Langmuir 1995,11, 2639-2647
2639
A Description of Adsorption in Activated Carbon Using a Hybrid Isotherm Equation H. D. Do and D. D. Do* Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia Received August 2, 1994. In Final Form: March 8, 1 9 9 P A hybrid isotherm equation for microporous solids was proposed in this paper by properly combining the Dubinin equation and the Nitta equation. The Nitta equation was incorporated to the “microporefilling” Dubinin equation t o accomplish three important goals, which the Dubinin equation fails t o fulfill: (1)t o provide the necessary Henry’s law at zero loading, demanded by thermodynamics, (2)t o understand the strength of adsorbate-adsorbate interaction in micropore filling mechanism in micropores of various sizes, and (3) to demarcate the critical pore size above which the micropore filling mechanism is not operative. It was found that this demarcation pore is not intrinsic but rather it depends on the specific adsorbate-adsorbent systems as well as the operating condjtions. For practical parameters use(, it was found that this demarcation pore size is in the order of 16 A, compared to the upper limit of 20 A set by the IUPAC.
Introduction An adsorption isotherm in microporous solids such as activated carbon is described well by the now well-known Dubinin equation, such as the Dubinin-Radushkevich equation or its generalized version the Dubinin-Astakhov equation. These equations are suitable to describe the isotherm of !olids having micropore size ranging from 4 to about 40 A. Although they are popular in their use to describe numerous systems with activated carbon in the literature, they suffer from one major handicap, that is it does not exhibit the Henry law isotherm at very low pressure, a requirement demanded by therm0dynamics.l Therefore, when using this Dubinin equation to predict multicomponent adsorption equilibria using the solution thermodynamics models, large errors may result. Recognizing the potential of the Dubinin equation because of its root in established potential theory of Polyani, various researchers have attempted to modify it, usually in an empirical way, to enforce the Henry law behavior at very low pressure. For example, Kapoor et aL2 used the following empirical equation:
Implicit in this equation is that the two modes of adsorption mechanisms, pore filling (Dubinin) and site adsorption (Henry law), are operative simultaneously at all range of pressure. Clearly this equation has a Henry law slope as the contribution of the Dubinin at zero pressure is zero. The coefficients /?I and /?z are obtained empirically, and Kapooret aL2and Kapoor andYang3 proposed the following functional form for these two parameters ~
~
~~~~~
* Author to whom all correspondence
should be addressed. Abstract published in Advance A C S Abstracts, May 15, 1995. (1) Talu, 0.;Myers, A. L. Rigorous thermodynamics treatment ofgas adsorption. MChE J. 1988,34, 1887-1893. ( 2 ) Kapoor, A.;Ritter, J. A.;Yang,R. T. On theDubinin-Radushkevich equation for adsorption in microporous solids in the Henry law region. Langmuir, 1989,5, 1118-1121. (3) Kapoor, A.; Yang, R. T. Correlation of equilibrium adsorption data of condensiblevapours on porous adsorbents. Gas Sep. Purifil989, 3, 187-192. @
0743-7463/95/2411-2639$09.00/0
where a is a constant determined by fitting eq 1 with experimental data. This value of a was found to fall in a very large range, ranging from 15 to about 10 000, a range too large to deduce any significant meaning of that parameter. In this paper, we will present another approach along the same line of enforcement of the Henry law at low pressure, and during the course of this analysis we will understand better the mechanism of adsorption in microporous solids from very low pressure to moderate pressures.
Theory We start the mathematical development by analyzing adsorption in a pore of half width r, and then generalize the model to microporous solids having a micropore size distribution F(r). Solids with Uniform Pore. For a given pore of half width r, the Dubinin-Astakhov equation describing the micropore filling of solids such as activated carbon can be written as follows4
where
A = RT In (Po/P) Here OD is the fractional loading at the gas phase pressure ofP, the subscript D is to denote the adsorption mechanism by Dubinin, and /? is the affinity coefficient of the adsorbate (=1for benzene). The exponent n has been found to vary from 1 to 3, although higher values have been reported by some authors, especially when dealing with highly uniform pores such as zeolites or carbon molecular sieve. When n = 2, the DA equation reduces to the usual form of the Dubinin-Radushkevich equation. The saturation pressure Po is a function of temperature and can be described by the Clausius-Clapeyron equation. The characteristic energy Eo is based on benzene adsorbate. (4) Rudizinski, W.; Everett, D. H. Adsorption ofgaseson heterogeneous surfaces; Academic Press: New York, 1992.
0 1995 American Chemical Society
Do and Do
2640 Langmuir, Vol. 11, No. 7, 1995 The latter is found by small angle X-ray scattering that it is a function of half width (assuming activated carbon pore has slit shape), r, as follows
E o ( r )= klr
(3)
where k = 12 kJ-nm/mol and r is in nm. Note the temperature independence of this characteristic energy. Although various forms for the relationship between Eo and r have been proposed in the literature, the above simple form is still the most popular, and it is used in this paper to illustrate the concept of our theory. It has been recognized for a long time that the Dubinin equation and its various versions do not satisfy the Henry law condition at zero loading, instead it gives a zero slope (zero Henry constant) at zero pressure. A number of approaches have been proposed in the literature to resolve this deficiency, such as the work of Kapoor and Yang3 as we have mentioned in the introduction. Recently, Shethna and Bhatia5 have developed a method by adding a Langmuir equation at low pressure and argued that for a pore of half width r there exists a threshold pressure such that Langmuir equation is valid below that pressure and the Dubinin equation holds above such pressure. They proposed the following for the Langmuir equation
where K is the usual affinity constant for the Langmuir equation and P* is P I P . At extremely low pressure we have the usual looking Henry law isotherm equation:
P limo, (r,T,P) = K(r,T)P-0 PO
(5)
It is noted that this Henry law equation has the saturation pressure appearing in the expression. This may seem improper because at extremely low pressure, the vapor pressure should not have any influence on the affinity between the adsorbate molecule and the solid surface atom and this affinity should be independent of pore size. It depends only on the system temperature as well as the specific pair of gas and solid. Thus, at low pressure, we should write lime, = K(T)P P-0
instead of the form of eq 5. As the pressure increases, the narrow confinement of the micropore (that is half width r ) starts to have an influence on the adsorption and in this paper we argue that it should affect through the interaction between the adsorbed molecules rather than the affinity between the adsorbate molecules and the solid. To this end, fundamental equations such as the Fowler-Guggenheim equation or the Nitta equation should serve this purpose well. The simplest form of Nitta equation6(onesite associated with one adsorbate molecule) is
where u is the molar interaction energy for adsorbate(5) Shethna, H. K.; Bhatia, s. K. Interpretation of adsorption isotherms at above-critical temperatures using a modified micropore filling model. Langmuir, 1994, 10, 870-876. (6) Nitta, T.; Shigetomi, T.; Kuro-oka M.; Katayama, T. An adsorption isotherm of multi-site occupancy model for homogeneous surface. J. Chem. Eng. Jpn. 1984, 17, 39-44 (1984).
adsorbate interaction which is assumed to be a function of micropore half width. The subscript N denotes that the Nitta mechanism operating. One should note that the Nitta equation is simply the Langmuir equation with lateral i n t e r a c t i ~ n.~ . ~ Ifthere are more than one site occupied by one adsorbate molecule, the equation proposed by Nitta et aL6 is
In this paper, we will assume that there is only one site associated with one adsorbate molecule (i.e., m = 11,K(T) is a function of temperature only and the interaction energy, u, is expected to increase with decreasing half width r as it makes good sense that the strength of the adsorbate interaction increases with the constriction of the surrounding microenvironment. For such a case, the relevant equation valid below a certain threshold pressure is the Nitta equation (eq 7). One also notes that the Nitta equation gives the correct Henry law isotherm at very low pressure. Although the Nitta equation was developed for flat and homogeneous surface, we use such equation to show the influence of the adsorbate-adsorbate interaction and how this would affect the transition of one mechanism of adsorption to another. In this paper, the Nitta equation is applicable in the very low pressure range where the molecular density of the adsorbed species is very low, one would then expect the molecules are located such that their interaction with each other is one-dimensional, similar to that on a flat surface at very low density. The heterogeneity is assumed to rise from the micropore size distribution, and this will be taken into account via the interaction energy, which is assumed to be a function of pore half width. This will be dealt with in the section of solids with micropore size distribution later. Using the approach of Shethna and Bhatia,5we assume that there exists a threshold pressure P,(r,T) such that when pressure is less than this threshold pressure the Nitta equation will apply, and above that pressure the Dubinin equation will hold. At the threshold pressure, we require that the fractional loading and its gradient with respect to pressure calculated by the Nitta equation must be the same as those calculated by the Dubinin equation in order to maintain smooth continuity between the two adsorption mechanisms, that is
and
(12) where etis the fractional loading at the threshold pressure. (7) Fowler, R. H.; Guggenheim, E. A. Statistical thermodynamics; Cambridge University Press: Cambridge, 1952. (8) Ross, S.; Winkler, W. J. Colloid Sci. 1955, 10, 319.
Langmuir, Vol. 11, No. 7, 1995 2641
Isotherm Equation for Microporous Solids For a given T, K(T), n, B, Eo, and Po,eqs 10 and 11are the required two equations for the interaction energy, u(r,T), and the fractional loading, et, at which the Nitta mechanism is switched to the Dubinin pore filling mechanism. Combining the two equations, eqs 10 and 11,we obtain the following nonlinear algebraic equation in terms of the threshold fractional loading, 8,
(E)L RT r
(13)
in which we have replaced E&) by k f r. Once the threshold fractional loading Otis calculated from the above equation, the threshold pressure and the interaction energy are calculated from
Pt =Poexp{ -
gk($]vn}
)(
)
n In e,(l - 20), 1 - n In 8, - In(?) n(1 In 0,) - 1 ( 1 - e,)2
+
-
Figure 1shows schematically the range of validity of two adsorption mechanisms. Given a microporous solid having a micropore size distribution, for micropores having half width less than rm , the adsorption process starts with the Nitta mechanism and then switches to the Dubinin micropore filling at the threshold fractional loading (i.e., threshold pressure). For micropores having half width larger than rm ,the mechanism operative over the whole range of pressure is the Nitta mechanism. Let ro (ro r,) be the minimum micropore half width. When the pressure (P)is greater than the minimum threshold pressure, the range of micropore half width for the Dubinin mechanism operative is ro P,
{
(23)
The above equation describes the adsorption equilibria in pore of half width of r . For describing the adsorption equilibria in microporous solids having a micropore size distribution, we assume that the pores are patchwise distributed, that is pores of the same half width are grouped into one patch; with this assumption of the solid topography, the overall adsorption isotherm for solids
Do and Do
2642 Langmuir, Vol. 11, No. 7, 1995
r
2.0
I
I
1.5
F(r) 1 .o
then when the pore is filled to a certain extent such that the adsorbate-adsorbate interaction is strong enough to induce the pore filling mechanism, hence the switch over from Nitta to Dubinin mechanism.
0.5
0.0
,
2.0
2
3
I
I
Result and Discussion To illustrate our theory, we use Benzene/Ajax activated carbon as our model system. The Henry constant K(T) is obtained from the equilibrium data at low pressure, and it is tabulated in Table lfor four different temperatures, 303,333,363, and 473 K. Also included in the table are the values of vapor pressure at those temperatures. The product KPO behaves in the same way as K(T), as temperature increases it decreases. If K and Po are assumed to obey the following expressions
F(r)l.o 0.5
K = K, e x p g )
0.0 2 2.0
Table 1. Value of Langmuir Affinity and Vapor Pressure of Benzene at Various Temperatures 303K 333K 363 K 473 K 350 932 10248 P(Torr) 104 2.038 x 5.211 x K(T0r1-l) 0.097 1.932x 4.857 2.089 KP 10.09 6.76
I
I
I .5
F(r)l.o 0.5
0.0 I
l-0
r- 2
r
: 3
3
Figure 1. Adsorption regime in different pore size ranges and different pressure ranges. having a micropore size distribution of F(r) and the smallest pore as TO are
The equation for overall adsorption isotherm must be solved numerically. The theory of adsorption mechanism can now be summarized graphically in Figure 1. It indicates that the Nitta mechanism is operative at all pressures in pores larger than r, , where the interaction between adsorbed molecules is so small to induce the pore filling mechanism. However, for pores of less than r,, the mechanism is started with a Nittamechanism, Le., surface adsorption with adsorbate-adsorbate interaction, and
PO =
3
P,O exp(- -
where Q is the heat of surface adsorption in the monolayer and AH,is the heat of vaporization, then the product of K and PO can be written as
Thus, the observed decrease of KPO with temperature implies that the heat of adsorption is greater than the heat of vaporization, which is as expected. For the Dubinin exponent n = 2, ,8 = 1,and K = 12000 J n d m o l , Table 2 shows values of the threshold fractional loading (eq 13), the threshold pressure (eq 14) and the interaction energy (eq 15) at T = 303 K for micropore having half width ranging from 0.65 to 2 nm. It is reminded that the threshold fractional loading and pressure in the table are the values at which the Nitta adsorption mechanism is switched to the Dubinin micropore filling mechanism. The interaction energy for the Nitta isotherm obtained in this case is negative, indicating the attraction of adsorbed molecules within the confinement of the micropore. The magnitude of the interaction energy is very high, which might be unrealistic but this high magnitude can be regarded as a reason for the inception of the micropore filling mechanism. The interaction energy decreases with a decrease in characteristic energy (i.e., larger micropore), implying lesser interaction among the adsorbed molecule as the pore gets larger. When the characteristic energy is greater than 7440 J/mol, or the micropore size is greater than 1.613 nm there is no solution for the threshold fractional loading, physically implying that the attraction is so weak that the Dubinin pore filling mechanism cannot occur beyond this pore size, but rather the Nitta mechanism remains valid beyond this pore size for the whole range of pressure. If the smallest micropore half width is TO = 0.65 nm, then the minimum threshold pressure is P,= 1.45 x Torr and the maximum threshold pressure is P, = 4.476 Torr, corresponding to r, = 1.613nm. What we note from this table is that the threshold fractional loading &(r) decreases with decreasing micropore half width. Thresh-
Isotherm Equation for Microporous Solids
Langmuir, Vol. 11, No. 7, 1995 2643
Table 2. Values of Threshold Fractional Loading, Threshold Pressure, and Interaction Energy at Different Micropore Half Width for BenzeneIAjax Activated Carbon System micropore half width r (nm) 0.65 0.7504 0.9502 1.21 1.4 r, = 1.6130 1.7030 2.0000
characteristic
threshold fractional
threshold
interaction
energy EO(JlmoU 18462 15992 12629 10000 8569
loading Bt 1.4526 x 10-21 1.0062 x 4.3160 x 6.2964 x 5.6568 x
pressure Pt (Torr)
energy uIRT -3.2422 x lozo -4.5704 x 1014 -9.9367x loo7 -5.7475 1003 -4.5606 x loo1
7440
0.32156
7046 6000
no solution no solution
9.3505 x 6.5492 x 2.8977 x 4.5205 x 10-O4 4.5313 x 4.4761
-0.27272
no solution no solution
no solution no solution
0.4
DUBININ Mechanism
WRT) elt -0.47 -0.46 -0.43 -0.36 -0.26 -0.088 no solution ~
no solution
I
0.3
8,
NITTA
0.2
Mechanism
T=473 T=363 T=333 T=303
K K K K
0.1
0.0 1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
r(nm> DUBININ Mechanism
Figure 3. Plot ofthreshold fractional loading versus micropore
half width with temperature as parameter. NITTA Mechanism
PA,,
TO
r*
rm
1*8
micropore half width
Figure 2. Plot of fractional uptake and pressure versus micropore half width. Illustration of switching mechanisms.
old fractional loading approaches its smallest value 8, (=1.4526 x at smallest micropore half width ro (=0.65 nm) and its maximum value 8, (= 0.32156) at the maximum demarcation micropore half width rm (1.613 nm).
The theory of adsorption mechanism can be best demonstrated by the illustration of Figure 2 where the threshold fractional loading and the threshold pressure versus micropore half width are plotted. When the gas phase pressure P is less than the minimum threshold pressure P, , or the fraction loading 6 is less than the minimum threshold fractional loading OE, the Nitta mechanism completely controls the adsorption over the whole micropore range. When the pressure reaches the
minimum threshold pressure P, or the fractional loading reaches the minimum threshold fractional loading e,, the onset of the Dubinin mechanism is started. For gas phase pressures greater than the minimum threshold pressure and less than the maximum threshold pressure P , the Dubinin mechanism is operative in the pore size ranging from ro to rt(P) and Nitta mechanism is operative above that range. The threshold micropore half width rt(P) increases with pressure until the pressure reaches the value ofP, . At pressure P =P , ,the threshold micropore half width attains its maximum value r,. When the pressure goes beyond the value of P , , the Dubinin mechanism is operative in the range ro < r < r, and the Nitta mechanism is operative in the range above r, . Let us consider a micropore having a half width rA (ro rA < r,). When the gas phase pressure in this pore is PA1 , and PAIis less than PM (where PAZis the threshold pressure at rA, i.e., PAZ= Pt(rA)),then the mechanism in this pore at this pressure is the Nitta mechanism. When the gas phase pressure in this pore increases to P M ,then the Dubinin mechanism starts to take over the Nitta mechanism in this pore at this pressure. From pressure PAZand beyond, the Dubinin mechanism is the only mechanism in this micropore. For a micropore with half width rg greater than the maximum Dubinin’s micropore half width r, ,the Nitta mechanism is the only mechanism in this pore at all pressure. Threshold fractional loading versus micropore half width is plotted in Figure 3 at various temperatures. All the curves cease at different maximum threshold fractional loading. The higher the temperature, the higher the maximum threshold fractional loading 6,(T). The temperature dependenceof 8,(T) is described in eq 19,where 8, is a function of Henry constant K(T) and vapor pressure
Do and Do
2644 Langmuir, Vol. 11, No. 7, 1995 2e+007
7
Oe+000 -2e+007
e5 b
-4e+007
-6e+007 -8e+007
- 1 e+008 0.6
1 .o
0.8
0.6 I O - ~ I O - ~100 1 0 ' lo2 l o 3 l o 4 P(Torr)
1.2
r(nm>
2e+000I T=303K, 1
Oe+000 -2c+OOO
z
\ -4e+000
5 -6e+000
t
1.610
t
r,(nm) 1.605
-8e+000 -le+001 ~
~~
1.2
1.6
1.4
1.600
0.0e+000
lo3
lo2
104
P(Torr) 1.615
-5.0e-001
e= \
10'
1 00
r(nm>
- 1 .Oe+000
1.610
i
I
k-1 I
/K=0.097 ~ = 3 0 Torr-' 3~
1
/
3 -1,5e+000
I 1.600
-2.5e+000
I
I
1.605
1.610
1.615
rbm> Figure 4. Plot of interaction energy versus micropore half width with temperature as parameter.
P(T)and both of them are a function of temperature. The region to the left of the curves in Figure 3 is controlled by the Dubinin mechanism, while that to the right is controlled by the Nitta mechanism as we have discussed in Figure 2. Interaction energy among adsorbed molecules versus micropore half width at various temperatures (303,333, 363,473 K ) are plotted in Figure 4. Because of the wide range of the magnitude of the interaction energy, we plot it in three different scale ranges. For pores having small half width, the interaction energy is very large and changes dramatically with a slight change in half width. The change of interaction energy with micropore half width is more gradual at large pore half width and then finally when the micropore half width is beyond the value of maximum Dubinin's micropore half width rmthere is no solution of the interaction energy. From the Figure 4b
1.600
'
I
4 5 P(Torr) Figure 5. Plot of threshold micropore half width versus pressure with temperature as parameter.
2
we note that throughout the range ofmicropore half width (1.27 nm < r < 1.54 nm) the interaction energy uIRT for the low temperature case (T= 303 K ) is stronger than that for the high temperature case (T= 473 K ) . This indicates the stronger attraction of adsorbate molecules in that micropore range at low temperatures. When the micropore half width approaches closer to the value of r,(T) (i.e.,r > 1.54nm), there is a crossoverbetween curves of different temperature. In Figure 5 we plot the threshold micropore half width versus pressure, which demarcates the pore ranges for Dubinin mechanism and the pore range for Nitta mechanism, for different temperatures. As gas phase pressure increases, the threshold micropore half width increases. Moreover, for the low temperature case the threshold micropore half width approaches its maximum value (i.e., r,(T)) at a lower pressure than that for the high tem-
Isotherm Equation for Microporous Solids
Langmuir, Vol. 11, No. 7, 1995 2645 le+00
2' I
1
I
I
I
500
h
Ll
k
8
le-11
W
a"
300
300
350
400
450
T(OK)
le-03
Table 3. Values of demarcation micropore half width and maximum threshold pressure at various temperature 333K
363 K 5.211 x 1.6109 84.354
I
450
Figure 7. Plot of minimum threshold pressure versus temperature with the smallestmicropore half width as parameter.
parameter.
303K
I
400
500
Figure 6. Plot of maximum micropore half width for Dubinin's mechanism versus temperature with Dubinin exponent as
K(T0m-l) 0.097 1.932 x 1.6116 r,(nm) 1.613 P, (Torr) 4.4761 23.286
I
350
le-13
473 K 2.038 x 1.610 1809.3
perature case (Figure 5a). Also the maximum Dubinin's micropore half width r, is larger for the low temperature case (Figure 5b). It means that the Dubinin mechanism is more significant (i.e. operative over a larger pore range) when the temperature is lower as one would physically expect. Table 3 lists the demarcation half width and the maximum threshold pressure a t different temperatures. In Figure 6 we plot the demarcation micropore half width rm (eq 18) versus temperature with Dubinin exponent n as parameter. It is seen that the maximum micropore half width for the Dubinin mechanism decreases with increasing temperature. The demarcation micropore half width is also higher for solid with more heterogeneous nature (i.e., low value of Dubinin's exponent) than that for solid with less heterogeneous nature (i.e., high value of Dubinin's exponent). Therefore for solids with the same minimum micropore half width ro, the one with the most heterogeneous internal structure (lowest value of n ) and a t the lowest temperature will have the largest micropore range for Dubinin mechanism. The threshold pressures at the smallest pore or the pressure a t which the onset of the Dubinin's mechanism P, is started are plotted versus temperature with the smallest micropore half width ro and with heterogeneous parameter (Dubinin exponent) as parameters in Figures 7 and 8, respectively. The pressure range (0 < P < P,)in which the Nitta adsorption mechanism is operative prior to the onset of the Dubinin micropore filling is smallest for the solidwith a smallest minimum micropore half width (Figure 7). This pressure range is also smaller if the solid becomes more heterogeneous (i.e., low value of Dubinin's exponent, see Figure 8). Table 4 demonstrates the effect ofthe Dubinin exponent on the significance ofthe Dubinin mechanism at T = 303 K. In Figure 9a we plot the pore size distribution of a solid with the structural parameters as ro = 1.609 nm, P = 2.0
n
L
le-23
Ll
€
0
d
3
/
le-43 le-53 300
350
400
450
500
T(OK) Figure 8. Plot of minimum threshold pressure versus temperature with Dubinin exponent as parameter. Table 4. Values of Demarcation Micropore Half Width and Minimum Pressure for Dubinin Isotherm at Different Values of the Dubinin Exponent n
2.0 1.8 1.6 1.4 1.2 2.069 2.642 1.613 1.687 1.816 0.590 x 10-14 0.258 x P,(Torr) 3.253 0.198 0.462 x Significance of Dubinin's Mechanism r,(nm)
-
nm, and u = 0.2 nm. At T = 303 K and with the Dubinin exponent as n = 1.9, the maximum micropore half width for the Dubinin mechanism is rm= 1.646 nm. The domain of the Dubinin mechanism is insignificant as compared to the domain of the Nitta mechanism as clearly shown by Figure 9a. The overall fractional uptake and contribution of the Dubinin and Nitta isotherms are plotted in Figure 9b. As we would expect, the contribution of the Dubinin mechanism is very minimal. The Nitta isotherm almost coincides with the overall isotherm. For the case of Dubinin domination, we plot the same pore size distribution as shown in Figure 9a but with low Dubinin exponent n = 1.21. The maximum micropore half width for the Dubinin mechanism is r, = 2.598 n m The Dubinin mechanism dominates most of the pores (Figure loa). The corresponding overall fractional uptake and the contribution of Dubinin and Nitta isotherm are
Do and Do
2646 Langmuir, Vol. 11, No. 7, 1995
a
2.5
2.5
2.0
rm=1.646nm r =2.0nm
A j/
1.5
10
=0.2nm
I
0
1 .o
8
I
2.0
1.5
a
F(r)
I
F(r) 1 .o
Nitta
0.5
0.5 Dubinin d o m a i n
0.0
rm
1.5
2.5
2.0
3.0
0.0 1.5
r(nm>
b
2.5rm
2.0
3.0
r(nm>
1.0
vera11 isother Nitta isotherm
0.8
6 o.6 0.4
tI
\
Dubinin isotherm
I
ro=1.609nm rm=1.646nm =0.2nm
0.2
Dubinin isotherm
0.0
0
20
\. \
'
40
,
60
0.0 80
100
P(Torr) Figure 9. (a) Pore size distribution for the case of Nitta domination. (b) Plot of fractional uptake and the individual contributions of Dubinin, Nitta filling for the case of Nitta domination. plotted in Figure lob, where we see that the overall isotherm is mainly dominated by the Dubinin isotherm. We see that in Figure 9 the Dubinin exponent is n = 1.9 and the overall isotherm is dominated mostly by the Nitta mechanism, and in Figure 10 where the Dubinin exponent is n = 1.21(i.e., the solidis more heterogeneous), the overall isotherm is dominated mostly by the Dubinin mechanism. If the Dubinin exponent is between 1.21and 1.9, the overall isotherm will be dominated equally by the two mechanisms.
' I 0
20
40
60
80
100
P(Torr) Figure 10. (a) Pore size distribution for the case of Dubinin domination. (b) Plot of fractional uptake and the individual contributions of Dubinin, Nitta filling for the case of Dubinin domination. energy between adsorbed molecules is strong enough and that the Nitta equation is valid throughout the entire pressure range in pores where the interaction is so weak to induce the micropore filling.
Acknowledgment. The financial support by ARC is gratefully acknowledged. Glossary
Conclusion We have presented in this paper a new hybrid isotherm to describe adsorption equilibria in homogeneous as well as heterogeneous microporous solids. This is achieved with a proper combination of the Nitta and the Dubinin isotherms. The new isotherm has the correct Henry law limit as the pressure approaches zero, removing the deficiency of the Dubinin equation having zero slope at low pressure. The important points arising from this analysis are that the micropore filling is merely the extension of the surface adsorption when the interaction
Dubinin parameter defined in eq 2 parameter defined in eq 12 maximum adsorbed concentration characteristic energy of Dubinin isotherm pore size distribution empirical constant of Dubinin isotherm Langmuir affinity coefficient for Nitta isotherm, eq 8 Dubinin variable exponent pressure
Langmuir, Vol. 11, No. 7, 1995 2647
Isotherm Equation for Microporous Solids
Po Pm
U
vapor pressure threshold pressure at maximum Dubinin's micropore half width threshold pressure defined in eq 14 minimum pressure for Dubinin isotherm dimensionless pressure isosteric heat of adsorption pore half width minimum micropore half width threshold micropore half width demarcation micropore half width gas constant temperature interaction energy
Greek Symbols empirical coefficient of Kapoor and Yang formula affinity factor of Dubinin isotherm empirical coefficient of Kapoor and Yang formula enthalpy of vaporization nondimensional isotherm root of eq 17 nondimensional Dubinin isotherm root of eq 19 nondimensional Nitta isotherm threshold fractional loading root of eq 21 LA940612V
