Langmuir 1994,10,4244-4249
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Theoretical Basis for the Potential Theory Adsorption Isotherms. The Dubinin-Radushkevich and Dubinin-Astakhov Equations S. G. Chen and R. T. Yang* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received May 16, 1994. In Final Form: August 11, 1994@ An isotherm equation is derived for adsorption of gases and vapors on microporous and mesoporous solids from statistical meclianical principles. The adsorbed phase is assumed to be a two-dimensionalfluid subjectedto a force field represented by a mean potential (45.). It is shown the heretofore empirical DubininAstakhov (D-A) equation and Dubinin-Radushkevich (D-R) equation (i.e., the potential theory) are approximated forms of this isotherm. For adsorption in micropores and mesopores,the fractional adsorption (0) is much greater than the relative pressure (PlPo); the general isotherm is thereby reduced to the D-A and D-R equations. From the approximated forms of the general isotherm, it is shown that the exponent n in the D-A equation is related to the degree of pore filling at the reference state ( 7 0 ) ; as a consequence it depends on the adsorbate as well as the pore structure of the adsorbent. Moreover, the characteristic energy of adsorption ( E )in the D-A and D-R equations is proportional to the mean potential (45.). Thus, the dependence ofE on pore size can be obtained directlyfrom first principleswithout resorting to empirical correlations. The low-pressurelimit of the general isotherm is Henry's law. It is shown that from Henry's constant, i.e., one adsorption data point, it is possible to calculate the heat of adsorption.
Introduction For physical adsorption of gases and vapors on microporous solids, the Dubinin-Polanyi theory of micropore filling has been of long-standing importance for both data correlation and gaining physical insight. In this theory, the distribution function of micropore filling, 8, is expressed by
8 = f(A/E,n)
(1)
whereA = -RTln(P/Po) and is the differential molar work of adsorption, E is the characteristic energy of adsorption, and n is a dimensionless parameter. The simple reason for grouping ME is to make it dimensionless.' By empirically matching the Weibull distribution function to (1 - €9,the resulting equation is the Dubinin-Astakhov (D-A) isotherm1S2
e = exp[
-($r] I")&(= exp[
(2)
adsorption systems involving only van der Waals forces, and the D-R equation is particularly useful for adsorption on activated carbon. Despite many attempts that have been made, no satisfactory theoretical basis has been developed for the D-R and D-A equation^.',^ Temperature invariance was the basic premise in formulating the D-R and D-A equations. However, it was noted by Stoeckli (as well as Dubinin and coworkers)8-10that the characteristic energy of adsorption (EO)decreases with increasing differential work of adsorption (A). This observation led StoeckligJoto postulate that the D-R equation applies only to structurally homogeneous systems, i.e., micropores with the same dimension. Consequently, the isotherm takes the form (4)
where g(E) is the energy distribution function of the micropores, and empirically, E is related to pore dimension by9,lO
where EOis the characteristic energy of adsorption for a E = Xlx (5) reference vapor and is termed similarity coefficient. where x is the half-width of the slit pore and X is an Similarly, for a Gaussian distribution function, one gets the empirical Dubinin-Radushkevich (D-R) i ~ o t h e r m ~ - ~ empirical constant. The basis for eq 5 and other forms for the E(x)relationship have been discussed exten~ively.~J'-~~ In this work, an isotherm equation for adsorption is 8 = exp - (3) derived from statistical mechanical principles. For appreciable degrees ofpore filling, e.g., 8 =- 0.1,this isotherm equation is reduced to the D-A and D-R equations. For The D-R and D-A isotherm equations have been used 8 0, this isotherm equation is reduced to Henry's law. with much success to correlate a large amount of adsorption data.lv5+ These equations generally apply well to (7) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous
[
-
Abstract published inAduanceACSAbstract, October 15,1994. (UDubinin, M. M. Progress in Surface and Membrane Science; Cadenhead,A., Ed.;Academic Press: New York, 1975;Vol. 9, pp 1-70. (2) Dubinin, M. M.; kstakhov, B. A. Izv. Akad. Nauk SSSR, Ser. Khim. 1971,5. (3) Dubinin, M. M.; Radushkevich, L. V. Dokl. Acad. Nauk SSSR 1947, 55, 331. (4) Radushkevich, L. V. Zh.Fiz. Khim. 1949,23, 1410. (5)Dubinin, M. M. Chem. Rev. 1D60, 60, 235. (6) Dubinin, M. M. Chemistry and Physics of Carbon;Walker, P. L., Jr., Ed.; Marcel Dekker: New York, 1966; Vol. 2, pp 51-120. @
Solids; Elsevier: Amsterdam, 1988; Chapter 2. (8) Rudzinski, W.;Everett,D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992; Chapters 5 and 6. (9) Stoeckli, H. F. J . Colloid Interface Sci. 1977, 59, 184. (10) Dubinin, M. M.; Stoeckli, H. F. J . Colloid Interface Sci. 1990, 75 . -, .?A - -. (11) McEnaney, B. Carbon 1987,25, 457. (12) Dubinin, M. M. Carbon 1989,27,457. (13) Stoeckli,H. F.; Ballerini, L.; De Bernardini, S.Carbon 1989,27, mi ""_. (14) Stoeckli, H. F. Carbon 1990,28, 1. ( 1 5 )Jaroniec, M.;Choma, J.; Lu, X. Chem.Eng. Sci. 1991,46,3299.
0743-7463/94/2410-4244$04.50/00 1994 American Chemical Society
Zsotherm Equation for Adsorption
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From this isotherm equation, we gain further insights into the parameters in the D-A and D-R equations and the E(x) relationship.
The Helmhotz free energyAa,pressureP", and chemical potential ,usfor the above system are
+
As = -kT In ZS, = k T In Am N@- kT In Q& (10) Derivation of Adsorption Isotherm Consider a system of N fluid molecules exposed to a a In Q& force field within a given pore size and geometry. Each P = - ( E ) TJV =kT(--+-)TJV (11) molecule in this force field has a potential energy of @h), where rk is the position coordinate of molecule k in the system. The partition function TNof the canonical a In Q& ensemble of the system can be expressed as16-18 ,us = = k T In A' @ - kT(-) aN T,Vs (12)
av
(g)T,vs+
where qrand qvare the rotational and vibrational partition functions of the N fluid molecules and are separated from the translational partition function and the superscript s denotes adsorbed phase. d is a constant indicating the dimensions of the space; for three-dimensional space d is 3 and for two-dimensional space d is 2. For adsorption in micropores, however, it is possible that for a certain range of pore sizes, the value of d may lie between 2 and 3. U(r,) is the potential energy between molecules i and j at distance ry and @(rk)is the potential energy of molecule k at rk, which arises from the force field in the pore. Vs is the volume of the system, d f l = drl drz ... d r for ~ all positions, y = 1/(kT) and
A = hl(2nmkT)1/2
(7)
is the de Broglie thermal wavelength, h is Plank's constant, m is the molecule weight, k is Boltzmann's constant, and T is the absolute temperature. The potential energy is a function of position in the pore. As an approximation, however, we assume that the force field may be approximated by a mean value, @, for all adsorbate molecules in the pore. This is not a bad approximation for the major range of 8 for most gas-sorbent systems, and is certainly a good one for molecules in the same "layer'). For porous materials, this assumption would apply to micropores and would not apply as well to mesopores. Consequently
where Q& is expressed by .l
where the superscript s, again, denotes the adsorbed phase. For the two-dimensional case, the pressure P" is the spreading pressure, and the volume Vs is actually the surface area. From the configurational integral, Q&, the equations above yield the Helmholtz free energy, As, pressure, Pa, and chemical potential, ,us. For confined pore spaces, the value ofd should be close to 2. We may then construct the configurational integral Q& through a two-dimensional fluid equation of state. Hendersonlg proposed a two-dimensional equation of state for the hard disk, which is a modification of the scaled-particletheory (SPT)20and takes the following form
N
where 7 = ( n l 4 ) ~u ~is ~the ~ diameter ; of the hard disk and ea= NN". Thus, 7 represents the fraction of the surface that is covered. The virial coefficients generated by eq 13 are in good agreement, up to the seventh virial coefficient, with the results of computer simulation.21 Hence eq 13 will be used as the reference system for our real fluid. Equations for more complicated fluids such as planar hard-dumbbells and planar hard-convex are also a ~ a i l a b l e . ~ ~ , ~ ~ For real fluids, one may introduce a perturbation term into eq 13 as for the three-dimensional equations of state, e.g. the van der Waals equation of state:
where a(T)is function of T only. The physical meaning of the second term is the same as that of the second term in the van der Waals equation of state. From eqs 10-14, we have
+ g9 (1 -
and is termed configurational integral. In the above equations, qr and q. are the internal partition functions of the system. If we assume that they do not change during the adsorption process (which is a good assumption), then they have no effect on the equilibriumofthe adsorbent-adsorbate system so we may drop them in the ensuring derivations.
and the chemical potential ,us
(16) Reed, T. M.;Gubbins, K. E. Applied Statistical Mechanics; McGraw-Hill: New York, 1973. (17) Lee, L.L.Molecular Themdvnumics ofNonidea1Fluids;Butterworths: Sroneham, MA,1988. (18) Steele, W.A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974; Chapter 4.
(19) Reiss, H.; Frisch, H. L.;Lebowitz,J. L. J. Chem.Phys. 1959,31, 369. (20) Ree, F. H.; Hoover, W. G. J . Chem. Phys. 1967, 6,4181. (21) Henderson, D.Mol. Phys. 1975,30,971. (22) Boublik, T.Mol. Phys. 1975,29, 429. (23) Boublik, T.Mol. Phys. 1988, 63, 685.
ln(eaA2)- 1 - g 7 ln(1 - 11)
L-
Chen and Yang
4246 Langmuir, Vol. 10, No. 11, 1994
The particular form of the perturbation term (in eq 14) is chosen so it leads to a simple expression, ag,in eq 16. For the bulk (fluid)phase, if the ideal gas law applies, one obtains16-ls Ag = -kT In 2% = -kTN[ln(ggA3) - 11 pg =
where K and n are constants resulting from the approximation and have no physical meaning. The approximation made in eq 24 is purely mathematical. The goodness of the approximation will be discussed further. Substituting eq 24 into eq 22, we have
(17)
(g)T,vg
= kT ln(ggA3)
where all the symbols have the same meanings (as in eqs 10 and 12) and the superscript g denotes the bulk (fluid) phase. Because pg = pus,from eqs 16 and 18, we have
In 8 - F(qO)K(-ln 8)1/n= nI):(
(25)
For adsorption in microporous and mesoporous materials at subcritical temperatures, as the relative pressure (PIPO) is increased, the micropores start to fill quickly at very low PIPo values, usually around This is seen in a large amount of experimental and theoretical m0dels.~5~ Thus, ~ ~ 8 >> PIPo. This is also valid for mesopores which are filled at PIP0 in the neighborhood of as predicted by the Kelvin equation. A good approximation is, therefore, Iln 81
