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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
