Fractal dimension of microporous carbon on the basis of the Polanyi-Dubinin theory of adsorption. Part 3: Adsorption and adsorption thermodynamics in the micropores of fractal carbons

COLLOIDS AND ELSEVIER

Colloids and Surfaces A: Physicochernicaland EngineeringAspects 136 (1998) 245-261

A

SURFACES

Fractal dimension of microporous carbon on the basis of the Polanyi-Dubinin theory of adsorption. Part 3: Adsorption and adsorption thermodynamics in the micropores of fractal carbons Artur P. Terzyk *, Piotr A. Gauden, Gerhard Rychlicki, Roman Wojsz Department of Chemistry, Faculty of Adsorption and Catalysis, N. Copernicus University, Gagarina 7, 87-100 Torun, Poland Received 9 May 1997; accepted 2 November 1997

Abstract

Fundamental thermodynamic relations are formulated based on the equation of physical adsorption on microporous fractal solids, proposed previously and derived from the Polany~Dubinin theory of volume filling of micropores. A new adsorption isotherm and corresponding adsorption heat equations are verified using the experimental data published by Dubinin and Polstyanov of benzene and cyclohexane adsorption and adsorption heat on three microporous carbons. The obtained average correlation coefficients are compared with those from the original Dubinin-Astakhov (DA) equation. The correlation between the theoretical and experimental data is satisfactory, especially in the range of relative adsorptive pressures for which, following Stoeckli, the potential theory is accepted as appropriate. It is shown that, for nearly all the cases, micropore volumes are similar to those obtained from the original DA equation. Fractal dimensions calculated from adsorption data of both sorbates on the same carbon are practically equal and can be treated as constants that characterize the micropores of a solid. Average pore diameters, calculated from the obtained fractal dimensions, and also minimal and maximal pore widths are similar to those determined by the methods proposed by other authors, especially those obtained using the equation of McEnaney, developed from the analysis of SAXS data. The explanation of why the approximate adsorption isotherm equation proposed by Avnir and Jaroniec cannot be applied for the correct determination of a micropore fractal dimension is given. © 1998 Elsevier Science B.V.

Keywords: Adsorption; Active carbon; Adsorption thermodynamics; Fractal dimension; Microporosity; Potential theory; Pore diameter

1. Introduction

It is well known that the structural and energetic heterogeneity of an adsorbent, for example of an

* Corresponding author. 0927-7757/98/$19.00 © 1998Elsevier Science B.V. All rights reserved. PH S0927-7757 (97)00359-2

activated carbon, are closely connected and that different attempts have been carried out to separate them in a theoretical description of adsorption [ 1]. One of the new possibilities to take the structural heterogeneity of a solid into account is offered by fractal geometry, and an adsorption isotherm measurement can be applied for the determination

246

A.P. Terzyk et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245 261

of the fractal dimension [2]. Although in the case of nonporous fractal solids (where surface fractal dimension should be taken into consideration) analogs of Henry's [3], F H H [4,5] or BET [6,7] adsorption isotherm equations can be used for this purpose, the postulated adsorption mechanisms make it impossible to apply them to the correct description of adsorption in micropores (and for

the evaluation of pore fractal dimension). To take the mechanism of micropore volume-filling into account in fractal theory, Avnir and Jaroniec [8] published an approximated solution of the global adsorption isotherm equation for microporous solids in which the fractal pore-size distribution proposed by Pfeifer and Avnir [9], and the Dubinin-Radushkevich [10] adsorption isotherm

0,4

<>

031 llJ~ / ? O') ¢¢') I:::

7 o=:

0 . 2 - -

JJ

0.15

~

0.1

o.o5-

f I;

----

,'

,'

-

O -

FRDA

F_XP

-

-

DA

/

CCp = 0,9998

0.00

I

0.000

'

0.0 0.00

I 0.02

I

'

I

0.002

I

'

0.004

'

0.04

I

'

I

0.006

I 0.06

'

'

0.008

I 0.08

I

0.010

'

I 0.10

P/Ps Fig. 1. The comparison of experimental (points) and theoretical (lines) adsorption isotherms for the system: benzene-carbonAG at 353.15 K. FRDA: fractal analog of the DA equation; DA: original DA equation, CCp=0.9998 - the case for which only adsorption isotherm was approximated (see the text).

A.P. Terzyk et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245 261

equation was applied. Although the final solution presented by those authors (equivalent to F H H fractal analog) is simple to use in calculations, we have shown recently that an analytical and complete solution of this problem leads to completely different results [ 11 ]. Jaroniec et al. [ 12], based on the above-mentioned isotherm formula, derived also the corresponding thermodynamics of adsorption on microporous fractals; however until now they have not verified the obtained equations using experimental data. It is well known that adsorption isotherm measures the affinity between a sorbate and a sorbent; however both the heat and the entropy contribute to this affinity [13]. Thus, to check the correctness of the proposed adsorption model it is necessary to verify experimentally not only the adsorption isotherm but also the adsorption heat equation. Young and Crowell [14] pointed out that frequently the single experimental adsorption isotherm data can be approximated by a few models that are often based on some contrary assumptions (for example the type I adsorption isotherm shape

247

can be generated by the Langmuir and/or DR model). In this paper the new adsorption isotherm and adsorption heat equations are applied to the description of adsorption and adsorption heat data on microporous carbons. The comparison of the obtained results with the original DubininAstakhov (DA) [15] equation is presented, and the average micropore diameters from new theory are compared with the ones calculated using the methods proposed by other authors.

2. Theory As reported previously, from the detailed solution of the so-called 'global' adsorption isotherm in which the potential theory equations, together with the fractal pore size distribution proposed by Pfeifer and Avnir are used, it arises that the fractal analog of the DA isotherm can be written as

45-\ 4O

BENZENE CARBONAG

\ ',

FRDA

0

35

~ ~ , ~ .

-

-

....

-

EXP OA CCq= 0.4786

3o

-J

25

I

2O

15

Io

' 0.00

I 0.05

'

I

'

0.10

I

'

0.15

W

[cm3/g

I 0,20

'

I 0.25

'

I 0.30

]

Fig. 2. The comparison of experimental and theoretical adsorption heat data for the system from Fig. 1 (symbols as in Fig. 1). Experimental data are given with error bars.

248

A.P. Terzyk et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 136 (1998) 245 261

0.35 t <>

0.30 t /

/O

0.25 --[

e--.-i

O)

<> / /

0.20 0.20

E i....a

0.15

/

0.15

0.10

0.10 ..q

,,o CYCLOHEXANE CARBON AG FRDA EXP

0.05

DA

0.05 0.00

' 0000

0.00 '

).00

I

0.02

'

I

'

I

0.002

i

0.04

'

0.004

'

I 0.008

i

0.06

'

'

I

'

0.008

i

0.08

I 0.010

'

J

0.10

P/Ps Fig. 3. The comparison of experimental and theoretical adsorption isotherms for the system: cyclohexane-carbon AG at 353.15 K.

[16,17]

)

T is t e m p e r a t u r e , 2; is an i n c o m p l e t e g a m m a f u n c t i o n a n d n is an e q u a t i o n parameter. M o r e o v e r

p=

3-D t3-D max

--

~ 3-D "~min

(la)

and where

0 is the

AG=kTln(ps/p),

degree o f pore filling, A = R is the universal gas c o n s t a n t ,

# = (/~c)-"

(lb)

A.P. Terzyk et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245-261

249

45 m

CYCLOHEXANE CARBONAG I

40-

FRDA

35-

O

_~

3o-

-.1 i

~

25-

20-

15

' 0.00

0.05

I

'

I

o lo

'

o15 W

I 020

'

I o2~

[cm3/g]

Fig. 4. The comparison of experimental and theoretical adsorption heat data for the system from Fig. 3.

where D is the pore fractal dimension for pores with minimal and maximal slit half-widths between Xminand x . . . . /3 is the affinity coefficient and ~cis constant. As was shown previously [17] if n = 2 the fractal DR equation is obtained. The main properties of the both fractal analogs (FRDR and FRDA) together with the application for the description of experimental adsorption data are shown elsewhere [16, 18,19], however, the equations were not verified thermodynamically. In our previous papers we presented the adsorption enthalpy equation derived from Eq. (1), assuming that an incomplete gamma function is independent of temperature [ 11, 17]. However, the simultaneous description of the adsorption isotherm and adsorption heat data leads to a conclusion that it is necessary to take into consideration the temperature dependence of the incomplete gamma function in Eq. (1). To do this the differential adsorption potential distribution F(A) can be written as [12] F(A) -

dF* (A) dA

-

d0(A) dA

(2)

It was shown, that the differential 'pure' molar enthalpy of micropore filling is connected with F(A) by the equation [12]: (3)

AH= -A kOTlo

F(A)

where ~ is the coefficient of an adsorbate (with molar volume V) thermal expansion given by

~=(~ln

V)

_dlnV

(4)

However, assuming that the main condition of the potential theory i.e. temperature invariance of the 'characteristic curve' (~A/?T)o =0 is fulfilled, we obtain:

6H= -A

~O(A)

F(A)

(5)

For the fractal analog of the DA equation

A.P. Terzyk et al. / Colloids Surfaces A. Physicochem. Eng. Aspects 136 (1998) 245-261

250

0 . 5 ~

0,4

--

0,3

--

<>

<>

O Q35

-

0.30

0.25

0.2-

0.20

BENZENE

CARBON SA 0.15

FRDA EXP

0.10

DA 0.1

-0,05

0,00

l

I

0.0

l 0.00

I 0.02

'

I

0.002

0.00,

'

'

0.004

i 0.04

I

'

0.006

'

I

'

0.008

I 0.06

'

I 0.010

I 0.08

P/Ps Fig. 5. The comparison of experimental and theoretical adsorption isotherms for the system: benzene-carbon SA at 353.15 K.

(Eq. (1) using Eq. (2) we can write

+ P A.-~#(D_3)(pA.)(D-3-.)/. 11

- F(A) = p- (#A') ('-3)/" n

(An

x ( A n - 1n#Xnax ~/Xnax~)(3

\ _

- D-n)/n

exp(A'#xnax)

A"-ln#X~in(A#"x~in) (3-0-")/") exp(A"#x~in)

--~;(3-~--n--D, Xnin#An)]

(6)

251

A.P. Terzyk et al. / Colloids Surfaces A." Physicochem, Eng. Aspects 136 (1998) 245 261 45-L

BENZENE CARBONSA

40--

O

E

._1 I

T

FRDA

-

o__

35

30

25

20

'

).00

I

0.05

'

I

I

0.10

'

0.15

W

I

0.20

'

I

0.25

[cma/g]

Fig. 6. The comparison of experimental and theoretical adsorption heat data for the system from Fig. 5.

Combining Eqs. (6) and (5) gives the differential 'pure' molar enthalpy of adsorption. The parameters obtained from Eqs, (1), (5) and (6) can be used for the characterization of porosity and for the calculation of average pore diameters of adsorbents using the equations of the square root of the mean value and the average value of x [20,21]: Xmax

which leads to

3-D (r 4-°-1

.~2=Xmin ( ~ - - ~ ) \ ~ 1 )

(10)

Otherwise Xmax

f xfl (x) dx = 1

(1 1)

Xmin

21=( f XZfx(x)dx) 1/2

(7)

Xrnin

and r=Xmax/Xmi n

which gives

(12)

where pore size distribution function is given by

(3-D)l/2 (rS-°-l) 1/2 x, =Xmin

(8)

and

f l ( X ) = p X 2-D

(13)

They can be compared with the average pore diameters, obtained from the parameter E0 of the original DA equation, using the relations proposed by McEnaney [22,23]:

Xmax

"~2= f Xfl (X) Xmin

dx

(9)

22 = 4.691 e x p ( - 0.0666Eo)

(14)

2 2 = 6 . 6 - 1.79 In Eo

(15)

252

A.P. Terzyk et aL / Colloids Surfaces A." Physicochern. Eng. Aspects 136 (1998) 245 261

0.5 "--

0.4

0.3

--

f

E

0.3--

-

0.2-0.2-

CYCLOHEXANE CARBON SA FRDA 0.1 0,1

EXP

m

DA

o.o

'

I

0.000

0.0

' 0.00

I 0.02

'

0.002

'

I

'

I

0.004

I

/

0.04

'

0.006

I

'

0.06

I

'

0.008

I 0.08

I

0.010

'

I 0.10

pips Fig. 7. The comparison of experimental and theoretical adsorption isotherms for the system: cyclohexane-carbon SA at 353.15 K.

and Stoeckli [24,25]:

et al. [21] can also be calculated by the equation:

2~ = 16.5/E0

(16)

2if= 18/Eo

(17)

22 = 10.8/(E 0 - 11.4)

(18)

Based on the average pore diameters, fractal dimension from the relation p r o p o s e d by Jaroniec

D= 6.44-6.17~

(19)

3. Approximation of the experimental data Before we discuss the obtained parameters and correlation coefficients (CCs) we should point out

A.P. Terzyk et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245-261

that it is possible to obtain high CC values based only on the approximation of adsorption isotherm results. This case is shown in Fig. 1 where, despite the high correlation between experimental and theoretical adsorption isotherm data, the calculated theoretical heat significantly differs from the experimental one. However, the simultaneous approximation of adsorption and adsorption heat data can lead to lower average CC values owing mainly to the cumulation of errors (it was shown by Gregg and Wheatley that the error of 1% in the pressure determination results in approximately 1.7 kJ mol 1 error in isosteric adsorption heat [26]). Based on the discussion of errors presented by Dubinin and Polstyanov [27-29], we assumed the maximal pressure and isosteric adsorption heat determination errors as 6% and 5% respectively. It should also be mentioned, that potential theory of adsorption is not a universal one [30], and our model, as a version of this theory should describe only the cases for which the Dubinin-Astahov equation is valid. There are also different opinions on the range of applicability of this equation. That is why we calculated CC values for two ranges of relative pressures:

253

1 x 10-6-0.1 (proposed in Ref. [31]) and 1 x 10-6-0.01 (proposed in Ref. [25]). The experimental data (published by Dubinin and Polstyanov) of adsorption and isosteric adsorption enthalpy of benzene and cyclohexane at 353.15K on the carbons, AG, SA and SK, [27-29] were approximated. They were measured using typical gravimetric apparatus. The equilibrium pressure was measured using a McLeod manometer and cathetometer for lower and higher pressures, respectively (the details can be found in Ref.[27]). They were chosen because the benzene adsorption data on carbons SA and SK have often been presented as the typical data for which DA theory is valid; however, as was pointed out by Dubinin himself, the correlation between DA and experimental data for SK carbon is worse [32]. The optimalization procedure used in the work presented here was analogous to that applied in Ref. [33]. Based on Eqs. (1), (5) and (6), the correlation coefficients of adsorption isotherm (CCp) as well as of adsorption heat (CCq) were calculated and the function (1-(CCpCCq) 2) was minimalized. The same procedure was used for the original DA adsorption

45-CYCLOHEXANE

CARBONSA tI

FRDA

T

40

_o_ _ E×~

O

E 35 J I

o" 30

25

'

0.00

I

0.05

'

I

0.10

'

I

0.15

'

I

0.20

W [cm3/g]

'

I

0.25

I

'

0.30

I 0.35

Fig. 8. The comparison of experimental and theoretical adsorption heat data for the system from Fig. 7.

254

A.P. Terzyk et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245-261 0.6

s-oO

0.5

/ 0.4

0.48 0.40

E

0.35

0 ' - - ' 0.3

0.30

/

/ j

J

0.25

0.2

0.20 --

BENZENE CARBON SK

0.15

-i

0.10

-i'

FRDA

EXP -

-

DA

-

0.1 0.05 0,00 -~ 0.000

0.0

0.004

I

I 0.00

0.002

0.02

0.006

0.008

I 0.04

0.06

=

0.010

I 0.08

P/Ps Fig. 9. The comparison of experimental and theoretical adsorption isotherms for the system: benzene-carbon SK at 353.15 K. Table 1 The parameters of the original DA equation obtained from simultaneous description of the adsoptinn and the adsorption heat data Carbon Adsorbate

Wo (cm3 g-l) E0 (kJ mole ~) n

AG AG SA SA SK SK

0.40416 0.35881 0.43211 0.43245 0.62030 0.63562

Benzene Cyclohexane Benzene Cyclohexane Benzene Cyclohexane

17.36065 16.01582 27.09652 24.84258 20.38497 18.10285

2.17378 2.00196 2.8048 2.03494 2.3396 1.72208

i s o t h e r m a n d h e a t equations. T h e n u m e r i c a l p r o g r a m s in F o r t r a n 77, p u b l i s h e d previously, [16,19] were m o d i f i e d a n d used for this purpose. The c o m p a r i s o n o f e x p e r i m e n t a l a n d theoretical isot h e r m s a n d heats is presented in Figs. 1-12 a n d the o b t a i n e d p a r a m e t e r s o f D A a n d its fractal a n a l o g ( F R D A ) , Eq. (1), are shown in Tables 1 a n d 2. In Figs. 13-15 the pore-size d i s t r i b u t i o n s c a l c u l a t e d f r o m Eq. (13) for b o t h s o r b a t e s are shown. In Table 3 we present C C values between e x p e r i m e n t a l a n d theoretical results for the b o t h equations. In Table 4 the p o r e d i a m e t e r s c a l c u l a t e d

A.P.

Terzyket aL / Colloids Surfaces A: Physicochem.Eng. Aspects 136 (1998) 245 261

45 - -

BENZENE CARBONSK ~T

40-

d~ O

255

_?

_

FRDA EXP

35

E ._~

30

.J

|

O"

25

20

15

' 0.00

I

'

I

i

I

0.10

0.05

'

I

0.15

W

'

0.20

I

'

0.25

i 0.30

[cm3/g]

Fig. 10. The comparison of experimental and theoretical adsorption heat data for the system from Fig. 9. Table 2 The parameters of the FRDA equation obtained from simultaneous descriptions of the adsorption and the adsorption-heat data Carbon

Adsorbate

Wo (cm3 g- 1)

D

n

Xmi n

(nm)

AG AG SA SA SK SK

Benzene Cyclohexane Benzene Cyclohexane Benzene Cyclohexane

0.39427 0.34768 0.47460 0.43679 0.80625 0.70179

2.505 2.451 2.890 2.915 2.300 2.297

2.1652 2.1037 2.0697 1.9882 2.1672 2.0015

0.520 0.590 0.505 0.461 0.450 0.456

Xmax(rim) 1.100 1.051 0.525 0.500 1.200 1.100

Table 3 The correlation coefficients for two relative pressure ranges (up to 0.01 and 0.1) between theoretical and experimental data for the adsorption isotherm (CCp)and adsorption heat (CCq) together with average correlation coefficients (CC.v) for DA and FRDA equations Carbon

Adsorbate

DA

FRDA

0.01

AG AG SA SA SK SK

Benzene Cyclohexane Benzene Cyclohexane Benzene Cyclohexane

0.l

0.01

0.1

CCp

CCq

CCav

CCp

CCq

CCav

CCp

CCq

CCav

CCp

CCq

CC,~

0.9464 0.9689 0.9975 0.9903 0.9927 0.9877

0.9765 0.9875 0.9778 0.9904 0.9836 0.9597

0.9613 0.9843 0.9876 0.9904 0.9881 0.9736

0.9851 0.9972 0.9970 0.9925 0.9963 0.9943

0.9826 0.9847 0.9778 0.9904 0.9837 0.9597

0.9838 0.9909 0.9874 0.9915 0.9899 0.9768

0.9957 0.9938 0.9936 0.9900 0.9961 0.9855

0.9785 0.9826 0.9923 0.9937 0.9951 0.9522

0.9870 0.9882 0.9891 0.9919 0.9956 0.9687

0.9923 0.9947 0.9948 0.9909 0.9940 0.9846

0.9848 0.9851 0.9923 0.9937 0.9951 0.9522

0.9885 0.9899 0.9935 0.9923 0.9945 0.9683

256

A.P. Terzyk et al. / Colloids Surfaces A. Physieochem. Eng. Aspects 136 (1998) 245-261 0.6

0,- 1 ..

/?" /

0,5

0.4

O')

¢,

0,4O

¢O

E

O

/

0.3

#

0.3

0.2 ~

0.2

CYCLOHEXANE CARBON SK FRDA

~

0.1 -

EXP

-

DA

-

0.1 0.0

'

0.001

o.o

' 0.00

I

I

'

I

0.002

*

0.02

I

'

0.004

'

0.04

I

'

I

0.006

I

'

0.06

'

0.008

I

I

0.010

'

0.08

I 0.10

pips Fig. 11. The comparison of experimental and theoretical adsorption isotherms for the system: cyclohexane-carbon SK at 353.15 K.

Table 4 Average pore diameters X (nm) obtained from FRDA parameters and from the equations proposed by other authors Carbon

Adsorbate

Eq. (8)

Eq. (10)

Eq. (14)

Eq. (15)

Eq. (16)

Eq. (17)

Eq. (18)

AG AG SA SA SK SK

Benzene Cyclohexane Benzene Cyclohexane Benzene Cyclohexane

0.810 0.821 0.515 0.480 0.836 0.787

0.792 0.810 0.515 0.480 0.837 0.764

0.738 0.807 0.386 0.448 0.603 0.702

0.745 0.817 0.347 0.425 0.602 0.708

0.475 0.515 0.304 0.332 0.405 0.456

0.518 0.526 0.332 0.362 0.442 0.497

0.906 1.170 0.344 0.402 0.601 0.806

Numbers of the equations are the same as those in the text

A.P. Terzyk et aL/ Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245 261

257

45--

t T tI

40

O

35-

I

30-

E

N

CYCLOHEXANE CARBONSK - ~

FRDA EXP

cr

25-

.L ~

'

20

I

0.00

'

I

0.05

'

I

0.10

'

'

~

'

0.15

I

0.20 W

'

I

0.25

'

I

0.30

'

I

0.35

0.40

[cm3/g]

Fig. 12. The comparison of experimental and theoretical adsorption heat data for the system from Fig. 11.

2.6 m CARBONAG ~ .

2.4

'

BENZENE

2.2 vX

2.0

1.8

1.6

1.4

~ 0.50

i

i

I

~

I

I

i

'

I

~

i

i

I

'

I

'

I

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 x

[nm]

Fig. 13. The pore size distributions calculated using Eq. (13) for the carbon AG.

~

i 1.05

i

i 1.10

A.P. Terzyk etaL/ColloidsSurfaces A: Physicochem. Eng. Aspec~ 136 (1998)245-261

258 55--

-

CARBON SA

50

<; _

BENZENE

+

CYCLOHEXANE

45

40 - -

35 ~

30

-

20

~

I

,.46 0.47

,

I 0.47

,

I

,

i

0.48

,

0.48

I

i

0.49

I

,

I

,

0.49

0.50

x

[nm]

I 0.50

,

I

I

0.51

I

I

I

0.51

,

0.52

I,

I

0.52

0.53

,

I 0.53

Fig. 14. The pore size distributions calculated using Eq. (13) for the carbon SA.

1.8 -CARBON SK

1.7

1.6

1.5

1.4

1.3

1.2

1.1

' 0.4

I 0.5

'

I 0.6

'

I

i

I

0.7

0.8

x

~

I 0.9

'

I

'

1.0

[nm]

Fig. 15. The pore size distributions calculated using Eq. (13) for the carbon SK.

I 1.1

'

I 1.2

A.P. Terzyk et aL/ Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245-261

259

relations correlate better with experimental data, however, the differences in CC between DA and FRDA are not significant. It also should be noted that, in the case of cyclohexane adsorption on carbon SK, both equations give the lowest CC. That is why the most significant differences in pore volumes obtained from DA and FRDA equations are observed only for that system. For the other systems similar micropore volumes from DA and FRDA equations were obtained. It is also very important to admit that, for the same carbon sample, the fractal dimension obtained using two different sorbates is practically the same and can be treated as a constant that characterizes the micropores of the adsorbent. Another important feature of the parameters obtained from the FRDA equation is the correlation between the average pore half-widths calculated using Eqs. (8) and (10) and those calculated using equations proposed by other authors, especially McEnaney. It should be noted that the empirical equation of McEnaney (Eq. (15)), based on SAXS data, correlates the average pore width with the characteristic energy of the original DA equation, and for the calculation of the average pore width from the FRDA

from Eqs.(7)-(19) are included, whereas in Eqs. (8) and (10) we used the data from Table 2. And finally, in Table 5 the fractal dimensions obtained from the FRDA equation are compared with those from Eq. (19) using two average pore diameters calculated from Eqs. (14)-(18) and from the Avnir and Jaroniec equation. To use this equation, the plots In W-ln(ln(ps/p)) (where ps and p are the saturation vapour pressure and the equilibrium pressure of the adsorbate, respectively) were drawn (one of them is shown in Fig. 16) and linearized in the range of the relative pressures 0.05-0.1 [81.

4. Discussion

From Figs. 1-12 and Table 3 it arises that for all the investigated cases in the low-pressure region (up to the relative pressures 0.01 ) the fractal analog of DA and corresponding adsorption heat equations describe experimental data with a higher average CC value than original DA relations. At higher relative pressures there are only two cases (cyclohexane: carbons AG and SK) where DA -0,75 -1,00

10=21391

-1,25 -1,50 -1,75 -2,00 -2,25 -2,50

Linear Regression:

l•

Y=A+B*X

nn m

-2,75 -3,00 -3,25 -3,50 -

Param

Value

sd

A B

-0,2233 0,06288 -0,86119 0,04704

=h • •

R =-0,98684

•

-3,75 -4,00 0,50

'

I

I

I

I

I

I

0,75

1,00

1,25

1,50

1,75

2,00

'

I

I

2,25

2,50

'

i

2,75

In(In(Ps/P)) Fig. 16. The linearization of the data for the system: benzene-carbon AG using the Avnir and Jaroniec adsorption isotherm equation in the range of relative pressure 0.05-0.1. R is the correlation coefficient and W is the volume of liquid-like adsorbate present in the micropores.

A.P. Terzyket al. / ColloidsSurfacesA: Physicochem.Eng.Aspects136 (1998)245~61

260

Table 5 The comparison of the fractal dimensions obtained from the FRDA equation with those calculated from the Avnir-Jaroniec equation and Eq. (19) Carbon

Adsorbate

FRDA

A-J

Eq.(19)Eq.(14)

Eq.(19)Eq.(15)

Eq.(19)Eq.(16)

Eq.(19)Eq.(17)

Eq.(19)Eq.(18)

AG AG SA SA SK SK

Benzene Cyclohexane Benzene Cyclohexane Benzene Cyclohexane

2.505 2.451 2.890 2.915 2.300 2.297

2.139 2.312 2.784 2.804 2.430 2.601

1.886 1.459 4.059 3.673 2.717 2.106

1.840 1.395 4.299 3.819 2.727 2.072

3.508 3.262 4.561 4.391 3.943 3.628

3.241 2.973 4.391 4.204 3.716 3.372

0.850 -0.778 4.317 3.961 2.732 1.469

Equation numbers are the same as those used in the text.

equation we use the fractal dimension and minimal and maximal half-width o f the slits (not the energy), obtained f r o m the approximation o f experimental adsorption and heat data (see Eqs. (8) and (10)). Finally we wish to c o m m e n t on some results from Table 5. It is seen that the empirical relationship proposed by Jaroniec et al. (Eq. (19)) mainly leads to nonphysical fractal dimension values. The approximate solution o f the global adsorption isotherm given by Avnir and Jaroniec (Eq. (1) for n = 2 is the analytical solution o f it) leads to different fractal dimensions than the ones obtained from Eq. (1). Moreover, they are not constant for the same c a r b o n and different adsorbates. To explain the obtained differences we can write the equation o f Avnir and Jaroniec in the f o r m

O=P = -( Fm(A~2- ) ~ D - 3 ) / 2 2

Number of points

B

D

R

3 4 6 7 9 10 11

-0.614 -0.633 -0.687 --0.691 -0.770 -0.819 -0.867

2.386 2.367 2.313 2.309 2.230 2.181 2.139

0.9999 0.9996 0.9970 0.9978 0.9926 0.9884 0.9968

written in a similar form:

(20)

where F is a g a m m a function. These authors assumed that Eq. (20) can be written as [8]

0 = kln(ps/p) D- 3

Table 6 The results of the D calculation from the A-J equation using the data from Fig. 16 for different relative pressure ranges

0 ; a [ l n ( p s / p ) ] D- 3

(24)

where G is given by

(21 )

where K is the constant which characterizes the adsorbate-adsorbent system and is given by

K= P F ( ~ - )

(22)

However, the analytical solution (Eq. (1) can be

The c o m p a r i s o n o f Eqs. (22) and (25) explains the obtained differences between D calculated from Eq. (1) and f r o m Eq. (21). It is seen that K in Eq. (21) cannot be treated as a constant, because,

A.P. Terzyk et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 136 (1998) 245-261

in practice, it d e p e n d s on the relative pressure ( b y the value o f A). This is s h o w n in Table 6 where we present the results o f D c a l c u l a t i o n f r o m Avnir a n d J a r o n i e c e q u a t i o n using the e x p e r i m e n t a l d a t a f r o m Fig. 16 a n d t a k i n g different relative pressure ranges for the l i n e a r i z a t i o n (i.e. different n u m b e r o f e x p e r i m e n t a l points). It is seen t h a t the o b t a i n e d fractal d i m e n s i o n s c h a n g e f r o m 2.13 to 2.38. T a k i n g this into a c c o u n t for a m i c r o p o r e fractal d i m e n s i o n o f 2 < D < 3, for this case we observe changes o f a b o u t 30% in D. W h a t is o b s e r v e d is a higher relative pressure, the lower the D value, so the differences are n o t caused by an e x p e r i m e n t a l error. T h e y are the results o f g r a d u a l changes in the K value with relative pressure.

5. Conclusions T h e p r e v i o u s l y p r o p o s e d fractal a n a l o g o f D A a d s o r p t i o n i s o t h e r m e q u a t i o n describes the experimental data of adsorption and of adsorption heat with a better average c o r r e l a t i o n coefficient t h a n in the D A e q u a t i o n b u t in the relative pressure r a n g e u p to 0.01 (i.e. p r o p o s e d b y Stoeckli). T h e m i c r o p o r e v o l u m e s o f the investigated c a r b o n s are similar to those o b t a i n e d f r o m the original D A e q u a t i o n . T h e o b t a i n e d fractal d i m e n s i o n is constant for the given c a r b o n so it can be t r e a t e d as a p a r a m e t e r which characterizes its m i c r o p o r o u s structure. T h e c a l c u l a t e d p a r a m e t e r s o f the F R D A e q u a t i o n are a c c e p t a b l e as far as the c u r r e n t k n o w l edge o f the m i c r o p o r o u s c a r b o n s is concerned. This can be c o n c l u d e d f r o m the c o r r e l a t i o n between the o b t a i n e d average p o r e widths a n d these c a l c u l a t e d f r o m the e q u a t i o n o f M c E n a n e y . The equation proposed by Avnir and Jaroniec leads to a different fractal d i m e n s i o n t h a n F R D A e q u a t i o n does. M o r e o v e r the fractal d i m e n s i o n for the same a d s o r b e n t d e p e n d s on the type o f the a d s o r b a t e . The o b t a i n e d differences between the e q u a t i o n s are c a u s e d by the incorrect a s s u m p t i o n o f i n d e p e n d e n c e o f K f r o m relative pressure.

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