On the properties of adsorption caused by the "Thermal Disorder" on the surface of a crystal

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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

T R A N S L A T I O N 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 )

b y F. Fo Volkenshtein Translated by Eo Rabkin OTTAWA May, 1950

T i t l e

r

NATIONAL RESEAFtCB COUNCIL O F CANADA

O t t a w , Canada

TRANSLATION TT-126

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

Zhurnal F i z i c h e s k o i Khfmii Vol. 2 3 , Noo 8 , 1949

Acadsiny of Scfancas, U,S,S,R,

ON SOW PROPERTIES OF ADSORPTION CAUSED BY THE

"THERMAL DISORDlB'"ON THE SURFACE OF A CRYSTAL by

F. F. Volkenshtein

T r a n s l a t e d by E s t h e r Rabkin

SUMMARY

The a d s o r p t i o n 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 w i t h tern- pera.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 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 k i n e t i c s of adsorp- t i o n a t d e f i n e d 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 s o - c a l l e d " 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 a b s e n t . An a d s o r p t i o n i s o - therm of the type Q

-

p h a s been obtained. 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 was found t o be a f u n c t i o n of t h e 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 Ibdefectth we mean any d i s -

d6f ec t s p r e s e n t i n a r e a l l a t t f c e , 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 n u c l e u s . Cracks, p a r t i c l e s of f o r e f gn s u b s t a n c e 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 t y p e , The s u p f a c e s 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 c a s e 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 c r o s c o p f c d e f e c t s : ( 1 ) a h o l e praoduced by t h e d i s a p p e a r a n c e 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,

D e f e c t s of t h e s e types a r e s c h e m a t i c a l l y shown i n Figure 1. We must n o t e 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 t h e 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 d e f e c t c

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 m i c r o - d e f e c t s , i g n o r i n g t h e macroscopic d i s t o r t i o n s , We w i l l t h u s 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 b e i n g f i x e d i n s i d e a c r y s t a l : t h e y p o s s e s s d e f i n i t e m o b i l i t y , The m o b i l i t y of t h e 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 a l o n g 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 t h e 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 t h e y approach 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 each o t h e r . 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

c o n s t r u c t e d 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 t h e y 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 o r t o t h e m e t a l l i c i o n s between t h e

n a t u r a l p o s i t i o n s , 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 r e p u l s e d 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 , p o s s e s s i n g 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 toge-

t 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 h o l e c o n s t i t u t e f o r m a t i o n s of a d i f f e r e n t type, p o s s e s s i n g 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, t h e same a s g e n e r a l r e a c t i o n s . These " r e a c t i o n s " , 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 . A c r y s t a l l i n e l a t t i c e can b o t h produce and absorb d e f e c t s . 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 g i v e n kind under g i v e n c o n d i t i o n s p o s s e s s e s a d e f i n i t e average l i f e d u r a t i o n . It can d i s a p p e a r and reappear.

We w i l l assume t h a t d e f e c t s do n o t d i s a p p e a r beyond t h e l i m i t s of t h e l a t t i c e and t h a t they do n o t e n t e r i t from

t h e o u t s i d e , I n t h f s case a t e q u i l i b r i u m 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 p e r 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 d by t h e e q u i l i b r i u m c o n d i t i o n s . The e q u i l i b ~ i u m c o n c e n t r a t i o n changes with temperature. I n a d d i t i o n , i t i s dependent on 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 b e 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 ) . T h i s i s connected w i t h 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 e q u i l i b r i u m c o n c e n t r a t i o n s a I n o t h e r words, we w i l l ignore t h e 'frozent"(metastable) s t a t e s of t h e 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 w i l l 1 be r e f e r r e d t o a a

d i s o r d e r i n t h e c r y s t a l , 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 s m a l l * n o t h e r words f t i s assumed t h a t the t o t a l c o n c e n t r a t i o n of a l l d e f e c t s i s s m a l l 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 , I n f a c t when t h e number of d e f e c t s i s comparable w i t h t h e number of nuc e i we can n o t 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 tem- p e r a t u r e we w i l l name a s t h e 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 - o r d e r . 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

,

super- imposed 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 s t o i c h i o m e t r y p o s s e s s i n g , f o r example, an e x c e s s of metalloid may c o n t a i n 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 v e r y beginning, The number of t h e s e d e f e c t s a t z e r o 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 addi-

t 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 s t o i c h i o m e t r y 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 super- i m p o s i t i o n 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 tem- p 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 c o n s i d e r a b l y 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 n e g l e c t e d a s compared w i t h 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 t h e s u r f a c e 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 m i c r o - d e f e c t s of v a r i o u s types. Thus, t h e s u r f a c e 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 a d s o r p t i o n ,

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 a d s o r p t f o n 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 h e t e r o g e n e i t i e s of t h e

s u r f a c e a r e such a d s o r p t i o n 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 t h e 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 s u r f ace a s a d s o r p t i o n c e n t r e s . 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 d o e s 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

s u r f a c e from t h e i d e a l s t a t e determines t h e a d s o r p t i o n c a p a c i t y .

We must n o t e t h a t t h e p r e s e n c e o f d e f e c t s on t h e s u r f a c e d o e s 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 e n e r g e t i c , E n e r g e t i c h e t e r o g e n e i t y p r e s u p p o s e s 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 b e 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 b e 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

a n 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 f o l l o w - i n g p r o p e r t i e s a r e a s s i g n e d t o t h e a d s o r p t i o n c e n t r e s :

(1) It i s assumed t h a t t h e 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 c o m p l e t e l y d e t e r m i n e d by t h e h i s t o r y of t h e s u r f a c e .

( 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 adsorp- t 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 f r e e .

It must be noted t h a t t h e conception r e g a r d i n g t h e a d s o r p t i o n 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 concep- t i o n . 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 assump- t 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 a d s o r p t i o n theory, L a t e r i t wf 11 be shown how t h e b a s i c a d s o r p t i o n 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 a d s o r p t i o n c e n t r e s , L e t 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 c o n s i d e r

t h a t t h e c o n c e n t r a t f o n of d e f e c t s A i n c r e a s e s w i t h t h e tem- p 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 =

-,

s o t h a t

X C

NA

&

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 c o n c e n t r a t i o n 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

t h e d i s o r d e r a t a temperature T ; moreover, t h e number X ex- p 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

( N A

-

X )

-

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 Z = Y

-

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 s u r f a c e and may b e used a s i t s c h a r a c t e r i s - t i c 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 ) c a s e s : (1) X = 0 o r

Z

= Y, T h i s 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 a b s e n t ) .

(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

a b s e n t ) .

We must note t h a t the g e n e r a l t h e o r i e s of adsorp- t 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 a d s o r p t i o n c e n t r e s , remain t r u e o n l y 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 a n a l y s i n g 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 c o n s i d e r t h a t each a d s o r p t i o n c e n t r e A may a c c e p t one and only one g a s molecule, s o t h a t t h e

a d s o r p t i o n 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 a d s o r p t i o n 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 o The f r e e c e n t r e s

A , 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 s u r f a c e b u t they a r e d e f e c t s of d i f f e r e n t t y p e s , 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 , E v i d e n t l y NB i s t h e number of g a s molecules adsorbed p e r u n i t s u r f a c e . The r e a c t i o n of a d s o r p t i o n and d e s o r p t i o n 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. T h i s 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 t h e r e v e r s e d i r e c t i o n , We w i l l d e s i g n a t e by q t h e h e a t of t h e r e a c t i o n , For an e q u i l i b r i u m s t a t e we have: N A O N G

-

_-

O C 9 where

ce

= ocoe NB

- Q

_{( 3 )}

Here NG i s t h e c o n c e n t r a t i o n of g a s 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 a d s o r p t i o n c e n t r e s ( f r e e + occupied) we may r e w r i t e t h e c o n d i t i o n ( 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) :

X < N < Y . ( 1 1 ) 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 e q u a t i o n (39) t ~ a n s f o r m s i 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 pro- ceeding 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 . I g n o r i n g

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 w i t h 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 t h e y 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 s u r f a c e 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 c o n c e n t r a t i o n 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 A ( 6 )

We w i l l c o n s i d e r , 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 e x p e n d i t u r e 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 e q u i l i b r i u m we must add t o equa- t i o n ( 5 ) t h e following equation:

..

u N~ _where

_/3

_{= p 0 e}

-

kT

-

=

p.

N~ 0 We now have: Y = NA

+

NB

+

NC, X = 0 , s o t h a t equation ( 7 ) may be r e w r i t t e n thus:

N

-

NB

=

P o

Y - N

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 t h e monmolecular

r e a c t i o n ( 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 i m p u r i t y a r e d i s t r i b u t e d . Let Y be the s u r f a c e c o n c e n t r a t i o n of such atoms. 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 t e d s t a t e ( e l e c t r o n e x c i t a t i o n ) . We

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 w i l l

c o n s i d e r 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 e x c i t e d atoms of t h e i m p u r i t y , 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 a n o t h e r p o s s i b l e c a s e , Let us

assume t h a t a l o n g 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. L e t u s assume t h a t u i s t h e energy

of d f s s o c f a t i o n . 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 f o r m a t i o n of a d e f e c t C.

Thus t h e f o l l o w i n g r e a c t i o n t a k e s p l a c e

C Z A A D , ( 8 )

proceedfng sfmultaneously wf t h r e a c t i o n (2). A t a s t a t e of e q u i l f b r f u m 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

e q u a t i o n t

N~ . N ~ =

/s,

_where

_p

₌

_pee

_-3%

o

N~ ( 9 )

VJe have here: Y = NA

+

NB + NC, X = NA + NB

+

ND9

so t h a t equatf on ( 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~molecular r 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 s o u r c e s of thermal 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 . We w i l l a n a l y s e 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 c o n s t r u c t e d 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 w i t h 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 c o n s i d e r , 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 t h e "excess" atoms of a m e t a l M,

which b e shown and designated i n f i g u r e (2) by t h e symbol A,

of the c r y s t a l . , Let X 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 a t T = 0 . 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 a d s o r p t i o n 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 a d s o r p t i o n and d e s o r p t i o n a r e s c h e m a t i c a l l y 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 . I n 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 a n o t h e r type: empty m e t a l l o f d p o s i t i o n s , 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 m e t a l l o f d s i t e , myst b e 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 d e s i g n a t e d 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 d e s i g n a t e 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 n a 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 t h e 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 r e - s u l t af such a d f s s o c f a t i o n i n t h e s u r f a c e 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 ( d e f e c t '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 , I n o t h e r words, d u r i n g h e a t i n g t h e 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

T h i s r e a c t i o n i s s c h e m a t i c a l l y 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 p l a y : t h e recombination of de- f 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 d e f e c t s . A ,

B, C , D, corresponding t o some temperature T a r e connected by t h e e q u a t i o n s ( 3 ) and ( 9 ) or,which i s t h e same t h i n g , by e q u a t f o n s ( 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 t h e number Y = X + Z c h a r a c t e r i z e s t h e d e g r e e 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 c a s e d e f e c t s C may be completely a b s e n t ( 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 a d s o r p t i o n c e n t r e s i n t h i s

c a s e 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 d u r i n g h e a t i n g exclu- s i v e l y a t t h e expense of t h e decomposition of d e f e c t s C ,

W e w i l l analyse another example of a bimolecular r e l a t i o n s h i p .

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 n e g a t i v e 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 i m p u r i t y 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 d e s i g n a t e d by t h e symbol

C , and t h e i o n i z e d by t h e symbol D o An e l e c t r o n which h a s d e p a r t e d 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 i v i ty of the c r y s t a l , Thus atoms of an i m p u r i t y appear t o be a r e s e r v o i r which sup- p 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 t h e 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 i o n 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 t h e l a t t i c e . I n Figure 3 they a r e d e s i g n a t e d 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 d e s i g n a t e d by t h e symbol B. I n o t h e r words, t h e a d s o r p t i o n 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 s t 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 p r o c e s s 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 e q u a t i o n s

(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 t h e 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 + N _{B 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 h e r e i n t h e same sense a s i n t h e t h e o ~ y of semf-conductors, "Holeft means the absence of an

4, The Isotherm and t h e D i f f e r e n t i a l Heat of Adsorp- t i o n by Taking I n t o Account t h e 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 a d s o r p t i o n , i t h a s no e f f e c t on the a d s o r p t i o n equilibrfum, Thus, from t h e viewpoint of e q u i l i b r i u m , 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 p a r a - 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 c a s e 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 e q u i l i b r i u m 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 e q u a t i o n s ( S f ) and ( g q ) , We w i l l rewri t e t h e s e equations t h u s :

We dl1 analyse e q u a t i o n ( 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 n o t e t h a t i f i n this e x p r e s s i o n we assume

Z = 0, which means t h a t t h e thermal d i s o r d e r i s d i s r e g a r d e d , t h e n we o b t a i n from ( 1 1 ) ; N = X , a s i t should b e ,

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 s c h e m a t i c a l l y shown i n Figure 4a. A s t h e s u r f a c e 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 a d s o r p t i o n c e n t r e s N = NA

+

NB i n a r e a s e a )

*

from a c e r t a i n minimum v a l u e ( 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 a d s o r p t i o n t h e r e a r i s e a d d i t i o n a l a d s o r p t i o n 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 t h e p o i n t A i n F i g u r e 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 c a s e T = 0; curve 4 corresponds t o the o t h e r l i m i t i n g c a s e , 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 e q u i l i b r i u m ( 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 s o l v i n g 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 follow- i n g e x p r e s s i o n f o r t h e isotherm:

We have h e r e designated:

It must be noted t h a t a t p

+

w (independent of t h e value f o r

T ) e x p r e s s i o n ( 1 2 ) g i v e s NB

+

Y ( s a t u r a t i o n ) .

If we n e g l e c t t h e 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 = Y) then t h e e x p r e s s i o n (12) i s transformed i n t o t h e g e n e r a l

*

as i t should b e ,

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 ( 1 2 may be approximated a s follows: a ) I n t h e r e g i o n of "smallr' p r e s s u r e s , a t Y p & 14;

r

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

<

y p

<

t

P

Y

C ) I n t h e r e g i o n of t b l a r g e v p r e s s u r e s , a t 1

<<

jj

<

d p :

%a, we o b t a i n t h e g e n e r a l Henry law a t t h e begin- n i n g of the isotherm, which i s d f s p l a c e d by t h e r e l a t i o n s h i p

%-fi

on t h e 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 n o t e t h a t t h e r e l a t i o n s h i p NB

-

fi

i n t h e g e n e r a l t h e o r f e s of a d s o r p t i o n may be obtained a s a r e s u l t of a heterogeneous s u r f a c e ( e x p o n e n t f a l 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 a d s o r p t i o n . 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 s a r e ignored and the d i s s o c i a t i o n of excess mole- c u l e s i s absent. Here t h e r e l a t i o n s h i p

~ ~ a f i

h a s a com- p l e t e l y d i f f e r e n t o r i g l n : i t 1 3 determined by t h e i n c r e a s e

i n t h e number of a d s o r p t i o n c e n t r e s o r i g i n a t e d from t h e sur- f 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 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 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 a d s o r p t i o n c e n t r e s , of which NB c e n t r e s a r e occupied by adsorbed mole- cule 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

-

X

c e n t r e s have a thermal o r i g i n , I n o r d e r t o produce a thermal a d s o r p t i o n c e n t r e , i t i s n e c e s s a r y 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 t h e s u r f a c e , 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 o b t a i n :

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

a t NB = 0 t o some minimum v a l u e $ =

$Ifn

at

NB = Y , he p o s i t i o n of t h e p o i n t s Qa, and i n Figure 5

i s dependent on t h e magnitude of t h e parameter 2, 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 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 Sn 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

Gin

d i s - p l a c e s f a s t e r t h a n t h e p o i n t

Gax,

SO that t h e curve s t r a i g h t e n s

out. 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 have

h,

=

Gin

= q o

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 a b s e n t . .n

The relatYonsh2.p between Q and NB f o r t h e c a s e 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 a d s o r p t f o n c e n t r e s N d o e s 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 orsf g 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 t h e n 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 , I n t h f s

c a s e 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 m e t a l , b u t i n our model we a r e d e a l i n g w i t h 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 w i t h

t h e 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

e l e c t r o n s , Thus, i n t h e model of Breger and Z h u k h o v i t s k i i 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 = oC. I n t h f s e q u a t i o n we w f l l assume t h a t

NGNA

)>

N N B I which means t h a t we w i l l n e g l e c t d e s o r p t i o n

a s compared t o a d s o r p t i o n , 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. L e t t f n g ko = x v N G P we may r e w r i t e equation 1 5 a s f o l l o w s :

where NA 1s a f u n c t i o n of N B o

We w f l l assume t h a t t h e 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 s o r d e r i s governed by a bimolecular r e a c t f on ( 8 ) , t h e n 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 s o r d e r i s d e s c r i b e d by a monomolecula~ Peac- t i o n ( 6 ) then t h e s t a t e of e q u f l i b r i u m may be expressed by e q u a t i o n s ( 7 ) o r (7') from which we o b t a i n :

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

c a s e s 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 t h e s e c a s e s ,

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

1 s g i ven by e x p r e s s i on ( 1 7 ) .

If i n ( 1 7 ) 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 ( 1 7 ) g i v e s

NA = Y

-

NBQ I n t h f a case we o b t a i n t h e g e n e r a l Langmufr k i n e t i c s : '

a s should be expected,

Conversely, if we n e g l e c t t h e 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. I n f a c t , i n t h e r e g i o n of "high" temperatures, assuming

P

>

NB, we have

from ( 1 7 ' ) : NA = Y

-

NB which a g a i n 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 r e g i o n

of s u f f i c i e n t l y temperatures. I n t h i s r e g i o n , i f we N

f o l l o w i n g 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 equa- t f o n ( 1 6 ) w e o b t a i n :

NB N

-

2

( 1

-

) =

-

i$ a t ,

Y

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 ( 2 1 ) g i v e s

We w f l l now r e t u r n t o the c a s e i n which t h e r e l a - t i o n s h i p between NA and

%

i s g i v e n 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

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 ) c o f n c l d e s with ( 1 9 ) . In t h f s case we o b t a i n t h e Langmuir k f n e t i e u , I n t h e r e g i o n of sufficiently low temperatures,

when c g 1, w e have

u

I n t h i s c a s e 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 w i t h t h e presence on t h e c r y s t a l s u r f a c e of a p o t e n t f a l b a r r i e r wfth a h e i g h t u , hence only t h o s e gaseous molecules which have a k i n e t i c energy f n a d i r e c t 1 on normal t o t h e s u r f a c e l a r g e r t h a n 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 n e a r 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 p r o p o r t i o n 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 mole- c u l e s remains c o n s t a n t ; 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

--

p r o p o r t i o n 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 c o n s t a n t ,

We must n o t e t h a t t h e t h e o r y of a c t i v a t e d adsorp- t i o n may be c o n s t r u c t e d without t h e concept of t h e p o t e n t i a l a r e I n 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 con-

s 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 c o n s i d e r t h a t n o t 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 e x c f - t a t l ~ n ( a c t i v a t i o n ) of t h e a d s o r p t f on c e n t r e . The a d s o r p t i o n I n t h f s c a s e 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 r e a c t f o n s , T h i s p i c t u r e completely a g r e e s wfth o u r s , I n which a d s o r p t i o n proceeds without a c t i v a t i o n , b u t t h e number of

a d s o r p t i o n c e n t r e s does n o t remain c o n s t a n t , 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 d e s c r i b e d by a monomolecular r e l a t i o n s h f p C A , t h e i n c r e a s e i n t h e number of c e n t r e s w i t h 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 t h e

" p a s s f v e v s t a t e C I n t o t h e " a c t i v e " s t a t e A o Such a t r a n s i t i o n

f s connected ~ 4 t h an expenditure of energy u and may be t r e a t e d a s an t O e x c i t a t i o n T ' of t h e a d s o r p t i o n c e n t r e , I n t h i s t r e a t - 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 ( d o e s n o t

change tvi t h tempera'tur e ) however, t h e number of '4active'b

c e n t r e s A , c a p a b l e f o r a d s o r p t i o n , I n c r e a s e s 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 e i 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 d u c t i v 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 , Z h u k h o v i t s k i i , Zhuro F i z , Khfm,, 2 1 - 9

423, ( 1 9 4 7 ) ,

3, Lennard-Jones, Trans, F a r , Soc,

-

28, 333 (1932).

ion R - ion M + a t o m M gaseous molecule o s o o o o o

0

ion R - o o o ~ s o o o s s e ~ e ~ o o o o o o o o o o o 0 0 0 0 0 0 € 3 e 0 e ~ 0 0 ~ ~ 0 0 0 0 0 0 0 0 s s o 0 ~ o e ~ e ~ e o o o o o o o ~ o o c c o a o o o c o c e e g o e l o o o o o o o o o o ion M + a t o m M

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 foreign a t o m in a normal s i t e .

3 . Ions with anomalous c h a r g e s .

a t o m of a n impurlty ion of a n impurity gaseous molecule F i g . 2 F i g . 3

a) - a)

Fig. 4

I .

0 x+p