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On the properties of adsorption caused by the "Thermal Disorder" on
the surface of a crystal
Volkenshtein, F. F.
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NATIONAL RESEARCH COUNCIL OF CANADA
TRANSLATION
TT-126
ON SOME P R O P E R T I E S OF ADSORPTION CAUSED B Y THE
"THERMAL DISORDERt' ON THE SURFACE OF A CRYSTAL
( 0 Nekotorykh Ocobennostyakh Adsorp t s i i Obyslovlennykh
"Teplovym Besporyadkon" na Poverkhnosti K r i s t a l a )
by
F. Fo Volkenshtein
Translated by
Eo Rabkin
OTTAWA
May,
1950
NATIONAL RESEAFtCB COUNCIL O F CANADA
O tt a w ,
TRANSLATION
Canada
TT-126
Titler
On Some Ppopeptie s of Adsorption Caused
by t h e wThermal Disorder" on t h e Surface
of a C r y s t a l
By:
F, F, Volkenshtein
Reference :
Zhurnal F i z i c h e s k o i Khimii, Vol, 23,
No. 8, 1949, p a 917, Akademiya Nauk,
U,S.S,R,
T r a n s l a t e d by:
E s t h e r Rabkin
Read by:
S, C , Liang
Zhurnal Fizicheskoi Khfmii Vol. 23, Noo 8 , 1949
Acadsiny of Scfancas, U,S,S,R,
ON SOW PROPERTIES OF ADSORPTION CAUSED BY THE
"THERMALDISORDlB'"ON
THE SURFACE OF A CRYSTAL
by
F. F. Volkenshtein
Translated by
Esther Rabkin
SUMMARY
The adsorption of gas molecules on t h e s u r f a c e of a
c r y s t a l when t h e number of adsorbed c e n t r e s v a r i e s with ternpera.ture has been analysed,
Adsorption c e n t r e s a r e t r e a t e d
a s d e f e c t s of t h e s u r f a c e , t h a t i s , a s l o c a l d i s t o r t i o n s i n
the p e r i o d i c s t r u c t u r e of t h e l a t t i c e .
The k i n e t i c s of adsorp-
t i o n a t defined c o n d i t i o n s a r e found t o be e x a c t l y t h e same
a s f o r t h e case of t h e so-called " a ~ t i v a t e d 'a~d s o r p t i o n , a l -
-
though t h e a c t i v a t i o n b a r r i e r i s absent.
therm of the type Q
An a d s o r p t i o n i s o -
p h a s been obtained.
The d i f f e r e n t i a l
h e a t of adsorption was found t o be a f u n c t i o n of the f i l l i n g
i n , although t h e s u r f a c e i s e n e r g e t i c a l l y homogeneous and t h e
r e a c t i o n s between t h e adsorbed molecules a r e ignored.
1,
Disorder I n s i d e a C r y s t a l
A r e a l c r y s t a l d i f f e r s from an i d e a l c r y s t a l by t h e
presence of d e f e c t s .
By t h e word Ibdefectthwe mean any d i s -
t o r t i o n i n the p e r i o d i c s t r u c t u r e of t h e l a t t i c e .
Among
d6f ec t s p r e s e n t i n a r e a l l a t t f ce, we must df s t i n g u f s h
between t h e macroscopic and microscopPc d e f e c t s ,
By macroscopic d e f e c t s we w f l l assume a d l s t o r t i o n over a r e g i o n , t h e dfmensfons of which ape c o n s i d e r a b l y
g r e a t e r t h a n the dimensfons of an i n d i v i d u a l c r y s t a l l i n e
nucleus.
Cracks, p a r t i c l e s of f o r e f gn substance and a l l
t y p e s of macsoacopfc embeddfngs a r e d e f e c t s of t h i s type,
The supfaces and r i b s of t h e c r y s t a l f t s e l f may b e looked
upon a s macroscopPc d e f e c t s which d i s t o r t t h e s t r i c t p e r i o d i c s t r u c t u r e of an i n f i n i t e i d e a l l a t t i c e ,
By mfcroscopic d e f e c t s we w f l l assume such a
d i s t o r t i o n whfch by f t s dimensf ons f s of t h e same o r d e r of
magnitude a s a n f n d f v i d u a l c r y s t a l l i n e n u c l e u s ,
Thus, f n
t h e case of a mfcroscopic d e f e c t , t h e p e r i o d i c s t r u c t u r e
of t h e c r y s t a l f s p r a c t f c a l l y r e - e s t a b l i s h e d a t a d i s t a n c e
of s e v e r a l l a t t f ce c o n s t a n t s ,
We w i l l n o t e t h e f o l l o w i n g t y p e s of mf croscopfc
d e f e c t s : ( 1 ) a h o l e praoduced by t h e disappearance of an
atom o r i o n from t h e i d e a l l a t t i c e ; (2) a n e u t r a l atom o r
i o n of the l a t t f c e l o c a t e d between t h e normal p o s i t i o n s ;
(3) an i o n I n a h e t e r o p o l a r l a t t i c e found i n a normal posf-
t i o n b u t c a r r y i n g an abnormal charge; ( 4 ) a f o r e i g n atom
l o c a t e d between t h e normal p o s i t i o n s ; ( 5 ) a f o r e i g n atom
l o c a t e d In t h e normal p o s f t i o n , t h a t I s , r e p l a c i n g a
n a t u r a l atom of t h e l a t t f ce,
Defects of t h e s e types a r e schematically shown i n
Figure 1. We must note t h a t each d e f e c t produces around
i t s e l f a c e r t a i n deformation of t h e l a t t i c e which i s n o t
shown i n t h e f i g u r e .
S t r i c t l y speaking, we should imagine
the t o t a l r e g i o n i n which t h e l a t t i c e i s deformed a s a
defectc
From now on we w i l l l i m i t o u r s e l v e s t o t h e a n a l y s i s
of micro-defects, ignoring t h e macroscopic d i s t o r t i o n s ,
We
w i l l thus d e a l w i t h an i d e a l i z e d p i c t u r e of a r e a l c r y s t a l .
We w i l l note t h a t , s t r i c t l y speaking, micro-defects
can n o t be considered a s being f i x e d i n s i d e a c r y s t a l :
possess d e f i n i t e m o b i l i t y ,
they
The m o b i l i t y of the d e f e c t s i s a
r e s u l t of t h e p e r i o d i c s t r u c t u r e of t h e l a t t i c e .
The d i s -
placement of a d e f e c t along t h e l a t t i c e demands some energy
of a c t i v a t i o n ; t h a t i s , i t i s connected with t h e overcoming
of p o t e n t i a l b a r r i e r s , t h e hei@;ht of which i s determined by
t h e n a t u r e of the d e f e c t , t h e s t r u c t u r e of t h e l a t t i c e and
t h e d i r e c t i o n of motion of t h e d e f e c t .
Another g e n e r a l p r o p e r t y of micro-daf e c t s i s t h e
presence of i n t e r a c t i o n s between them, which i s n o t i c e a b l e
when they approach each o t h e r .
each o t h e r .
D e f e c t s may a t t r a c t o r r e p u l s e
Thus, f o r example, i n a h e t e r o p o l a r l a t t i c e
constructed from i o n M+ and 'R
t h e empty m e t a l l i c p o s i t i o n s
r e p u l s e each o t h e r , b u t they a r e a t t r a c t e d t o t h e empty
m e t a l l o i d p o s i t i o n s or t o t h e m e t a l l i c i o n s between t h e
natural positions,
An e l e c t r o n i n such a l a t t i c e , which we
must imagine a s a n e u t r a l s t a t e of t h e i o n M+, i s a t t r a c t e d
t o t h e empty m e t a l l o i d p o s i t i o n and i s repulsed from an
empty m e t a l l i c p o s i t i o n (1).
When two o r s e v e r a l d e f e c t s a r e combined t o g e t h e r
a new d e f e c t , possessing d i f f e r e n t p r o p e r t i e s , i s formed.
Thus, f o r example, m e t a l l o i d and m e t a l l i c h o l e s joined toget h e r , o r an e l e c t r o n a t t r a c t e d by a m e t a l l o i d hole c o n s t i t u t e
formations of a d i f f e r e n t type, possessing p r o p e r t i e s d i f f e r e n t from t h e i r components analysed i n d i v i d u a l l y .
Thus, i n s i d e a r e a l c r y s t a l l i n e l a t t i c e we have a
c h a r a c t e r i s t i c "chemistry of d e f e c t s M . "Reactionsm between
d e f e c t s may be exothermic o r endothermic, the same a s g e n e r a l
reactions.
These "reactions",
t h e same a s g e n e r a l r e a c t i ons,
may proceed w i t h o r without a c t i v a t i o n , depending on t h e
n a t u r e of t h e r e a c t i n g d e f e c t s .
b o t h produce and absorb d e f e c t s .
A c r y s t a l l i n e l a t t i c e can
Thus, f o r example, i n an
i d e a l l a t t i c e , the displacement of an atom from a p o s i t i o n
i n t o t h e space between t h e n a t u r a l p o s i t i o n s c o n s t i t u t e s an
example of a r e a c t i o n which produces a d e f e c t .
A d e f e c t of a given kind under given conditions
possesses a d e f i n i t e average l i f e duration.
It can disappear
and reappear.
We w i l l assume t h a t d e f e c t s do n o t disappear beyond
t h e l i m i t s of the l a t t i c e and t h a t they do not e n t e r i t from
the outside,
I n t h f s case a t equilibrium t h e r e a r i s e a s
many d e f e c t s of a given kfnd a s t h e number t h a t d i s a p p e a r s
per u n i t time i n a u n i t
f volume of t h e c r y s t a l ,
The con-
c e n t r a t i o n of d e f e c t s i n t h i s case i s d e t e ~ m i n e dby t h e
equilibrium conditions.
The e q u i l i b ~ i u mconcentration
I n a d d i t i o n , i t i s dependent on
changes with temperature.
t h e h i s t o r y of t h e sample.
I n a given sample a t a given
temp r a t u r e t h e e q u i l i b r i u m concentpation may be changed
under the i n f l u e n c e of e x t e r n a l f o r c e s ( i l l u m i n a t i o n , e l e c t r i c f i e l d and o t h e r s ) .
This i s connected with t h e f a c t ,
t h a t t h e e x t e r n a l f o r c e s charge t h e r a t e s of t h e r e a c t i o n s ,
i n which t h e s e d e f e c t s p a r t i c i p a t e .
From now on when we speak of t h e c o n c e n t r a t i o n of
d e f e c t s we w i l l mean equilibrium c o n c e n t r a t i o n s a
words, we w i l l ignore t h e 'frozent"(metastable)
In other
s t a t e s of
the l a t t i c e .
The t o t a l number
f d e f e c t s of a l l t y p e s contained
i n a u n i t volume of a c s y s t a
disorder i n t h e crystal,
w i l l 1 be r e f e r r e d t o a
a
It i s assumed t h a t t h e t o t a l d i s -
order i s s u f f i i e n t l y small* n o t h e r words f t i s assumed
t h a t the t o t a l concentration of a l l d e f e c t s i s small a s
compared t o t h e number of n u c l i i n a u n i t volume. Otherwise
we would s t e p out beyond t h e frame of our p i c t u r e ,
In f a c t
when t h e number of d e f e c t s i s comparable with t h e number of
nuc e i we can not speak of t'micro-defects",
t h e summation of
which i n t h i s case must be looked upon a s a fGmacro-defectw,
I n a t o t a l d i s o r d e r we must d i s t i n g u i s h between
t h e h i s t o r i c a l and t h e thermal p o r t i o n s of t h e d i s o r d e r ,
The p o r t i o n of t h e d i s o r d e r which i s r e t a i n e d a t zero temp e r a t u r e we w i l l name a s the h i s t o r i c a l ( i r r e v e r s i b l e ) d i s order.
B y t h e thermal d i s o r d e r , which i s of a r e v e r s i b l e
c h a r a c t e r , we w i l l assume an a d d i t i o n a l d i s o r d e r , superimposed on t h e i n i t i a l h i s t o r i c a l d i s o r d e r by h e a t i n g .
Thus, f o r example, a l a t t i c e of type M, Rr with a
d i s t o r t e d stoichiometry p o s s e s s i n g , f o r example, an excess
of metalloid may contain empty m e t a l l i c p o s i t i o n s r i g h t from
t h e very beginning,
The number of t h e s e d e f e c t s a t zero
temperature c h a r a c t e r i z e s t h e h i s t o r i c a l d i s o r d e r ,
During
h e a t i n g , m e t a l l i c i o n s of t h e l a t t i c e move from t h e n a t u r a l
p o s i t i o n s i n t o t h e space between t h e p o s i t i o n s , hence addit i o n a l empty m e t a l l i c p o s i t i o n s a r e formed.
The degree of
d i s t o r t i o n of t h e stoichiometry d u r i n g h e a t i n g does n o t
change, b u t t h e g e n e r a l d i s o r d e r i n c r e a s e s due t o t h e superimposition of the thermal on t h e h i s t o r i c a l df s o r d e r ,
The r e l a t i o n s h i p between t h e h i s t o r i c a l and thermal
d i s o r d e r s i s n a t u r a l l y dependent on t h e h i s t o r y and t h e temp e r a t u r e of t h e sample,
I n some c a s e s t h e h i s t o r i c a l d i s -
o r d e r predominates considerably over t h e thermal d i s o r d e r .
I n t h i s case t h e t o t a l d i s o r d e r i s p r a c t i c a l l y unchanged
with temperature.
I n o t h e r c a s e s , on t h e c o n t r a r y , t h e
h i s t o r i c a l d i s o r d e r may be neglected a s compared with t h e
thermal d i s o r d e r .
I n t h e s e c a s e s p r a c t i c a l l y t h e whole
d i s o r d e r i s of a h e a t o r i g i n .
2.
Disorder on t h e Surface of a C r y s t a l and
I t s Function i n Adsorption
On the surface of an i d e a l i z e d r e a l c r y s t a l , the
same a s i n s i d e of a c r y s t a l , we d e a l w i t h micro-defects of
v a r i o u s types.
Thus, t h e surface of a r e a l c r y s t a l i s
c h a r a c t e r i z e d by a d e f i n i t e degree of d i s o r d e r .
The laws
con t r o l l i n g t h i s d i sorder a r e r e f l e c t e d i n t h e behaviour
of adsorption,
I n f a c t t h e a d s o r p t i o n of g a s molecules t a k e s
p l a c e , a s i s known, a$ i n d i v i d u a l adsorptfon c e n t r e s , t h e
number of which, g e n e r a l l y speaking, may n o t be l a r g e i n
comparf son with t h e t o t a l number of t h e s u r f a c e atoms,
According t o Taylor, t h e geometric heterogenei t i e s of t h e
s u r f a c e a r e such adsorption c e n t r e s .
On our i d e a l i z e d sur-
f a c e , t h e micro-def e c t s appear t o be such h e t e r o g e n e i t i e s .
Remaining i n the frame of the Taylor assumptions, we may
t r e a t micro-defects p r e s e n t on t h e surf ace a s adsorption
centres.
From t h i s p o i n t of view t h e i d e a l s u r f a c e does
n o t g e n e r a l l y adsorb,
The degree of d e v i a t i o n of t h e r e a l
surface from t h e i d e a l s t a t e determines t h e adsorption
capacity.
We must n o t e t h a t t h e p r e s e n c e of d e f e c t s on t h e
s u r f a c e does n o t by i t s e l f mean t h a t t h e h e t e r o g e n e i t y i s
energetic,
E n e r g e t i c h e t e r o g e n e i t y presupposes t h e p r e s e n c e
of v a r i o u s t y p e s of a d s o r p t i o n c e n t r e s ,
However, i f t h e
a d s o r p t i o n c e n t r e s w i l l be d e f e c t s of o n l y one d e f i n i t e t y p e ,
c h a r a c t e r i z e d by t h e same h e h t a d s o r p t i o n q , t h e n our s u r f a c e from an a d s o r p t i o n p o i n t of view w i l l be e n e r g e t i c a l l y
homogeneous.
From now on, f o r s i m p l i c i t y , we w i l l d e a l w i t h
an e n e r g e t i c a l l y homogeneous s u r f a c e c o n s i d e r i n g t h a t t h e
a d s o r p t i o n of a g a s molecule may t a k e p l a c e n o t on any d e f e c t
of t h e s u r f a c e , b u t o n l y on d e f e c t s of a d e f i n i t e t y p e .
I n t h e g e n e r a l t h e o r i e s of a d s o r p t i o n , t h e following propertie s are assigned t o t h e adsorption centres:
(1) It i s assumed t h a t the number of a d s o r p t i o n c e n t r e s
on t h e s u r f a c e i s a c o n s t a n t v a l u e , c h a r a c t e r i s t i c f o r a
g i v e n s u r f a c e and that i t does n o t change with t e m p e r a t u r e ,
T h i s number i s completely determined by t h e h i s t o r y of t h e
surf ace.
( 2 ) F u r t h e r , i t i s assumed t h a t t h e a d s o r p t i o n c e n t r e s
a r e l o c a l i z e d on t h e s u r f a c e .
They a r e immobileo The h i s -
t o r y of t h e a d s o r p t i o n c e n t r e s d o e s n o t change with t i m e s o
t h a t we d e a l w i t h a s o - c a l l e d f r o z e n d i s t r i b u t i o n of adsorpt i o n c e n t r e s on t h e s u r f a c e ,
(3)
F i n a l l y , i t i s assumed t h a t t h e number of c e n t r e s
does n o t change wl t h coverage.
I n o t h e r words, t h e t o t a l
number of c e n t r e s i s independent of t h e f a c t of how many of
them a r e occupied by adsorbed molecules and how many remain
free.
It must be noted t h a t t h e conception r e g a r d i n g
t h e adsorption c e n t r e s a s d e f e c t s of t h e s u r f a c e does n o t i n
t h e l e a s t support t h e s e assumptions.
The r e v e r s e i s t r u e :
a l l t h r e e assumptions a r e i n c o n t r a d i c t i o n with t h f s conception.
Thus, i d e n t i f y i n g t h e a d s o r p t i o n c e n t r e s w i t h t h e
d e f e c t s of t h e s u r f a c e , we must d i s r e g a r d t h e t h r e e assumpt i o n s enumerated above, which a r e t h e b a s i s of t h e g e n e r a l
adsorption theory,
Later i t wf 11 be shown how t h e b a s i c
adsorption laws change.
We w P l l d e s i g n a t e by A t h e d e f e c t s which a c t a s t h e
adsorption c e n t r e s ,
Let NA be t h e c o n c e n t r a t i o n of t h e s e
d e f e c t s on t h e s u r f a c e a t a temperature T .
We w i l l consider
t h a t t h e concentratfon of d e f e c t s A i n c r e a s e s with t h e temp e r a t u r e from some minimum value NA = X a t T = 0 t o some maxi-
m value NA = Y a t T =
-,
so t h a t
X CNA & Y.
(1)
We w i l l agree t o c h a r a c t e r i z e t h e d i s o r d e r on t h e
s u r f a c e by a concentration of d e f e c t s A, s e t t i n g a s i d e t h e
o t h e r d e f e c t s which may be p r e s e n t on t h e s u r f a c e and which
may e n t e r i n t o r e a c t i o n s w i t h d e f e c t s A .
Then, NA e x p r e s s e s
the d i s o r d e r a t a temperature T ; moreover, t h e number X exp r e s s e s t h e h i s t o r i c a l p o r t i o n of the d i s o r d e r , and t h e number
- X) -
(NA
Z =Y
-
t h e thermal p o r t i o n of t h e d i s o r d e r .
The number
X i s t h e d-ifference between t h e maximum and minimum
d i s o r d e r s r e a l i z e d on t h e s u r f a c e .
This number remains con-
s t a n t f o r a given surface and may be used a s i t s c h a r a c t e r i s tic
0
We w i l l d i s t i n g u i s h between two p a r t i c u l a r ( l i m i t i n g ) cases:
(1) X = 0 o r Z = Y,
This i s t h e case when t h e t o t a l
d i s o r d e r i s of a thermal o r i g i n ( t h e h i s t o r i c a l d i s o r d e r i s
absent).
(2) X = Y or Z = 0.
This i s t h e case when t h e t o t a l
d i s o r d e r i s of a h i s t o r i c a l o r i g i n ( t h e thermal d i s o r d e r i s
absent).
We must note t h a t the g e n e r a l t h e o r i e s of adsorpt i o n d e a l w i t h t h e second of t h e s e two p a r t i c u l a r c a s e s ,
Thus, t h e s e t h e o r i e s , based on a c o n s t a n t (unchanging
tem-
p e r a t u r e s ) number of adsorption c e n t r e s , remain t r u e only
while t h e thermal d i s o r d e r can be considered s u f f i c i e n t l y
small a s compared t o t h e h i s t o r i c a l .
I n the p r e s e n t paper
we a r e analysing a more g e n e r a l c a s e , when t h e thermal and
the h i s t o r i c a l d i s o r d e r s a r e comparable i n magnitude,
Our
formulae may be transformed i n t o t h e g e n e r a l formula of t h e
Langmuir theory f o r t h e l i m i t i n g case Z = 0.
We w i l l consider t h a t each adsorption c e n t r e A
may a c c e p t one and only one gas molecule, s o t h a t t h e
adsorption c e n t r e on which a gas molecule i s found i s
incapable of f u r t h e r a d s o r p t i o n and f a l l s out of t h e p l a y ,
Such an occupied adsorption c e n t r e , t h a t i s , u n i t e d with
t h e adsorbed gas molecule, we w i l l d e s i g n a t e by t h e symbol
B , i n d i s t i n c t i o n from t h e f r e e c e n t r e
A,
A o
The f r e e c e n t r e s
a s w e l l a s t h e occupied c e n t r e s B , c o n s t i t u t e d e f e c t s
on t h e surface b u t they a r e d e f e c t s of d i f f e r e n t types,
W e w i l l d e s i g n a t e by NB the c o n c e n t r a t i o n of t h e B d e f e c t s ,
Evidently NB i s t h e number of gas molecules adsorbed p e r
u n i t surface.
The r e a c t i o n of a d s o r p t i o n and desorption
i s expressed i n t h e form:
A
+
G & B 9
(2
where G i s t h e symbol f o r t h e g a s molecule.
This r e a c t i o n
i s exothermic i n t h e forward d i r e c t i o n and endothermic i n
the reverse direction,
the reaction,
NAONG
NB
--
We w i l l d e s i g n a t e by q t h e h e a t of
For an equilibrium s t a t e we have:
O C 9
where
-Q
ce = ocoe
(3)
Here NG i s t h e concentration of gas molecules i n t h e gas
phase :
Adapting t h e d e s i g n a t i o n N = NA + NB
where,
obviously, N i s t h e t o t a l number of adsorption c e n t r e s
( f r e e + occupied) we may r e w r i t e t h e condition ( 3 ) thus:
Taking i n t o account r e a c t i o n (2) we must r e w r i t e
i n s t e a d of ( 1):
(11)
X < N < Y .
I n t h e p a r t i c u l a r case, when the t o t a l d i s o r d e r
i s of a p u r e l y h i s t o r i c a l o r i g i n we have X = N = Y and
equation (39) t ~ a n s f o r m si n t o t h e g e n e r a l L a n m u i r e q u i l i brium equation:
3,
Tlie Appearance and Disappearance of
Adsorption
Centres
--- -- -
-
- -
- -
- -
-
-
D e f e c t s A , which a c t a s t h e a d s o r p t i o n c e n t r e s ,
p a r t i c i p a t e not only i n t h e r e a c t i o n s of a d s o r p t i o n and
d e s o r p t i o n ( 2 ) b u t a l s o i n a number of o t h e r r e a o t i o n s proceeding on t h e s u r f a c e simultaneously w i t h r e a c t i o n (2),
These simultaneous r e a c t i o n s may vary depending on t h e
n a t u r e of t h e d e f e c t s p r e s e n t on t h e s u r f a c e .
Ignoring
t h e s e d e f e c t s means a r e t u r n t o the Langmuir t h e o r y o Below
we dl1 analyse two of t h e s i m p l e s t c a s e s , when d e f e c t s A
r e a c t with t h e o t h e r d e f e c t s on t h e s u r f a c e according t o
monomolecular and bimolecular r e l a t i o n s ,
A s t o d e f e c t s B,
we must consider t h a t they p a r t i c i p a t e only i n r e a c t i o n ( 2 ) .
If we would a s s i g n t o t h e s e d e f e c t s the c a p a c i t y t o p a r t a k e
i n any o t h e r r e a c t i o n s , then t h i s would mean a t r a n s i t i o n
t o a heterogeneous s u r f a c e , and we would go beyond the frame
of our theory.
Let u s Imagine t h a t along with d e f e c t s A end B
t h e surface a l s o c o n t a i n s d e f e c t s of a t h i r d type which we
w i l l d e s i g n a t e by C , and t h e concentration of which a t a
temperature T we w i l l d e s i g n a t e by NC,
Vje wf 11 assume t h a t
t h e d e f e c t s C and A a r e i n t e r t r a n s f o r m a b l e ,
C S
(6)
A
We w i l l consider, t h a t the tramlbrmation of a d e f e c t C i n t o
a d e f e c t A demands a c e r t a i n expenditure of energy u,
Thus, d e f e c t s A take p a r t i n two simultaneous
r e a c t f o n s ( 2 ) and ( 6 ) .
A t equilibrium we must add t o equa-
t i o n ( 5 ) the following equation:
..
/3 = p 0 e - kT
u
N~
= p.
N~
where
We now have: Y = NA + NB + NC,
0
X = 0, so that
equation ( 7 ) may be r e w r i t t e n thus:
N
-
NB
Y - N
=
Po
From t h e two equations ( 3 ) and ( 7 ) t w o unknowns
may be determined: the t o t a l number of adsorbed c e n t r e s N
and the t o t a 1 number of adsorbed molecules NB.
We w i 11 analyse an example of the monmolecular
reaction ( 6 ) .
We w i l l assume t h a t on t h e s u r f a c e of a c r y s t a l ,
atoms of a f o r e i g n impurity a r e d i s t r i b u t e d .
s u r f a c e concentration of such atoms.
Let Y be the
During h e a t i n g a por-
t i o n of the atoms of t h e impurity i s t r a n s f e r r e d from t h e
norlnaP i n t o the excf ted s t a t e
(electron excitation).
w i l l d e s i g n a t e by u tho energy o r e x c f t a t i o n .
We
We w i l l
consider t h a t t h e a d s o r p t i o n c e n t r e s a r e e x a c t l y t h e s e
e x c i t e d a t m s of t h e impurf t y .
I n corrospondence with
t h e d e s i g n a t i o n s adopted above, we w i l l d e s i g n a t e by C t h e
atoms of t h e impurity found i n t h e normal s t a t e , by A t h e
excited
atoms of t h e impurity, by B t h e a toms of t h e impu-
r i t y connected w i t h t h e adsorbed molecules,
We have two
sfmuBtaneously proceeding r e a c t i ons ( 2 ) and (6).
The con-
d i t i o n s of equflibroium s t a t e s a r e expressed by t h e e q u a t i o n s
(31) and (7s).
We w i 11 analyse another p o s s i b l e case,
Let us
assume t h a t along wfth t h e d e f e c t s A ( f r e e a d s o r p t i o n c e n t r e s )
and d e f e c t s B (occupied a d s o r p t i o n c e n t r e s ) two o t h e r t y p e s
of d e f e c t s C and D a r e p r e s e n t on t h e s u r f a c e of a c r y s t a l ,
which do n o t d i r e c t l y p a r t i c i p a t e i n a d s o r p t i o n .
Vfe w i l l
assume, however, t h a t t h e d e f e c t s C a r e capable of b r e a k i n g
up i n t o d e f e c t s A and D.
of d f s s o c f a t i o n .
L e t u s assume t h a t u i s t h e energy
We w i l l d s o assume t h a t a r e v e r s i b l e pro-
c e s s f s a l s o pogsfble: t h e recombination of d e f e c t s A and D
wfth the formation of a d e f e c t C.
Thus t h e following r e a c t i o n t a k e s p l a c e
C Z A AD,
proceedfng sfmultaneously wf t h r e a c t i o n (2).
(8)
A t a s t a t e of
equilfbrfum we must add t o e q u a t i o n (3') t h e f o l l o w i n g
equation t
N~ . N ~ =
N~
/s,
VJe have here:
so t h a t equatf on
where
p
=
pee -3%
Y = NA + NB + NC,
o
(9)
X = NA + NB + ND9
( 9 ) may be r e w r i t t e n thus2
From equatfons (3') and ( 9 ' )
t h e unknowns N and NB
can be determined,
I n t h e case of the bimolecular r e a c t i o n ( 8 ),analysed
h e r e , a s i n the case of t h e mon~molecularr e a c t i o n ( 6 ) , ad$ o r p t i o n c e n t r e s A ' l o r i g i n a t e " from d e f e c t s C,
During h e a t i n g
t h e number of d e f e c t s A i n c r e a s e s a t t h e expense of t h e d i s appearance of d e f e c t s C,
Thus, t h e r e a c t i o n s ( 6 ) o r ( 8 )
appear t o be sources of thermal d i s o r d e r on t h e s u r f a c e of a
crystal.
We w i l l analyse t h e example of a bimolecular r e a c t i o n
( 8 ) ,.
We w i l l assume, t h a t we d e a l w i t h a h e t e r o p o l a r
c r y s t a l constructed from i o n s M+ and ,'R
i n which t h e s t o i -
chiometrfc r a t i o i s d i s t o r t e d t o some degree.
A s an example,
we s h a l l d e a l with a c r y s t a l which h a s a s t o i c h i o m e t r i c excess
of a
metal.
We a f l l consider, t h a t on t h e s u r f a c e of such
a c r y s t a l a r e d i s t r i b u t e d the "excess" atoms of a metal M,
which b e shown and designated i n f i g u r e (2) by t h e symbol A,
These "excesst' atoms can be t r e a t e d a s d e f e c t s on t h e s u r f a c e
of the c r y s t a l . ,
defects a t T = 0.
Let X be t h e concentration of these
We w i l l consider t h a t t h e a d s o r p t i o n
c e n t r e s a r e d e f e c t s of t h f s [and only t h i s ) type.
An
adsorption c e n t r e , connected w i t h a gaseous molecule, f s
designated I n FOgzxre 2 by the symbol B.
The r e a c t i o n s of
adsorption and d e s o r p t i o n a r e schematically shown i n
Figure 2 by t h e arrow3 1 and 2 r e s p e c t i v e l y .
In a d d i t i o n ,
we w i l l assume t h a t i n the s u r f a c e l a y e r of a c r y s t a l t h e r e
a r e a l s o contained d e f e c t s of another type: empty m e t a l l o f d
positions,
It must be noted t h a t one of the m e t a l l i c i o n s
the l a t t i c e , d i r e c t l y nefghbouring w i t h such an empty
metallofd s i t e , myst be i n a n e u t r a l s t a t e ( i n o t h e r words
f t must take an e x t r a e l e c t r o n ) ,
D e f e c t s of t h i s type
(empty m e t a l l o i d s i t e connected wf t h an atom of t h e m e t a l )
a r e designated f n Figure 2 by t h e symbol C.
The concentra-
t i o n of such d e f e c t s a t T = 0 we w i l l designate by Z o These
d e f e c t s - a c c o r d i n g t o our assumption do n o t p a r t i c i p a t e
d i . ~ e c t l y , f na d s o r p t i o n o The d e f e c t s A and C ensure a s t o i chiometric excess of a metal i n t h e s u r f a c e l a y e r of t h e
crystal a t T = 0 ,
During h e a t i n g , t h e atoms of the metal connected
with t h e empty m e t a l l o f d s i t e s , d i s s o c i a t e from the s u r f a c e
of t h e l a y e r and t o t h e s u r f a c e of t h e c r y s t a l ,
A s a re-
s u l t af such a d f s s o c f a t i o n i n t h e surface l a y e r t h e r e remains
an empty m e t a l l i c sf t e , connected wf t h t h e empty m e t a l l o i d
s i t e (defect
'd)
i n Figure 2), and on t h e s u r f a c e of the
c r y s t a l t h e r e appears an "excesst4 m e t a l l i c atom A ,
In
o t h e r words, during h e a t i n g the d e f e c t C b r e a k s up t o form
a d e f e c t A and a d e f e c t
B o This r e a c t i o n i s schematically
shown i n Figure 2 by t h e arrow 3,
The r e v e r s e p r o c e s s
simultaneously comes i n t o play: t h e recombination of def e c t A and D which b r i n g s about t h e formation of a d e f e c t
C,
T h i s r e a c t i o n of t h e recombination i s shown i n Figure 2
by t h e arrow 4.
The equilfbrfuln c o n c e n t r a t i o n s of t h e defects. A ,
B, C , D, corresponding t o some temperature T a r e connected
by t h e equations ( 3 ) and ( 9 ) or,which i s t h e same t h i n g ,
by equatfons ( 3 ' ) and ( g 9 ) .
Here t h e number X c h a r a c t e r i z e s
t h e i n i t i a l ( h i s t o p i c a l ) d i s o r d e r on t h e s u r f a c e of a c r y s t a l ,
and the number Y = X + Z c h a r a c t e r i z e s t h e degree of d i s t o r t i o n of t h e stofchiometry of t h e s u r f a c e l a y e r of t h e c ~ y s t a l ~
I n a p a r t i c u l a r case d e f e c t s C may be completely absent
(2 = 0 ),
I n t h i s case t h e t o t a l d i s o r d e r i s of a p u r e l y
h i s t o r i c a l o r i ~ i n :t h e number of adsorption c e n t r e s i n t h i s
case w i l l n o t change ~ 5 t h
temperature (Langmuir t h e o r y ) ,
In another p a r t i c u l a r case a t T = 0, t h e d e f e c t s of type A
may be completely absent (X = O ) o
I n t h i s case t h e adsorp-
t i o n c e n t r e s A appear on t h e s u r f a c e during h e a t i n g exclus i v e l y a t the expense of t h e decomposition of d e f e c t s C ,
and the d i s o r d e r i s of a p u r e l y thermal o r i g i n ,
W e w i l l analyse another example of a bimolecular
relationship.
We dl1 assume t h a t on t h e s u r f a c e of an i o n i c
c r y s t a l , atoms of an imgurity a.re d i s t r i b u t e d .
The concen-
t r a t i o n of theso atoms we w i l l d e s i g n a t e by 2.
A s an
example, we s h a l l assume t h a t t h e s e atoms a r e l o c a t e d on
t h e negative i o n s of t h e l a t t i ce a s shown i n Figure 3,
During h e a t i n g t h e atoms of t h e impurity i o n i z e , and t h f s
i o n i z a t i o n i n c r e a s e s with temperature.
In Figure 3 the
n e u t r a l atoms of t h e impurity a r e designated by t h e symbol
C , and t h e ionized by t h e symbol D o An e l e c t r o n which h a s
departed from an atom of t h e impurity becomes the c o l l e c t i v e p r o p e r t y of t h e s u r f a c e l a y e r of t h e l a t t i c e ( f r e e
e l e c t ~ o n ) , Free e l e c t r o n s , t h e c o n c e n t r a t i o n of which
i n c r e a s e s with temperature, f o m an e l e c t r o n g a s , which
determines t h e e l e c t r o - c o n d u ~ t vi i ty of the c r y s t a l ,
Thus
atoms of an impurity appear t o be a r e s e r v o i r which supp l i e s t h e conducting e l e c t r o n s .
The appearance of a f r e e
e l e c t r o n i n t h e s u r f a c e l a y e r of t h e l a t t i c e means a
n e u t r a l i z a t i o n of one of t h e i o n s M+ of the s u r f a c e l a y e r .
The d i splacement of an e l e c t r o n along t h e s u r f a c e means
t h e displacement of t h e n e u t r a l s t a t e M from one ion M+
t o t h e neighbourfng ion M+.
We w i l l consider t h a t adsorp-
t i o n c e n t r e s a r e e x a c t l y t h e se n e u t r a l i z e d m e t a l l i c atoms
of the l a t t i c e .
I n Figure 3 they a r e designated by t h e
symbol A ; an adsorgtf on cenbre, which h a s taken onto i t s e l f
a gas molecule and I s connected wf t h i t , i s designated by
t h e symbol B.
I n other words, t h e adsorption c e n t r e s i n
our model a r e tho f r e e e l e c t r o n s ,
(The conceptf on regard-
i n g e l e c tro-conductivity a s a d s o r p t i on c e n t r e s h a s been
p o s t u l a t e d b y 0, M, Todes.)
The i o n f z a t f o n p r o c e s s of a n e u t r a l atom of t h e
impurity C mag be t r e a t e d a s a simultaneous appealaance of a
f r e e e l e c t r o n A and a tihole'!* D , connected with an atom of
t h e impurity.
Along with t h i s p ~ o c e s st h e r e t a k e s p l a c e
t h e r e v e r s i b l e n e u t r a l i z a t i o n process of t h e impuri t y f on,
c o n s i s t i n g of a recombination of an e l e c t r o n w i t h a "holefY,
The condf t i o n s of equi l i b r i u m a r e expressed by t h e equations
(3') and ( 9 0 f o r which, however, i t must be assumed t h a t
X = 0 and Y = Z , ( t h e absence of t h e h i s t o r i c a l d i s o r d e r ) ,
I n f a c t , t h e number of "holesft f s e q u a l t o the t o t a l number
of e l e c t r o n s , t r a n s f e r r e d t o t h e c o l l e c t i v e s t a t e ,
These
c o l l e c t i v e e l e c t p o n s a r e composed of e l e c t r o n s which remain
f r e e , and of e l e c t r o n s which e n t e r i n t o a bond with gaseous
molecules, t h u s f a l l i n g out of a c t i o n ,
We have
%
= N A + NB 9
from which i t f o l l o w s t h a t X = 0 a n d Y = Z,
4t
The term "holeff i s used here i n t h e same sense a s i n t h e
t h e o ~ yof semf-conductors,
"Holeft means the absence of an
electron,
4,
The Isotherm and the D i f f e r e n t i a l Heat of Adsorpt i o n by Taking I n t o Account the Thermal Disorder
It can be e a s i l y shown t h a t , although t h e thermal
d i s o r d e r due t o t h e monanolecular r e a c t i o n ( 6 ) e x e r t s an
e f f e c t on t h e k i n e t i c s of adsorption, i t has no e f f e c t on
the a d s o r p t i o n equilibrfum,
Thus, from the viewpoint of
equilibrium, r e a c t i o n ( 6 ) i s of no i n t e r e s t ,
I n t h i s para-
graph we w i l l l i m i t o u r s e l v e s t o t h e case when t h e thermal
d i s o r d e r can be expressed by t h e bimolecular r e a c t i o n (81,
I n t h i s case t h e equilibrium c o n c e n t r a t i o n s ND and N = NA+NB
a r e determined from t h e equations ( S f ) and ( g q) , We w i l l
rewri t e these equations thus :
We dl1 analyse equation ( l o b ) ,
Solving t h e equa-
t i o n with r e s p e c t t o N we o b t a i n N a s a f u n c t i o n of T
and^^:
We must note t h a t i f i n this expression we assume
Z = 0, which means t h a t t h e thermal d i s o r d e r i s disregarded,
then we o b t a i n from ( 1 1 ) ; N = X , a s i t should be,
The r e l a t i onship between N and NB ( a t a g i ven T =
c o n s t a n t ) i s schematically shown i n Figure 4a.
A s the surface
becomes covered with adsorbed molecules ( a s NB i n c r e a s e s ) ,
t h e t o t a l number of adsorption c e n t r e s N = NA + NB inareasea)
*
I n t h e o r i g i n a l t h e s i g n i s +,
from a c e r t a i n minimum value ( a t VEj = 0 ) up t o a maximum
value N = Y ( a t NB = Y ) ~ Thus, i n t h e p r c e s s of adsorption
t h e r e a r i s e a d d i t i o n a l adsorption c e n t r e s ,
A s t h e tempera-
t u r e i n c r e a s e s the p o i n t A i n Figure 4a d i s p l a c e s upwards
while t h e p o i n t B remains f i x e d
.
The family of curves N = N( NB )
,
corresponding t o v a r i o u s values of T , a r e shown i n Figure 4b.
The curves a r e numbered i n t h e order of i n c r e a s i n g T o
Curve 1
corresponds t o t h e l i m i t i n g case T = 0; curve 4 corresponds t o
the o t h e r l i m i t i n g case, T =
- 0
We w i l l d e r i v e an equation f o r t h e isothermo For
t h i s purpose we w i l l r e t u r n t o t h e equation of equilibrium ( l o & ) ,
S u b s t i t u t i n g i n t h i s equation t h e
expressions ( 4 ) and ( 1 1 ) and
solving t h i s equation w i t h r e s p e c t t o NB, we o b t a i n t h e following expression f o r the isotherm:
We have here designated:
It must be noted t h a t a t p
T ) expression ( 1 2 ) g i v e s NB
+w
+Y
(independent of t h e value f o r
(saturation).
If we n e g l e c t the thermal d i s o r d e r i n comparison with
t h e h i s t o r i c a l , assuming Z = 0 ( o r , which i s t h e same t h i n g ,
X =
*I n
Y) then the expression (12) i s transformed i n t o the g e n e r a l
the o r i g i n a l t h e s i g n i s
+ 0
as i t should be,
If, however, we n e g l e c t t h e historical d i s o r d e r a s
compared with t h e thermal d i s o r d e r , assuming X = 0, and i f we
a l s o consider t h a t Z = Y i s s u f f i c i e n t l y l a r g e , so t h a t
Z = Y )) p , then formula (12 may be approximated a s follows:
a ) In t h e region of "smallr' p r e s s u r e s , a t Y p &
b ) 1n t h e r e g i o n of "averagem p r e s s u r e s , a t 1
C)
I n t h e region of t b l a r g e vp r e s s u r e s , a t 1
%a,
14; r
< y p<P
t
<< jjY < d p :
we o b t a i n the g e n e r a l Henry law a t t h e begin-
ning of the isotherm, which i s dfsplaced by t h e r e l a t i o n s h i p
on t h e
%-fi
mfddle of t h e isotherm, and which a g a i n i n i t s
t u r n i s transformed i n t o s a t u r a t i o n .
We must note t h a t t h e r e l a t i o n s h i p NB
- fi
i n the
g e n e r a l t h e o r f e s of adsorption may be obtained a s a r e s u l t of
a heterogeneous s u r f a c e (exponentfal d i s t r i b u t i o n f u n c t i o n ) o r
a s a r e s u l t of i n t e r a c t i ons between adsorbed molecules o r ,
f i n a l l y , by assuming t h a t the e x c e s s molecules d % s s o e i a t e
d u r i n g adsorption.
Generally t h e isotherm N B d
6
indf c a t e s
t h a t one of t h e s e t h r e e c o n d i t i o n s i s p r e s e n t ,
I n our case,
however, t h e s u r f a c e i s known t o be homogeneous ( a d s o r p t i o n
c e n t r e s of one type o n l y ) , t h e i n t e r a c t i o n s of adsorbed
~ o l a c u l e sa r e ignored and the d i s s o c i a t i o n of excess molec u l e s i s absent.
Here t h e r e l a t i o n s h i p
pletely different origln: i t
13
~ ~ ah afs ai com-
determined by t h e i n c r e a s e
i n t h e number of adsorption c e n t r e s o r i g i n a t e d from t h e surf a c e being covered a s a r e s u l t of t h e thermal d i s o r d e r ,
We dl1 now t u r n t o t h e c a l c u l a t i o n s of the d i f f e r e n t i a l h e a t of a d s o r p t i o n Q,
For t h i s we w i l l determine t h e
energy W of a c r y s t a l which h a s on i t s s u r f a c e N adsorption
c e n t r e s , of which NB c e n t r e s a r e occupied by adsorbed molecule s and NA c e n t r e s a r e f r e e .
O f the t o t a l number N c e n t r e s ,
X c e n t r e s have a h i s t o r P c a l o r i g i n , and t h e remaining N
c e n t r e s have a thermal o r i g i n ,
-X
I n o r d e r t o produce a thermal
adsorption c e n t r e , i t i s necessary t o use up a q u a n t i t y of
energy u; on t h e o t h e r hand, a combination of each gaseous
molecule with an a d s o r p t i o n c e n t r e r e l e a s e s an energy q.
Thus, we w i l l have:
w = UCN
- X) -
q
~
~
.
Here f o r zero energy we t a k e t h e energy of a system which h a s
no adsorbed molecules and which h a s no thermal d i s o r d e r on
the surface,
We must n o t e t h a t t h e s e l e c t i o n of the zero
p o i n t i s n o t e s s e n t i a l f o r t h e c a l c u l a t i o n s of Q,
For tk.e d i f f e r e n t i a l h e a t of a d s o r p t i o n we obtain:
S u b s t i t u t i n g (Il), we o b t a i n
If we n e g l e c t t h e thermal d i s o r d e r , assuming
Z =
0,
then e x p r e s s i on (14) g i ves Q = q = c o n s t a n t , a s should be
expected,
The v a r i a t f o n of Q w i t h NB i s due t o t h e thermal
d i s o r d e r and i s t h e more pronounced t h e l a r g e r i s Z o
The curve Q = Q(NB) i s s c h e m a t i c a l l y shown i n Figure
5,
A s the s u r f a c e i s covered, Q d e c r e a s e s from some maximum
value Q =
&ma
NB = Y ,
he p o s i t i o n of t h e p o i n t s Qa,
a t NB = 0 t o some minimum v a l u e $ =
and
i s dependent on t h e magnitude of t h e parameter 2,
$Ifnat
i n Figure 5
By decseas-
i n g Z , t h a t i s , a s t h e magnitude of t h e h i s t o r i c a l d i s o r d e r i n
Sn
t h e g e n e r a l d i s o r d e r i n c r e a s e s , t h e p o i n t s Qmax and Qmin
Figure 5 a r e d i s p l a c e d u ~ w a r d s ,moreover t h e p o i n t
p l a c e s f a s t e r than the p o i n t
out.
have
Gax, SO
Gin d i s -
that the curve s t r a i g h t e n s
I n t h e limf t a t Z = 0 ( a p u r e l y h i s t o r i c a l d i s o r d e r ) we
h, = Gin =
qo
We obtained a d e c r e a s e i n t h e d i f f e r e n t i a l h e a t with
coverage, although t h e s u r f a c e i s e n e r g e t i c a l l y homogeneous
and t h e f n t e r a c t i o n s between adsorbed molecules a r e absent.
.n
I n the o r i g i n a l 'the s i g n i s +,
The relatYonsh2.p between Q and NB f o r t h e case which we have
analysed, i s s t i p u l a t e d by t h e f a c t t h a t t h e t o t a l number o f
adsorptfon c e n t r e s N does n o t remain c o n s t a n t b u t i n c r e a s e s
a s NB f n c r e a s e s ,
The a d s o r p t i o n c e n t r e s on t h e surf ace of a
c r y s t a l a r e t r e a t e d h e r e a s a c h a r a c t e r i s t i c of t h e coverage
of g a s on t h e orsfg i n a l s u r f a c e , t h e c o n c e n t r a t i on of t h e
c e n t r e s i n c r e a s i n g t o g e t h e r w i t h an i n c r e a s e i n NB,
and t h e
energy change of t h i s i n c r e a s e should be taken i n t o account
when s a l c u l a t f n g t h e d i f f e r e n t i a l h e a t of a d s o r p t i o n ,
I n p a r t i c u l a r , i f we assume t h e model analysed a t
t h e end of s e c t i o n 3 i n which t h e f r e e e l e c t r o n s of a c r y s t a l
a r e t h e a d s o r p t i o n c e n t r e s then t h e "gas of a d s o r p t i o n centres1'
may be considered a s an e l e c t r o n g a s i n a c r y s t a l ,
In thf s
case our a n a l y s i s a g r e e s w i t h t h a t by Breger and Zhukhovf t s k i d
( 2 ) , who i n t h e c a l c u l a t i o n s of t h e d i f f e r e n t i a l h e a t of t h e
a d s o r p t i o n took i n t o account t h e change of energy of an e l e c t r o n gas d u r i n g a d s o r p t i o n ,
The d f f f e r e n c e i s t h a t B r e g e ~
and Zhukhovitskii have analysed an e l e c t r o n g a s i n a metal,
b u t i n our model we a r e d e a l i n g with an e l e c t r o n g a s I n a
semi-conductor,
I n t h e model of Breger and Z h ~ k h o v i t s k f f ~ t h e
same i n our model, each adsorbed molecule i s connected with
the s u r f a c e of a c r y s t a l by means of a l a t t i c e e l e c t r o n ,
hence t h i s e l e c t r o n f a l l s out from t h e t o t a l f a m i l y of f r e e
electrons,
Thus, i n t h e model of Breger and Zhukhovitskii t h e
f r e e e l e c t r o n s of a c r y s t a l a r e t r e a t e d a s a d s o r p t i o n c e n t r e s ,
t h e same a s f n our model.
5,
The E f f e c t of t h e Thermal Disorder on t h e
K i n e t i cs of Adsorptf on
For t h e r a t e of a d s o r p t i o n we have
where
GC " / q r=
NGNA
)> N N B I which means t h a t we
oC.
I n t h f s equation we w f l l assume t h a t
a s compared t o a d s o r p t i o n ,
w i l l neglect desorption
This c o n d i t i o n i s f u l f i l l e d a t
s u f f i c i e n t l y small NB o r s u f f i c i e n t l y l a r g e q.
Lettfng
ko = x v N G P we may r e w r i t e equation 15 a s f o l l o w s :
where NA 1s a f u n c t i o n of
NBo
We w f l l assume t h a t the s t a t e of e q u i l i b r i u m i s
r e t a i n e d f o r the a d s o r p t i o n c e n t r e s ,
If the thermal df sorder
i s governed by a bimolecular r e a c t f on ( 8 ) , then t h e s t a t e of
e q u i l i b r i u m may be expressed by e q u a t i o n s ( 9 )
o r ( 9 , ) from
which we o b t a i n ( s e e ( 1 1 ) ) :
If t h e thermal df sorder i s described by a monomolecula~ Peac-
t i o n ( 6 ) then t h e s t a t e of equflibrium may be expressed by
equations ( 7 ) o r (7') from which we obtain:
By s u b s t i t u t i n g ( 1 7 ) o r (18) i n t o (161, w e can
f i n d t h e r e q u i r e d r e l a t i o n s h i p between
%
and t f o r t h e
cases of bimolecular and monomolecular r e a c t i o n s r e s p e c t i vely,
Vie dl1 analyse both of these cases,
F f r s t , we wf 11 analyse t h e c a s e when t h e r e l a t f on-
shfp be tween NA and NB 1s g i ven by e x p r e s s i on ( 1 7 ) .
If i n (17) we assume t h a t
Z = 0 ( o r X = Y-), which
means t h a t we n e g l e c t t h e thermal d i s o r d e r , then (17) g i v e s
NA = Y
- NBQ
kinetics:
I n t h f a case we o b t a i n the g e n e r a l Langmufr
'
a s should be expected,
Conversely, if we neglect the h i s t o r i c a l d i s o r d e r
assuming i n ( 1 7 ) X = 0 ( o r 2 = Y ) , then i n t h i s case ( 1 7 ) w i l l
become:
Expression (179) may be considerably s i m p l i f i e d f o r the two
l i m i t i n g cases: f o r the case of s u f f i c i e n t l y "high" and f o r
t h e case of s u f f i c i e n t l y "low" temperatures.
region of "high" temperatures, assuming
from ( 1 7 ' ) : NA = Y
- NB
P
In f a c t , i n t h e
>
NB,
we have
which again l e a d s t o t h e formqla ( 1 9 ) .
The d e v i a t i o n from t h e Langmuir k i n e t i c s appear i n the region
of s u f f i c i e n t l y
consider t h a t
temperatures.
N
/3 4 N ~ ? ,
I n t h i s region, i f we
expression (17 ) assumes t h e
following sfmple form:
Substf t u t f n g (20) i n t o ( 1 6 ) and i n t e g r a t i n g equat f o n ( 1 6 ) w e obtain:
NB
2
Y
(1
- N)
=
- i$a t ,
where the f o l l o w f n g deaf g n a t i on i s assumed:
A t NB
/g
Y (21) gives
We w f l l now r e t u r n t o the case i n which t h e r e l a t i o n s h i p between NA and
%
i s given by e x p r e s s i on ( 1 8 ) ,
S u b s t i t u t i n g ( 1 8 ) i n t o ( 1 6 ) we o b t a i n :
from h e r e , assumfng t h e d e s i g n a t f o n
we o b t a i n
I n t h e r e g f o n of s u f f f c f e n t l y h i g h temperatures,
when
p
9 1, we
have according t o ( 2 3 ) : k = ko and formula
( 2 4 ) cofncldes with ( 1 9 ) .
I n t h e r e g i o n of sufficiently low temperatures,
kfnetieu,
when
In t h f s case we o b t a i n t h e Langmuir
c g
1, w e have
u
I n t h i s case we o b t a i n k f n e t f c s t y p i c a l f o r t h e s o - c a l l e d
" a c t i v a t e d " adsorptf on,
For t h e theory of a c t f v a t e d a d s o r p t i o n t h e o r i g i n
of formulae ( 2 4 ) and ( 2 5 ) i s g e n e r a l l y connected with t h e
presence on t h e c r y s t a l surface of a p o t e n t f a l b a r r i e r wfth
a height u , hence only those gaseous molecules which have a
k i n e t i c energy f n a d i r e c t 1on normal t o t h e s u r f a c e l a r g e r
than t h e energy u (3,4) w i l l r e a c h t h e s u r f a c e ,
However, in
our ease t h e r e f s no such p o t e n t i a l b a r r i e r near the s u r f a c e
of the c r y s t a l and formulae ( 2 4 ) and ( 2 5 ) have a d i f f e r e n t
o r i g i n , I n t h e concept of the a c t i v a t f o n p o t e n t i a l b a r r i e r
--
t h e number of gas molecules f a l l f n g on t h e s u r f a c e f n ~ e a s e s
kT
wfth t h e temperature i n proportion t o a m u l t f p l f e r e
,
while t h e number of adsorptfon c e n t r e s a c c e p t i n g t h e s e molec u l e s remains constant; conversely, f n t h f s concept, t h e
number of a c c e p t i n g c e n t r e s i n c r e a s e s
--
P$
t h temperature i n
U
proportion t o t h e m u l t i p l i e r e kT, while t h e number of f a l l f n g
molecules remains p r a c t i c a l l y constant,
We must note t h a t t h e theory of a c t i v a t e d adsorp-
t i o n may be constructed without t h e concept of t h e p o t e n t i a l
a
r
e
In f a c t , we w i l l assume t h a t t h e r e a r e no p o t e n t i a l
b a r r i e r s , and t h a t t h e s u r f a c e of a c r y s t a l c o n t a i n s a cons t a n t ( n o t changing w i t h temperature) number of adsorp t i o n
c e n t r e s , however, we w i l l consider t h a t not a l l t h e centres;
a r e capable t o adsorb, b u t only those c e n t r e s yrrhfch a r e found
f n an e x c i t e d ( a c t i v e ) s t a t e ,
Let u be t h e energy of excf-
t a t l ~ n( a c t i v a t i o n ) of t h e adsorptf on c e n t r e .
The adsorption
I n t h f s case befng a r e a c t f o n of combfnation between a gas
molecule and an adsorp t f on c e n t r e , wf 11 r e q u i r e an a c t i v a t i on
energy u.
Thus we have i n the c a s e of a heterogeneous r e -
a c t i o n a s f t u a t i o n w e l l known i n t h e chemistry of homogeneous
reactfons,
This p i c t u r e completely a g r e e s wfth o u r s , I n which
adsorption proceeds without a c t i v a t i o n , b u t the number of
adsorption c e n t r e s does not remain constant, and i n c r e a s e s
w f t h temperature ( t h e r m a l d i s o r d e r ! ) ,
I n f a c t , f n our case
when t h e appearance and df sappe arance of adsorp t i on c e n t r e s
i s described by a monomolecular r e l a t i o n s h f p C
A,
the
i n c r e a s e i n t h e number of c e n t r e s with temperature may be
t r e a t e d a s a t r a n s i t f o n of a d s o r p t i o n c e n t r e s from the
"passfvev s t a t e C I n t o t h e " a c t i v e " s t a t e
Ao
Such a t r a n s i t i o n
f s connected ~ 4 t han expenditure of energy u and may be t r e a t e d
a s an tOexcitationT'of t h e a d s o r p t i o n c e n t r e ,
In t h i s treat-
ment the t o t a l number of c e n t r e s remains c o n s t a n t (does n o t
change
tvi t
centres A,
h tempera'tur e ) however, t h e number of '4active'b
capable f o r adsorption, Increases w i t h h e a t i n g
a t t h e expense of a d e c ~ e a s ei n t h e number of 'spassive'k
c e n t r e s C which d o n o t d i r e c t l y p a r t i c i p a t e i n a d s o r p t i o n ,
The a:lt;lor wishes t o e x p r e s s h i s g r a t i t u d e t o
C , Z, RoginskSi f o r a number of v a l u a b l e remarks d u r i n g t h e
d i s c u s s i o n of t h i s work,
S e q t t o t h e e d i t o r J u l y 1 4 , 1948,
Academy of S c i e n c e s U,S,S.R,
I n s t i t u t e of P h y s i c a l Chemistry
D i v i s i o n of C a t a l y s i s and Topochemistry
Moscow,
REFERENCES
1,
F ,F0 V o l k e n s t e i n , ' L ~ l e c t r o - ~ o n dtui cv i t y of Semi -Condue t o r sLb
Chapter 1, S t a t e P u b l S s h e r s , 1947,
2,
A,Kh, Breger and A , A ,
Zhukhovitskii,
Zhuro F i z , Khfm,, 2
-9 1
423, ( 1 9 4 7 ) ,
3,
Lennard-Jones,
3,
F,F, V o l k e n s t e i n ,
-
Trans, F a r , Soc, 28, 333 (1932).
-
Zhur, F i z , K h i m , 21, 163 (1947) ,
o o o ~ s o o o s s e ~ e ~ o o o o o o o o o o o
o sooo 000000€3e0e~00~~0000000
o o 0 s s o 0 ~ o e ~ e ~ e o o o o o o o ~ o o
ccoaooococeegoeloooooooooo
1. Vacant s i t e .
4 . A forel,rn atom i n the i n t e r s t i c e s .
2 . Neutral atom in the i n t e r s t l c e s . 5 . A foreigna t o m in a normal s i t e .
3 . Ions with anomalous c h a r g e s .
0 ion R ion R ion M +
atom M
gaseous
molecule
Fig. 2
ion M +
atom M
a t o m of a n
impurlty
ion of a n impurity
gaseous
molecule
Fig. 3
a)
-
Fig. 4
0
I
.
x+p
Fig. 5
a)