Kinetic Models in Heterogeneous Catalysis - ACS Publications

(4) To what extent can kinetics furnish unambiguous answers to certain fundamental ... has expressed an extreme position on the entire matter: "'Mecha...
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2 Kinetic Models in Heterogeneous Catalysis S O L W. W E L L E R

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Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, Ν . Y. 14214

The

diverse

objectives monly muir,

meanings

used and Freundlich,

concentrations isotherm, reactive

models are reviewed.

theoretically and

derivable

problems

Temkin—attempt

arise

with

apparent

and abnormal

coefficients.

A few procedural

whose primary primary

In the

modelling

suggestions

interest is reaction

surface Langmuir

of

from

adsorption

dependence

approaches

interest is reactor design

com­

adsorption

values

temperature

Various

years have been taken to kinetic catalysis.

to relate

enhanced

negative

varied

Three

isotherms—Lang­

to gas phase composition.

mixtures,

coefficients, sorption

of "kinetic models" and the

in establishing

in the in

of

ad­

past

10

heterogeneous

are offered for

those

and for those

whose

mechanism.

' T ^ h e subject of k i n e t i c m o d e l s i n heterogeneous catalysis is a t r o u b l i n g A

one. A r i g o r o u s t h e o r y of heterogeneous c a t a l y t i c k i n e t i c s is analogous

to a G r e e k t r a g e d y : the p r o t a g o n i s t is d o o m e d to f a i l u r e , a n d t h e course of t h a t f a i l u r e is k n o w n to the observers f r o m the start. T h e roots of t h i s p r e o r d a i n e d d o o m f o r p r a c t i c a l systems are t w o - f o l d . F i r s t , the surface of a p r a c t i c a l catalyst is not u n i f o r m , g e o m e t r i c a l l y or e n e r g e t i c a l l y ; t h e adsorbate m o l e c u l e s m a y i n t e r a c t b o t h to p r o d u c e surface complexes a n d also to g i v e a d s o r p t i o n energies that d e p e n d o n surface coverage;

n e i t h e r of these effects is q u a n t i t a t i v e l y p r e d i c t a b l e

t o d a y f r o m first p r i n c i p l e s f o r a n y a r b i t r a r y system. A s a result, w e h a v e n o w a y of k n o w i n g i n a d v a n c e , for a n y system, w h i c h of s e v e r a l different isotherms is q u a n t i t a t i v e l y v a l i d .

M o r e is s a i d a b o u t different i s o t h e r m s

later, a n d I p o i n t out h e r e o n l y that e a c h of t h e m d e p e n d s o n t h e u n v e r i fiable

a s s u m p t i o n of some m o d e l a b o u t t h e u n i f o r m i t y of t h e surface or

t h e d i s t r i b u t i o n of energies. T h e s e c o n d root of the t r a g e d y lies i n the n e e d to use a d s o r p t i o n isotherms at a l l . I n h o m o g e n e o u s k i n e t i c s , a n d often i n e n z y m a t i c k i n e t i c s , 26 In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

2.

WELLER

Kinetic

27

Models

one c a n measure d i r e c t l y the concentrations of the r e a c t i n g species.

With

the a d d e d a s s u m p t i o n that the mass a c t i o n l a w is a p p l i c a b l e to e l e m e n t a r y r e a c t i o n steps, w e c a n t h e n p i e c e out a r e a c t i o n m e c h a n i s m w h i c h c a n b e e x a m i n e d for consistency w i t h the g l o b a l k i n e t i c s . I n heterogeneous catalysis w e g e n e r a l l y are i g n o r a n t of the surface concentrations.

Even

as p o w e r f u l a t o o l as m e a s u r e m e n t of the I R a b s o r p t i o n spectra of a d s o r b e d species, for e x a m p l e , m a y give i n f o r m a t i o n a b o u t entities w h i c h are present i n l a r g e c o n c e n t r a t i o n o n the surface b u t m a y b e i r r e l e v a n t to the r e a c t i o n m e c h a n i s m .

[ T h e c o n c e p t of a "most a b u n d a n t surface

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i n t e r m e d i a t e " ( m a s i ) , discussed b y B o u d a r t i n his recent a n d excellent review ( I ) , mediates.]

is c o n c e r n e d

w i t h surface species t h a t are, i n fact, i n t e r -

W e t h e n a d d to this p r o b l e m of i n d i r e c t estimate of surface

concentrations the i m p l i c i t a s s u m p t i o n that the s i m p l e mass a c t i o n l a w a p p l i e s to e l e m e n t a r y surface reactions.

T h e l i t e r a t u r e is r e p l e t e w i t h

c l a i m s of r a t e - l i m i t i n g surface reactions that are t h i r d or f o u r t h o r d e r — a result w h i c h w o u l d m a k e a h o m o g e n e o u s k i n e t i c i s t t r e m b l e . N e v e r t h e l e s s , the reactions occur, w e c a n m a k e some sense out of t h e k i n e t i c s , a n d w e h a v e some f e e l i n g a b o u t t h e g e n e r a l p r i n c i p l e s t h a t u n d r l y heterogeneous k i n e t i c s . W h a t is the r a t i o n a l m a n to do? has b e e n a s p e c t r u m of possible approaches, a l l of w h i c h h a v e a n d examples of w h i c h I w i l l give. those

who

There

advocates,

N e a r one e n d of t h e s p e c t r u m are

c o r r e c t l y c l a i m t h a t f u n d a m e n t a l u n d e r s t a n d i n g is c l e a r l y

p r e f e r a b l e to r a w e m p i r i c i s m a n d w h o also b e l i e v e that there is a d e q u a t e reason to a c c e p t the q u a n t i t a t i v e v a l i d i t y of some i s o t h e r m (say, L a n g m u i r - H i n s h e l w o o d ) a n d of the mass a c t i o n p r i n c i p l e , a n d t h a t i t is possible to establish the r e a c t i o n m e c h a n i s m ( or at least the r a t e - l i m i t i n g s t e p ) t h r o u g h consistency w i t h the o b s e r v e d k i n e t i c s . N e a r the

other

e n d of the s p e c t r u m are those w h o c l a i m t h a t n o n e of t h e theories has q u a n t i t a t i v e v a l i d i t y a n d t h a t t h e i m p o r t a n t t h i n g i n p r a c t i c e is to o b t a i n the most c o n v e n i e n t e m p i r i c a l k i n e t i c e q u a t i o n t h a t p r o v i d e s a d e q u a t e fit to observations a n d that p e r m i t s d e s i g n of c o m m e r c i a l reactors i n t e n d e d to operate w i t h i n the r a n g e of parameters studies.

The

intermediate

positions are n u m e r o u s . O n e p o p u l a r one, at least a m o n g chemists t u r n e d kineticists, is to assert that some i s o t h e r m , say L a n g m u i r - H i n s h e l w o o d , gives correct q u a l i t a t i v e insights a l t h o u g h i t m a y l a c k q u a n t i t a t i v e v a l i d i t y , that u s e f u l forms of the g l o b a l rate expression m a y b e d e d u c e d

on

the basis of this a s s u m p t i o n , b u t t h a t b y n o means is the r e a c t i o n m e c h a n i s m to b e b e l i e v e d as p r o v e d s i m p l y f r o m consistency w i t h k i n e t i c d a t a . A n o t h e r i n t e r m e d i a t e p o s i t i o n , i n t r i n s i c a l l y a t t r a c t i v e , is to say t h a t w e w i l l progressively complicate

(as n e e d e d ) the s i m p l e s t t h e o r e t i c a l ex-

p r e s s i o n {e.g., of the L a n g m u i r - H i n s h e l w o o d t y p e ) , i n c o r p o r a t e

those

changes r e q u i r e d b y heterogeneity or i n t e r a c t i o n , a n d suffer as a necessary e v i l the r e s u l t i n g c o m p l i c a t i o n s i n the final rate l a w . T h e d i f f i c u l t y

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

28

CHEMICAL

REACTION ENGINEERING

w i t h this, of course, is t h a t w e don't k n o w a priori

REVIEWS

what complications

are q u a n t i t a t i v e l y justified for o u r system. P e r h a p s the best w a y to p r o c e e d h e r e is to t r y to a n s w e r s o m e f u n d a m e n t a l questions. ( 1 ) S o m e d e f i n i t i o n of terms is a p p r o p r i a t e . w e mean by "kinetic model"?

S p e c i f i c a l l y , w h a t do

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( 2 ) W h y do w e w i s h to e s t a b l i s h the k i n e t i c m o d e l ? objectives as engineers or e n g i n e e r i n g scientists?

W h a t are o u r

( 3 ) S i n c e w e n o r m a l l y cannot d i r e c t l y d e t e r m i n e the surface c o n centrations of r e a c t i n g species, w h a t is the r a n g e of v a l i d i t y of the v a r i o u s a d s o r p t i o n isotherms t h a t relate surface concentrations to the o b s e r v a b l e gas-phase p a r t i a l pressures? ( 4 ) T o w h a t extent c a n k i n e t i c s f u r n i s h u n a m b i g u o u s answers to c e r t a i n f u n d a m e n t a l questions c o n c e r n i n g the n a t u r e of the catalyst surface a n d the d e t a i l e d course of the surface reactions w h i c h are a c t u a l l y occurring? Kinetic

Models

T h e r e are at least t w o i n t e r p r e t a t i o n s of this expression to b e f o u n d i n the l i t e r a t u r e : ( a ) T h e a c t u a l m e c h a n i s m , at the m o l e c u l a r l e v e l , b y w h i c h the c h e m i c a l r e a c t i o n occurs. ( b ) A c o n v e n i e n t a n d r e a s o n a b l e r e p r e s e n t a t i o n of the r e a c t i o n w h i c h , a l t h o u g h n o t i n g e n e r a l u n i q u e , is at least consistent w i t h k n o w n d a t a a n d p e r m i t s b o t h i n t e r p o l a t i o n a n d some e x t r a p o l a t i o n . [ K n o z i n g e r et al. (2) h a v e r e c e n t l y d r a w n even m o r e e x p l i c i t d i s t i n c t i o n s i n the f o l l o w i n g d e f i n i t i o n s : " M e c h a n i s t i c m o d e l : a r e a c t i o n s c h e m e w h i c h c a n b e i n t e r p r e t e d as a possible m o l e c u l a r m e c h a n i s m , t h e i n t e r m e d i a t e species a n d active sites of w h i c h c a n b e o b s e r v e d d i r e c t l y or m u s t b e p o s t u l a t e d o n the g r o u n d s of e x p e r i m e n t a l e v i d e n c e . K i n e t i c e q u a t i o n : a r a t e e q u a t i o n t h a t is d e d u c e d for a g i v e n m e c h a nistic model. K i n e t i c m o d e l : a p u r e l y f o r m a l r e a c t i o n scheme w h o s e i n t e r m e d i a t e s a n d a c t i v e sites are not i n t e r p r e t e d as a n y r e a l c h e m i c a l species. F o r m a l k i n e t i c e q u a t i o n : a rate e q u a t i o n that is d e d u c e d for a g i v e n kinetic model."] T h e difference b e t w e e n the t w o i n t e r p r e t a t i o n s is n o t t r i v i a l since i t is a l w a y s d e s i r a b l e i n science to d i s t i n g u i s h b e t w e e n r e a l i t y a n d p l a u s i b i l i t y or, i n other w o r d s , b e t w e e n necessity a n d consistency.

I n o n e case,

e s t a b l i s h i n g the k i n e t i c m o d e l is c o n s i d e r e d the e q u i v a l e n t of e s t a b l i s h i n g a u n i q u e t r u t h c o n c e r n i n g the w a y i n w h i c h the r e a c t i o n a c t u a l l y occurs. I n the other, o n l y consistency a n d c o n v e n i e n c e are i n v o l v e d . B o u d a r t

(3)

has expressed a n extreme p o s i t i o n o n the e n t i r e m a t t e r : " ' M e c h a n i s m ' or ' m o d e l ' c a n m e a n a n a s s u m e d r e a c t i o n n e t w o r k , or a p l a u s i b l e s e q u e n c e of steps for a g i v e n r e a c t i o n , o r a p o s t u l a t e d s t e r e o c h e m i c a l p a t h d u r i n g

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

2.

WELLER

Models

29

the course of a n i s o l a t e d step.

S i n c e methods of i n v e s t i g a t i o n a n d goals

Kinetic

are so u t t e r l y different i n the s t u d y of n e t w o r k s , sequences a n d steps, the w o r d s ' m e c h a n i s m ' or m o d e l ' s h o u l d be a v o i d e d .

T h e y have acquired

the b a d c o n n o t a t i o n associated w i t h i r r e s p o n s i b l e or v a i n s p e c u l a t i o n , l a r g e l y to describe achievements that v a r y w i d e l y i n s o p h i s t i c a t i o n . " A t this p o i n t i t is u s e f u l to ask w h y w e w i s h to e s t a b l i s h the k i n e t i c m o d e l for a surface r e a c t i o n . I f t h e reason is to p r o v e c o n c l u s i v e l y the m o l e c u l a r m e c h a n i s m b y w h i c h the r e a c t i o n is t r u l y o c c u r r i n g o n the catalyst surface, t h e n t h e response m u s t p r o b a b l y be t h a t ( 1 ) the m e c h a ­

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n i s m of

no

s o l i d - c a t a l y z e d r e a c t i o n is a d e q u a t e l y

m o l e c u l a r l e v e l , e v e n for as s i m p l e a case as H — D 2

2

understood

at

the

exchange, a n d ( 2 ) of

a l l the s o p h i s t i c a t e d approaches t h a t h a v e b e e n a p p l i e d to t h e e s t a b l i s h ­ i n g of c a t a l y t i c r e a c t i o n m e c h a n i s m s , k i n e t i c s is a m o n g the least u s e f u l i n p r o v i d i n g u n a m b i g u o u s answers.

M u c h m o r e u s e f u l , i n different sys­

tems, are t e c h n i q u e s s u c h as i s o t o p i c tracer studies, i s o t o p i c r a t e effects, d e t e r m i n a t i o n of s t o i c h i o m e t r i c n u m b e r , i n v e s t i g a t i o n of the stereochemis­ t r y of c o m p l e x

r e a c t i o n p r o d u c t s , a n d i n f r a r e d a b s o r p t i o n spectra

of

a d s o r b e d species, w h i c h h a v e a l l g i v e n s o m e i n s i g h t as to the r e a c t i o n mechanism. O n the other h a n d , i f w e w i s h a k i n e t i c m o d e l for less a m b i t i o u s reasons—to

b e consistent w i t h r a t e d a t a , to p e r m i t reactor d e s i g n , to

suggest n e w experiments b a s e d o n p r e d i c t i o n s of the m o d e l , to c o n t r i b u t e q u a l i t a t i v e i n s i g h t i n t o a possible r e a c t i o n p a t h — t h e n m a n y

approaches

are possible a n d p l a u s i b l e . Adsorption

Isotherms

M o s t of us go t h r o u g h the f o l l o w i n g steps i m p l i c i t l y or e x p l i c i t l y , i n d e r i v i n g a rate e q u a t i o n o n the basis of a p a r t i c u l a r k i n e t i c m o d e l f o r a surface-catalyzed reaction: ( a ) C h o o s e a p a r t i c u l a r surface r e a c t i o n as the r a t e - l i m i t i n g step. ( F o r s i m p l i c i t y , this d i s c u s s i o n is l i m i t e d to cases w h e r e s u c h a p r o c e d u r e is justifiable—i.e., w h e r e a l l h e a t a n d mass transfer steps a n d a l l a d s o r p t i o n a n d d e s o r p t i o n processes are r e l a t i v e l y r a p i d ; o n l y one step i n a n o v e r a l l sequence is rate l i m i t i n g , etc. ) A s a n e x a m p l e , i n a r e a c t i o n 2 A -> Β + C , w e m a y w i s h to c o n s i d e r t h e i m p l i c a t i o n of a s s u m i n g t h a t ( 1 ) r e a c t i o n occurs o n l y b e t w e e n m o l e c u l e s of c h e m i s o r b e d A , a n d ( 2 ) t h e r a t e l i m i t i n g step is r e a c t i o n b e t w e e n t w o m o l e c u l e s of A a d s o r b e d o n adjacent sites. ( b ) A s s u m e t h a t a c o n v e n t i o n a l m a s s - a c t i o n l a w applies to s u c h surface reactions, w h e r e the G u l d b e r g - W a a g e " a c t i v e masses" are p r o ­ p o r t i o n a l to the surface concentrations ( o r f r a c t i o n a l c o v e r a g e ) . ( c ) A s s u m e that some i s o t h e r m e q u a t i o n c o r r e c t l y relates the surface c o n c e n t r a t i o n of a n y species to the o b s e r v a b l e p a r t i a l pressures of a l l species i n the a m b i e n t b u l k gas.

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

30

C H E M I C A L

R E A C T I O N

E N G I N E E R I N G

R E V I E W S

( d ) D e d u c e the c o r r e s p o n d i n g rate e q u a t i o n r e l a t i n g the k i n e t i c s to t h e o b s e r v a b l e p a r t i a l pressures.

global

A l t h o u g h the f o l l o w i n g discussion w i l l focus l a r g e l y o n step c, b r i e f c o m m e n t s a b o u t a a n d b are a p p r o p r i a t e . surface

reactions

L a n g m u i r (4, 5)

i n terms

of

F i r s t , i t is c o m m o n to treat

the f u n d a m e n t a l

and Hinshelwood

concept advanced

that the molecules

(6)—i.e.,

w h i l e a d s o r b e d , w i t h c o v e r a g e not exceeding

a monolayer.

by

react

Langmuir

h i m s e l f , i n his e x t r a o r d i n a r y 1921 p a p e r o n the P t - c a t a l y z e d o x i d a t i o n of C O and H

(5),

2

suggested

a n alternate p o s s i b i l i t y : that r e a c t i o n

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o c c u r d i r e c t l y o n c o l l i s i o n of a gas m o l e c u l e w i t h a n a d s o r b e d or a t o m .

T h i s concept was later d e v e l o p e d

may

molecule

b y R i d e a l a n d E l e y (7,

8)

a n d has b e e n v a r i o u s l y l a b e l l e d t h e L a n g m u i r - R i d e a l , R i d e a l - E l e y , E l e y R i d e a l , or " d i v e - b o m b " m e c h a n i s m .

T h e resultant rate equations

significantly f r o m the L a n g m u i r - H i n s h e l w o o d f o r m .

differ

T h e r e are f e w , i f

any, u n a m b i g u o u s examples w h e r e this m e c h a n i s m is operative.

[Never-

theless, S i n f e l t has h a d great p r a c t i c a l success i n a p p l y i n g a m i l d l y c o m p l i c a t e d v e r s i o n of hydrogenolysis

this m e c h a n i s m

i n the treatment of

o v e r s u p p o r t e d metals.

hydrocarbon

H i s m o d e l is d e r i v e d f r o m that

u s e d e a r l i e r b y T a y l o r a n d co-workers at P r i n c e t o n , cf. Réf. 10.1 ever, the M a r s - v a n K r e v e l e n ( 9 ) t r a n s i t i o n m e t a l oxides

How-

a p p r o a c h to o x i d a t i o n reactions

over

does i n fact i n v o k e the n o t i o n of c y c l i c

redox

r e a c t i o n of the catalyst d i r e c t l y o n c o l l i s i o n of a gaseous m o l e c u l e

( sub-

strate or 0 ) 2

w i t h the ( o x i d i z e d or r e d u c e d ) surface.

This

approach

has a c h i e v e d c o n s i d e r a b l e p o p u l a r i t y ; its f u n d a m e n t a l f o r m u l a t i o n a n d l i m i t a t i o n s are c o n s i d e r e d i n some d e t a i l later. W i t h respect to i t e m b , i f the b i n d i n g energies of a d s o r b e d

mole-

cules are n o t i d e n t i c a l , there is no reason to expect e q u a l r e a c t i v i t y of those molecules.

T h e d i r e c t result is that the a s s u m p t i o n of a surface

mass-action rate l a w is w i t h o u t ' t h e o r e t i c a l justification for systems n o n - u n i f o r m energetics.

of

T o p u t the m a t t e r another w a y , the surface rate

constant k, n o r m a l l y a s s u m e d to be o n l y a f u n c t i o n of t e m p e r a t u r e , w i l l also be a f u n c t i o n of surface coverage i n the g e n e r a l case. W e n o w consider i t e m c a n d i n q u i r e : w h a t is the p r o p e r e q u i l i b r i u m a d s o r p t i o n i s o t h e r m to be u s e d i n d e v e l o p i n g a rate e q u a t i o n ?

Again

f o r s i m p l i c i t y , the discussion is l i m i t e d to a d s o r p t i o n w i t h o u t d i s s o c i a t i o n . B e f o r e a t t e m p t i n g a n answer, w e r e v i e w some salient characteristics of t h e L a n g m u i r , F r e u n d l i c h , a n d T e m k i n isotherms a n d the assumptions involved i n their derivation. r e l e v a n c e to m o n o l a y e r Langmuir Isotherm.

These

are s i n g l e d out because of

their

chemisorption. T h e L a n g m u i r isotherm, initially derived

the basis of k i n e t i c arguments (11),

on

c a n b e d e r i v e d e q u a l l y w e l l f r o m the

statistical t h e r m o d y n a m i c s a p p r o p r i a t e to e q u i l i b r i u m i n i d e a l l o c a l i z e d monolayers

(12).

However,

i n either case the v a l i d i t y of the

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

simple

2.

WELLER

Kinetic

31

Models

i s o t h e r m depends o n the satisfaction of a set of postulates, some of w h i c h are m o r e easily g r a n t e d t h a n others as b e i n g true for r e a l systems.

Re­

gardless of a n y questions c o n c e r n i n g t h e q u a n t i t a t i v e v a l i d i t y of a l l t h e postulates, L a n g m u i r ' s concepts are of f u n d a m e n t a l i m p o r t a n c e to a l l subsequent t h e o r e t i c a l w o r k i n the

field.

Some of the k e y assumptions a n d results of the e l e m e n t a r y e q u a t i o n

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are: ( 1 ) A d s o r p t i o n occurs o n a finite n u m b e r of e q u i v a l e n t sites o n a u n i f o r m surface. ( 2 ) E a c h site c a n adsorb one a n d o n l y one gas m o l e c u l e , a n d the a d s o r p t i o n is l o c a l i z e d . T h i s i m p l i e s l o c a l i z e d m o n o l a y e r s . If the m o l e ­ cules w e r e t o t a l l y free to m o v e over the surface w i t h n o m u t u a l i n t e r ­ a c t i o n , t h e y w o u l d b e h a v e as a t w o - d i m e n s i o n a l p e r f e c t gas. F r e e d o m to m o v e , b u t w i t h v a r y i n g degrees of m u t u a l i n t e r a c t i o n , w o u l d c o r r e ­ s p o n d to the a p p r o p r i a t e l y i m p e r f e c t , t w o - d i m e n s i o n a l gas. ( 3 ) T h e a d s o r b e d m o l e c u l e s d o n o t i n t e r a c t , a n d t h e i r energies are i n d e p e n d e n t of the presence or absence of a d s o r b e d m o l e c u l e s o n n e i g h ­ b o r i n g sites. ( 4 ) I f t w o or m o r e species of gas m o l e c u l e s are present, t h e y w i l l c o m p e t e for a d s o r p t i o n o n t h e fixed n u m b e r of e q u i v a l e n t sites. ( 5 ) F o r a n y gaseous species i , the c o r r e s p o n d i n g e q u i l i b r i u m a d ­ s o r p t i o n constant K i is the r a t i o of a n a d s o r p t i o n r a t e constant fc a n d a d e s o r p t i o n rate constant k . K is i n t r i n s i c a l l y p o s i t i v e . M o r e o v e r , K i = e x p C - A G i V R T ] = e x p [ + A S i ° / R ] e x p [ - A H ° i / R T ] , w h e r e AG ASi°, a n d ΔΗι° are the free energy, e n t r o p y , a n d e n t h a l p y change o n a d s o r p t i o n . ( 6 ) F o r a n y gaseous species i at a p a r t i a l pressure p the f r a c t i o n of sites c o v e r e d b y a d s o r b e d i (θι) increases l i n e a r l y w i t h p at sufficiently l o w values of p\\ conversely, θι becomes i n d e p e n d e n t of p\ at sufficiently h i g h values of p . in

in

{

U

u

x

{

T h e p r o b l e m s w i t h the q u a n t i t a t i v e , l i t e r a l a p p l i c a t i o n of the s i m p l e L a n g m u i r i s o t h e r m h a v e b e e n presented i n m a n y places.

Only a few

a d d i t i o n a l c o m m e n t s are m a d e here, chiefly i n c o n n e c t i o n w i t h the t e m ­ p e r a t u r e d e p e n d e n c e a n d the s i g n of the constants K i . P a r e n t h e t i c a l l y , L a n g m u i r was one of his o w n most severe critics. I n his 1938 F a r a d a y L e c t u r e (13)

he observes, i n d i s c u s s i n g his w o r k o n the a d s o r p t i o n of

c e s i u m o n tungsten,

. . t h e p h y s i c a l assumptions u n d e r l y i n g this f a c t o r

( 1 — θ ) [ i n the e q u a t i o n for the a d s o r p t i o n rate] are v e r y i m p r o b a b l e . . . . E v e n i f there w e r e n o m o b i l i t y at a l l , w e cannot j u s t i f y the f a c t o r ...

i f the sites are closely adjacent to one another.

I n this case i t is

m o r e r e a s o n a b l e to assume t h a t ( 1 — θ ) s h o u l d b e r e p l a c e d b y ( 1 — β ) 4

B o u d a r t , i n 1956 (14)

a n d since, has p r o p e r l y e m p h a s i z e d the i m ­

p o r t a n c e of e x a m i n i n g the t e m p e r a t u r e d e p e n d e n c e of the e q u i l i b r i u m a d s o r p t i o n constants K i i n o r d e r to establish t h a t the heats a n d entropies of a d s o r p t i o n are reasonable. H e q u o t e d t h e e x a m p l e of stibine d e c o m p o ­ s i t i o n o n a n t i m o n y , w h e r e the k i n e t i c d a t a are w e l l fitted b y the L a n g m u i r e q u a t i o n for a single site m o d e l , r = kKp/(l

+ Kp).

Experimentally, Κ

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

32

CHEMICAL

REACTION ENGINEERING

REVIEWS

is essentially i n d e p e n d e n t of t e m p e r a t u r e , i m p l y i n g that AH ~ 0, a n u n ­ reasonable

result.

The

deduced

AS

a

is also unreasonable.

Boudart's

suggestion that the " p a r a d o x of heterogeneous k i n e t i c s " c a n a c c o u n t for the t e m p e r a t u r e i n d e p e n d e n c e of Κ i n v o l v e s the i d e a that, a l t h o u g h the surface coverage decreases w i t h i n c r e a s i n g t e m p e r a t u r e , sites of h i g h e r adsorption energy

( i n a n e n e r g e t i c a l l y heterogeneous system)

o p e r a t i v e at h i g h e r t e m p e r a t u r e .

become

T h e q u a l i t a t i v e a r g u m e n t is p l a u s i b l e ,

b u t there are difficulties i n d e v e l o p i n g q u a n t i t a t i v e r i g o r : ( a ) the a c t u a l d i s t r i b u t i o n of energetics is a l w a y s q u a n t i t a t i v e l y u n k n o w n ; a n d ( b )

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w o u l d b e c o i n c i d e n t a l t h a t these t w o effects of opposite

it

sign should

h a p p e n to just b a l a n c e . A n o t h e r e x a m p l e i l l u s t r a t i n g strange b e h a v i o r of a n a d s o r p t i o n c o n ­ stant is the f a s c i n a t i n g d i s p r o p o r t i o n a t i o n of p r o p y l e n e to b u t e n e e t h y l e n e over t u n g s t e n o x i d e - s i l i c a . B o t h L u c k n e r et al. (15) kudur and Thodos

(16)

and

and Hatti-

agree t h a t a d u a l - s i t e surface r e a c t i o n is rate

c o n t r o l l i n g a n d t h a t the i n i t i a l rates are g i v e n b y t h e s t a n d a r d d u a l - s i t e Langmuir-Hinshelwood equation: r

Q

where K

3

and p

3

=

Cp /(1+#3P ) 3

2

3

2

are the a d s o r p t i o n constant a n d p a r t i a l pressure, respec­

t i v e l y , of p r o p y l e n e .

T h e w o r k of H a t t i k u d u r a n d T h o d o s is p a r t i c u l a r l y

c o n v i n c i n g because of the u n u s u a l l y large r a n g e of p a r t i a l pressures i n v e s ­ t i g a t e d ( m o r e t h a n 3 0 - f o l d for p r o p y l e n e ) . T h e u n e x p e c t e d result is that, as b o t h sets of investigators agree, K a c t u a l l y increases w i t h i n c r e a s i n g t e m p e r a t u r e whereas expects a decrease.

one

H a t t i k u d u r a n d T h o d o s r e p o r t a v a l u e of Δ Η

+ 12,140 c a l / g - m o l e for the a d s o r p t i o n of p r o p y l e n e . n e r et al. do not f a l l o n a straight l i n e for In K

3

3

normally α

=

( T h e data of L u c k ­

vs. 1/T.)

Now

endo-

t h e r m i c a d s o r p t i o n is possible ( 1 7 ) , b u t there are v e r y f e w a u t h e n t i c a t e d examples a n d these seem to i n v o l v e either m o l e c u l a r h y d r o g e n or m o l e c u ­ l a r o x y g e n (18).

Since the free e n e r g y of a d s o r p t i o n m u s t be n e g a t i v e ,

endothermic adsorption implies that T A S

a

> ΔΗ . &

Thomas and Thomas

suggest, for h e u r i s t i c purposes, t w o w a y s of h a v i n g a n e n t r o p y increase o n a d s o r p t i o n r a t h e r t h a n the e x p e c t e d decrease. sociative

adsorption

with

complete

T h e first i n v o k e s d i s ­

two-dimensional

m o b i l i t y of

the

a d s o r b e d fragments; t h e s e c o n d is that for some reason the e n t r o p y of t h e s o l i d itself increases m o r e t h a n t h e e n t r o p y of the a d s o r b e d decreases.

gas

F o r present purposes w e note o n l y t h a t the first suggestion

w o u l d m e a n g i v i n g u p the L a n g m u i r i s o t h e r m , for w h i c h i m m o b i l e a d ­ s o r p t i o n is p o s t u l a t e d ; m o r e o v e r , the p i c t u r e seems i n t u i t i v e l y i m p r o b a b l e for p r o p y l e n e d i s p r o p o r t i o n a t i o n . T h e s e c o n d suggestion raises a difficult t h e o r e t i c a l p r o b l e m , w h i c h I b e l i e v e n o one has q u a n t i t a t i v e l y a t t e m p t e d to date.

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

2.

WELLER

Kinetic

33

Models

T h e last p o i n t to b e m e n t i o n e d relates to the s i g n of the K's.

I n the

s t a n d a r d a p p l i c a t i o n of L a n g m u i r - t y p e isotherms, those k i n e t i c m o d e l s are d i s c a r d e d that l e a d to rate equations for w h i c h a n y K i d e t e r m i n e d f r o m e x p e r i m e n t a l r a t e d a t a turns out to b e significantly negative. years ago it was n o t e d (19)

Some

that the l i m i t e d w o r k a v a i l a b l e o n m i x e d

a d s o r p t i o n f r o m a p o t e n t i a l l y r e a c t i v e gas m i x t u r e i n d i c a t e d t h a t e n h a n c e d , rather t h a n c o m p e t i t i v e , a d s o r p t i o n m i g h t b e a c o m m o n nomenon

a n d t h a t force-fitting of " e n h a n c e d

phe-

adsorption" data into a

L a n g m u i r f o r m guarantees a n e g a t i v e v a l u e for at least one K i i n t h e

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r a n g e of c o n d i t i o n s s t u d i e d . I n the i n t e r v e n i n g years m o r e examples of a d s o r p t i o n f r o m r e a c t i v e mixtures h a v e b e e n e x a m i n e d , a n d the e n h a n c e d a d s o r p t i o n b e h a v i o r for at least one ( sometimes b o t h ) of the constituents does seem to b e the g e n e r a l p a t t e r n . T h e b e h a v i o r of H - C O m i x t u r e s 2

o n C o a n d F e F i s c h e r - T r o p s c h catalysts, Z n O , a n d n o b l e metals has b e e n r e c e n t l y r e v i e w e d b y G u p t a et al. (20). adsorption from C H - 0 3

of 0

2

6

2

Gérai et al. (21),

i n a s t u d y of

on C u O and C u 0 , report enhanced adsorption 2

f r o m the m i x t u r e a n d decreased a d s o r p t i o n of C H ; at a l l t e m p e r a 3

6

tures a n d f r o m a l l m i x t u r e s the t o t a l a d s o r p t i o n was m o r e t h a n a d d i t i v e for b o t h C u O a n d C u 0 . 2

I n t u i t i v e l y w e c a n u n d e r s t a n d the reason f o r e n h a n c e d

adsorption,

w h e n i t occurs, i n terms of t h e f o r m a t i o n of a surface c o m p l e x w h i c h is m o r e strongly a d s o r b e d t h a n either c o m p o n e n t s i n g l y . D i r e c t e v i d e n c e , f r o m I R spectra, c a l o r i m t e r y , a n d other t e c h n i q u e s , is a v a i l a b l e for s u c h complexes i n a n u m b e r of cases (see,

for e x a m p l e , R e f s .

21-24).

It has b e e n suggested t h a t i f a l l the other c r i t e r i a r e q u i r e d f o r q u a n t i tative v a l i d i t y of t h e L a n g m u i r - H i n s h e l w o o d t h e o r y w e r e satisfied, w e c o u l d t a k e i n t o a c c o u n t e x p l i c i t l y the f o r m a t i o n of s u c h surface complexes, w i t h the a d d i t i o n of another e q u i l i b r i u m constant, a n d m a i n t a i n the f o r m of the L a n g m u i r expressions.

U n f o r t u n a t e l y , i t is difficult e x p e r i m e n t a l l y

to establish the existence, n a t u r e , a n d surface e q u i l i b r i u m constants for s u c h complexes i n a q u a n t i t a t i v e w a y , a n d v e r y f e w kineticists i n t e r e s t e d i n m e c h a n i s m h a v e t a k e n the t r o u b l e to investigate this p r o b l e m . Freundlich and Temkin Isotherms. S i n c e the several d e r i v a t i o n s of the F r e u n d l i c h a n d T e m k i n isotherms are s u m m a r i z e d i n v a r i o u s reports (25, 26, 27, 28),

the details are not r e p e a t e d here. R e a d e r s n o t f a m i l i a r

w i t h S i p s ' elegant p a p e r m a y b e interested to k n o w that h e has u s e d Stieltjes transforms to solve the i n v e r s e of the u s u a l p r o b l e m — i . e . , to d e d u c e the p o s s i b l e d i s t r i b u t i o n functions f o r a d s o r p t i o n energy t h a t are consistent w i t h the e x p e r i m e n t a l i s o t h e r m Θ

(ρ).

A U d e r i v a t i o n s of these isotherms u t i l i z e the L a n g m u i r result f o r a d s o r p t i o n o n surface sites of one p a r t i c u l a r a d s o r p t i o n energy, a n d i n this sense the L a n g m u i r i s o t h e r m is a f u n d a m e n t a l starting p l a c e . w i s h to m a k e a f e w p o i n t s :

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

I do

34

CHEMICAL

REACTION ENGINEERING REVIEWS

(1) T h e F r e u n d l i c h a n d T e m k i n isotherms a r e n o less ( a n d n o m o r e ) t h e o r e t i c a l l y justified t h a n t h e L a n g m u i r i s o t h e r m . T h e b a s i c c o n c e p t of l o c a l i z e d , m o n o l a y e r c h e m i s o r p t i o n is r e q u i r e d for a l l . D i f f e r ­ ent isotherms r e s u l t a c c o r d i n g to t h e assumptions m a d e a b o u t t h e heat of a d s o r p t i o n , Δ Η , a n d t h e site d i s t r i b u t i o n f u n c t i o n , Ν(ΔΗ ), among other things. T h e L a n g m u i r i s o t h e r m r e q u i r e s the a s s u m p t i o n t h a t —ΔΗ = constant. T h e F r e u n d l i c h i s o t h e r m c a n b e d e d u c e d f r o m t h e a s s u m p ­ tions that Ν = ae~ o a n d t h a t θ ( o r ρ ) is s m a l l ; [cf. the c o m m e n t b y T h o m a s a n d T h o m a s ( R e f . 18, p . 44): " T h e F r e u n d l i c h e q u a t i o n is . . . n o l o n g e r to b e r e g a r d e d as m e r e l y a c o n v e n i e n t f o r m of r e p r e s e n t i n g t h e L a n g m u i r e q u a t i o n at i n t e r m e d i a t e values of Θ. M o r e o v e r , t h e m e t h o d of d e r i v a t i o n disposes of t h e c r i t i c i s m that the F r e u n d l i c h e q u a t i o n p r e ­ dicts a p r o g r e s s i v e l y i n c r e a s i n g coverage w i t h i n c r e a s i n g p r e s s u r e : the i s o t h e r m is e x p e c t e d to b e v a l i d o n l y at l o w coverages."] T h e T e m k i n e q u a t i o n is o b t a i n e d i f t h e a d s o r p t i o n heat decreases l i n e a r l y ( b e t w e e n m a x i m u m a n d m i n i m u m v a l u e s ) w i t h surface coverage θ a n d i f θ is i n a m i d d l e range. α

η

α

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AII/AII

( 2 ) W e d o n o t k n o w , f o r a n y a r b i t r a r y system, w h i c h assumptions a b o u t t h e d i s t r i b u t i o n functions for e n e r g y are t r u e or w h i c h isotherms are therefore t h e o r e t i c a l l y v a l i d . [ I n this context, C l a r k ( R e f . 26, p . 57) has c l e a r l y stated a d u a l d i f f i c u l t y : " I t s h o u l d b e e m p h a s i z e d . . . t h a t agreement b e t w e e n t h e t h e o r e t i c a l isotherms w i t h d i s t r i b u t i o n functions d e t e r m i n e d b y t h e v a r i o u s procedures . . . a n d e x p e r i m e n t a l isotherms does n o t guarantee that t h e true p h y s i c a l p i c t u r e has b e e n d i s c o v e r e d . . . . A n o t h e r difficulty is t h e i n h e r e n t i n s e n s i t i v i t y of the t h e o r e t i c a l iso­ t h e r m to the f o r m of the d i s t r i b u t i o n f u n c t i o n w i t h i n the a c c u r a c y of experimental data."] (3) R e s t r a i n t is therefore a p p r o p r i a t e i n insistence that a n y one i s o t h e r m has u n i q u e q u a n t i t a t i v e v a l i d i t y i n a p p l i c a t i o n s to c a t a l y t i c kinetics. Rate Equations and Kinetic Models: An Eclectic

Review

T h i s section contains a s a m p l i n g f r o m the enormous

literature on

c a t a l y t i c k i n e t i c s p u b l i s h e d d u r i n g t h e last 10-15 years.

T h e samples

w e r e chosen

p a r t l y to i l l u s t r a t e the s p e c t r u m of approaches

u s e d to

i n t e r p r e t k i n e t i c d a t a a n d p a r t l y to i n d i c a t e some areas of s u b s t a n t i a l disagreement. Ammonia Synthesis: an Example of Ambiguity.

B o u d a r t ' s 1972 r e ­

v i e w ( 1 ) contains a n excellent discussion of this system as e x e m p l i f y i n g "two-step

c a t a l y t i c reactions."

M y p u r p o s e here is o n l y to a d d to t h e

v a r i o u s l y p r o p o s e d rate equations some recent results of B r i l l (29) a n d to i n d i c a t e t h e difficulty, even i n this m o s t - s t u d i e d of c a t a l y t i c reactions, of d r a w i n g conclusions about s u c h b a s i c questions as ( a ) is t h e surface o f a n i r o n catalyst u n i f o r m or heterogeneous, a n d ( b ) i f the surface is u n i ­ f o r m , is Ho a d s o r b e d m o l e c u l a r l y o n a single site or d i s s o c i a t i v e l y o n d u a l sites.

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

2.

WELLER

Kinetic

35

Models

T h e classic T e m k i n - P y z h e v e q u a t i o n uses the T e m k i n i s o t h e r m ( i m ­ p l y i n g a heterogeneous surface ) for the surface c o n c e n t r a t i o n of n i t r o g e n as r e l a t e d to the

fictitious

n i t r o g e n p a r t i a l pressure t h a t w o u l d b e i n

e q u i l i b r i u m w i t h the a c t u a l h y d r o g e n a n d a m m o n i a p a r t i a l pressures. F o r the rate of the f o r w a r d r e a c t i o n , the T e m k i n - P y z h e v treatment gives: / P H , ' V

7

VPNIII / 2

T h e constant i m u s t b e d e t e r m i n e d ; to agree w i t h e x p e r i m e n t , i t is often Downloaded by UNIV OF PITTSBURGH on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch002

chosen to be b e t w e e n 0.5 a n d 0.6. T h e 1942 p a p e r of B r u n a u e r et al. (27), its

w h i c h is m o r e e x p l i c i t i n

d e r i v a t i o n s , also starts w i t h the a s s u m p t i o n of

l i n e a r decrease i n

a d s o r p t i o n heat w i t h i n c r e a s i n g c o v e r a g e ( h e t e r o g e n e i t y )

a n d ends w i t h

a g e n e r a l i z e d " T e m k i n i s o t h e r m " t h a t is v a l i d for the e n t i r e r a n g e

of

a d s o r p t i o n , whereas the o r i g i n a l T e m k i n i s o t h e r m is v a l i d o n l y i n the m i d d l e range. Temkin-Pyzhev.

D i s s o c i a t i v e a d s o r p t i o n of No is a g a i n a s s u m e d , as i n F o r the f o r w a r d r e a c t i o n o n l y the rate expression at

l o w surface coverage (0 )

becomes:

N

k pyi

=

2

V If a =

Vu^

)

1 ( a r b i t r a r y c h o i c e ), this reduces to : r

=

k ps αα Κ 2

/

B o u d a r t has s h o w n (14, 30)

ρχηλ

2

that the a s s u m p t i o n of a h o m o g e n e o u s

surface ( L a n g m u i r m o d e l ) a n d dissociative a d s o r p t i o n of n i t r o g e n as the r a t e - d e t e r m i n i n g step leads to the synthesis e q u a t i o n : r

which

is i d e n t i c a l w i t h

=

kp __ N2

the e q u a t i o n

deduced

by

the p r o c e d u r e

B r u n a u e r et al. for a heterogeneous surface ( p r o v i d e d a =

of

1 ).

S t i l l m o r e r e c e n t l y , B r i l l a n d co-workers h a v e r e p o r t e d i n t e r e s t i n g results : ( a ) F i e l d e l e c t r o n m i c r o s c o p i c studies s h o w that N adsorption occurs p r e f e r e n t i a l l y o n (111) faces of i r o n a n d v e r y m u c h less o n (100) a n d (110) faces (31). 2

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

36

CHEMICAL

R E A C T I O N ENGINEERING REVIEWS

( b ) O n the basis of field i o n mass s p e c t r o m e t r i c studies, N H a p pears to be the first p r o d u c t f o r m e d o n exposure of i r o n to N — H (32). 2

2

2

( c ) I R studies of F e - M g O catalyst exposed to N - H at 4 1 0 ° C i n d i c a t e the p r e s e n c e of p a r t i a l l y h y d r o g e n a t e d N m o l e c u l e s (i.e., u n d i s s o c i a t e d ) h a v i n g b o n d s s i m i l a r to those h y d r a z i n e ( 3 3 ) . 2

2

2

O n this basis, B r i l l d e r i v e d a rate expression for a m m o n i a synthesis u n d e r the assumptions t h a t the surface is h o m o g e n e o u s b u t t h a t the r a t e - d e t e r m i n i n g step

is the n o n - d i s s o c i a t i v e

a d s o r p t i o n of

molecular

n i t r o g e n o n a s i n g l e site (-29). F u r t h e r m o r e , he shows t h a t a set of exp e r i m e n t a l i n t e g r a l c o n v e r s i o n d a t a t a k e n at 340 ° C is satisfied b y this Downloaded by UNIV OF PITTSBURGH on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch002

rate expression w i t h a s t a n d a r d d e v i a t i o n of db 1.5% i n the rate constant. T h e same d a t a , fitted b y the e q u a t i o n of O z a k i et al. for d i s s o c i a t i v e a d sorption on

d u a l sites, gives

a n almost i d e n t i c a l s t a n d a r d d e v i a t i o n ,

± 1.4%, i n the c o r r e s p o n d i n g rate constant. H o p e f u l l y f u r t h e r i n g e n i o u s experiments w i l l s h e d f u r t h e r l i g h t o n the reaction mechanism

( a l t h o u g h after 40 years one m a y

p e s s i m i s m ) b u t i n this s p e c i a l case, at least, the expressions

fitting

approach

of g l o b a l rate

seems u n l i k e l y to differentiate k i n e t i c m o d e l s

that

differ

r a d i c a l l y i n t h e i r assumptions. The

P a r a - O r t h o - H y d r o g e n Conversion: an Example of Simplicity.

p a r a m a g n e t i c m e c h a n i s m for the p a r a - o r t h o - h y d r o g e n shift r e a c t i o n at l o w temperatures s h o u l d constitute the simplest c h e m i c a l e x a m p l e of a c a t a l y z e d r e a c t i o n . H u t c h i n s o n et al. (34)

c o n d u c t e d a c a r e f u l s t u d y of

the a p p r o a c h to e q u i l i b r i u m , f r o m b o t h sides of the e q u i l i b r i u m c o m p o s i t i o n , over f e r r i c oxide g e l at 7 6 ° K . T h e results are d i s c o n c e r t i n g .

They

m a y b e s u m m a r i z e d as f o l l o w s : (1) W i t h expression has face r e a c t i o n , T h i s simplifies

the L a n g m u i r - H i n s h e l w o o d a p p r o a c h , the o v e r a l l rate the same f o r m regardless of w h i c h step ( a d s o r p t i o n , suror d e s o r p t i o n ) or c o m b i n a t i o n of steps is rate l i m i t i n g . c r i t i c a l testing of the a p p r o a c h .

( 2 ) T h e d e d u c e d L a n g m u i r - H i n s h e l w o o d rate constant is different d e p e n d i n g o n w h e t h e r one approaches e q u i l i b r i u m f r o m the o r t h o - r i c h or p a r a - r i c h side. T h i s result w o u l d v i o l a t e the p r i n c i p l e of m i c r o s c o p i c r e v e r s i b i l i t y a n d indicates that no s i m p l e L a n g m u i r - H i n s h e l w o o d m o d e l c a n be a p p l i c a b l e i n this system. (3) T h e discrepancy between theory and experiment can be res o l v e d q u a l i t a t i v e l y i f one postulates that the a c t i v a t i o n energies for a d s o r p t i o n a n d d e s o r p t i o n c h a n g e w i t h coverage of the catalyst surface. H o w e v e r , s u c h a postulate is t a n t a m o u n t to g i v i n g u p the q u a n t i t a t i v e v a l i d i t y of the L a n g m u i r - H i n s h e l w o o d m o d e l . Dehydration of Ethanol to Ether. O n e of the most g r a t i f y i n g p u b l i c a t i o n s i n c a t a l y t i c kinetics is the p a p e r of K a b e l a n d J o h a n s o n ( 3 5 )

on

the v a p o r - p h a s e d e h y d r a t i o n of e t h a n o l to d i e t h y l ether over the a c i d f o r m of D o w e x 5 0 — a s u l f o n a t e d s t y r e n e - d i v i n y l b e n z e n e c o p o l y m e r .

The

authors f o u n d t h e i r k i n e t i c d a t a to b e consistent w i t h the e q u a t i o n p r e -

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

2.

Kinetic

WELLER

37

Modek

d i e t e d f r o m a s i m p l e L a n g m u i r m o d e l i n w h i c h the r a t e - l i m i t i n g step is a surface r e a c t i o n b e t w e e n a d j a c e n t l y a d s o r b e d e t h a n o l m o l e c u l e s :

7'

=

[l+K P +K Pw+K P ]> A

A

w

Fl

F

the subscripts A , W , a n d Ε r e p r e s e n t i n g e t h a n o l , w a t e r , a n d ether, r e s p e c ­ t i v e l y . T h i s itself is p l a u s i b l e , b u t i t is not the reason f o r the e x c e p t i o n a l

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interest of the w o r k . T h e authors p r o c e e d e d to do that w h i c h is s e l d o m d o n e — n a m e l y , to d e t e r m i n e K , K , a n d K A

w

E

directly and independently

f r o m a d s o r p t i o n isotherms ( L a n g m u i r ) of the i n d i v i d u a l p u r e c o m p o ­ nents. T h e e x t r a o r d i n a r y result was the agreement b e t w e e n the t w o sets of a d s o r p t i o n constants—one set d e t e r m i n e d b y fitting of t h e k i n e t i c d a t a , the other set f r o m the i n d i v i d u a l isotherms. A t 1 2 0 ° C , for e x a m p l e , K , A

K , and K w

E

d e t e r m i n e d f r o m the k i n e t i c s w e r e 3.4, 7.0, a n d ~ 0.

The

c o r r e s p o n d i n g values f r o m the isotherms w e r e 2.5, 7.6, a n d ~ 0. T h e experience w i t h D o w e x 50 does not p e r m i t extension to another c o m m o n catalyst for a l c o h o l d e h y d r a t i o n — a l u m i n a . K n o z i n g e r a n d P i n e s h a v e c o n d u c t e d the most extensive research o n the m e c h a n i s m of a l c o h o l reactions over a l u m i n a . K n o z i n g e r et al. earlier d a t a (36) method.

have recently analyzed

(2)

o n the k i n e t i c s of ether f o r m a t i o n b y a g r i d search

T h e e x p e r i m e n t a l d a t a c o u l d not b e fitted b y equations

based

o n the e x p e c t e d m e c h a n i s m . O n the other h a n d , five f o r m a l k i n e t i c e q u a ­ tions w e r e d e v e l o p e d w h i c h describe the d a t a e q u a l l y w e l l ( w i t h m e a n error i d e n t i c a l to the m e a n e x p e r i m e n t a l e r r o r )

a n d w h i c h cannot

d i s t i n g u i s h e d . A l l five give c o m p a r a b l e a c t i v a t i o n energies.

be

T h e authors

c o n c l u d e : " A t least for the ether f o r m a t i o n o n a l u m i n a . . . . , it therefore seems u n r e a l i s t i c to use k i n e t i c analysis for the e l u c i d a t i o n of the m o l e c u ­ lar m e c h a n i s m of the r e a c t i o n . T h e o n l y possible result is a p u r e l y f o r m a l d e s c r i p t i o n of the r e a c t i o n rate as a f u n c t i o n of the p a r t i a l pressures of a l c o h o l , ether, a n d w a t e r . " Hydrogénation of Cyclopropane.

A s e c o n d e x a m p l e of a r e a c t i o n

i n w h i c h agreement is c l a i m e d b e t w e e n the a d s o r p t i o n constant f r o m a rate e q u a t i o n

(Langmuir form)

deduced

and that measured from

a d s o r p t i o n i s o t h e r m is the recent p a p e r of S r i d h a r a n d R u t h v e n

the (37).

T h e k i n e t i c s for c y c l o p r o p a n e hydrogénation over f o u r s u p p o r t e d n i c k e l catalysts at 6 0 ° C w e r e f o u n d to be best fitted b y the s i m p l e e x p r e s s i o n : r =

k K

cVc

w h e r e s u b s c r i p t c represents c y c l o p r o p a n e . determined K

c

(l+K )cPc

1

T h e values of the k i n e t i c a l l y

for the four catalysts r a n g e d f r o m 2.12 to 3.71 a t m " ; the

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

1

38

CHEMICAL

v a l u e of K

REACTION ENGINEERING REVIEWS

c a l c u l a t e d f r o m the i s o t h e r m data of B e n s o n a n d K w a n

c

(38)

was r e p o r t e d to be 2.5 a i m , i n g o o d agreement. - 1

S e v e r a l aspects of the w o r k l e a d one to b e less t h a n t o t a l l y s a n g u i n e a b o u t the r e s u l t s : ( 1 ) T h e a c t u a l plots of r vs. p are s i g m o i d a l — a result w h i c h , as the authors a p p r e c i a t e d , is not t r u l y c o m p a t i b l e w i t h the a b o v e rate e q u a t i o n . c

( 2 ) N o t e r m i n v o l v i n g p appears i n the " a d s o r p t i o n d e n o m i n a t o r " despite the fact ( w h i c h the authors note ) that h y d r o g e n is m o r e s t r o n g l y a d s o r b e d o n n i c k e l t h a n is c y c l o p r o p a n e . n

( 3 ) T h e c o m m e r c i a l N i - S i 0 - A L 0 catalyst s t u d i e d b y B e n s o n a n d K w a n was o b t a i n e d f r o m a different source ( A t l a n t i c R e f i n i n g C o . , vs. H a r s h a w ).

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2

3

( 4 ) B e n s o n a n d K w a n fitted t h e i r o w n rate d a t a for c y c l o p r o p a n e h y d r o g e n to a p o w e r rate l a w , fitted t h e i r o w n a d s o r p t i o n d a t a to F r e u n d l i c h isotherms a n d p r o p o s e d t h a t the r a t e - d e t e r m i n i n g step was a surface reaction between adsorbed cyclopropane and an adsorbed hydrogen atom. Statistical Model Building. I n 1962 a s e m i n a l p a p e r b y B o x a n d H u n t e r (39)

described an extremely powerful, iterative m o d e l - b u i l d i n g

m e t h o d i n v o l v i n g m i n i m i z a t i o n of the residuals of a d i a g n o s t i c p a r a m e t e r . H u n t e r a n d M e z a k i (40),

i n a p p l y i n g this a p p r o a c h , discussed the results

of a n e x p e r i m e n t a l s t u d y of m e t h a n e o x i d a t i o n over P d - A l 0 . 2

3

A frac-

t i o n a l f a c t o r i a l d e s i g n was u s e d for the e x p e r i m e n t a l r u n s . A n a l y s i s of these d a t a w a s c o m p l e t e d i n a subsequent a r t i c l e b y K i t t r e l l et al.

(41).

T h e rate expression ( H o u g e n - W a t s o n t y p e ) d e v e l o p e d to fit the e x p e r i m e n t a l d a t a of H u n t e r a n d M e z a k i a d e q u a t e l y was Pan,

Voi

K i t t r e l l et al. w e r e c a r e f u l to p o i n t out that . . . "no c l a i m is m a d e

con-

c e r n i n g the m e c h a n i s m of the r e a c t i o n or e v e n the uniqueness of the m o d e l w h i c h has b e e n set f o r t h as a d e q u a t e l y d e s c r i b i n g the e x p e r i m e n t a l data."

T h i s caveat seems a p p r o p r i a t e since one w o u l d h a v e difficulty

b e l i e v i n g i n the p h y s i c a l r e a l i t y of the r e a c t i o n m e c h a n i s m w h i c h , b y a s t r a i g h t f o r w a r d L a n g m u i r - H i n s e l w o o d a p p r o a c h for a t h i r d - o r d e r r e a c tion between a methane molecule and two adsorbed oxygen

molecules,

w o u l d l e a d to the p r o p o s e d rate e q u a t i o n . Some d i c h o t o m y

of t h i n k i n g nevertheless appears to exist

the p r o p o n e n t s of this a p p r o a c h .

B o x a n d H i l l (42),

among

i n a paper entitled

" D i s c r i m i n a t i o n A m o n g M e c h a n i s t i c M o d e l s , " p r o p o s e to " d i s c o v e r mechanism

the

for a p a r t i c u l a r p h e n o m e n o n l e a d i n g to a specific m a t h e m a t i -

cal model. . . .

T o d i s c r i m i n a t e a m o n g these

[possible m e c h a n i s m s ]

a

s e q u e n t i a l p r o c e d u r e is d e v e l o p e d i n w h i c h c a l c u l a t i o n s m a d e after e a c h e x p e r i m e n t d e t e r m i n e the most d i s c r i m i n a t o r y process c o n d i t i o n s for use i n the next e x p e r i m e n t . "

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

2.

WELLE R

Kinetic

39

Models

T h e p r o p o s e d use of s o p h i s t i c a t e d c o m p u t e r analysis to establish or to d i s p r o v e a m e c h a n i s m has b e e n the subject of c o n s i d e r a b l e criticism.

recent

B o u d a r t ( I ) , for e x a m p l e , i n c o m m e n t i n g o n a s t a t i s t i c a l r e -

analysis b y L o g a n a n d P h i l i p (43)

of m e c h a n i s t i c d a t a b y O z a k i et

al.

o n a m m o n i a synthesis, observes " T h e r e is a l w a y s a r e a l d a n g e r i n kinetics to treat d a t a w i t h a p o w e r f u l m e t h o d of analysis w h i c h m a y be far better t h a n the d a t a themselves." A l l a r a a n d E d e l s o n (44) stronger

position

i n a discussion

of

have taken an even

p a r a m e t e r i z a t i o n techniques

in

k i n e t i c s : " T h e a b i l i t y to fit a set of n u m b e r s to a f u n c t i o n a l f o r m r e s e m -

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b l i n g c h e m i c a l k i n e t i c equations cannot of itself establish the v a l i d i t y of the m o d e l . . . . W e

believe

that it is most

i m p o r t a n t to d i s t i n g u i s h

b e t w e e n the c o r r e l a t i v e v a l u e of p a r a m e t e r fits as o p p o s e d to the p r e d i c t i v e c a p a b i l i t i e s of a t r u l y f u n d a m e n t a l m o d e l . . . . W e

especially

object to the p u b l i c a t i o n of these parameters as rate constants, a t e r m w h i c h i m p l i e s a f u n d a m e n t a l p r o p e r t y of the reaction, since this o n l y f u r t h e r aggravates a n a l r e a d y c o n f u s e d s i t u a t i o n i n the l i t e r a t u r e . " A s a p a r e n t h e t i c c o m m e n t , the a u t h o r feels that a f u n d a m e n t a l diffic u l t y is not that statistical analysis is a n i n a p p r o p r i a t e l y p o w e r f u l t o o l for d i s c r i m i n a t i n g b e t w e e n a n u m b e r of p o s t u l a t e d models.

T h e difficulty is

i n d e c i d i n g w h e t h e r a n y of the models has a s o u n d t h e o r e t i c a l basis. C h a r l e s W a r e (45)

has c a l l e d the author's attention to the a p p l i c a t i o n

of a m o d i f i e d B o x - H u n t e r a p p r o a c h to p r o c e e d f r o m s t a t i s t i c a l l y d e s i g n e d , i s o t h e r m a l l a b o r a t o r y d a t a to successful p r e d i c t i o n of the

performance

a n d t e m p e r a t u r e d i s t r i b u t i o n i n a d i a b a t i c units. I n b r i e f , a p o w e r rate l a w suggested b y l i t e r a t u r e i n f o r m a t i o n is used as a first a p p r o x i m a t i o n . T h e analysis of l a b o r a t o r y d a t a indicates a possible n e e d for m o d i f y i n g the f o r m of the rate l a w , a n d i t e r a t i o n of the entire process finally results i n a rate l a w s a t i s f y i n g the statistical c r i t e r i a i m p o s e d . T h e e m p i r i c a l rate l a w is satisfactory for d e s i g n i n g c o m m e r c i a l units o p e r a t i n g w i t h i n the range of variables s t u d i e d , a n d m e c h a n i s t i c conclusions are not d r a w n . Hydrogénation of

Propylene and Isobutylene.

An

exceptionally

c a r e f u l a n d fine s t u d y of olefin hydrogénation over P t - A l 0 2

l i s h e d b y Rogers et al. (46)

3

was p u b -

i n 1966. T h e e x p e r i m e n t a l k i n e t i c d a t a h a v e

b e e n a n a l y z e d not o n l y b y the authors b u t also i n at least three subseq u e n t papers. T h e final rate e q u a t i o n of Rogers et al. w a s :

= r

aK K PiV2 (i+K +K y 1

lPl

/

2

2V2

,

βΚ Κ