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Derivation of the Equations for Isotherm Curves of Adsorption on Microporous Gel Materials Freddy Romm* Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel Received September 20, 1994. In Final Form: April 24, 1996X A new method of building isotherm curves for adsorption on silica and alumina gels, using the statistical polymer method, has been proposed. This method considers the chemisorption as the result of a chemical reaction between the adsorbent and the polymeric adsorbate, leading to the change of the weight distribution of multicomponent polymers, that is described directly by the equations of the multicomponent version of the statistical polymer method. The physisorption is considered as the formation of a weak complex between the adsorbent and the adsorbate inside micropores differing in the excess energy, that being described based on the micropores’ energy distribution found by the statistical polymer method. The estimated relative square dispersion is in most cases on the level of some percents. A widespread paradoxsnonzero adsorption at P f 0sis explained as the result not of chemisorption but of physisorption processes: more concretely, of the discrete form of the energetic distribution of micropores.
Introduction
Description of Gels as Polymers
The derivation of equations for isotherm curves of adsorption on gel materials like silica and alumina gels is a very complicated problem in the theory of adsorption. For several adsorbates, such isotherms comprise physisorption and chemisorption parts. Then, a very specific and problematic phenomenon found not only for polar vapor adsorbates but also for gases like nitrogen is sometimes the nonzero value of the amount of adsorbate (θ) at very low pressures, P f 0. That is traditionally attributed to chemisorption. Some authors tried to separate the contribution of chemisorption and physisorption parts, assuming the amount of chemically captured adsorbate to be a constant,1 whereas the physisorption part is treated by a semiempiric method. The most widespread method of the description of physisorption is that of Dubinin assuming a Gaussian (or quasiGaussian, in the Dubinin-Astakhov approach) distribution of micropores in energy (or size).2-5 Numerous researchers applied this approach to all kinds of microporous adsorbents, without taking into account the principal difference in the mechanisms of their formation. Nevertheless, it is obvious that materials of different origins cannot possess the same internal structure. Hence, the application of Dubinin’s approach either to gels or to carbons seems incorrect, at least, for one of these types of adsorbents and probably, even for both them.6 Thus, a correct method of building adsorption isotherms should take into account the specific features of formation of the adsorbent, comprising the type of the primary material and the technology conditions of its treatment. In this paper, the author proposes a new method to build isotherms of adsorption on gel materials, especially silica and alumina gels.
The method proposed by the author is based on the widespread polymeric model which considers gels as equilibrium (or close to equilibrium) mixtures of multicomponent branched cross-linked polymers stabilized by weak interactions and distributed in weight.7 Then, the main problem consists in the evaluation of the weight distribution of polymers. The problem of the evaluation of the weight distribution of polymers is classical for the physical chemistry of highmolecular-weight compounds. This problem has a general solution only in the case of irreversible polymerization, whereas gel systems much better correspond to the reversible case. The methods still used for the description of irreversible polymerization have not satisfied the general reversible case. However, the statistical polymer method proposed recently by the author allows the theoretical treatment of polymeric systems produced by reversible polymerization.8 In the multicomponent case, the main equations of the statistical polymer (SP) method are derived as follows: In a system containing M components, we consider an assemblage of polymers containing the same numbers
* Current address: Laboratory of Chemistry of Materials and Industrial Chemistry, Faculty of Sciences and Technics, University Jean Monnet in St. Etienne, Dr. Paul Michelon St. 23, 42023 SaintEtienne Cedex 2, France. E-mail:
[email protected]. X Abstract published in Advance ACS Abstracts, June 15, 1996.
where xi is the molar fraction of the i-th component in the liquid phase; if yi ) ni/n, xi ) yi/si. We use the Flory approximation, assuming the weight fraction of the i-th component in the n-mer (n > 1) does not depend on n. To describe such a structure, we employ some definitions from ref 8.
(1) Desai, R.; Hussain, M.; Ruthven, D. M. Can. J. Chem. Eng. 1992, 70, 699. (2) Gregg, S. G.; Sing, K. S. W. Adsorption, surface area and porosity; Academic Press: London, New York, 1982. (3) Dubinin, M. M. Carbon 1981, 19, 321. (4) Dubinin, M. M.; Kadlec, O. Carbon 1987, 25, 321. (5) Stoeckli, F.; Morrel, D. Chimia 1980, 34, 502. (6) Aharoni, Ch.; Romm, F. Langmuir 1995, 11 (5), 1744.
S0743-7463(94)00750-X CCC: $12.00
M
n1, n2, ..., nM (
nj ) n, ∑ j)1
ni g 0)
of monomeric units of all components, without rings (crosslinks), as an averaged polymeric structure. The specific interactions between the polymers and monomers can be characterized by the following parameter:
si ) lim nf∞
( ) ni xi n
(7) Iler, R. K. Chemistry of silica; Wiley-Interscience: New York, 1979. (8) Romm, F. J. Phys. Chem. 1994, 98, 5765.
© 1996 American Chemical Society
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We chose a monomer unit as basis.8 The bonds of the basis with other monomer units are characterized by the notion of a level of a unit, defined as the distance from the basis to the unit. The first level is the basis. The presence of the i-th component on the k-th level Rik(n) is defined as the number of units of the i-th component on the k-th level (k > 1). For k ) 1,
Ri1(n) )
ni
) yi
M
nj ∑ j)1
M
The presence on the k-th level (k > 1) is limited by the possibilities of the (k - 1)-th level to accept new units onto the k-th level. We define a vacancy on the k-th level as the possibility of the (k - 1)-th level to accept a unit onto the k-th level.8 We denote the number of vacancies on the k-th level as Vk(n). This is the sum of vacancies presented by all the components: M
Vk(n) )
Vjk(n) ∑ j)1
(1)
For a statistical polymer structure containing {ni}M units we can write the following equations: M
∑ Rik(n) ) ni k)1
(2)
Rik(n) ) 0 when k > n
(3)
If mi is the maximum number of possible branches of a unit of the i-th component (its functionality minus one), we can obtain the recurrent equations for the number of vacancies:
Vi2(1) ) (mi + 1)yi
(4)
Vi1(n) ) 0
(5)
M
Vik(n) ) miRi(k-1)(n) -
Rjk(n) Vj(k-1)(n) ∑ j)1
here a vacancy on the k-th level is occupied by a new unit because of the reaction of polymerization or delivered due to destruction. The kinetics of the direct reaction (eq 11) is determined by the concentration of polymers Pol(n), the concentration of monomers Monj, and the number of vacancies estimated from (8). The kinetics of the inverse reaction (eq 11) is defined by the concentration of polymers Pol(n+1) and by the number of units which have one single bond with the basis part of the polymer. We denote the units having one single bond with the polymer by the notion extreme units. Obviously, an extreme unit of the i-th component has mi vacancies. We denote the number of extreme units of the i-th type on the k-th level of the polymer as Uik(n). Now, we can write eq 11 in the following form: M
Vacik(n) +
yj Monj S Exi′k(n+1) ∑ j)1
(12)
where Exi′k is the extreme unit of the i-th component on the k-th level. We can obtain the recurrent formulas for Rik(n), Vik(n), and Uik(n). When a monomer contacts with the polymer Pol(n), that can occupy a vacancy of the i-th type at the probability P(n):
∑ ∑yjVjk
P(n) )
k)1 j)1
(13)
VΣ
(7) (8)
The probability of the acceptance of a monomer especially onto the k-th level is given by the following equation:
k)2
M M
Obviously, the possibilities for the reaction of polymerization with participation of the considered polymer containing n units are characterized by the number of vacancies. To describe the reaction of destruction, we will consider the following reactions: M
yj Monj S Pol(n+1) ∑ j)1
(9)
Pol(n1) + Pol(n2) S Pol(n1+n2)
(10)
Pol(n) +
(11)
n+1 M
M
nj(mj - 1) + 2 ∑ Vk(n) ) ∑ j)1
yj Monj S Pol(n+1) ∑ j)1
Pol(n) O Vack(n) +
(6)
Vk-1(n)
Vik(n) ) 0 when k > (n + 1) n+1
to xi. According to the principles of chemical thermodynamics, we may limit the number of considered reactions by independent reactions. For the system considered, the reactions of type (9) form a complex of independent reactions, and we always may decompose every reaction in form (10) into some processes in form (9). Hence, we may use equations of type (9) as a complete description of the chemical equilibrium between the statistical polymers. The mechanism of a reaction of type (9) can be considered as follows:
where Pol(N) is the statistical polymer containing N units and Monj is a monomer of the j-th component. The probability that M belongs to the i-th component is equal
Pk(n) )
∑ ∑xiSijVjk i)1 j)1
(14)
VΣ
The change of the value of the presence of the i-th component on the k-th level because of the acceptance of a new monomer unit is
δRik(n) ) Rik(n+1) - Rik(n) ) Pik(n) )
yiVk(n) VΣ(n)
(15)
The change of the number of vacancies on the k-th level is
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Romm
Vik(n+1) - Vik(n) ) miδRi(k-1)(n) - δRik(n) ) miyiV(k-1)(n) - yiVk(n) VΣ(n)
lim nf∞
(16)
UiΣ(n) n
1
) Riyi ) 1+
K+ Kr(nf∞) )
(17) Ui1(1) ) yi
(18)
Ui1(2) ) yi; Ui2(2) ) yi
(19)
The recurrent equation for the total number of extreme units is given by
]
mi UiΣ(n+1) ) yi + UiΣ(n) 1 VΣ
[ ] mj
UΣ(n+1) ) 1 +
UjΣ(n) 1 ∑ V j)1
δRi(k+1)(n)
M
1m
(20)
VΣ(n)
The kinetics of reactions of type (9) is given by the following equations: M
∑ j)1
W-(n) ) K-UΣ(n)Cn
(21) (22)
where W+, W-, and K+, K- are the rates and the constants of the rates of the direct and the inverse reactions, respectively, Cn is the concentration of the n-mer (its mole fraction), and Ci1 is the concentration of monomers of the i-th component. For the equilibrium, one obtains
W+(n) ) W-(n+1) w
K+VΣ(n)
Cn+1 ) M
Cj1Cn ∑ j)1
K-UΣ(n+1)
) Kr(n)
(23) Taking into account the dependence of K+, K- or the temperature,8 one obtains
(
Kr(n) ) K0 exp -
)
∆E VΣ(n) RgT UΣ(n+1)
1
∑BsΣ(N)[VΣ(N) - s] ) 2VΣ(N)[N - 1 - B0Σ(N)] 2s)1 (27)
Σ
CjI
(26)
Now, we will consider the ring (cross-link) formation. First of all, we note that a ring is a link between two vacancies belonging to different monomeric units inside the same macromolecule. Hence, the total number of possibilities for ring formation is
where BsΣ(N) is the number of units possessing s-vacancies and B0Σ(N) ) UΣ(N). The rates of the reactions of ring formation and destruction are written as, respectively,
1
W+(n) ) K+VΣ(n)Cn
M
Rjyj ∑ j)1
Cr )
) (k+1)(n)
yj(mj - 1) ∑ j)1 K-
Besides, from (17) and (20) it follows:
∑ Uik(n) V k)1
M
M
miUik(n) Uik(n+1) - Uik(n) ) δRik(n) δRi(k+1)(n) Vk+1(n)
M
(25)
(yjmj) - 1 ∑ j)1
The change of the number of extreme units is
[
mi
(24)
here K0 ) K1/K2 and ∆E ) Ea1 - Ea2, where K1, K2 and Ea1, Ea2 are constant parameters and the energies of activation for the direct and inverse reactions, respectively. For enough large n, one may assume that Kr does not depend on n. Then
Wc+ ) Kc+Cr
(28)
Wc- ) Kc-Gr
(29)
where Gr(N) is the number of rings, whereas Kc+ and Kcare the constants of ring formation and destruction, respectively. In equilibrium:
Kcr )
Wc+ ) Wc- w Kc+Cr ) Kc-Gr
(30)
Kc+ Gr 2Gr ) ) Kc- Cr VΣ(N)[N - B0Σ(N) - 1]
(31)
In comparison with the solution without rings obtained in ref 7, the process of ring formation reduces the real number of vacancies and of extreme units:
VΣ(N) ) (m - 1)N + 2 - 2Gr
(32)
(c) (N) ) U(c) U(c+1) Σ Σ (N) - mUΣ (N) Gr(N)/VΣ(N) (33)
Energy Distribution For building theoretical isotherms of adsorption, the evaluation of the energy distribution of pores in the system is required.6 The method proposed above allows us to solve such a problem. We assume that the energy distribution of micropores does not differ from that of macromolecules. The physical reason for this assumption is that any divergence between these distributions would reduce the stability of the microporous material. Thus, we have first of all to evaluate the energy distribution of polymers. One can approximately estimate the potential energy of a polymer on the basis of the numbers of all kinds of vacancies:
E[{Ni}M] )
1 M
Ni ∑ i)1
M
jVjΣ(Nj) ∑ j)1
(34)
where i is the formal potential attributed to a vacancy of the i-th type. The values of ViΣ are estimated from eqs
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Langmuir, Vol. 12, No. 14, 1996 3493
4-8. Then, the energy distribution in the system is estimated based on eqs 4-8, whereas the maximum and minimum values of E for non-cross-linked polymers are given by
Emin ) Min{(mi - 2)i}
(35)
Emax ) Max{mii}
(36)
Obviously, an energy distribution of micropores in polymeric systems is discrete. For silica gel with a low content of water, one has assumed mSiO2 ) 3, mH2O ) 2, Emin ) 0, and Emax ) 3SiO2. For alumina gel, one has taken mAlO1.5 ) 3, mH2O ) 2, and AlO1.5 ≈ eSiO2. Chemisorption Chemisorption can take place only in systems satisfying the two following conditions: (1) the adsorbate molecules are able to considerably aggregate, and (2) the adsorbent forms specific bonds with the adsorbate. Describing chemisorption, we will employ the polymeric model of gels described in refs 7 and 9. Obviously, chemisorption is always accompanied by physisorption. A correct description of a gel-containing system like silica/alumina-water should contain parameters corresponding to both adsorptive processes. However, chemisorption gives one semiempiric parameter, whereas physisorption gives four (see below). The advantages of the five-parameter theoretical model over a fruitful five-parameter empiric equation would seem doubtable. Therefore, it is desirable to distinguish systems where chemisorption dominates physisorption and vice versa. The results of calculations presented below show that this is possible for numerous systems. One can cite one situation, at least, when chemisorption predominates: fresh gels. In the case of chemisorption (especially of water or alcohols), the adsorbate acts on the solid adsorbent and destroys it. Such a process has the following features: (1) The parameters characterizing the porous structure (size, energy distributions of pores) are not significant, because these are destroyed, and the adsorption does not differ from the absorption. (2) An illimited sorption of a vapor leads to the change of the aggregative state of the sorbent which becomes a true gel. The resulting isotherm can have a zone of the transition, more or less strike, from one aggregative state of the adsorbent to other one. Below, we consider only the case of low content of adsorbate. The start of the process is similar to physisorption because the micropores are not destroyed; therefore this part of the curve possesses some features of physisorption. In the case of chemisorption, the form of the energy distribution of micropores has no importance. The amount of adsorbate (water) is estimated from eqs 5-26. The amount of water in monomers is equal to C1xH2O, while that in n-mers equals nCnyH2O. Then, the amount of adsorbate is given by
Figure 1. Chemisorption of water on silica G-200-0 at 298 K.
of monomers, Cn and C0n are those of n-mers, respectively, and x0H2O and y0H2O are the initial mole fractions of water in monomers and in polymers, respectively. The relation of the values of xH2O and of PH2O is given by the equation of the pressure of saturated water vapor. Besides ∞
C1(1 - xH2O) + (1 - yH2O)
∑ nCn )
n)2
∞
C01(1 - x0H2O) + (1 - y0H2O)
∑ nC0n
n)2
The calculations have been carried out by means of the method of minimum squares, using the following optimization functional:6
Φ)
1
Ne
∑
4Nek)1
[
θT(Pk)
θek -
θek
]
2
θT(Pk)
where Ne is the number of treated experimental points, Pk is the relative pressure in the k-th experience, θek is the experimental result, and θT(Pk) is the theoretical result. The general procedure of computing does not differ from that described in ref 6. For a chemisorption process, we have only one characteristic parametersthe total molar fraction of the adsorbate in all the polymers, which is determined by the adsorbate selectivity (sa), the selectivity on the gel-forming component (sg), and the degree of polymerization (K0). Therefore, for the treatment of experimental data given for the same temperature, one may specify K0 ) 1; then only one varied parameter remains s′ ) sg/sa. Some typical isotherm curves of chemisorption are presented in Figures 1-3. The results of computing of parameters of some adsorptive systems, based on eqs 2-26 and using the data from refs 1 and 10-16, are given in Table 1. Since the proposed model contains one semiempiric parameter only, the obtained values of the relative square
θ(PH2O) ) C1xH2O - C01x0H2O + ∞
∑
n)2
n(yH2OCn - y0H2OC0n) (37)
where C1 and C01 are the current and the initial (before the start of the process of chemisorption) concentrations (9) Kester, F. L.; Lunde, P. J. AIChE 1973, 69, 87.
(10) Afzal, M.; Khan, M.; Ahmad, H. Colloid Polym. Sci. 1991, 269, 483. (11) Park, I.; Knaebel, K. S. AIChE J. 1992, 38, 660. (12) Jury, S. H.; Edwards, H. R. Can. J. Chem. Eng. 1971, 49, 663. (13) Naono, H.; Fujiwara, R.; Yagi, M. J. Colloid Interface Sci. 1980, 76, 74. (14) Naono, H.; Hakuman, M. J. Colloid Interface Sci. 1993, 158, 19. (15) Naono, H.; Hakuman, M. J. Colloid Interface Sci. 1991, 145, 405. (16) Munro, L. A.; Johnson, F. M. G. Ind. Eng. Chem. 1925, 17, 88.
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Figure 2. Chemisorption of water on silica at 298 K.
composition of the initial adsorbent (is it silica or alumina, does it contain additional salts, etc.); (2) the temperature of the measurement; and (3) the pretreatment of the initial adsorbent. The influence of additional substances can be estimated directly from eqs 5-26; of course, in this case M > 2. Nevertheless, if the sum concentration of additional salts is low enough, their influence can be neglected. The temperature influences the value of s' according to eq 24. The pretreatment of the adsorbent influences its ability to react with water. In the case if a fresh adsorbent is taken, the mass-transfer processes do not significantly limit the chemisorption. In the case of preheated adsorbents (which are used in most cases) composed of mechanically stable particles (in most studies, those are assumed to be spheres7), the chemisorption takes place only on the surface of these particles. Then, the real amount of the adsorbent able to react rapidly with water is much less than the formal one, and the equilibrium appearing in the measurement is not true. However, if the chemisorption is carried out for a long enough time, the initial bonds inside the particles of the adsorbent are broken, and then the resulting real amount of the reacting adsorbent is close to the formal one (found from the measurement of the total weight). Besides, the pretreatment can change the values of x0, y0, and C0, which depend on the partial pressure of water vapor. Physisorption We consider a physisorption process as the following pseudochemical reaction between the pore possessing the excess energy of formation and the gas G, without association of adsorbate molecules:
Por() + G S G‚Ads Figure 3. Chemisorption of water on silica-2 at 298 K. Table 1. Analysis of the Applicability of the One-Parameter Model of Chemisorption of Water on Gels to Some Systems Containing Freshly Formed Adsorbents
(38)
Equation 38 means that micropores are filled with the adsorbate progressively, not “by one stroke”. This assumption can be invalid at temperatures much lower than the critical one. Physisorption changes electric and specific quantum characteristics of interaction in micropores. It had been shown in ref 6 that the process of adsorption (eq 38) is described by the following equations:
adsorbent
T, K
s′
Φ
Silica G-200-0 Al.Alcan AA200X from 470C Alcan A300 Al-gel AD AD SG preheated at 473 K 673 K Silica-2 Silica-1 Silica-3 Silica-5 Silica-4 silica gel alumina dry activated alumina
298 299 297 298 323 298
17.0 231.0 8.81 0.117 2.99 69.4
9.6 × 10-3 0.0608 0.131 0.0286 0.0623 0.0182
298 298 298 298 298 298 298 313 298
0.760 0.758 0.197 0.760 0.758 138.0 31.8 23.9 4.70
0.134 0.0991 5.29 × 10-3 0.0343 0.0322 7.05 × 10-3 0.0201 0.0124 0.0974
Kr() ) ∆µG/S + ψ() γp ψ() exp ) C0 exp ) S exp (40) RgT RgT RgT
366 352 339 325 311 302
0.755 4.66 13.1 10.3 18.4 23.2
0.0841 0.101 0.0133 0.0599 0.0152 0.0527
where θ is the amount of the adsorbate G in all the micropores, X() is the fraction of the volume of pores possessing an energy which is occupied by the adsorbate, ∆µG/S is the change of the standard potential of G because of the contact with a standard pore, S ) exp[(∆µG/S + U0)/ RgT] Rg, is gas constant, Q0 is the normalizing coefficient, f() is the function of the energy distribution of micropores, min and max are the minimum and maximum possible values of , ψ is the additional potential because of entering G into the nonstandard pore, which can be written, in the simplest case, in the linear form:
dispersion Φ can be estimated as low enough that they can be considered as a serious support of cited assumptions. The formal value of s' (that is given in Table 1) is determined by the following factors: (1) the chemical
X() )
(
1 1 + Kr()PG
)
( )
∫
θ ) Q0
max
min
[1 - X()]f()d
(39)
( )
(41)
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Langmuir, Vol. 12, No. 14, 1996 3495
Table 2. Analysis of the Applicability of the Three-Parametric Model of Physisorption to Some Stabilized Adsorbents adsorbate/adsorbent
T, K
ζ, J/mol
K0
Ar/silica (for T ) 298) Ar/Silica G-200 (for T ) 298) C3H7OH/Silica G-200 (for T ) 298) benzene/Silica G-200 (for T ) 298) water/Alumina Alcan AA200X from 473 K water/Alumina Alcan AA200X from 623 K water/alumina Alcan ID880105/Alcan PSAI
77 77 77 77
1.51 × 103 8.82 × 102 2.23 × 103 2.23 × 103
1.93 4.64 2.15 2.15
0.235 0.286 0.447 0.152
0.017 0.021 0.011 5.6 × 10-3
299
7.21 × 103
2.30
0.113
0.0216
299
6.68 × 103
2.63
0.562
0.157
299 299 77 77 77 273 289.7 296 299 302.7 309 304.9 298 293.9 273 323 298 78 77 303 298 294 280
8.61 × 103 7.21 × 103 3.18 × 103 3.21 × 103 2.83 × 103 5.03 × 103 5.03 × 103 5.03 × 103 5.03 × 103 3.50 × 103 2.78 × 103 2.78 × 103 2.78 × 103 2.78 × 103 2.78 × 103 1.51 × 103 5.16 × 103 5.87 × 102 2.76 × 102 1.51 × 102 1.51 × 103 1.41 × 103 1.41 × 103
2.15 2.30 2.26 2.15 6.78 1.93 1.93 1.93 1.93 1.93 6.79 6.79 6.79 6.79 6.79 2.60 2.15 3.59 7.05 3.59 3.59 7.93 7.93
0.152 0.113 1.14 × -4 9.9 × 10-4 3.28 × 10-4 0.111 0.0227 0.0496 0.0227 1.00 × 10-3 0.115 0.0254 0.0262 0.181 0.113 0.342 0.115 0.342 0.106 0.342 0.342 0.169 0.222
0.0255 0.0138 0.0161 0.017 0.057 4.9 × 10-3 0.014 0.081 0.086 0.093 0.034 0.070 0.039 0.0206 9.3 × 10-3 0.034 0.044 0.034 0.078 5.3 × 10-3 0.034 0.136 0.0498
N2/SG dried N2/SG not dried N2/SG dried 1 h after preparation CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel water/DE water/silica N2/SG N2/MCM-41 water/SG TK 800 water/SG E benzene/silica benzene/silica
ψ ) U0 + γp
(42)
where U0 and γp have the sense of semiempiric parameters.6 For computer evaluations, it is reasonable to employ the following parameter:
{
γpSiO for silica ζ) γ 2 p AlO1.5 for alumina In the case of a discrete energy distribution, one writes the sum instead of the integral, and for a two-component polymeric system eq 34 is written as
θ ) Q0
∑
[1 - X(E({Nk}M)2))]f({Nk}M)2) (43)
{Nk}M)2
where the sum is taken over all energy states of the considered system and the value of E is given by eq 27. The function of the energy distribution is given by the number of microstates (monomeric units in polymers) possessing the energy E. Since the maximum number of vacancies per monomeric unit is realized in the case of monomers, adsorbate molecules are captured first of all by micropores with a higher content of monomers. Thus, one has obtained the general method of modeling of the physisorption in micropores at a specified energy distribution. It is interesting to notice that the amount of the adsorbed substance can significantly differ from zero at P f 0. This happens in polymeric adsorbents with a low degree of polymerization. In such systems, the fraction of monomers is high. If the micropore energy is determined by the averaged energy of the constituting macromolecules, the fraction of micropores possessing the maximum energy is significant. Therefore, the amount of adsorbate captured by such adsorbents at P f 0 differs from zero.
S
Φ
Thus, in contrast to the traditional approach, the proposed model considers the phenomenon of the nonzero adsorption at P f 0 as the result not of chemisorption but of physisorption; more concretely, of the discrete form of the energy distribution of micropores. For computing the parameters of systems where the adsorbate is vapor (water, etc.), the association of adsorbate molecules has been neglected. The results of calculations presented in Table 2 show that the relative square dispersion is low for the considered systems; hence, such an approximation is correct for some systems. Since the main information which must be obtained from the statistical polymer method for modeling of physisorption is the form of the energy distribution, one may formally specify the selectivities of the components; then, the only semiempiric parameter is K0, and one may employ the one-component model.8 Some typical isotherm curves of physisorption on gel materials are presented in Figures 4-6. The results of the calculations, on the basis of eq2-29 and 36 and using the data in refs 1, 10, 11, and 17, are given in Table 2. Analyzing the results presented in Table 2, one can notice: The formal constant of polymerization K0 changes from 1.93 to 7.93, whereas the spectrum of possible values of the analogous parameter for chemisorption s′ (see Table 1) is much larger. It seems to happen because the factors influencing chemisorption and analyzed above (first of all, the chemical composition and the pretreatment) are less important for physisorption. Additional substances (except polymers) do not signicantly change the energy distribution. The divergence between the chemical properties of silica and alumina, which is obviously very important in the case of chemisorption, is less significant for the form of the energy distribution and, hence, for the (17) Franke, O.; Schulz-Ekloff, G.; Rathousky, J.; Starek, J.; Zukal, A. J. Chem. Soc., Chem. Commun. 1993, 724.
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Table 3. Analysis of the Applicability of the Four-Parametric Dubinin’s Model of Physisorption to Some Adsorbents adsorbate Ar C3H7OH C6H6 water water water nitrogen nitrogen nitrogen CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel CH3COC2H5/Al-gel water/DE N2/SG N2/MCM-41 LWater/SG TK 800 LWater/SG E
adsorbent
T, K
G-200-0 G-200-0 G-200-0 alumina Alcan AA200X Alcan ID80105 Alcan PSAI SG dried at high T SG not dried SG prepared at pH)3.9, dried
77 77 77 473 279 279 77 77 77 273 289.7 296 299 302.7 309 304.9 298 293.9 273 323 78 77 303 298
Figure 4. Adsorption of water on alumina Alcan PSAI at 298 K.
C0
0(RgT)
γ(RgT)
Φ
1.00 6.05 46.4
A
3.41 0.151 2.15
1.85 1.85 3.16
6.63 0.151 0.215
2.0 × 10-2 4.2 × 10-2 0.032
3.53 × 10-3 46.4 0.125 3.53 × 10-3 0.0594 5.64 × 10-4 1.00 1.00 2.7 × 10-3 2.7 × 10-3 4.64 × 10-4 4.64 × 10-4 4.64 × 10-4 1.00 1.00 1.00 1.00 1.00 4.64 × 10-3 1.00 1.00
1.162 2.15 1.75 1.16 1.16 1.20 4.64 2.15 1.42 1.42 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15
1.85 3.16 1.70 1.85 1.85 1.29 3.16 3.16 1.70 1.70 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16
4.40 0.215 5.71 4.40 2.92 4.40 4.64 1.00 4.64 4.64 0.215 0.215 0.215 0.215 0.215 0.215 0.215 1.00 0.215 0.215 0.215
0.011 0.041 0.01 0.039 0.042 0.153 9.0 × 10-3 0.026 0.039 0.083 0.091 0.012 0.077 0.029 0.022 0.010 0.031 0.036 0.053 3.0 × 10-2 0.028
Figure 6. Adsorption of benzene on silica G-200 at 298 K.
forming the adsorbent are less important, because vacancies of a polymeric particle are located mostly on its surface. The values of ζ are on the level of RgT. That means that the energy of interaction of a vacancy with molecules of adsorbates is significant. This is probably because of the change of the electrostatic field and quantum characteristics in micropores near vacancies (which are, obviously, of electric origin) because of the presence of molecules of an adsorbate. For argon they are the minimum, while for nitrogen they are sometimes moderate. One can explain relatively high values of ζ for some systems with nitrogen by its large quadrupole moment. Though the proposed model contains three semiempirical fitted parameters, whereas that of chemisorption has one-parameter only, the values of the relative square dispersion are comparable. This seems to raise serious questions about the correctness of the proposed model, comprising the approximation of the separate descriptions of chemisorption and physisorption. Figure 5. Adsorption of water on silica gel TK 800 at 303 K.
physisorption. The pretreatment influences directly the value of K0 (which is determined by the temperature and by the partial pressure of water of the pretreatment), but the resulting form and weight distribution of particles
Comparison with the Results Obtained by the Dubinin Method To estimate the usefulness of the derived equations, the obtained above results have been compared to those
Adsorption on Microporous Gel Materials
Langmuir, Vol. 12, No. 14, 1996 3497
obtained by the Dubinin method assuming a Gaussian distribution of the energy of the micropores:6
[ ( )( )]
W ) W0 exp -B
T β
2
ln
P0 P
2
[ ( )]
) A exp -
- 0 RgT
2
(44) The distribution (eq 44) has been substituted into eqs 39-41. The results of the calculations are presented in Table 3. One can notice that the value of the dispersion is close to that obtained by the proposed method and given in Table 2, but it is due to employment of four empirical parameters: B, β, U0, and γp, whereas the proposed above model of physisorption employs three parameters (compare to Table 2). Thus, the model proposed above gives the same level of error of statistical treatment of experimental data as the Dubinin approach, while it employs fewer parameters. This result seems to be a principal advantage of the proposed method. Conclusions A new method of building isotherms of adsorption on silica and alumina gels, using the statistical polymer method, has been proposed. This method considers the
chemisorption as the result of a chemical reaction between the adsorbent and the polymeric adsorbate, leading to the change of the weight distribution of multicomponent polymers, that is described directly by the equations of the multicomponent version of the statistical polymer method. The physisorption is considered as the formation of a weak complex between the adsorbent and the adsorbate inside micropores differing in the excess energy, that is described based on the micropores’ energy distribution found by the statistical polymer method. The estimated relative square dispersion is in most cases on the level of some percents. That is close to the results obtained from the four-parameter Dubinin model, while the proposed model employs fewer fitted parameters (one for chemisorption, three for physisorption). A widespread paradoxsthe nonzero value of the amount of adsorbate at P f 0 s is explained as a result not of chemisorption but of physisorption processes: more concretely, of the discrete form of the energy distribution of micropores. Acknowledgment. I thank Prof. M. Folman (Department of Chemistry, Technion-IIT, Haifa, Israel) for useful consultations. LA940750K
