5606
Langmuir 1996, 12, 5606-5613
Potential Theory of Adsorption of Nonelectrolytes from Dilute Aqueous Solutions. Benzene Adsorption Vladimir Kh. Dobruskin st. Aiala 21, Beer-Yacov, 70300, Israel Received February 26, 1996. In Final Form: July 22, 1996X The equilibrium relationship is derived from the potential theory of adsorption, the concept of micropore volume filling, and assumptions that solute adsorption results in a solvent displacement from micropores and in a formation of an interface between an adsorbed phase in micropores and a solution in an extramicropore space. Water desorption from micropores is characterized by the affinity coefficient which is estimated from immersion heats. The energy of the interface formation is calculated from a model of active carbon structure. An isotherm equation shows the relationship between adsorption of the organic component in micropores, on the one hand, and parameters of active carbons (limiting adsorption volume, micropore volume, characteristic energy, immersion heats, interface area), parameters of solution components (solute and solvent affinity coefficients, interfacial tension, saturated concentration, molar volumes), and conditions of operation (temperature, equilibrium concentration), on the other hand. The correlation between adsorption from aqueous solutions and both the carbon crystallite sizes and the oxygen complexes on the carbon surface is predicted. The main results are discussed in terms of the Weibull distribution function, and the generalized equation is proposed. The theory has been successfully applied to published data for adsorption of benzene from aqueous solutions on active carbons. It is pointed out that there is an urgent need for more independent studies, and a number of suggestions for future work are given.
Introduction
f(A) )
A great many sorbents, of which active carbon (AC) is probably the most significant, have been widely studied and used for modern industrial purification processes. In spite of the significance of this phenomenon, the theoretical foundations of solute adsorption are still insufficiently solved in comparison with adsorption from a gas phase; this situation is due to the great complexity of the phenomenon. From the theoretical viewpoint, the isotherm equations for adsorption from dilute solutions may be obtained from the isotherm equations for gas adsorption. The important stages in the study of single-gas adsorption on microporous solids were made by Dubinin and Radushkevich,1 and Dubinin and Astakhov (D-A)2,3 proposed the general expression for the isotherm. In the Dubinin theory of single-gas adsorption, the degree of micropore filling, θ (fractional adsorption), is related to the limiting adsorption volume, W0, and the occupied adsorption volume, W, as follows:
θ ) W/W0
(1)
The fraction of unoccupied adsorption volume, (1 - θ) is considered to be the Weibull cumulative distribution function F(A)
[ ( )]
F(A) ) 1 - exp -
A βE0
n
(2)
that is associated with the Weibull density function4 f(A) X
Abstract published in Advance ACS Abstracts, October 1, 1996.
(1) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk. SSSR 1947, 55, 331. (2) Astakhov, V. A.; Dubinin, M. M.; Romankov, P. G. In Adsorbents, their Preparation, Properties and Application; Dubinin, M. M.; Plachenov, T. G., Eds.; Nauka: Leningrad, 1971; p 92 (Russian). (3) Dubinin, M. M. In Adsorption-Desorption Phenomena; Ricca, F., Ed.; Academic Press: London, 1972; pp 3-18. (4) Bury, K. V. Statistical Models in Applied Science; Willey: New York, 1975; Chapter 12.
S0743-7463(96)00175-8 CCC: $12.00
[ ( )]
n A An-1 exp βE0 (βE0)n
n
(3)
where A is the differential maximal molar work of adsorption equal with a minus sign to the variation in the Gibbs free energy of adsorption, E is the characteristic adsorption energy, β is the affinity or similarity coefficient, and n is the Weibull distribution parameter. The parameters of the distribution function are calculated with respect to the standard gas with β0 ) 1, which may be considered to be a probe of energy heterogeneity that provides information about adsorption sites. The subscript 0 denotes the parameters of the reference (standard) gas. According to Polanyi,5-7 adsorption potential is the work of adsorption which compresses the ambient gas until it condenses
A ) RT ln Ps/P
(4)
where P is the partial pressure in the gas phase and Ps is the saturated pressure of the liquid adsorbate. Dubinin and Stoeckli8,9 showed that the characteristic adsorption energy of an active carbon is defined by the average micropore size
E ) X/x
(5)
where x is the half-width of the split pore and X is the empirical constant. In the case of active carbons, n is a rule equal to 2, and the D-A equation finally takes the form
[ ( )]
θ ) exp -
A βE0
2
(6)
(5) Polanyi, M. Verh. Dtsch. Phys. Ges. 1914 , 16, 1032. (6) Polanyi, M. Trans. Faraday Soc. 1932, 28, 316. (7) Bakaev, V. A.; Steel, W. A. Adsorpt. Sci. Technol. 1993, 10, 123136 (special issue). (8) Stoekli, H. F. J. Colloid Interface Sci. 1977, 59, 184. (9) Dubinin, M. M.; Stoekli, H. F. J. Colloid Interface Sci. 1980, 75, 34.
© 1996 American Chemical Society
Benzene Adsorption from Dilute Aqueous Solutions
Langmuir, Vol. 12, No. 23, 1996 5607
The comprehensive review of Dubinin theory is given in a special issue dedicated to his memory.10 Adsorption from solutions differs from single-gas adsorption: (i) the liquid mixture contains at least two components, each of which is adsorbed in the porous volume; (ii) the adsorption volume of AC placed in a solution is always filled with solution components, and θ ) 1; (iii) the excess adsorption of solute molecules in a micropore space results in a solvent displacement from micropores. In many papers11-18 devoted to the adsorption from dilute solutions the authors applied the D-A gas/ solid adsorption isotherm by replacing the adsorbate pressure with the solute concentration. Such empirical treatment had resulted in the discovery of a number of important facts. It had been shown11-18 that for adsorption of difficultly soluble compounds from aqueous solutions the limiting adsorption volume, W0, is practically equal to the micropore volume, and adsorption may be described by eq 6, in which the “adsorption potential” is given by the expression
Ac ) RT ln Cs/C
reviewed by Kaganovski at al.12,13,15,16 The limiting adsorption volume, W0, for adsorption of benzene, nitrobenzene, dinitrobenzene, chlorobenzene, n-nitroaniline, n-nitrophenol, n-chlorophenol, and chloroform16 was found to be practically equal to the micropore volume, Vmi. The small difference between these values was explained by monolayer adsorption on the surface of mesopores. Kaganovski et al. came to the conclusion that the limiting adsorption of difficultly soluble organic compounds results in the complete displacement of water from micropores. Kaganovski’s theory of physical adsorption from aqueous solutions is not discussed here, and our references to his experiments should not be considered to be agreement with his theoretical treatment. Additional information about the equilibrium in the adsorbed phase may be obtained from the theory of physical adsorption of a multicomponent gas mixture.19 It was shown that a separation factor in the micropores is determined by the ratio rj of average residence times, τjoc and τjw, of the components in the occupied volume
(7)
where Cs is the saturated concentration of the organic component in water at a given temperature and C is the equilibrium concentration in water. The “characteristic adsorption energy”, E*, for liquid/solid adsorption is significantly less than that for adsorption from a gas phase. Nevertheless, this formal approach ignores the molecular mechanism of adsorption from solutions and may be considered to be just a verification of the applicability of the D-A equation as an empirical equation for data correlation. The purpose of the present work is to describe solute adsorption. We call a solution “dilute” when a solvent is present in a large excess compared with the solute saturated concentration; i.e., we are concerned here only with solutes that are difficultly soluble in water. Our consideration will be limited to the physical adsorption of nondissociating organic compounds on microporous solids. Weak electrolytes such as phenols, amines, or organic acids can also be treated as nondissociating compounds if adsorption occurs at pH values when their dissociation is suppressed. For dilute aqueous solutions with a nondissociating solute and weak water-solute interactions, an adsorbate’s accessibility to the micropore is consider to be similar to that for gas/solid adsorption.17.18 The equilibrium relationships for adsorption of nonelectrolytes from dilute aqueous solutions are derived from the potential theory, and further insight into adsorption from solutions is gained. Distribution in an Adsorbed Phase. Interface Energy A great deal of information on adsorption from water solutions on ACs with known porous structure was (10) Adsorpt. Sci. Technol. 1993, 10 (special Issue). (11) Stadnic, A. M.; Eltekov, Yu. A. Zh. Fiz. Khim. 1975, 49, 771. (12) Koganovski, A. M.; Klimenko, H. A.; Levchenko, T. M.; Roda, I. G. Adsorption of organic compounds from water; Khimia: Leningrad, 1990; pp 21-85 (Russian). (13) Koganovski, A. M.; Klimenko, H. A.; Levchenko, T. M.; Marutovski, R. M.; Roda, I. G. Purification and recycling of waste water; Khimia: Moscow, 1983; pp. 74-100 (Russian). (14) Eltekova, N. A.; Eltekov, Yu. A. Adsorpt. Sci. Technol. 1993, 10, 203-209. (15) Koganovski, A. M.; Levchenko, T. M.; Roda, I. G.; Marutovski, R. M. Purification of waste water by adsorption; Technika: Kiev, 1981; pp 11-13 (Russian). (16) Koganovski, A. M. Purification of waste water by adsorption and ion exchange; Naukova Dumka: Kiev, 1983; pp. 9-35 (Russian). (17) Jaroniec, M.; Choma, J.; Burakiewich-Mortka, W. Carbon 1991, 29, 1294-1296. (18) Choma, J.; Burakiewich-Mortka, W.; Jaroniec, M.; Gilpin, R. K. Langmuir 1993, 9, 2355-2561.
rj )
x
1 θ
{ [ (x ) ] [ (x )]}
mocβw exp(b2) exp mwβoc
ln
1 -b θ
xπb 1 - erf
ln
2
+
1 -b θ
(8)
where moc, mw and βoc, βw are the masses of the molecules and the affinity coefficients of the components of the adsorbed phase and b ) ∆βE0/2RT, where ∆β ) βoc - βw. The subscripts oc and w denote here an organic compound and water, respectively. Through the affinity of the characteristic curve, the greater difference in the adsorption energy at small θ results in the greater values of the separation factor. It will be shown further that βw ≈ 0.1. When the standard gas is benzene, the characteristic energy of an ordinary AC is close to 20000 J/mol and for a mixture of water and benzene at 298 K, b ) 3.63. For b >1.5 eq 8 shows a very strong dependence of the separation factor on the occupied micropore fraction.19 For this reason, if a complete displacement of water was observed at high solute concentration, it would be observed the more so for θs < 1. Here θs is related to the micropore volume occupied with a solute. These facts indicate that two phases coexist inside the AC placed in a solution; one phase is presented by the solute in the micropore space, and another phase is presented by the dilute solution in the extramicropore space. An adsorbed phase is considered to be similar to the normal liquid. Since an extramicropore space is constantly occupied by a liquid with another concentration, such a process inevitably leads to the interface formation between an adsorbed phase and a solution. For example, in the case of benzene adsorption, an interface exists between liquid benzene and a dilute solution with C < Cs ) 0.023 mol/L. Let surface area, Si, be an interface between micropores occupied with a solute and a solution. An interface between two phases in the state of equilibrium is associated with the interface energy
Es ) γSi
(9)
where γ is the interfacial tension. To estimate an interface area, a model of porous structure must be taken into consideration. Dubinin20,21 assumed an idealized slitlike model of a micropore limited (19) Dobruskin, V. Kh. Langmuir 1996, 12, 987-993. (20) Dubinin, M. M.; Fedoseev, D. V. Izv. Akad. Nauk SSSR, Ser. Khim. 1982, 2, 246-253. (21) Dubinin, M. M. Carbon 1992, 20 (3), 195-200.
5608 Langmuir, Vol. 12, No. 23, 1996
Dobruskin
in size with the radius of the round base, r, and the halfwidth of a flat slit, x. Virtually, we imagine the interface area as the opened surface of micropores positioned inside the stack of round graphite planes, the stack being plunged into a solution. The volume of the single micropore and its outer surface are 2πr2x and 4πrx, respectively. When AC adsorbs W cm3/g of a compound, the number of occupied micropores is equal to W/2πr2x. Hence,
Si ) 4πrx
W 2W ) 2 r 2πr x
(10)
It was assumed21 that under the condition of formation of micropores the ratio
δ ) r/x
(11)
may vary within a small range, and the mean value of δ was considered to be constant. Dubinin also found that δ )1.56.21 Taking into account this approximation and eq 5, one obtains the relationship between Si and the characteristic adsorption energy
Si )
2WE δX
(12)
An interfacial tension between a liquid adsorbate and a dilute solution may be estimated with Antonow’s rule:22
∆γ = γas - γoc
(13)
where γoc and γas are the surface tensions of the organic component and aqueous solution, respectively. The surface tension of aqueous solutions depends upon the solution concentration C and may be described by the Szyszkowsky equation:22
γas/γw ) 1 - B ln(1 + C/d)
(14)
where γw is the water surface tension and B and d are the constants. Since the surface tension of water is lowered by the addition of an organic solute, then, by the Gibbs equation, the solute must be adsorbed at the interface, and γas may be approximated by the surface tension of saturated aqueous solutions γass:
∆γ = γass - γoc
(15)
Finally, the surface energy, Es, associated with this interface may be expressed as follows
Es )
2WE∆γ δX
(16)
In the treatment developed here, we introduced the simplifying assumption: the fact that a surface tension depends upon the radius of curvature was ignored, and the values of γ were taken to be those for a bulk liquid.23 Affinity Coefficient of Water Desorption Dubinin and Serpinski24-27 had developed the concept of water adsorption as a result of the manifestation of (22) Adamson, A. W. Physical Chemistry of Surfaces; Mir: Moscow, 1979; pp 61 and 93 (Russian edition). (23) Steel, W. A. The interaction of gases with solid surfaces; Pergamon Press: Oxford, 1974; pp 245-269. (24) Dubinin, M. M.; Zaverina, E. D.; Serpinsky, V. V. J. Chem. Soc. 1955, 1760. (25) Dubinin, M. M. Porous Structure and Adsorption Properties of Activated Carbons; Military Press: Moscow, 1965; pp 63-69 (Russian). (26) Dubinin, M. M. Carbon 1981, 18, 355-364. (27) Dubinin, M. M.; Serpinsky, V. V. Carbon 1981, 19, 402.
hydrogen bonds. According to this theory, two kinds of adsorption centers are involved in water adsorption. The main causes of water adsorption are primary adsorption centers, i.e. oxygen surface complexes. Each adsorbed water molecule is a secondary adsorption center, which is also capable of forming hydrogen bonds with other water molecules. It was mentioned that micropores of AC in a solution are always filled with solution components and θ ) θs + θas ) 1, where θas is the fraction of micropores occupied with an aqueous solution. Since a liquid mixture is an almost incompressible condensed phase, the solute accumulation in micropores displaces an equivalent amount of water, and adsorption of 1 mol of a solute with the molar volume Vms leads to the displacement of
D ) Vms/Vmw
(17)
water moles, where Vmw is the molar volume of water and D is a coefficient of displacement in the adsorbed phase. For example, 1 molecule of benzene displaces 4.89 molecules of water in micropores, and for desorption to occur, about five molecules of water must be removed and different hydrogen bonds must be broken. As a result, the energy of desorption must be more than the energy of adsorption on the secondary centers and less than that on the primary centers. To describe water desorption, the integrated parameter of adsorption on different adsorption centers should be taken into consideration. It may be done on the basis of immersion heats, ∆hi, of AC into water and organic substances, which are related to the distribution of adsorption energies over adsorption sites as follows28
-∆hi )
∫01qnet(T,θ) dθ
(18)
Here qnet is the net adsorption heat. Stoeckli28,29 showed that the enthalpy of immersion of microporous solid into a liquid, ∆hi (J/g), may be evaluated by the expression
-∆hi )
βE0W0 (1 + RT)Γ(1 + 1/n) Vm
(19)
where R is the thermal expansion coefficient of the adsorbate and Γ is the tabulated gamma function. Let βdw ) β be a coefficient calculated from the enthalpy of immersion into water by eq 19. The objective now is to show that βdw may be taken to be an affinity coefficient of water desorption. In doing so we must be sure that the principles underlying eq 19 are observed: (i) eq 6 describes water adsorption; (ii) the characteristic curve of water adsorption in the coordinates A-θ shows an affinity relationship. Stoeckli et al.30 demonstrated that the water adsorption isotherm on AC can be decomposed into two contributions, each of which can be treated as an D-A isotherm. The initial section suggests the presence of sites with characteristic energies in the range 5-8 kJ mol-1; the second part of the isotherm suggests the presence of sites with the lower value of characteristic energies. This conclusion is compatible with the earlier model of Dubinin and Serpinski. Stoeckli and Jakubov31 showed that a plot of W/W0 vs RT ln P0/P leads to the unique characteristic curve for carbons, as was found in the case of organic (28) Stoekli, H. F. Izv. Akad. Nauk SSSR, Ser. Khim. 1981, 63. (29) Stoeckli, F. Adsorpt. Sci. Technol. 1993, 10, 3-17. (30) Stoeckli, F.; Currit, L.; Laederach, A.; Centeno, T. A. Chem. Soc. Faraday Trans. 1994, 90 (24), 3689-3691. (31) Stoeckli, F.; Jakubov, T. Chem. Soc. Faraday Trans. 1994, 90 (5), 783-786.
Benzene Adsorption from Dilute Aqueous Solutions
Langmuir, Vol. 12, No. 23, 1996 5609
vapors. The coefficient calculated from eq 19 characterizes not the primary or the secondary centers each taken separately but the average value over all adsorption centers. For this reason, it may be taken to be the affinity coefficient for water desorption. In contrast to the adsorption of organic compounds, βdw * βw, where βw is the affinity coefficient of water adsorption, and water adsorption on microporous carbons manifests a characteristic hysteresis loop. If the parameter n in eqs 2 or 3 remains invariable for different vapors, then2,3
(A/A0)θ ) E/E0 ) β
(21)
where Ad is the differential molar work required for water desorption. From the tabulated values of the gamma function, it appears that if n varies from 2 to 6, the numerical value of eq 19 increases by only 5%.31 A very slight dependence of the gamma function on n justifies the approximation (eq 21). Equation 21 establishes the relation between the energy of water desorption and the energy of adsorption of the reference compound in the same micropore. Another expression for βdw may be derived from the ratio of the enthalpies of immersion ∆hiw and ∆hi0
∆hiw βdwVm0(1 + RwT) ) ∆hio β0Vmw(1 + R0T)
(22)
∆hiw(1 + R0T) ∆hi0D(1 + RwT)
(23)
The thermal expansion coefficients of water and benzene at 298 K are equal32 to 0.207 × 10-3 and 1.237 × 10-3 grad-1, respectively, and eq 23 leads to
βdw ) 0.264∆hiw/∆hi0
(24)
Derivation of Equation of Adsorption from Aqueous Solutions Consider the change in the Gibbs free energy in the adsorbent-solution system. To fill a micropore with 1 mol of a solute at a given θ, the maximal molar work of adsorption supplied by the field of the adsorption forces is made up of the work of adsorbate compression, Ac, the work of desorption, Ad, of D water moles, the work required for the interface formation, Es, and the work required for overcoming the solute-solvent interaction, Aa,
-∆G ) A ) Ac + Ad + Es + Aa
(26)
The second term of eq 25, Ad, is determined by the affinity coefficient of water desorption (eq 21) as follows
Ad ) DβdwA ) DβdwRT ln Cs/C
(27)
For 1 mol of adsorbate, W in eq 16 is substituted by Vms, and the total change of the free energy is then made up as follows
[
∆G ) - (1 + Dβdw)RT ln
]
Cs 2VmsE∆γ + + ∆U C δX
(28)
The first and the second terms of eq 25, Ac and Ad, depend upon θs, while Es and Aa are constants. The first term in eq 28 is equal to the energy required both for the adsorbate compression from C to Cs and for the synchronous displacement of water from the adsorbed phase. This term, Acd, expresses the work related to the changes of concentrations in a solution.
Cs Acd ) (1 + Dβdw)RT ln C
(29)
Chen and Yang37 showed that for a gas/solid adsorption the characteristic energy is directly related to the mean potential (Φ) in the pore
KN0Φ ) βE0 ) E
Benzene is usually taken as the primary reference and by convention β0 ) 1. Taking into account eq 17, one has
βdw )
Aa ) ∆U
(20)
where the subscript 0 denotes the parameters of the reference substance. Despite the fact that n for organics and water has different values, the following approximation can be made from eq 20
(Ad)θ ) βdw(A0)θ
regular solutions; i.e., all conditions for forming an ideal solution are satisfied except that the interchange energy, ∆U, is not zero.33-36 As a result, we can take into account solute-solvent interaction by introducing the fourth term, Aa, which may be approximated by the interchange energy
(25)
where Ac and Es are given by eqs 7 and 16. Equation 7 is valid for an ideal solution. Deviations from the laws of perfect solutions are expressed formally by introducing the activity coefficient. In our case, since we examine only dilute solutions, we can consider them to be the (32) Chemical Engineers’ Handbook; Perry, J. H., Ed.; Khimia: Leningrad, 1969; Vol. 1, p 66 (Russian edition).
(30)
where K and N0 are constants. In contrast to adsorption from the gas phase, the mean energy arising from the force field in the pore and available for the adsorbate compression and water displacement, Ea, is less than E, i.e.
(
Ea ) E - (Es + Aa) ) E 1 -
)
2Vms∆γ ∆U δX E
(31)
To derive an equation analogous to the D-A expression, eqs 29 and 31 must be substituted in eq 6. Finally, we have for solute adsorption, a (mol/g), from water solution
{[ (
Cs W0 W0 C θ ) exp a) Vms s Vms 2Vms∆γ ∆U βE 1 δX E (1 + Dβdw)RT ln
]} 2
)
(32)
Equation 32 demonstrates the relationship between adsorption of the organic component in micropores on the one hand and the parameters of active carbons, the solution components, and the conditions of operation on (33) Guggenheim, E. A. Mixtures Clarendon Press: Oxford, 1952; p 29. (34) Prigogine, I. The molecular theory of solutions; North-Holland Publishing: Amsterdam, 1957; pp 5-20. (35) Shachporonov, M. I. Introduction in molecular theory of solutions; Gostechizdat: Moscow, 1956; p 220 (Russian). (36) Moelwyn-Hughes, E. A. Physical Chemistry; Foreign Press: Moscow, 1962; pp 676-734 (Russian edition). (37) Chen, S. G.; Yang, R. T. Langmuir 1994, 10, 4244.
5610 Langmuir, Vol. 12, No. 23, 1996
Dobruskin
Table 1. Structural Characteristics of Active Carbonsa adsorbent
origin
E, kJ/mol
Vmi, cm3/g
AG-3 (1st substructure) AG-3 (2nd substructure) AG-3 (total) BAC
coal coal coal birch
20.1 9.7
0.19 0.13 0.32 0.23
a
25.0
Vme, cm3/g
Vmc, cm3/g
x, nm
Sme, m2/g
ame, cm3/g
moxygen wt %
65b 57c
0.024 0.021
0.5 6.0
0.60 1.20 0.09 0.09
0.55 1.40
0.48
Data from ref 14 and ref 12 p 40. b Data from ref 12 p 31. c Data from ref 46 p 178.
the other hand. The value of βdw shows the influence of the carbon surface on adsorption from aqueous solutions. Chemical modification of the carbon surface by the oxidation increases the concentration of oxygen complexes and the heat of immersion into water.38-40 The equation predicts decreasing of solute adsorption on the oxidized AC through the increasing of ∆hiw and βdw. The term, 2Vms∆γ/δX describes an influence of the interface energy on solute adsorption. Its value is entirely determined by the average sizes of carbon crystallites, crystallite packing, and merging and depends only upon the geometrical structure of a carbon, being the characteristic parameter of the given AC. It is possible to define the formal coefficient, β*, as follows
β* )
(
)
2Vms∆γ ∆U 1 1d δX E 1 + Dβ w
(33)
The authors of the studies17,18 addressed to the correlation between adsorption phenomena at the gas/solid and liquid/ solid interfaces introduced the expression β* ) E*/E called the “interface affinity coefficient”. Equation 33 shows that β* depends on the properties of AC and a solution and appears not to be a measure of the similarity as is defined by eq 19. It is known that many real adsorbents may be characterized by bi- and polymodal energy distributions.41,42 In such cases, the experimental data cannot be described satisfactorily by the monomodal Weibull distribution. Dubinin used two-term expressions43 to determine the adsorption, a (mol/g), on microporous solids that possess bimodal structure:
a ) a1 + a2 )
[ ( )]
W01 A exp Vm βE1
2
+
[ ( )]
W02 A 2 exp Vm βE2 (34)
where a1 and a2 are the adsorption in the first and the second substructures and W01, W02, E1, and E2 are the limiting volumes of adsorption space and the characteristic energies of the first and the second substructures, respectively. Stoeckli8,33 suggested that the D-A equation applies only to a structurally homogeneous system, and agreement with experiments would be improved by introducing the distribution function to take account of the heterogeneous structure. But Dubinin44 showed that a two-term expression describes the experimental data with the same accuracy as Stoeckli’s equation. Just as (38) Dubinin, M. M.; Poliakov, N. S.; Petukhova, G. A. Adsorpt. Sci. Technol. 1993, 10, 17-26. (39) Carrott, P. J. M. Adsorpt. Sci. Technol. 1993, 10, 63-73. (40) Poliakov, N. S.; Petukhova, G. A.; Vnukov, S. P.; Shevchenko, A. O. Adsorpt. Sci. Technol. 1993, 10, 165-173. (41) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (42) See ref 22 pp 485-491. (43) Izotova, T. I.; Dubinin, M. M. Zh. Fiz. Khim. 1965, 39, 27962803. (44) Dubinin, M. M. In Adsorption in Micropores; Dubinin, M. M., Serpinski, V. V., Eds.; Nauka: Moscow, 1983; pp 186-192 (Russian).
for gas adsorption, when AC is characterized by the twoterm equation, this two-term expression must be applied to calculate the contribution of each substructure to the overall adsorption from solutions. Comparison with Experimental Data The method under consideration was applied to benzene adsorption14 from aqueous solutions at 298 K. For a water-benzene system the interchange energy is equal to zero,45 and eq 32 takes the form
{[
(
)
]}
∆hiw Cs 1 + 0.264D RT ln W0 ∆hi0 C a) exp Vm0 2Vm0∆γ E 1δX
(
)
2
(35)
Table 1 lists the parameters of the Russian active carbons employed. The experimental values of immersion heats were unknown and estimated proceeding from the relationship between the oxygen content and immersion heats.47 Kazmierczak et al. found an empirical linear correlation between ψ ) (∆hi0 - ∆hiw)/∆hi0 and the total oxygen content, Tox, of the ashless active carbons:
ψ ) 0.74 - 5.28Tox
(36)
Since the oxygen contents (Table 1) are related to the commercial carbons, it was assumed that only half of the oxygen is bounded to the surface complexes, the second part being bounded with ash elements. The ∆hiw/∆hi0 ratios calculated from eq 36 for AG-3 and BAC are equal to 0.27 and 0.42, and the βdw values (eq 24) are 0.071 and 0.111, respectively. Benzene adsorption on the mesopore surface was taken into consideration as follows. At a monolayer benzene thickness12 of 0.37 nm, the maximal adsorption on the mesopore surface, ame (cm3/g), is 0.37Sme. Approximating that adsorption in the mesopore does not differ significantly from that in large-sized supermicropores of the second substructure, one can consider the limiting adsorption volume of the second substructure to be equal to (W02 + ame). A more accurate calculation should be done on the basis of adsorption on the unporous surface. The results of the calculations are given in Tables 2 and 3. Despite the fact that in individual cases deviations of a reach 19%, on the whole agreement between the experimental and calculated values of adsorption is satisfactory. The value of δ in Table 2 was estimated proceeding from ∆γ at the interface between benzene and saturated benzene aqueous solution, i.e. ∆γ ) 34.1 erg/ cm2 (dynes/cm).48 Such an assumption leads to δ )1.3, which is 17% less than the δ ) 1.56 found by Dubinin.21 (45) Cantor, C. R.; Schimmel, P. R. Biophysical chemistry, Part 1; Mir: Moscow, 1984; p 265 (Russian edition). (46) Buturin, G. M. High porous carbonaceous materials; Khimia: Moscow, 1976; p 178. (47) Kazmierczak, J.; Biniak, S.; Swiatkovski, A.; Radeke, K-H. J. Chem. Soc., Faraday Trans. 1991, 87 (21), 3557-3561. (48) . Chemical Handbook; Nikolski, B. P., Ed.; Khimia: Leningrad, 1962; Vol. 1, p 1026 (Russian).
Benzene Adsorption from Dilute Aqueous Solutions
Langmuir, Vol. 12, No. 23, 1996 5611
Table 2. Parameters Characterizing Benzene Adsorption from Aqueous Solution carbon
W0, cm3/g
interface area, m2/g
2Vms∆γ/δX
δ
βdw
β*
β*E, J/mol
AG-3, first substructure AG-3, second substructure AG-3, total structure BAC
0.19 0.16 0.33 0.23
488 162 650 740
0.387 0.387 0.387 0.385
1.3 1.3 1.3 1.3
0.071 0.071 0.071 0.111
0.457 0.457 0.457 0.398
9200 4440
tion23 occurring in mesopores. Kiselev50 proposed the general thermodynamic relationship
Table 3. Benzene Adsorption on the AG-3 experiment a, mmol/g C/Cs 0.000 394 0.0007 88 0.000 94 0.001 535 0.002 169 0.002 687 0.002 939 0.003 14 0.003 704 0.004 425 0.004 492 0.005 274 0.006 385 0.007 542 0.008 788 0.009 028 0.009 369 0.010 927 0.012 52 0.017 19 0.021 095 0.033 827 0.036 547 0.056 704 0.083 464 0.142 011 0.236 983 0.277 989 0.361 397 0.486 872
0.024 0.052 0.054 0.099 0.128 0.179 0.189 0.192 0.245 0.276 0.283 0.308 0.297 0.324 0.388 0.403 0.403 0.468 0.505 0.693 0.808 1.154 1.245 1.626 2.033 2.501 3.008 3.078 3.402 3.403
calculated values a i, a2, a, mmol/g mmol/g mmol/g 0.025 0.053 0.064 0.103 0.141 0.170 0.184 0.194 0.222 0.257 0.259 0.294 0.339 0.382 0.425 0.433 0.444 0.492 0.537 0.652 0.733 0.940 0.976 1.188 1.380 1.638 1.858 1.917 2.003 2.079
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.002 0.003 0.005 0.011 0.018 0.051 0.060 0.140 0.267 0.555 0.953 1.091 1.317 1.547
0.025 0.053 0.064 0.103 0.141 0.170 0.184 0.194 0.222 0.257 0.259 0.294 0.340 0.383 0.427 0.435 0.446 0.495 0.542 0.663 0.751 0.991 1.036 1.328 1.647 2.193 2.812 3.008 3.320 3.626
9940
deviation, % 4.4 2.3 18 3.9 10.2 -5 -2.8 1.1 -9.2 -7 -8.3 -4.5 14.3 18.2 10 7.9 10.7 5.8 7.2 -4.4 -7.1 -14.1 -16.8 -18.3 -19 -12.3 -6.5 -2.3 -2.4 6.6
From eqs 5 and 11, the average radius of carbon crystallites is given as
r ) δX/E
(37)
For the first and the second substructures of AG-3 and for BAC, we obtain crystallite sizes (diameters) of 1.6, 3.2, and 1.25 nm, respectively. If δ ) 1.3, the interface areas for BAC and AG-3 are equal to 740 and 650 m2/g, respectively. For δ ) 1.56, these values reduce to 615 and 542 m2/g. If one proceeds from δ ) 1.56, he should assume that ∆γ ) 41 erg/cm2. It must be admitted that there is not enough data for a well-defined choice of the values of ∆γ and δ, and only their ratio ∆γ/δ can be calculated. Further experimental study in this direction is desirable. Further Discussion and Conclusions Interface Energy. The problems of interface formation and interface energy are related to the fundamental problem of the boundary between true and colloid solutions. A general prerequisite for the stable existence of an interface between two phases is that the free energy of formation of the interface be positive.49 In a narrow space of the micropore which can accommodate three to four molecules, the concept of surface tension loses its physical meaning. But for mesopores this conception undoubtedly can be used. Thus, the concept of surface tension is applied in the theory of the capillary condensa(49) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1990; p 1.
γ dS ) ∆µ da
(38)
where γ is an interfacial tension of the liquid/vapor interface, dS is the surface that disappears when a pore is filled by capillary condensation, ∆µ is the change in chemical potential, equal to RT ln P/P0, and da is the number of moles of liquid taken up by the pore. A mesopore surface, Sme, may be calculated51,52 from the integral form of eq 38
γSme )
∫aa A da e
(39)
b
where ab and ae are the adsorption values related to the beginning and the end of the capillary condensation. According to Kiselev, the work required for an interface formation, γSme, is supplied by the adsorption potential. This equation was thoroughly verified and recommended for a calculation of mesopore area.25 If adsorption occurs in a solution, an interfacial energy will be associated with the interface between occupied micropores and the extramicropore space; this situation is due to the difference in a solute concentration and micropore “ability to promote state of matter which cannot exist in the external world of homogeneous bulk matter”.53 The model of AC structure employed in this study is generally sufficiently realistic for use in adsorption problems. This is the same model21,44 that was employed for deriving expression 5. It should be emphasized that the physical meanings of Si in eq 10 and of Sme in eq 39 are quite different. An interface area is an area dividing two occupied volumes, while Sme is the surface of a pore skeleton covered with a monolayer film. The comparison of the total geometric surface of the micropore walls29 Smi ) Vmi/x and interface area (eq 10) leads to
Si ) Smi/δ
(40)
The theory predicts that, for AC samples with the same value of E, the carbon with the minimum interface area or the maximum size of crystallites manifests the best adsorption properties. Weibull Distribution in Solutions. It is expedient to discuss eq 32 proceeding from the properties of the distribution function. In general, the Weibull density distribution function, fw, over the Weibull variable, χ, is described by scale, shape, and location parameters,4 σ, λ, and µ, respectively.
fw(χ,µ,σ,λ) )
(σλ)(χ -σ µ)
λ-1
2
[ (χ -σ µ) ]
exp -
(41)
Here µ < χ; -∞ < µ < ∞; 0 < σ, λ. The expected value of (50) Kiselev, A. V. Usp. Khim. 1945, 14, 367. (51) Kiselev, A. V.; Mikos, N. N. Zh. Fiz. Khim 92, 1948, 22, 10431057. (52) Kiselev, A. V. In Method of investigation of the porous structure of highly dispersed substances; Academy Press: Moscow, 1953; p 86 (Russian). (53) Nicolson, D. J. Chem. Soc., Faraday Trans. 1996, 92, 1-10. (54) Flood, E. A. In The Solid-Gas Interface; Flood, E. A., Ed.; Mir: Moscow, 1970; pp. 62-75 (Russian edition).
5612 Langmuir, Vol. 12, No. 23, 1996
Dobruskin
the variable χj is given by the expression
χj ) σΓ(1 + 1/λ)
(42)
βE0 ) σ; n ) λ; µ ) 0; χ ) A
(43)
Although we referred to RT ln Cs/C as a variable, the expression Acd ) (1 + Dβw)RT ln Cs/C may equally well be thought of as a variable. In this case, eq 45 transforms into
For gas adsorption
Experimental data show that the shape parameter, n, for adsorption from gas and liquid phases is invariable. The distribution of a carbon adsorption volume over the adsorption potential is determined by its porous structure. From a consistent theoretical treatment of gas/ solid and liquid/solid isotherms one expects the same result to be obtained. Although the properties of AC are invariable, nevertheless, the distribution functions derived from the isotherms will be deferent. For gas/solid adsorption, the distribution of the actually empty volume (1 - θ) will be found, while for adsorption from solutions one will observe the micropore volume occupied with a solution (1 - θs). Since the scale parameter determines the average value of the variable (eq 42) and the work supplied by the adsorption field is being partly consumed for an interface formation and for overcoming of solutesolvent interaction, σ in a solution is diminished to σ ) β(E - 2VmsE∆γ/δX - ∆U). Besides, the distributions under discussion are defined with respect to different variables. Substitution of RT ln P0/P by the variable RT ln Cs/C leads to a reduction of the scale parameter to σ/(1 + Dβw); the latter is due to synchronous displacement of water from the adsorbed phase. As a result, in the case of liquid/solid adsorption, one has
σ ) E* )
(
)
2Vms∆γ ∆U βE 1d δX E 1 + Dβ w
(44)
A liquid/solid isotherm is displaced to the right by the relative concentration with respect to the gas/solid isotherm, and the average value of RT ln Cs/C over the range 0 < θ < 1 is less than the average value of the adsorption potential. Finally, the Weibull density function of the distribution of the adsorption volume over Ac takes the form
fw(Ac) )
[
βE 1 + Dβdw
{[
exp -
n 2Vms∆γ ∆U 1δX E
)]
(
(
βE 1 + Dβdw
Acn-1 ×
n
Ac 2Vms∆γ ∆U 1δX E
]}
)
n
(45)
The latter expression is a generalized form of the Weibull distribution of the adsorption volume. One can consider a single-gas phase to be a specific solute of vacancies and the values of ∆γ, βd, and ∆Ha to be equal to zero. Equations 2 and 3 are associated by the relation
θ)
[
∫A∞f(T,θ) dθ ) exp -(βE1 ) 1
A
]
2
(46)
Equation 46 expresses the main postulate of the potential theory;53 i.e., the occupied volume is equal to the pore space in which A > A1. Equation 32 is associated with eq 45 in the same manner
W0 a) Vms
fw(Acd) )
n 2Vms∆γ ∆U βE 1 δX E
[ (
[( (
exp -
)]
n
Acdn-1 ×
)) ]
Acd 2Vms∆γ ∆U βE 1 δX E
n
(48)
where
(
β* ) 1 -
)
2Vms∆γ ∆U δX E
and
(
E* ) Eβ 1 -
)
2Vms∆γ ∆U δX E
is the scale parameter. Conclusion. In this article we are concerned only with those aspects of equilibrium that are relevant to adsorption from aqueous solutions with complete displacement in the adsorbed phase. In our treatment only a complete displacement imposes a very useful simplification which consists of smearing out the equilibrium in the adsorbed phase. There is no particular theoretical significance when solvent adsorption occurs. We also started from another very strong simplification that only adsorption of nonelectrolytes would be discussed. In the case of adsorption of electrolytes, the energy of ion repulsion in micropores and the energy associated with the charged interface should be considered. These factors induce further reduction of the scale parameter and isotherm displacement to the right by concentrations with respect to adsorption of the same nondissociating compound. The numerous examples of such displacement are given by the pH influence on adsorption of weak electrolytes.12,13 Generally, the properties of an interface will be affected by a change in either of the two phases involved. It may be concluded that the model proposed here manages to take into account many of the principal factors involved in liquid/solid adsorption and holds great promise for future generalization. But any complications of the proposed theory are expedient only after the independent verification of eq 32. It should be mentioned that examination of experimental data for model testing showed that every set of experimental data was incomplete. Eltekov’s study was used for this purpose because his experiments were based on ACs with known porous structures and reported in tabulated forms. Nevertheless, the values of βdw had to be estimated indirectly because of the lack of ∆hiw and ∆hi0. Further studies based on ACs with well-known porous structures and immersion heats are desirable: (i) experiments with oxidized carbons to verify the contribution of ∆hiw, ∆hi0, and βdw; (ii) experiments with adsorption of different solutes on the same active carbon to verify the contribution of the interfacial tension and, probably, solute-solvent interactions. Glossary
∫A fw(Ac) dAc ∞ c
(47)
A
differential molar work of adsorption; adsorption potential
Benzene Adsorption from Dilute Aqueous Solutions Aa Ac, Ad Acd a a1, a2 ab, ae ame B b C, Cs D d E E* Es Ea ∆hi ∆hi0, ∆hiw K m N0 n qnet P, Ps r rj R Si Smi, Sme T Tox
work of overcoming of solute-solvent interaction work of solute compression; work of water desorption energy of both the adsorbate compression and water displacement solute adsorption (mol/g) adsorption in the first and the second substructures adsorption values related to the beginning and the end of the capillary condensation maximal adsorption on the mesopore surface constant, eq 14 constant, eq 8 equilibrium and saturated concentration in water coefficient of displacement in the adsorbed phase constant, eq 14 characteristic adsorption energy Weibull distribution parameter for adsorption from solutions surface energy associated with an interface energy available for the adsorbate compression and water displacement immersion heat (J/g) immersion heat into a reference substance (benzene) and water constant, eq 30 molecule mass constant, eq 30 Weibull distribution parameter net differential adsorption heat partial and saturated pressures in the gas phase radius of the round base of the crystallites ratio of average residence times gas constant interface between micropores occupied with a solute and a solution surface of the micropore walls; mesopore surface absolute temperature total oxygen content
Langmuir, Vol. 12, No. 23, 1996 5613 ∆U Vmi(me,mc) Vms, Vmw W W0 X x
interchange energy volumes of pores molar volume of a solute and water occupied adsorption volume limiting adsorption volume empirical constant in eq 5 half-width of the split pore
Greek Symbols R β, βdw β* Γ δ µ λ σ γoc, γas γass θ θsθ as χ χj ψ Φ
thermal coefficient of limiting adsorption affinity coefficient; affinity coefficient of water desorption coefficient that is equal to E*/E the gamma function constant, eq 11 the Weibull location parameter the Weibull shape parameter the Weibull scale parameter surface tensions of the organic component and aqueous solution surface tension of saturated aqueous solutions degree of micropore filling fraction of micropore occupied with a solute and a solution the Weibull variable expected value of the Weibull variable coefficient in eq 36 mean potential in a pore
Subscripts as aqueous solution ass aqueous saturated solution I immersion; interface m molar value mi, me, mc micro-, meso-, and macropore 0 parameters of the standard substance 1, 2 number of a substructure (eq 34) s solute w water Superscript d
water desorption LA960175F
