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Volume 38A, number 6 PHYSICS L E T T E R S 13 March 1972
E S T I M A T I O N O F T H E A D S O R P T I O N S I T E D I S T R I B U T I O N F R O M T H E S E C O N D A N D T H I R D G A S - S O L I D V I R I A L C O E F F I C I E N T S
R. W~LAD YSI~AW Department of Physical Chemistry, Institute of Chemistry UMCS, Lublin, Nowotki 10, Poland
Received 14 January 1972
A new method is proposed of estimating the topological structure of the adsorption sites, based on the virial expansion for adsorption isotherm.
The v i r i a l t r ea tmen t of the physica l adsorp- tion of gases has been successfu l ly used to de- sc r ibe p rope r t i e s of many adsorpt ion sys tems. However, the theore t ica l inves t igat ions based on this desc r ip t ion fo rma l i sm, which have been made so far, concern only the case of so-ca l led "non- local ized" adsorpt ion, i.e. the case when full mobil i ty of the adsorbed pa r t i c l e s may be assumed.
Adsorpt ion sys t ems with "localized" pa r t i c l e s have been cons idered using other descr ip t ion f o r m a l i s m s [1-3] which have a number of se r ious weaknesses in compar i son with the v i r ta l for- mal i sm. For ins tance, there exis t diff icult ies in taking into account co r re l a t ions between the adsorbed pa r t i c l e s [4,5].
The reason , for which the v i r ta l t r ea tmen t was not used in the descr ip t ion of "localized" adsorpt ion sys tems seems to be r a the r of a h is - tor ica l nature. The v i r i a l f o rma l i sm was p r e - viously used in the desc r ip t ion of bulk phases , cons is t ing of non- loca l ized par t ic les . Next it was in a na tu ra l way extended, to desc r ibe non- local ized par t i c le sys t ems in an ex te rna l field, which in pa r t i cu l a r may be provided by the ad- so rben t sur face [6].
Consider now the adsorpt ion surface with w ident ical adsorpt ion s i tes having adsorpt ion e n e r - gy equal to e. Assume next that each adsorpt ion si te has ~ nea re s t ne ighbours -adsorp t ion s i tes , where ~ will not in genera l be an in teger , be- cause of the usual defects in the c rys ta l p lanes of adsorbents . Let U denote the in terac t ion energy between two pa r t i c l e s adsorbed on two n e a r e s t - neighbour s i tes . We shal l not make any a s sump- tion concern ing the values for w, e, ~ or U. These values will occur as the r e s u l t of our m e a s u r e m e n t s and calculat ions. Let ~V be the number of the adsorbed molecules . Then [7]
= ~ V ( b j - b j ) z J , (1) j>~l
where bj and bj concern the s i tuat ions m the p resence , and in the absence of the adsorpt ion forces , respect ively . Here the absence of the adsorpt ion forces should be in te rpre ted as the p resence of the same number w of adsorpt ion s i tes , each having the adsorpt ion energy e = 0.
The c lus te r in tegra ls ~* and b~ a re connected J
with appropriate configurat ion in tegrals . For example,
V b 1 = Z 1, V b 1 = Z 1 (2)
* * * 2 (Z1)2 . Vb 2 = Z 2 - (Z1) , Vb 2 = Z 2 -
Neglecting b 2 in compar i son with b~ for the po- s i t ions on the adsorpt ion s i tes , and taking into account the nea res t -ne ighbour in te rac t ion only, we get:
N= w ~ e x p [ - c / k T ] } z + ¢o ~ e x p [ - 2 e / k T ] } (3}
× ~ e x p [ - v / k T ] - 1}z 2 + . . .
At smal l dens i t ies of the adsorbate phase the ac- tivity can be identified with the densi ty of the ad- sorbate phase. Then we get explici t formulae for the second B2s and the third B3s v i r ia l coef- f ic ients , respect ive ly ,
B2s = w exp [ - e / kT] (5)
S3s = w~{exp [-2E//kT]}{exp [ - v / k T] - 1}.
These cofficients can be eas i ly obtained exper i - mental ly , for example by gas chromatography [8].
F rom the plot lnB2s agains t 1 / T one can ob- tain both w, and E. After inse r t ing these values
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Volume 38A, number 6 P H Y S I C S L E T T E R S 1 3 M a r c h 1972
Table 1 Resul ts of the au thor ' s mea s u r em en t s by using 4.000g of the s i l icagel produced by Schuchardt in Muenchen, Wes t -Germany , and carbon t e t rach lo r ide produced by
P. O. Ch. Gliwice, Poland
t empe ra tu r e B2s B 3 s (°C) (1) (lZ/mole)
90.4 0.19174 - 128.769 100.0 0.14007 - 51.345 110.4 0.10758 - 32.198 120.7 0.08317 - 20.291 130.3 0.06328 - 10.117
in to eq. (5) one g e t s B 3 s a s a t w o - p a r a m e t e r f u n c t i o n of t e m p e r a t u r e , w i t h m e a n w h i l e unknown p a r a m e t e r s ~ and U. We f ind t h e s e p a r a m e t e r s by the " b e s t - f i t " p r o c e d u r e , w i th r e s p e c t to the v a l u e s of B 3 s found e x p e r i m e n t a l l y .
To i l l u s t r a t e t h i s m e t h o d , m e a s u r e m e n t s h a v e b e e n m a d e of the s e c o n d , and the t h i r d v i r i a l c o e f f i c i e n t s fo r c a r b o n t e t r a c h l o r i d e a d s o r b e d by s i l i c a g e l . The c h r o m a t o g r a p h i c c o l u m n c o n t a i n e d s i l i c a g e l p r o d u c e d by S c h u c h a r d t in M u e n c h e n , W e s t G e r m a n y .
By u s i n g the e x p e r i m e n t a l d a t a p r e s e n t e d in t a b l e 1, the f o l l o w i n g v a l u e s h a v e b e e n o b t a i n e d fo r w, ~, ~ a n d U:
w = 3.201 × 108 , ~ = - 3 9 9 2 c a l / m o l e
= 7 .430 , U = - 1 . 3 1 0 c a l / m o l e .
The m o s t i n t e r e s t i n g a r e the two l a s t v a l u e s , a s d e s c r i b i n g the t o p o l o g i c a l s t r u c t u r e of the a d s o r p - t i on s i t e s .
One m a y e x t r a c t a d d i t i o n a l l y i n t e r e s t i n g i n - f o r m a t i o n a b o u t the a v e r a g e d i s t a n c e r b e t w e e n two n e a r e s t n e i g h b o u r - a d s o r p t i o n s i t e s , w h e n c o n s i d e r i n g the t h e o r y of the t h i r d - o r d e r i n t e r - a c t i o n ( two a d s o r b e d m o l e c u l e s p l u s s u r f a c e ) , in
the f o r m d e v e l o p e d by S inanog lu and P i t z e r [9]. A s s u m i n g the a d s o r b e d m o l e c u l e s to l i e a t the s a m e d i s t a n c e f r o m the a d s o r b e n t s u r f a c e ( i .e . n e g l e c t i n g the o s c i l l a t i o n e f f e c t s ) , one o b t a i n s two p o s s i b l e v a l u e s f o r r , a t w h i c h U = 0. T h e s e a r e r ~ 0 . 9 5 r o , o r r ~ 1 . 4 t o , r o b e i n g the s l o w - c o l l i s i o n d i a m e t e r of the a d s o r b e d m o l e c u l e in the b u l k p h a s e .
The f i r s t v a l u e m u s t r a t h e r be e x c l u d e d for the fo l l owing g e o m e t r i c a l r e a s o n . A s s u m i n g t h a t the d i s t a n c e b e t w e e n two n e a r e s t s i t e s f r o m the s u r r o u n d i n g s of a p a r t i c u l a r m o l e c u l e i s the s a m e a s b e t w e e n t h i s p a r t i c u l a r m o l e c u l e and the m o l e c u l e s f r o m the s u r r o u n d i n g s , and i s e q u a l to a b o u t t o , one g e t s ~ = 6 a s the h i g h e s t p o s s i b l e v a l u e fo r ~. T h u s , t h e r e r e m a i n s the v a l u e r = 1 . 4 t o , w h i c h i s e q u a l to 8.5 A , w h e n u s i n g the
v a l u e fo r t o , o b t a i n e d by L a u g e r [10].
References
[1] For the older l i t e ra tu re see the monograph of A. R. Mi l le r , The adsorpt ion of gases on solids (Cambridge, Univers i ty P r e s s , 1949}.
[2] The more r ecen t l i t e ra tu re is given in the mono- graph of D. M. Young and A. D. Crowell , Physical adsorpt ion of gases (London, But terworth , 1962).
[3] K. F. Wojciechowski, Acta Phys. Poloniea 27 (1965) 893.
[4] P. Kaste le i jn , Phys iea 22 (1956) 397. [5] L. E. Ballentine and D. D. Bet ts , Physica 28 {1962)
1077. [6] S. Ono, J. Phys . Soc. Japan 6 (1951) 10. [7] R. A. P ie ro t t i , Chem. Phys. Le t te rs 2 (1968) 385. [8] W. Rudzinski , Z. Suprynowicz and J. Rayss , Study
on the possibi l i ty of de terminat ion of the th i rd gas - solid v i r ia l coefficient by gaschromatography, J. Chromatog. , to be published.
[9] O. Sinanoglu and K. P i t z e r , J. Chem. Phys. 32 (1960) 1279.
[10] P. Lauger , Helv. Chim. Acta 45 (1962) 1109.
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