On the Interactions Between Adsorbed Molecules in Connection with the Theory of Adsorption on Heterogeneous Surfaces

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NATIONAL RSSEARCH COUNCIL OF CANADA DIVISION OF INFORM.ATION SERVICES

T e c h n i c a l T r a n s l a t i o n No, TT-120

ON TKE INTERACTIONS BETVIEEN ADSORBED MOLECULES I N CONNECTION lVITH THE THEORY OF ADSORPTION ON HETEROGENEOUS SURFACES

( K Voprocy o V z a l m o d e i s t v i i Adsorbirovannykh blolekul v Svyaei s T e o r i e i A d s o r p t s i i Na Neodnorodnykh ~ o v e r k h n a s t y a k h ) by F, F, Volkenshtein 25 March, 1950 T h i s r e p o r t may n o t be p u b l i s h e d i n whale o r i n p a r t w i t h o u t t h e w r i t t e n c o n s e n t of t h e N a t i o n a l Research Council.

NATIONAL RESEARCH LABORATORIES Ottawa, Canada

TECHNICAL TRANSLATION NO 4

TT-

120

Divi sion of Information Servioes

On the Interaction between Adsorbed Molecules in Connection with the

Theory of Adsorption on Heterogeneous Surfaces

By: F. F o Volkenshtein

References: Zhurnal Fizicheskoi Khimii

CJ,

of Phys. Chem.)

Vol, 21, No. 2, 1947, pp, 163-178 Translated by: Esther Rabkin

Zhurnal F i z i c h e s k o i l i h i m i i , Vol. 21, No, -2, 1947

ON THE INTERACTIONS BET!!FIEEN ADSORBED MOLECULES I N CONNECTION

WITH THE THEORY OF. ADSORPTION ON !ETEROGENEOUS SURFACES

by

F. F. Volkenshteln T r a n s l a t e d by Esther Rabkin

1, Formulation of t h e problem

The elementary theory of molecular a d s o r p t i o n developed by Langmuir

"

' i s based on two assumptionsr-

1. The surf ace of t h e adsorbent i s considered uniform from t h e energy p o i n t of view; t h a t i s , t h e h e a t of adsorption Q i s assumed uniform throughout t h e s u r f a c e of t h e adsorbent: Q = constant.

2, The adsorbed molecules do n o t i n t e r a c t between them- s e l v e s .

The f a c t t h a t t h e experimental a d s o r p t i o n i s o - therms i n most c a s e s do n o t agree w i t h t h e t h e o r e t i c a l Langmuir isotherms n e c e s s i t a t e s a r e - a n a l y s i s of t h e .

o r i g i n a l assumptions i n t h e Langmuir theory. F i r s t , t h e development of t h e theory i s p o s s i b l e i n two d i r e c t i o n s : -

1. I t i s p o s s i b l e t o d i s r e g a r d t h e f i r s t simplifying assumption of Langmuir and r e t a i n t h e second.

r e t a i n t h e f i r s t .

For t h e f i r s t t h e r e a r e a t p r e s e n t e number of'

(2)

t h e o r e t i c a l works In t h e s e t h e i n t e r a c t i o n s between t h e adsorbed molecules a r e ignored, but t h e h e t e r o g e n e i t y of t h e s u r f a c e i s taken i n t o account. This h e t e r o g e n e i t y

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

p

(Q) :-

where dS i s the s e c t i o n of the s u r f a c e f o r which the h e a t of adsorption i s found i n the i n t e r v a l from Q t o Q 3. dQ,

Depending on t h e form assumed f o r t h e f u n c t i o n

P ( Q )

we o b t a i n a d e f i n i t e form f o r t h e isotherm 0 ( p )

.

The s o l u t i o n of t h e r e v e r s e problem i s of

i n t e r e s t , t h a t i s , t o f i n d such a s u f f i c i e n t l y simple mathe- m a t i c a l method by means of which we could f i n d from t h e

experimentally obtained isotherm 0 = 8 ( p ) the corres- ponding d i s t r i b u t i o n f u n c t i o n

P

=

P

(G.\,

t h a t f s, we could determine t h e heterogeneity of t h : s u r f a c e . This problem has

(3)

r e c e n t l y been solved by C,Z, Roginskl

,

who found simple .approximate formulae which permit t o f i n d t h e f u n c t i o n

p

=

P

( Q ) from a given e m p i r i c a l f u n c t i o n 8 = 0 ( p ) ,

and t h e r e v e r s e ,

We w i l l adopt the Roginskf method only f o r s u f f i c - i e n t l y wide d i s t r i b u t i o n s , f o r which Q

max

-

%in

9

9''

-

3

-

i n s i d e which P f 0, while o u t s i d e of t h i s

interval'.^=

0 4 )

For the second assumption t h e r e a l s o e x i s ; a number

( 5 )

of c a l c u l a t i o n s

.

I n t h e s e c a l c u l a t i o n s t h e s u r f a c e l a assumed t o be homogeneous, but t h e molecules adsorbed on t h e surface a r e considered t o be i n t e r a c t i n g . Depending on t h e law of i n t e r a c t i o n , t h a t i s , on t h e f orm of the h n c t i o n

CfJ = ( r ) , c o n s i s t i n g of t h e i n t e r a c t i o n energy between

two molecules a s a f u n c t i o n of t h e d i s t a n c e between them (assuming t h a t the i n t e r a c t i o n f orcev a r e c e n t r a l f o r c e s ) , we o b t a i n one o r other form f o r t h e isotherm 8

=

8 ( p ) .

I t i s of i n t e r e s t t o f i n d a s u f f i c i e n t l y convenient method, which would permit t o determine t h e form of t h e f u n c t i o n

=

9

( r ) f ~ o m t h e experimentally obtained isotherm

8 = Q ( p )

,

and conversely, by a given law of i n t e r a c t i o n

9

=

u;

( r ) , t h e corresponding isotherm 8 = Q ( p ) . To

f ind these formulae comprises t h e problem of t h i s paper. Since one and t h e same isotherm 0 = 8 ( p ) may be obtained a s a r e s u l t of a d e f i n i t e s e l e c t i o n of t h e d i s t r i b u t i o n f u n c t i o n

p

= f (Q: ( b y ignoring t h e i n t e r - a c t i o n of t h e molecules), and t h e same a s a r e s u l t of a d e f i n i t e s e l e c t i o n of t h e i n t e r a c t i o n law =

qD

( r ) (by ignoring t h e h e t e r o g e n e i t y of t h e s u r f a c e ) , then from t h e experimental isotherms alone i t i s n o t p o a s i b l e t o decide d e f i n i t e l y which one of t h e two assumptions i n t h e Langmufr t h e o r y should be disregarded and which one should be r e t a i n e d ,

I n connection w i t h t h i s , i t i s of i n t e r e s t t o f i n d such formulae which would connect t h e f u n c t i o n CfJ ( r ) w i t h t h e corresponding f u n c t i o n P (Q), and t h i s would p e r m i t t o determine t h e e q u i v a l e n t law of i n t e r a c t i o n ( f ~ o m t h o iso-

therm p o i n t of view) f o r t h e d i s t r i b u t i o n a c c o r d i n g t o t h e h e a t of a d s o r p t i o n , o r t h e converse, To f i n d t h e s e formulae i s a l s o a problem of t h i s paper.

The problems p r e s e n t e d h e r e belong t o t h e phenom- e n o l o g i c a l t h e o r y of a d s o r p t i o n . I n t h e p h y s i c a l t h e o r y of a d s o r p t i o n , which h a s n o t y e t been c o n s t r u c t e d , t h e f u n c t i o n s

P

( Q ) and

cf.

( r ) must, of c o u r s e , be f i x e d by t h e p h y s i c a l model i t s e l f , and cannot b e s e l e c t e d on t h e b a s i s of t h e form

of t h e experimental isotherms,

2. O r i g i n a l assumptions

1% w i l l assume a p l a n e s u r f a c e f o r t h e a d s o r b e n t . We w i l l assume t h a t a l l t h e adsorbed molecules, which we w i l l assume t o be i d e n t i c a l , a r e uniformly d i s t r i b u t e d on t h i s s u r f a c e . It must be noted, t h a t i n t h e p r e s e n c e of r e p u l s i o n f o r c e s between the adsorbed molecules such a uniform d i s t r i - b u t i o n c o r r e s p o n d s t o a minimum of energy, t h a t i s , i t appears

t o b e a more f a v o u r a b l e d i s t r i b u t i o n from t h e e n e r g y p o i n t of view. I n t h e c a s e of t h e a t t r a c t i o n f o r c e s a uniform d i s t r i b u t i o n corresponds t o an e x c i t e d s t a t e : t h e molecules i n t h i s c a s e p o s s e s s a tendency t o c o n t r a c t ,

We w i l l denote by

.&

t h e d i s t a n c e between t h e s u r f a c e of t h e adsorbent and t h e a d s o r p t i o n l a y e r , and by r t h e s h o r t e s t d i s t a n c e between two neighbouring adsorbed molecules ( F i g s . 1 & 2 ) , Tle must n o t e t h a t tho magnitude of

,k

i s i n t h e o r d e r of t h e diameter of t h e molecules. Let u s assume t h a t :

8

-

i s t h e degree of f i l l i n g - i n of t h e s u r f a c e . 2

M

-

i s t h e number of adsorbed molecules ( p e r 1 cm ) ,

No

-

i s t h e l a r g e s t n b e r of molecules which can be

adsorbed by 1 capof the adsorbent s u r f a c e .

q

-

i s t h e q u a n t i t y of t h e adsorb d substance

₈

( p e r 1 cm ),

q*

-

i s t h e l a r g e s t q u a n t i t y whi h c o u l d be adsorbed _{on 1 cm}

8

.

r

-

i s t h e s h o r t e s t d i s t a n c e between two adsorbed molocules,

r

-

i s t h e minimum d i s t a n c e t o which two adsorbed 0

molecules may approach.

Between t h e s e v a l u e s t h e f o l l o w i n g obvious r e l a t i o n - s h i p s e x i s t : -

-

The parameter r o may be named t h e " e f f e c t i v e diametertt of t h e moleculeo

We w i l l f u r t h e r assume t h a t t h e i n t e r a c t i o n f o r c e a between t h e adsorbed molecule and a molecule of t h e adsorbent, a s w e l l a s between two adsorbed molecules, a r e c e n t r a l f o r c e s . Thus, we exclude from t h e a n a l y s i s t h e c a s e s i n which i n t e r - a c t i o n i s dependent on t h e mutual o r i e n t a t i o n of t h e molecules,

a s w e l l a s the c a s e i n which t h e I n t e r a c t i o n f o r c a s a r e of a chemical n a t u r e , t h a t Is, p o s s e s s t h e p r o p e r t y t o be s a t u r a t e d .

Each adsorbed molecule l o c a t e d on t h e s u r f a c e of t h e a d s o r b e n t i n t e r a c t s , s t r i c t l y speaklng, w i t h a l l t h e mole- c u l e s i n t h e l a t t i c e of t h e a d s o r b e n t , a n d w i t h a l l t h e o t h e r adsorbed molecules. I t s energy

@

i s t h u s composed of two p a r t s : -

where

(4)

i s t h e energy of i n t e r a c t i o n w i t h the l a t t i c e

0

of t h e a d s o r b e n t ,

@

t r )

i s t h e energy of i n t e r a c t i o n w i t h t h e o t h e r adsorbed m o l e c i ~ l e s .

The term r$

(f

), s t r i c t l y speaking, i s dependent

0

on t h e p o s i t i o n of our adsorbed molecule i n r e l a t i o n t o t h e c o r n e r s of t h e l a t t i c e , t h a t i s , dependent on i t s co- o r d i n a t e s y, z ( F i g , 2), b e i n g a p e r i o d i c f u n c t i o n of t h e s e c o o r d i n a t e s . However, we w i l l n e g l e c t t h i s r e l a t i o n s h i p , t h a t i s , we w i l l n e g l e c t t h e " h e t e r o g e n e i t y " of t h e s u r f a c e , c o n d i t i o n e d by t h e c r y s t a l l i n e s t r u c t u r e of t h e adsorbent i t s e l f , c o n s i d e r i n g

@

(1)

t o be uniform f o r a l l t h e p o i n t s 0 on t h e s u r f a c e .

moreover,

where g and g2 a r e whole numbers. Here

g

( r ) i s t h s

1

i n t e r a c t i o n e n e r g y be tween two adsorbed molecules found a t a d i s t a n c e r from each o t h e r , By f i n d i n g a s o l u t i o n f o r t h i s f u n c t i o n we by t h e same means would f i n d a ~ o l u t i o n f o r t h e "law of i n t e r a c t i ~ n ' ~ ,

It i s e v i d e n t t h a t t h e a b s o l u t e magnitude of t h e energy

-

c o n s i s t s of t h e h e a t of a d s o r p t i o n Q, r e l a t i n g t o one molecule :

I n f a c t , Q a x p r e s s e s t h e e n e r g y whiclz must be used

up i n o r d e r t o remove one adsorbed molecule beyond t h e

l i m i t s of t h e a d s o r p t i o n l a y e r i n t o i n . f i n i t y , i f we o n l y

n e g l e c t (which we w i l l do from now o n ) t h e deformation of t h e l a t t i c e ~ i n d t h e r e d i s t r i b u t i c m of molecules a l o n g t h e s u r f ace of t h e a d s o r b e n t , which r e s u l t s from t h e desorp1;ion of each i n d i v i d u a l molecule.

Thus we seq t h a t t h e s u r f a c e , u n i f o r m l y covered w i t h adsorbed molecules i n t ; e r u c t i n g betwoen each o t h e r , can be r e g a r d e d a s a homogeneous s u r f a c e , b u t f o r which t h e

magnitmude of t h e h e a t of a d s o r p t i o n h a s changed. To t h e h e a t of a d s o r p t i o n Q0 =

-

@,,

c o r r e s p o n d i n g t o a f r e e s u r f a c e , we add t h e term

O Q

=

-

c o r r e s p o n d i n g t o t h e degree of

f i l l i n g - i n , which may be e i t h e r p o s i t i v e o r n e g a t i v e ,

depending whether t h e i n t e r a c t i o n f o r c e s between t h e adsorbed molecules a r e f o r c e s of a t t r a c t i o n o r r e p u l s i o n .

3. On the p o s s i b l e o r i g i n of t h e a c t i v a t i o n b a r r i e r

We w i l l assume t h a t one of t h e adsorbed molecules h a s escaped t h e a d s o r p t i o n l a y e r and h a s reached a d i s t a n c e

x from t h e s u r f a c e of t h e a d s o r b e n t , a s shown by a d o t t e d l i n e on F i g , 1. We w i l l a l s o assume t h a t t h e o t h e r adsorbed molecules remain i n t h e i r p o s i t i o n s , I n t h i s c a s e t h e energy

of our molecule may be e x p r e s s e d i n t h e f o l l o w i n g form:-

where @ A X ) i s t h e i n t e r a c t i o n energy of our molecule w i t h t h e l a t t i c e of t h e a d s o r b e n t ,

A+

( r , x) i s t h e i n t e r a c t i o n energy of our molecule w i t h t h e remaining molecules of t h e a d s o r p t i o n l a y e r . It i s e v i d e n t t h a t : A t x

= A

t h e e x p r e s s i o n ( 6 ) i s transformed i n t o (4) and t h e e x p r e s s i o n ( 5 ) i n t o ( 2 ) . Two t y p e s of r e l a t i o n s h i p s p o s s i b l e between

@

0

the absence of a d s o r p t i o n , Curve 2 e x p r e s s e s t h e c a s e sf a d s o r p t i o n , Adsorption i s d e t e c t e d by t h e p r e s e n c e of a minimum on t h e curve. The v a l u e x =

k

determines t h e

p o s i t i o n of t h i s minimum. The d e p t h of t h e minimum expres- s e s t h e h e a t of a d s o r p t i o n Qo f o r a homogeneous f r e e s u r f a m ,

Two t y p e s of r e l a t i o n s h i p p o s s i b l e between and x a r e shown i n F i g o 4, If t h e f o r c e s a c t i n g between t h e adsorbed molecules a r e r e p u l s i o n f a r c e s , t h e n we o b t a i n concave curves of Type 1; i n t h e c a s e s of a t t r a c t i v e f o r c e s we have convsx c u r v e s of Type 2, F a m i l i e s of c u r v e s cor- responding t o v a r i o u s v a l u e s of t h e parameter r a r e shown

i n t h e f i g u r e s ,

The maximum on t h e c u r v e s of Type 1 and t h e minimum on t h e c u r v e s of Type 2 a p p e a r s a t x

=A.

The h e i g h t of t h e maxi- mum o r t h e depth of t h e minimum i n c r e a s e a s r d e c r e a s e s ,

t h a t i s , a s t h e f i l l i n g - i n Q i n c r e a s e s .

By a d d i n g t h e c u r v e s of Type 1 (from F i g , 3 )

w i t h t h e c u r v e s of Type 2 (from F i g , 4 ) ( t h e s e c u r v e s a r e once more shown i n F i g o 5 ) we can o b t a i n a curve which c r o s s e s t h e a b s c i s s a ( c u r v e 3 of F i g o 5 ) : e s t a b l i s h i n g R

p o t e n t i a l b a r r i e r b e f a r e t h e p o t e n t i a l d i p ,

Thus an a c t i v a t i o n b a r r i e r may appear a s a r e s u l t of t h e i n t e r a c t i o n s be tween two adsorbed molecules, i f t h e

i n t e r a c t i o n f o r c e s a r e f o r c e s ol"repulsion, Such i s t h e p o s s i b l e n a t u r e of t h e a c t i v a t i o n b a r r i e r , a r i s i n g on t;he s u r f a c e of t h e a d s o r b e n t , The h e i g h t of' t h e b a r r i e r , c o n s i s t i n g of t h e a c t i v a t i o n e n e r g y E, i n c r e a s e s a s t h e p a r a m e t e r r d e c r e a s e s , t h a t i s , a s t h e s u r f a c e b e g i n s t o f i l l i n , t h e depth of t h e p o t e n t i a l p i t e x p r e s s i n g t h e h e a t of a d s o r p t i o n Q

d e c r e a s e s . Thus w i t h a change i n t h e q u a n t i t y of adsorbed molecules b o t h E and Q change i n o p p o s i t e d i r e c t i o n s o

The c u r v e s c@ ( x ) f o r v a r i o u s B a r e shown i n F i g o 60 A t some v a l u e 8 = 8 t h e minimum of t h e c u r v e 0

@

( x ) i s t a n g e n t t o t h e a b s c i s s a . Thus B i s t h e r o o t 0 of t h e e q u a t i o n :

Q

+

a~

( Q ) = 0, 0 A t 8

4

8 t h e h e a t of a d s o r p t i o n i s p o s i t i v e ( Q > 0 ) g 0

a t 8

>

8 i t becomes n e g a t i v e

(Q

<o)

, The a d s o r p t i o n ,

0

however, c o n t i n u e s a l s o a t n e g a t i v e v a l u e s o f

Q,

and i t d i s c o n t i n u e s o n l y when 43 r e a c h e s t h e v a l u e 8 = 1, which c o r r e s p o n d s t o t h e d i s a p p e a r a n c e of' the minimum on t h e curve

($

(x). The d i s a p p e a r a n c e of t h e minimum a p p e a r s t o be a condition from which t h e paraametors N and r o a r e

0

determined, The v a l u e Q = 1 i s by no means proof t h a t t h e s u r f a c e i s completely f i l l e d i n , b u t i t may be a n a l y s e d a s such, i f we a s s i g n t o t h e molecules i n s t e a d of t h e i r t r u e d i a m e t e r a c e r t a i n e f f e c t i v e d i a m e t e r r

.

4. The d e t e r m i n a t i o n of isotherms corresponding t o a given

_--

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

We w i l l assume t h a t t h e a d s o r p t i o n l a y e r i s i n e q u i l i b r i u m with t h e gaseous phase. W e w i l l d e s i g n a t e by n l t h e number of t h e adsorbed molecules, and by n2 t h e number of desorbed molecules p e r second from t h e s u r f ace of 1 cm 2 of t h e adsorbent,

It i s obvious t h e n t h a t 2

where = 1/4 f l r 2 which i s t h e " e f f e c t i v e a r e a U of 1

molecule, and m i s t h e mass of 1 molecule,

We w i l l consfder t h e c o e f f f c i e n t of r e f l e c t i o n

cc

a s being e q u a l t o u n i t y , T h i s means t h a t each gaseous molecule f a l l i n g on t h e s u r f a c e of t h e adsorbent i s t r a p p e d by i t .

We w i l l a s s m e t h a t t h e c o e f f i c i e n t 9

-

t'frequency of v i b r a t i o p s " of t h e adsorbed molecule

-

f s independent of

t h e degree of f i l l i n g - i n Q, I n o t h e r words, we w i l l n e g l e c t t h e e f f e c t of i n t e r a c t i o n on t h e magnitude of t h i s c o e f f i c i e n t .

Expressions ( 7 ) d i f f e r from the corresponding e x p r e s s i o n s i n t h e Langmuir t eory, f i r s t , by t h e presenae

-

A Q ~ Q )

presence of a f a c t o r e

--

.,

Both t h e s e f a c t o r s appear

a s a r e s u l t of t a k i n g t h e i n t e r a c t i o n i n t o account. They c h a r a c t e r i s e t h e t r a n s i t i o n from the p o t e n t i a l curve 2 t o t h e p o t e n t i a l curve 3 i n Fig. 5, It i s e x a c t l y these two t r a n s i t i o n s which determine the i n t e r a c t i o n s i n t h e frame of our model.

From the condition of equilfbrium

-

nl

-

"2 we o b t a i n where Qo a = kTy

-

m

From formula ( 8 ) we can determine t h e form of t h e isotherm by a given law of i n t e r a c t i o n

:+

( r )

.

I n f a c t , according t o ( 1 ) :

A Q

( Q ) =

-

Ir 0 0 0 ( 9 )

S u b s t i t u t i n g ( 9 ) i n t o (€I), we can r e - w r i t e formula ( 8 ) thus:

Solving t h i s e q u a t i o n with r e s p e c t t o 8 we o b t a i n t h e r e q u i r e d e q u a t i o n f o r t h e isotherm$

It must be noted, t h a t i r r e s p e c t i v e of t h e law of i n t e r a c t i o n , we always have:

"9

=

4Q

Q = 1, I . e .

r

=

r

0' moreover,

49,

can be e i t h e r p o s i t i v e o r n e g a t i v e :

>

0 f o r t h e c a s e of r e p u l s i o n f o r c e s , L 0 f o r t h e c a s e of a t t r a c t i o n f o r c e s . I n a d d i t i o n , t h e f u n c t i o n

641

( r )

may be c o n s i d e r e d a s a monotonous f u n c t i o n of

r ,

Thus, when 8 changes from

8 = 0 t o 8 = 1, t h e second term of t h e left-hand s e c t i o n of equation ( 1 0 ) monotonously changes from 0 t o

Ago,

while t h e f i r s t term monotonously changes from

+

00 t o

-

QC>

Thus, i n t h e r e g i o n of ''very weak", a s w e l l a s i n t h e

r e g i o n of "very strong'' f i l l i n g - i n ( t h a t i s , a t 8 = 0. and

O = 1 ) we may n e g l e c t t h e second' term i n comparison w i t h

t h e f i r s t .

1 - 8 b

T h i s gives:

_In

_{' g} = . I n

-

P

Me o b t a i n e d t h e Langmuir isotherm, In t h e r e g i o n of naverage" f i l l i n g - i n t

1

s o ,

i, e. 8 -;s

conversely, we nay n e g l e c t t h e f i r s t term i n oomp&ko$son with t h e second,

This gives:

It i s of i n t e r e s t t h a t t h e isotherm ( 1 0 ) i s transformed i n t o t h e Langmuir isotherm, not o n l y a t small f i l l i n g - i n , which i s n a t u r a l , b u t a l s o a t s u f f i c i e n t l y l a r g e f i l l i n g - i n , i n the r e g i o n near t o s a t u r a t i o n where t h e law of i n t e r a c t i o n again appears t o be obscure, The law of i n t e r a c t i o n i s

expressed only i n t h e middle s e c t i o n of t h e isotherm, The r e g i o n of "average1' f i l l i n g - i n , f o r which formula (11) holds,

i s determined by t h e conditionr

As an example, we w i l l determine t h e isotherm eor- responding t o the power l a w of i n t e r a c t i o n :

where

4: ),O f o r the case of t h e f o r c e s of r e p u l s i o n , and

L O f o r the case of t h e f o r c e s of a t t r a c t i o n , According t o (3):

S u b s t i t u t i n g ( 1 4 ) i n t o (11) we obtafn:

We must n o t e , t h a t i n s t e a d of t h e Isotherm ( 1 5 ) Temkin ( 2 ) obtained t h e Williams isotherm f o r t h e power law of i n t e r a c t i o n ( 1 3 ) ;

It can be shown ( s e e Appendix) t h a t f o r t h e c a s e of t h e power law of i n t e r a c t i o n t h e Williams isotherm can

be deduced a s a p a r t i c u l a r c a s e , from t h e g e n e r a l formula

( l o ) ,

A s a n o t h e r example, we w i l l determine t h e isotherms corresponding t o t h e e x p o n e n t i a l law of i n t e r a c t i o n :

Cp

Cr) = ot e

- P r

P

-

(16) where a g a i n & d. 0 f o r t h e c a s e of t h e f o r c e s of repulsion, OC

>

0 f o r the c a s e of the f o r c e s of a t t r a c t % : a n , If w e consider t h a t o n l y t h e d i r e c t l y neigh-

bouring molecules a r e i n t e r a c t i n g molecules, t h a t i s , i f we r e t a i n i n t h e sum (3) o n l y t h e terms f o r which

-

16

-

and n e g l e c t a l l t h e o t h e r s , we w i l l have: S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o formula ( l l ) , we o b t a i n : P r o _P 4 0 0 -

.T

= a & 6 r where

5, The d e t e r m i n a t i o n of t h e law of i n t e r a c t i o n from experimental 'isotherms

We have used formula (8) i n t h e p r e v i o u s s e c t i o n f o r t h e d e t e r m i n a t i o n of t h e i s o t h e r m s by a g i v e n law of i n t e r a c t i o n . It may a l s o be used f o r t h e s o l u t i o n of t h e r e v e r s e problem; f o r t h e d e t e r m i n a t i o n of t h e l a w of i n t e r - a c t i o n corresponding t o a given form of isotherm,

Let u s assume an e m p i r i c a l e q u a t i o n f o r t h e isotherm: Q = Q ( p ) S o l v i n g t h i s e q u a t i o n w i t h r e s p e c t t o ( p ) we w i l l have: P = p C Q ) * * o a ( U ) 4t 2 It should b e cl = ( p r o )

.

In t h e o r i g i n a l p u b l i c a t i o n , t h e form appearing above was given,

S u b s t i t u t i n g (18) i n t o ( 8 ) we f l n d s

I n t h e r e g i o n of " a v d ~ a g e f i l l i n g - i n " , t h a t i s , t h o s e v a l u e s of 9

,

f o r which

.,

we o b t a i n

T h i s i s t h e r e q u i r e d formula, by means of which we can accomplish t h e t r a n s i t i o n from t h e f u n c t i o n 8 ( p ) t o t h e f u n c t i o n ( P ) ~ A s a n example, we will a n a l y s e t h e F r e u n d l i c h isotherm: Thu s : S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o (211, having p r e v i o u s l y r e p l a c e d i n i t B by (

2

l2

D We o b t a i n : r where

Such a law of i n t e r a c t i o n w i t h t h e assumption t h a t t h e s u r f a c e of t h e adsorbent i s homogeneous l e a d s t o t h e Freundlich isotherm; and t h a t t h e d e v i a t i o n of t h e Freundlich isotherm from t h e Langmuir isotherm i s

e n t i r e l y conditioned by t h e i n t e r a c t i o n of t h e molecules, We o b t a i n f o r c e s of r e p u l s i o n which d e c r e a s e with d i s t a n c e

according t o a l o g a r i t h m i c law, The r e l a t i o n s h i p between

A

and

r

f o r v a r i o u s n [ a c c o r d i n g t o formula (22)) i s s c h e m a t i c a l l y shown i n F i g , 7, PJe must once more n o t e , t h a t formula ( 2 2 ) i s o n l y t r u e when c o n d i t i o n ( 2 0 ) i s

f u l f i l l e d , t h a t i s , a t n o t very small and n o t v e r y l a r g e r ,

A s another example we wf 11 analyse t h e l o g a r i t h m i c i,sotherm From h e r e

AQ

p = B e , Assuming Q

=(2)2

and s u b k t i t n t i n g i n t o ( 2 1 ) we f i n d : where

We o b t a i n a square law of i n t e r a c t f o n , corresponding t o a r e p u l s i o n a t p

>

B ( s i n c e i n t h i s case A

>

O), and an at-

( 2 4 ) and ( 2 3 ) agree With t h e formulae of t h e p r e v i o u s s e c t i o n ,

6 , The d e t e r m i n a t i o n of t h e law of i n t e r a c t i o n b y a a i v e n d i s t r i b u t i o n f u n c t i o n , and t h e r e v e r s e

Let u s assume t h a t t h e s u r f a c e or' t h e a d s o r b e n t

f a he terogeneoua from t h e e n e r g y p o i n t of view. Vie w i l l c h a r a c t e r i s e t h i s h e t e r o g e n e i t y by a d i s t r i b u t i o n l'unction

p

CQ)

.

If t h e molecules a r e consl.dered t o be i n t e r a c t i n g and t h e d i s t r i b u t i o n surf i c i e n t l y wide ( b y comparison w i t h

kT), t h e n t h e i s o t h e r m @ ( p ) may be e a s i l y c a l c u l a t e d by

t h e given d i s t r i b u t i o n f u n c t i o n p ( Q ) w i t h t h e a i d of t h e Roginskf formula (3)

where

The r e v e r s e problem, c o n s i s t i n g i n f i n d i n g t h e d i s t r i b u t i o n f u n c t i o n PC&) by a given i s o t h e r m ( p ) , may be solved w i t h a i d of t h e formula:

I t must be n o t e d t h a t one and t h e saEe i s o t h e r m can b e o b t a i n e d , on one hand, a s a r e s u l t of c o n s f d e r i n g t h e h e t e r o g e n e i t y and i g n o r i n g t h e i n t e r a c t i o n , and on t h e o t h e r hand, a 3 a r e s u l t of c o n s i d e r i n g t h e I n t e r a c t i o n and i g n o r i . n g t h e h e t e r o g e n e i t y , Thus, f o r each d i s t r i b u t i o n f u n c t i o n

p

CQ)

t h e r e e x i s t s some, e q u i v a l e n t .from t h e p o i n t of view of t h e i s o t h e r m , law of i n t e r a c t i o n

(P

(r)

,

and t h e r e v e r s e . TJe w i l l e s t a b l i s h a c ~ n n e c t i o n between t h e f i ~ n c t i o n P ( & )

and tf ( r )

.

We w i l l f i n d f o r m u l a e which would parmi t t o

accomplish t h e t r a n s i t i o n from a given d i s t r i b u t i o n P ( Q )

t o t h e corresponding law of i n t e r a c t i o n

(P(r),

and t h e r e v e r s e t r a n s i t i o n

-

from a g i v e n law of i n t e r a c t i o n @(r)

t o t h e c o r r e s p o n d i n g d i s t r i b u t i o n f u n c t i o n P ( Q ) .

We w i l l a t t e m p t f i r s t a s o l u t i o n of t h e d i r e c t problem: t h e d e t e r m i n a t i o n of t h e law o f i n t e r a c t i o n

y ( r )

by a given d i s t r i b u t i o n

,&(Q).

For t h i s we w i l l u s e

formula ( 2 5 ) .

On t h e b a s i s of (1) and ( 1 1 ) we have:

S u b s t i t u t i n g ( 2 8 ) i n t o ( 2 5 ) , we f i n d : .

2 Qmax

(+)

=

S

P ( Q ) dQo

.

. * ( 2 9 )

Q O

-n9

The right-hand p a r t of equation ( 2 9 ) r e p r e s e n t s some f u n c t l o n of A

a.

Solving equation ( 2 9 ) with r e s p e c t t o

A:$,

we o b t a i n

AC#l

a s a f u n c t i o n of r, t h a t i s , t h e law of i n t e r a c t i o n which i n t e r e s t s u s :

A s an example, we w i l l determine t h e law of i n t e r a c t i o n corresponding t o a uniform d i s t r i b u t i o n : P l a c i n g ( 3 0 ) under the i n t e g r a l s i g n of (29), i n t e g r a t i n g and s u b s t i t u t i n g the l i m i t s , w e o b t a i n : from which: where t

nip

= oc,

+ q ,

r \ e w i l l now t u r n our a t t e n t i o n t o t h e s o l u t i o n of t h e r e v e r s e problem: t h e d e t e r m i n a t i o n . of t h e d i s t r i b u t i o n

function

P ( Q )

by a given law of interaction

(P(r).

For this we w i l l use forqula ( 2 7 ) which we will re-write

thus:

Substituting (28) into this expression:

where the f'u~ction r

=

r

( ) can be considered as a

known function, so long a s the law of interaction is known, that is, the function

As an example, we will determine the distribution

corresponding to the power law of interaction (13); w e have:

I n t h i s c a s e (32) gives: 2

_c

_{2 -2} 2 r o i

p ( e )

=

; a m i s -

I ! / 3 ( 4 : ~ ) ~ / / 3

-

J

n:p=

Q~

-

e

o r where

I n t h e t a b l e shown below, we have compiled t h e v a r i o u s laws of d i s t r i b u t i o n , t h e corresponding laws of

i n t e r a c t i o n c a l c u l a t e d b y formula (29), and t h e corresponding i s o t h e r m s c a l c u l a t e d by formula ( 2 5 ) o r ( 1 1 ) .

The p h y s i c a l sense of t h e laws of i n t e r a c t i o n n o t e d i n t h e second column of t h e t a b l e remains, of oourse,

j u s t a s obscure a s t h e p h y s i c a l s e n s e f o r t h e d i s t r i b u t i o n f u n c t i o n s n o t e d i n t h e f i r s t column, which a r e g e n e r a l l y used a s o p e r a t o r s i n t h e t h e o r y of heterogeneous a u r f a c e a . The problem of t h e p r e s e n t work i s t o l i m i t o u r s e l v e s o n l y

t o t h e e s t a b l i s h m e n t of a f o r m a l connection between t h e f u n c t i o n s p

( Q )

and

7

( r ) , The q u e s t i o n r e g a r d i n g t h e

p h y s i c a l o r i g t n of t h e d i s t r i b u t i o n f u n c t i o n s and t h e laws nf

i n t e r a c t i o n i s a problem i n t h e p h y s i c a l t h e o r y of a d s o r p t i o n , t h e s o l u t i o n of vhich would p e r m i t t h e choice f o r onch

c o n c r e t e c a s e between two .of t h e a l t e r n a t i v e hypotheses (heterogeneous s u r f a c e s i n t h e absence of i n t e r a c t i o n , o r

homogeneous surf ace8 i n the presence of i n t e r a c t i o n ) , which remain e q u a l l y important i n t h e frame of pheno- menological theory.

We must note t h a t , assuming a c o e f f i c i e n t H s u f f i c i e n t l y l a r g e ( t h a t i s ,

f l

s u f f i c i e n t l y small), i n the hyperbolic d i s t r i b u t i o n ( t h e t h i r d l i n e of t h e Table), we can write f o r the corresponding law of i n t e r a c t i o n $

where 3

Assuming a c o e f f i c i e n t

H

s u f f i c i e n t l y l a r g e ( t h a t is,oC2

s u f f i c i e n t l y small) i n the exponential law of d i s t r i b u t i o n ( f o u r t h l i n e of the Table), we can w r i t e f o r t h e law of i n t e r a c t i o n corre spondlng t o t h i s d i s t r i b u t i o n s

where

It

i s

evident (and t h i s i s q u i t e an i n t e r e s t i n g case) t h a t a uniform, an hyperbolic and an exponential d i s t r i b u t i o n a l l l e a d ( a t s u f f i c i e n t l y l a r g e H) t o exactly the same square law of i n t e r a c t i o n .

7. The c o n n e c t i o n between 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 , a s a f u n c t i o n of f i l l i n g - i n , and t h e l a w of i n t e r a c t i o n We w i l l r e t u r n t o t h e o r i g i n a l e q u a t i o n ( 2 ) which we w i l l w r i t e t h u s : m e r e Q =

-

-

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 f o r homogeneous s u r f a c e s b y t a k i n g i n t o account t h e i n t e r a c t i o n s ( r e l a t i v e t o 1 molecule) ;

&=

q0 -

t h e d i f f e r e n t i a l h e a t f o r a homogeneous f r e e s u r f a c e , t h a t i s , i n t h e absence of i n t e r a c t i o n s ( a l s o r e l a t i v e t o 1 mol.ecule)g

f i ~

=

a

$'

-

t h e change i n t h e d i f f e r e n t i a l h e a t brought about by t a k i n g i n t o account t h e i n t e r a c t i o n s , The p r e s e n c e of t h e term

A Q

i n e x p r e s s i o n (34) i n d i c a t e s t h a t Q i s dependent on t h e d e g r e e of f i l l i n g - i n The form of t h i s r e l a t i o n s h i p i s determined by t h e form of t h e f u n c t i o n ( r ) t h a t i s , by t h e law of i n t e r - a c t i o n between the adsorbed m o l e c u l e s o I n f a c t , on t h e b a s i s of (10) we can r e - w r i t e ( 3 4 ) a s f o l l o w s :

Thus, f o r example, i n t h e c a s e of t h e power l a w of i n t e r - a c t i o n ( 1 3 ) , we have, a c c o r d i n g t o ( 1 4 ) :

Here belong t h e van Cer Waalsq i h t o r a c t i o n

(a

>

0 . P = 61,

t h e Coulombic i n t e r a c t i o n ( O C

L

0,

P

=

l ) , t h e c a s e f o r d i p o l e s o r i e n t e d i n p a r a l l e l (cl:

>

0 , .

,5

= 3 ) , and o t h e r s .

The c u r v e s Q = Q ( 8 ) f o r v a r i o u s ac and

/3

a r e s c h e m a t i c a l l y shown i n F i g , 8 , A t cT; = 0, which c o r r e s p o n d s t o t h e absence of i n t e r a c t i o n , we have a h o r i z o n t a l straight l i n e Q Q

.

A t GC

<

0, t h a t i s , i n t h e c a s e of a t t r a c t i o n ,

0 0

t h e curve Q = Q [ 8 ) i s l o c a t e d above t h e s t r a i g h t l i n e

Qo Qo; a t OG

7

0, t h a t i s , i n the c a s e of r e p u l s i o n , t h e curve i s l o c a t e d below t h e s t r a i g h t l i n e Q, Qo. I n f a c t ,

when t h e a d s o r p t i o n molecules a t t r a c t each o t h e r , t h e b r e a k i n g away of each adsorbed molecule i s hindered by t h e presence of neighbours, and t h e r e f o r e , t h e d i f f e r e n t i a l h e a t must i n c r e a s e d u r i n g t h e f i l l i n g - i n of t h e s u r f a c e ; i f , however, between t h e molecules t h e r e a r e f o r c e s of r e p u l s i o n , then t h e d i f f e r e n t i a l h e a t must d e c r e a s e a s t h e s u r f a c e i s f i l l e d , s i n c e t h e p r e s e n c e of neighbours a t the adsorbed molecule

w i l l , i n t h i s c a s e , h e l p i t t o break away from t h e a d s o r p t i o n l a y e r , The c r o s s i n g with t h e a b s c i s s a of the curve & = & ( 8 )

I n F i g . 8 (which i s p o s s i b l e a t q;

>

0 ) corresponds i n F i g , 6 t o a displacement of t h e minimum of t h e curve

9

=

.$

(X)

from t h e r e g i o n of t h e n e g a t i v e i n t o t h e r e g i o n of t h e p o s i t i v e

9,

The p r e s e n c e of an i n f l e x i o n on the curve Q =

Q

( 8 )

,

observed sometimes i n experiments, i s i n t h e frame of the a c c e p t e d model, an i n d i c a t i o n of t h e change i n

t h e law of i n t e r a c t i o n d u r i n g t h e t r a n s i t i o n f r m small r

t o l a r g e r ( f o r example, t h e change of t h e index

/9

i n the power law ( 13) )

,

The presence of a maximum or a minimum i n d i c a t e s t h e t r a n s i t i o n of t h e f o r c e s of a t t r a c t i o n i n t o f o r c e s of r e p u l s i o n , and conversely ( t h e change i n s i g n

f o r t h e c o e f f i c i e n t cC i n t h e power law of i n t e r a c t i o n ( 1 3 ) ) , I t must be noted t h a t formula (35) completely

a g r e e s with t h e formula f o r 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 , deducible from t h e t h e o r y of heterogeneous s u r f a c e s . I n

the Roginski method t h e d i f f e r e n t i a l h e a t i s determined a s t h a t value of Q which corresponds t o t h e value 8 = 1/2 i n t h e Langmufr formula: '

1

GI =

Q

Therefore,

This i s t r u e f o r t h e r e g i o n of "average f i l l i n g - i n t ' . Our

formula (35) cofncides with t h e Roginskf formula (361, which

and, s u b s t i t u t i n g ( 1 1 ) i n t o ( 3 5 ) :

I n c o n c l u s i o n , I wish t o e x p r e s s my g r a t i t u d e t o a c o r r e spondent-member of t h e Academy of S c i e n c e s , U.S.S,R., C,Z, Roginski, f o r t h e i n t e r e s t h e h a s shown i n

t h i s work.

Sent

t o t h e E d i t o r

A p r i l 9 t h , 1946. Academy of Soienaes, I n s t i t u t e of P h y s i c a l Chemistry U.S.S,R,

Moscowo I n t h e c a s e of t h e power law of i n t e r a c t i o n ( 1 3 ) t h e g e n e r a l formula ( 1 0 ) on t h e b a s i s of ( 1 4 ) assumes t h e f ormt

In

t h e r e g i o n of s m a l l f i l l i n g - i n , t h a t i s a t g ., / 1 9 t h i s giveso

where If we a l s o consider t h a t : t h i s mean8: thus, i n s t e a d of ( b ) we w i l l have: P

o n

-5

= C ₁ Q + c ₂

( a >

where 2 1 c1 =

$

t i n b ) / QO

';

The isotherm (d) e x p r e s s e s t h e well-known

Yilliams isotherm, deduced a l s o b y Temkin (3)

,

who based i t on the same power law of i n t e r a c t i o n (13), b u t assumed

a d i f f e r e n t model. The Temkin formula d i f f e r s from

formula ( d ) o n l y i n t h e value of t h e a o e f f i c l e n t s cl and c2.

is transformed into the W i l l i m s isotherm may be

interpreted thus:

This is possible at sufficiently small pressures if the forces of interaction are sufficiently large, that is, if

c

is sufficiently small.

1, Langmuir, 3, Am. Chem, Soc., 40, 1361, (1918).

2. Taylor and Zhurnal Fizicheskoi Khimii,

(J,

Phys, Jonesg

Chem, )

5,

181 (1936)

.

Langmuf r

,

3, Am. Chem. Soc., 40, 1403 (1918).

Zeldowltsch, Acta physicochimica URSS,

h,

961, (1935). Temkin, Zhurnal Fizicheskoi JChimii,

(J.

Phys.

Chem.)

15,

296, (1941).

3 . Roginski, Doklady Akademii Nauk,

-

45, 61, 66, 194,

206, (1944);

~ 6 t a Physicochimica URSS,

-

20, 227, ( 1945), 4, Roginski and A c t a physicochimlca URSS, 21,

-

519, (1946).

Tode s,

5, Frumkin, Trudy Khimicheskogo Insti tu ta Im., Karpova,

NO,

4, p e 56, (1925) a

Langmuir

,

J. Am, Chem, Soc,, 54, 2798, (1932). Langmuir, .Physa Rev., 44, 423, (1933).

Langmuf r, Zhurnal Fizicheskoi Khimii, (J Phys.

Chem. )

,

g,

16lS (1935)

want3 3 Proc, Roy0 Soc., A,

-

161, 127, (1937).

van €3 9 Proc. Cambr, Phil, Soc,,

-

34, 238, (1938).

Chang, Proc. Cambr, Phil, Soc.,

-

34, 224, (1938). Roberts, Proc. R a Soc., A,

161,

141, (1937).

Roberts, Proc, Cambr, Phil. Soc. 34, 399, (1938). Koboze-r and Zhurnal Fizicheskoi Khlmll, (J. Phya.

Gol'dfel~d,

Chem,,)

10,

612, (1937), Temkin, loco cit.

-il I-- Fig. 1 . Fig. 2 . 0

-

adsorbent molecules 9

-

adsorbed m o l e c u l e s 0

-

adsorbent m o l e c u l e s 9

-

adsorbed m o l e c u ~ e s Prrc. 4 Pnc. 5