi)


Additional Examples

Mole Change

l + Al Bl + Cl

r = k srCtKA(C A-C BCC/K)/(1 + ΣKiCi)2


Kinetic Groups

Yang/Hougen tables from Froment, G. F. and K. B. Bischoff, Chemical Reactor Analysis and Design, Wiley

Adsorption of A controlling kAAdsorption of B controlling kBDesorption of R controlling kRKAdsorption of A controlling with dissociation kAImpact of A controlling kAKBHomogeneous reaction controlling k


A R A R + S A + B R A + B R + S

Without dissociation ksrKA ksrKA ksrKAKB ksrKAKBWith dissociation of A ksrKA ksrKA ksrKAKB ksrKAKBB not adsorbed ksrKA ksrKA ksrKA ksrKAB not adsorbed, A dissociated ksrKA ksrKA ksrKA ksrKA


Exponents of Adsorption Groups


Adsorption of A controlling without dissociation n = 1Desorption of R controlling n = 1Adsorption of A controlling with dissociation n = 2Impact of A without dissociation A + B R n = 1Impact of A without dissociation A + B R + S n = 2Homogeneous reaction n = 0


A R A R + S A + B R A + B R + S

No dissociation of A 1 2 2 2Dissociation of A 2 2 3 3Dissociation of A(B not adsorbed) 2 2 2 2No dissociation of A(B not adsorbed) 1 2 1 2


Driving-Force Groups

Reaction A R A R + S A + B R A + B R + S

Adsorption of A controlling pA - pRK pA - pRpS

K pA - pRKpB

pA - pRpSKpB

Adsorption of B controlling 0 0 pB - pRKpA

pB - pRpSKpA

Desorption of R controlling pA - pRK

pApS

- pRk pApB- pR

KpApB

ps - pR

K

Surface reaction controlling pA - pRK pA - pRpS

K pApB - pRK pApB -

pRpSK

Impact of controlling 0 0 pApB - pRK pApB - pRpS

K(A not adsorbed)

Homogeneous reaction pA - pRK pA - pRpS

K pApB - pRK pApB - pRpS

Kcontrolling



Replacement In the General Adsorption Groups


(l + KApA + KBpB + KSpS + KRPR + KIpI )

Reaction A R A R + S A + B R A + B R + S

Where adsorption of A is rate KApRK

KApRpSK

KApRKpB

KApRpSKpBcontrolling, replace KApA by

Where adsorption of B is rate 0 0 KBpRKpA

KBpRpSKpAcontrolling, replace KSpS by

Where desorption of A is rate KKRpA KKRpApS

KKRpSpB KKRpApB

pscontrolling, replace KApA by

Where adsorption of A is rate KApRK

KApRpSK

KApRKpB

KApRpSKpBcontrolling with dissociation

of A, replace KApA by

Where equilibrium adsorptionof A takes place with dissociation of A, replace KApA KApA KApA KApAKApA byand similarly for other components adsorbed with dissociation

Where A is not adsorbedreplace KApA by 0 0 0 0and similarly for othercomponents that are notadsorbed


ADSORPTION/DESORPTION CONTROLLING

• SURFACE REACTION IN VIRTUAL EQUILIBRIUM

KSR = Πi Cil νil

1. Al + Bl Rl + Sl with Adsorption of "A" controlling

r = kaA CA Cl – kdA CAl = k 'A(CACl – CAl/KA)

CBl = KBCBCl

CRl = KRCRClCSl = KSCSClCAl = C RlCSl/(CBlKSR)

4 eqn, 4 unknowns

Used instead of final adsorption isotherm

r = kACt (CA - CRCS /CBK)

1+

CRCSCB

KAK + KBCB + K RCR + KS CS


Some Criteria for Parameter Estimation

1. Adsorption and Rate Constants be Statistically non-negative

2. E* > 0 ln k ↓ with 1/ T ↑ (Arrhenius)

3. Exothermic adsorption ln k ↑ with 1/ T ↑

Confidence intervals may make negative-value containingmodels acceptable.


Linearization of Model

rA = kr KA(pA - pR pS/K)(1+ KApA + KRpR + KSpS)2 y = a + bpA + c pR + dpS

y = pA - pRpS/KrA

a = 1/ kKAb = KA/ kKA

c = KR/ kKAd = KS/ kKA

This allows linear regression. However, this suffers from lack ostatistical rigor--y not truly dependent variable.


Catalysis in Series

A1 → A2 → A3

A1 strongly held, A2 weak, A3 not at all

r1 = k1K1A1/(1+ K1A1 –~ k1

r2 = k2K2A2 2 2 –~ k2K2A2

@ Max of A2 0 = r 1 - r2 ⇒ k1 = k2K2A2

or A2|Max = k1k2K2

r1 r2

Independent Rates:

)

/(1+ K )A


Coupled Rates

In reality, with coupling, r2 = k2K2A2/(1+K 2A2 + K1A1) –~ k2K2

K1 A2A1

A2 =

k1

k2

K1

K2 A1

orA2

C

A2I = K1A1 >>1

Gain in selectivity due to coupling• This is thermodynamic in nature

MAX

MAX

MAX

•


Analytical Rate Laws Derived from Mechanism BEFORE Use In Reactor Model

A Heterogeneous Chemistry (A B) Example

A + AA BB B +

l ll ll l

⇔⇔⇔

rA

A sr A B A sr

sr

BA

A sr

sr

AB

l A B K

K k k Kk K kK

KkK A

K kK

KkK B

=−

+ +

+ +

+

+ +

+

0

1 1 1 1 1 1 1( / )

rAsr A

A B

l k K A B KK A K B

=−

+ +0

1( / ) rA

B

A B

l k K A B KK A KK A

=−

+ +0

1( / ) rA

A

AB

l k A B KKK

B K B=

−

+ +

0

1

( / )

surface rxn control adsorption control

desorption control


Steady State Solution

A = R

Ksr

dCa/dt = r a

dCAl /dt = 0 = ra - rsr

dCRl/dt = 0 = rsr - rd

dCR/dt = r d

r = Ct (C A-CR/K)

1

KAksr + 1

kA + 1

KkR +

1KAksr

+ 1+KKkkR

ACA +

1

KAksr +

1+K srKsrkA

K RCR

A


Derivation of Rate Laws: Useful Tools The Steady State Approximation

k2


Derivation of Rate Laws: Useful Tools The Steady State Approximation


The Steady State Approximation Estimation of Relaxation Time


Summary of Relaxation Time Issues The Steady State Approximation


Closer Look at the Relaxation Time The Steady State Approximation

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Catalytic Kinetics: Cycles and the LHHW Formalism kinetics Physical adsorption Chemisorption From...

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