Langmuir 1996, 12, 987-993
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Physical Adsorption of a Multicomponent Mixture in Micropores V. Kh. Dobruskin st Aiala 21, Beer-Yacov, 70300, Israel Received July 3, 1995. In Final Form: October 23, 1995X This paper presents the theory of vapor-adsorbate equilibrium of a multicomponent mixture on microporous solids. The equilibrium relationships were obtained on the basis of the concept of micropore volume filling and the adsorbed-phase model, in which adsorbate-adsorbent interactions predominate over lateral interaction between adsorbed molecules. The predominant role of micropores is extended to the calculation of adsorbed-phase composition. The only single-gas isotherm parameters are incorporated into equations for calculation of separation factors, adsorption of components, and overall adsorption. It is shown that the separation factor in micropore space is a function of occupied micropore volume, affinity coefficients, adsorbent characteristic energy, temperature, and partial pressures. The theory has been successfully applied to published data for the adsorption of binary gas mixtures on activated carbons.
Introduction Although substantial progress has been recorded in the description of single-gas adsorption, the generalization of single-gas adsorption theory to multicomponent mixtures leads to great difficulties. Comprehensive reviews of the results in this field are presented in the monographs by Yang1 and Jaroniec and Madey2; numerous Russian studies are cited in a monograph by Keltzev3 and in books4,5 edited by Dubinin and Serpinski. The most significant advances in the study of singlegas adsorption on microporous solids were made by Dubinin and Radushkevich6 (D-R), and Dubinin and Astakhov7,8 (D-A) 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 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/βE)n}
(2)
that is associated with the Weibull density function9 f(A)
f(A) ) n/(βE)nAn-1 exp{-(A/βE)n}
(3)
where A is the differential molar work of adsorption, E is X Abstract published in Advance ACS Abstracts, February 1, 1996.
(1) Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, MA, 1987; Chapter 3. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988; Chapter 4. (3) Keltzev, N. V. Foundation of Adsorption Technology; Khimia: Moscow, 1984; Chapters 5 and 16 (Russian). (4) Tolmachev, A. T. In Adsorption in Micropores; Dubinin, M. M., Serpinski, V. V., Eds.; Nauka: Moscow, 1983; pp 26-45 (Russian). (5) Dubinin, M. M., Serpinski, V. V., Eds. Multicomponent physisorption; Nauka: Moscow, 1972; Chapter 1 (Russian). (6) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk. SSSR 1947, 55, 331. (7) 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). (8) Dubinin, M. M. In Adsorption-Desorption Phenomena; Ricca, F., Ed.; Academic Press: London, 1972; pp 3-18. (9) Bury, K. V. Statistical Models in Applied Science; John Wiley: New York, 1975; p 405.
0743-7463/96/2412-0987$12.00/0
the characteristic adsorption energy, β is the affinity or similarity coefficient, and n is the Weibull distribution parameter. In the case of activated carbons, n is usually equal to 2, and the D-A equation takes the final form
θ ) exp{-(A/βE)2}
(4)
The differential molar work of adsorption A is analogous to the adsorption potential in Polanyi’s theory.10,11 When the adsorption temperature, T, is below that of the adsorbate critical temperature, the adsorption potential may be evaluated by the expression
A ) RT ln P0/P
(5)
where P is the partial pressure in the gas phase and P0 is the saturated pressure of the liquid adsorbate. A comprehensive review of the Dubinin theory is given in a special issue dedicated to his memory.12 It would be natural to expect experimental data to be well represented by generalization of this theory to adsorption of gas mixtures. But direct extension of the D-R equation13,14 can hardly be considered to be successful. The authors of the generalization under discussion have arbitrarily postulated a correlation between the partial pressures of the gases in the adsorbed mixture and the saturated vapor pressures of normal liquid solutions. Their equation includes the liquid solution parameters and, consequently, cannot be applied and verified in cases of great practical importance, in which the adsorption temperatures employed are above the solution boiling point. Moreover, due to the inadequacy of the extended equation for solving the component adsorption, they applied the empirical Lewis method to calculate the adsorbed phase composition. However, the Lewis relationship by itself is applicable to the prediction of binary equilibrium.1,15 In the present work, equilibrium relationships for multicomponent adsorption are derived on the basis of Polanyi-Dubinin potential theory. The predominant role of the micropores is extended to the calculation of adsorbed (10) Polanyi, M. Verh. Dtsch. Phys. Ges. 1914, 16, 1032. (11) Polanyi, M. Trans. Faraday Soc. 1932, 28, 316. (12) Adsorpt. Sci. Technol., Special Issue 1993, 10. (13) Bering, B. P.; Serpinski, V. V.; Surinova, S. I. Dokl. Akad. Nauk SSSR 1963, 153, 129. (14) Bering, B. P.; Serpinski, V. V.; Surinova, S. I. Izv. Akad. Nauk SSSR, Ser. Khim. 1965, 769. (15) Ponec, V.; Knor, Z.; Cerny, S. Adsorption on Solid; Butterworths: London, 1974; p 397.
© 1996 American Chemical Society
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phase composition, and further insight into the separation factor is gained.
the binary mixture, we obtain for the ratio of the average residence times in an elementary volume, dW
Derivation of the Separation Factor Relationship for Binary Adsorption
τ1/τ2 ) ν2/ν1 exp[(q1 - q2)/RT]
According to Frenkel’s assumption,16 the average molecule residence time, τ, is given by the equation:
where subscripts 1 and 2 denote the parameters of the components 1 and 2, respectively. When deriving eq 10, we consider the adsorption behavior of each component of the binary mixture to be independent. The vibration frequency, ν, of the adsorbate-adsorbent bonds may be evaluated by means of the expression21
τ ) τ0 exp(∆E/RT)
(6)
where τ0 is a frequency factor and ∆E is an energy barrier that must be overcome for desorption to occur. τ is considered to be the reciprocal of the constant of the desorption rate, k17
τ ) 1/k
(7)
Then, application of the absolute rate theory for desorption leads to the following expression:18 #
k ) kBT/h Q /Qa exp(-∆E/RT)
ν ) (2πd0)-1[(f - 3)(s - 3) λ/m]0.5
(10)
(11)
where d0 is equilibrium bond length, f and s are Mee potential parameters, m is the molecule mass, and λ is the desorption heat. Then the ratio ν2/ν1 may be approximated by
ν2/ν1 = (m1/m2 × λ2/λ1)0.5
(12)
(8) and eq 10 can be to transformed into
where kB and h are Boltzmann’s and Planck’s constants and Q# and Qa are the partition functions of the activated complex and the adsorbed molecule. The contributions of the various terms to the partition function for adsorption states are discussed in a number of monographs.17-20 In the case of single-gas adsorption, the pre-exponent term may be approximated by the reciprocal of the vibration frequency, ν of the adsorbate-surface bond18,19
τ ≈ 1/ν exp(∆E/RT)
τ1/τ2 = (m1/m2 × λ2/λ1)0.5 exp(∆q/RT)
(13)
where ∆q is the difference between the net adsorption energies of the two components, into the volume, dW. The further simplifications following from Dubinin’s theory may be made in this equation. The theory gives the expression8 for the differential net heat of a single-gas adsorption
(9)
Equation 9 is usually applied to the estimation of a residence time on the solid surface, but it may be extended to equilibrium in a volume. The object now is to determine the parameters of eq 9 in the case of binary adsorption in micropores. Let us consider an occupied infinitesimal adsorption volume, dW with potential A, in equilibrium with the binary gas mixture. When adsorbed in a micropore, a molecule occupies an adsorption site with increased adsorption energy. This micropore effect is caused by the superposition of the adsorption forces generated at the opposite micropore walls. For desorption to occur, the adsorbed molecule must pass over an energy barrier on its path to the gas phase. Let us assume that the path of the molecule from micropore to gas phase passes through a stage of adsorption in the extramicropore volume, and this second adsorption state represents an activated complex. This hypothesis will be discussed further. The adsorption energy in the extramicropore volume is considered to be practically equal to the condensation latent heat.8 Therefore, the energy barrier ∆E between these two adsorption states is equal to the net differential adsorption8 heat q, i.e., ∆E ) q. In the case of multicomponent equilibrium, eq 9 must be fulfilled for each component. Proceeding just as in a single-gas case and applying eq 9 for both components of (16) Frenkel, Y. I. Kinetic Theory of the Liquid State; Akad. Nauk: Moscow, 1945 (Russian). (17) Adamson, A. W. Physical Chemistry of Surface; John Wiley: New York; 1976; pp 554-557. (18) Roberts, M. W.; McKee, C. S. Chemistry of the Metal-Gas Interface; Clarendon Press: Oxford, 1978; pp 266-267. (19) Jaycock, M. J.; Parfitt, G. D. Chemistry of Interfaces; John Wiley: New York, 1986; p 168. (20) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970; p 215.
q ) A + R T/2 (ln θ)-0.5
(14)
where R is the thermal coefficient of limiting adsorption. The contribution of the second term of eq 14 to q does not exceed 10%. Assuming the adsorption energy of each component to be independent of the presence of the other component, and applying eq 14 to both components of the binary mixture, we obtain for ∆q
∆q ) ∆A + T/2 [R1 (ln θ1)-0.5 - R2 (ln θ2)-0.5]
(15)
where ∆A ) A1 - A2 is the difference of the adsorption potential in the volume dW. Estimation of the term in brackets for actual values of R1, R2 and for values θ ranging from 0.2 to 0.9 shows that the second term contributes not more than a few percent. Therefore, eq 15 can be approximated by
∆q ) ∆A
(16)
Taking into account the similarity of the characteristic curves8 at constant θ, we have
∆A ) ∆βA
(17)
where ∆β ) β1 - β2 is the difference between the affinity coefficients of the components and A is the adsorption potential of the standard (reference) gas in the volume dW. An analogous simplification can be applied to the first term, (m1/m2 × λ2/λ1)0.5, of eq 13. The absolute values of the heats of adsorption and desorption are equal. Because variation of (λ2/λ1)0.5 in eq 12 is overshadowed by (21) Moelwyn-Hughes, E. A. Physical Chemistry; Pergamon Press: London, 1957; p 940.
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that of the exponential term, we may approximate the former by the constant term
(λ2/λ1)0.5 = (β2/β1)0.5
(18)
Substitution of (17) and (18) in (13) leads to
τ1/τ2 = (m1/m2 × β2/β1)0.5 exp(∆β A/RT)
(19)
This equation is valid for an infinitesimal adsorption volume. It will be convenient to denote this ratio as r ) τ1/τ2. It is obvious that r is a function of the adsorption potential A. To derive an analogous relation for a finite occupied volume, W, the distribution function of the micropore volume over adsorption potential should be taken into consideration. This leads to the following expression:
rj )
∫0wr dW
(20)
W
The value rj expresses the ratio of average residence times, τj1 and τj2, of the components, 1 and 2, occupying a volume, W. Replacing dW by W0dθ (eq 1), and introducing dθ from eq 4 and r from eq 19, we transform eq 20 to the following: rj ) (m1/m2 × β2/β1)0.5
∫
∞
A
W0 exp(∆β A/RT){2/(βE)2A exp[-(A/βE)2]} dA W
(21) Let us now discuss this equation. There are three different values, β, β1, and β2, in this relationship. The standard gas with β ) 1 may be considered to be a probe of energy heterogeneity that provides information about adsorption sites. Furthermore, the component adsorption energies are also determined with respect to this selected standard gas; the values W, E, and A and integration limits in eq 21 are related to the standard gas by eq 4. Taking account of eq 1, eq 21 becomes
rj ) (m1/m2 × β2/β1)0.5 θ-1
∫A∞ 2/E2 A exp[-(A/E)2 + ∆β A/RT] dA (22)
in which the lower limit, A, is related to θ by eq 4
A ) E(ln 1/θ)0.5
(23)
Let us introduce the notation
b ) ∆β E/2RT
(24)
After some mathematical processing22 (Appendix A), one obtains the following:
rj ) (m1/m2 × β2/β1)
0.5
θ
-1
×
exp(b ){exp{-[(ln 1/θ)0.5 - b]2} + π0.5b{1 - erf[(ln 1/θ)0.5 - b]}} (25) 2
Figure 1. Dependence of separation factor on micropore filling fraction.
number of molecules striking the surface.17 This relation is valid also for a binary mixture, and it may be extended to equilibrium in a volume. The value xj is equal to the ratio of the numbers of molecules n1 and n2 in this volume, i.e., xj ) n1/n2. Let surface area S be the interface between volume W and the gas phase. The number of molecules of each kind in the volume W is directly proportional to their average residence time in this volume and the number of molecules crossing the interface S. This leads to
xj ) n1/n2 ) τj1N1S/τj2N2S ) rjN1/N2
(26)
were Ni is the number of molecules of component i striking unit surface area per unit time. Using the expression Ni ) Pi/(2πmikBT)0.5 following from gas kinetic theory and eq 25, we transform eq 26 into
xj ) P1/P2(β2/β1)0.5 θ-1 × exp(b2){exp{-[(ln 1/θ)0.5 - b]2} + π0.5b{1 - erf[(ln 1/θ)0.5 - b]}} (27) In deriving eq 27, we ignored any difference in molecular sizes, but that will be taken into consideration later, when micropore filling by a binary mixture is discussed. Equation 27 shows the relationship between the molar ratio of adsorbed components in micropores xj, on the one hand, and affinity coefficients β1 and β2, adsorbent characteristic energy E, temperature, partial pressures P1 and P2, and occupied fraction of micropore θ, on the other hand. An expression for calculating the separation factor S (relative volatility) follows from eq 27:
S ) (β2/β1)0.5 θ-1 exp(b2){exp{-[(ln 1/θ)0.5 - b]2} + π0.5b{1 - erf[(ln 1/θ)0.5 - b]}} (28) When θ tends to unity and the occupied volume approaches the limiting occupied volume, eq 28 may be approximated by
S ) (β2/β1)0.5 {1 + π0.5b exp(b2)[1 + erf(b)]} (29)
The object now is to derive an expression for the component molar ratio xj in the occupied volume W. When adsorption occurs on the surface, the number of molecules in the adsorbed phase in the state of the equilibrium is equal to the product of the average residence time and the
If, simultaneously, b is small (b e 0.2), we have
(22) Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. Integrals and Series; Gordon Breach Science Publishers: New York, 1986; Vol. 1, p 139.
Figure 1 presents several members of the family of curves represented by eq 28. To show the relationship
S ) (β2/β1)0.5[1 + π0.5b(1 + b)]
(30)
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between occupied micropore fraction θ and the parameter b, the data are presented for the case when β20.5/β10.5 ≈ 1. For a relatively large value of b Figure 1 shows a very strong dependence of the separation factor on the occupied micropore fraction. When b ) 1.5, the separation factor (eq 28) exceeds 350 at θ ) 0.1 and S > 50 at θ ) 1, resulting in practically complete displacement of the component with a smaller affinity coefficient. When b tends to zero, the separation factor approaches unity and becomes independent of θ. In the case of activated carbons, the value of β can be approximated by the ratio of the parachor of the component, Pr, to that of the standard gas,23,24 Prst, and, consequently:
∆β ≈ ∆ Pr/Prst
(31)
where ∆Pr is the difference of parachors of the components. Equations 27 and 28 may be modified by expression 31 to take into account the influence of the chemical nature of the separation factor. When the standard gas is benzene, the characteristic energy, E, of an ordinary activated carbon is close to 20000 J/mol. For a mixture of saturated and unsaturated hydrocarbons with the same number of carbons at a temperature of about 300 K, the value b appears to be about 0.2; for a mixture of hydrocarbons differing by one CH2 group, b ≈ 0.8. For the first mixture, S is close to 1.5 and is practically independent of θ; for the second mixture, the separation factor is reduced from S ≈ 18 for θ ) 0.1 to S ≈ 5 for θ ) 1.0. For more accurate evaluation of the separation factor, one must apply the value of β calculated from individual isotherms. Derivation of Binary Adsorption Equilibrium Relationship Consider a binary gas in isothermal equilibrium with an adsorbent. Let χ1 and χ2 be the molar fractions in the adsorbed phase and γ1 and γ2 the molar fractions of the components in the gas phase. The composition of the gas phase is determined by the partial pressures P1 and P2, and γ1/γ2 ) P1/P2; the composition of the adsorbed phase is determined by the molar ratio of adsorbed components xj (eq 27), and for P1/P2 ) constant, xj is a function only of the occupied micropore fraction. If there is no volume change upon adsorption, the molar volume of the adsorbed phase Vm is defined in terms of pure component10,25 molar volumes V1 and V2
Vm ) xj(1 + xj)-1V1 + (1 + xj)-1V2
(32)
where the subscript m indicates mixture parameters, and molar fractions in the adsorbed phase are expressed with respect to xj. Let us assume there is no significant energy effect of component mixing in the micropore. We will discuss this assumption in detail later. To simplify, the components within the micropore are unmixed. In such a case the adsorbed phase may be characterized by the same parameters as in the case of adsorption of individual components. Maximal molar work of adsorption, Am, that must be supplied by the field of the adsorption forces is equal to the sum of the energies of compression of the independent components from equilibrium pressures P1 (23) Dubinin, M. M. Porous Structure and Adsorption Properties of Activated Carbons; Military Press: Moscow, 1965; p 12 (Russian). (24) Dubinin, M. M.; Zaverina, E. D. Zh.. Fiz. Khim. 1950, 24, 12621272. (25) Lee, A. K. K. Can. J. Chem. Eng. 1973, 51, 688.
and P2 to saturated pressures P01 and P02, respectively, and is given by the expression
Am ) γ1 RT ln P01/P1 + γ2 RT ln P02/P2
(33)
Let us assume that binary adsorption may be described by an equation analogous to the D-A equation. Proceeding just as in the case of single-gas adsorption, we can introduce an effective affinity coefficient of the mixture, βm to describe the characteristic mixture adsorption energy with respect to a reference vapor. Because of the additivity of energy values, this coefficient must be connected with an adsorbed phase composition. In the case of single-gas adsorption, when the fraction of occupied volume, θ ) 1/e, eq 4 leads to the following:
E ) (A)θ)1/e
(34)
where e is the base of natural logarithms. With this relation taken into account, the value of the effective affinity coefficient of the mixture, βm, may be determined in the terms of the adsorbed phase composition xjp at the characteristic point θ ) 1/e as follows
βm ) xp(1 + xjp)-1 β1 + (1 + xjp)-1 β2
(35)
where xjp is defined from eq 27 at θ ) 1/e:
xjp ) P1/P2(β2/β1)0.5 e-1 exp(b2){exp{-[1 - b]2} + π0.5b{1 - erf[1 - b]}} (36) And finally, the binary adsorption equilibrium relationship may be obtained by combining eqs 4, 33, and 35
θm ) exp{-(Am/βmE)2}
(37)
The value θm expresses the overall micropore occupied volume fraction and is found to be the fraction of micropore volume in which adsorption potential exceeds the value given by eq 33. The adsorption of the components (mol/g) is determined by their mole fractions in the adsorbed phase
a1 ) W0 θm Vm-1xj(1 + xj)-1; a2 ) W0θm Vm-1 (1 + xj)-1 (38) where the value xj is connected with θm by eq 27. The complete description of isothermal behavior of a binary gas mixture is represented by the surface (eq 37) in threedimensional space (θ, p1, p2). The multitude of parameters of this surface βmE is given by the relation βmE ) f(xp) determined by eq 35. It is known that many real adsorbents may be characterized by bi- and polymodal energy distributions.2,26 In such cases, the experimental data cannot be described satisfactorily by the monomodal Weibull distribution. Dubinin used two-term expressions27 to determine the adsorption, a (mol/g), on microporous solids that possess bimodal structure
a ) W01 V-1 exp{-(A/βE1)2 + W02 V-1 exp{-(A/βE2)2 (39) where W01 and W02 are the limiting volumes of adsorption space of the first and the second substructures and E1 and E2 are, respectively, their characteristic energies. Sto(26) Rudzinski, W.; Nieszporek, K.; Dabrowski, A. In ref 9, pp 3563. (27) Izotova, T. I.; Dubinin, M. M. Zh. Fiz. Khim. 1965, 39, 27962803.
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Table 1. Adsorption Parameters and Agreement with Experiment Data gas mixture
∆β
E, J/mol
standard gas
θm
S (eq 28)
S(experiment)
ethylene-acetylene propylene-propane butylene-butane propane-butane carbon dioxide-acetylene ethylene-acetylene (SG) ethylene-propane 1 atm ethylene-propane 2.25 atm ethylene-propane 7.4 atm
0.017 0.028 0.058 0.194* 0.027 0.162 0.252* 0.252* 0.252*
11170 13600 17390 17390 23000 11200 13600 13600 13600
acetylene propane benzene benzene benzene acetylene propane propane propane
0.34 0.69 0.93 0.88 0.26 0.32 0.61 0.73 0.89
1.1 1.1 1.4 4.1 1.4 2.7 12.3 10.8 8.9
1.4 0.9 1.5 4.37 1.8 3.0 13.6 10.2 8.4
Table 2. Separation Factor for Propane-Butane System on AG2 activated carbon structure first substructure second substructure AG2 (total structure)
W0, cm3/g
E, kJ/mol
xjp
θm
S (calculation)
S (experiment)
0.22 0.09 0.31
21.1 10.0
9.97 2.68
0.92 0.69 0.85
5.69 2.21 4.68
4.4
eckli28,29 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 Dubinin30 and Eltekov31 showed that a two-term expression describes the experimental data with the same accuracy as Stoeckli’s equation. Just as for single-gas adsorption, when activated carbon is characterized by the two-term equation, this two-term expression must be applied to calculate the contribution of each substructure to the separation factor and overall adsorption. The latter values will be determined as a weighted sum of these two contributions, each being calculated by means of eqs 28 and 37
SΣ ) (W01S1 + W02S2)/W0
(40)
θmΣ ) (W01θm1 + W02θm2)/W0
(41)
where W01 + W02 ) W0 and subscript Σ indicates the total bimodal structure of activated carbon. Comparison with Experimental Data The method under consideration was applied to several binary systems investigated by Lewis et al.32-34 and Keltzev.35 The comparison of calculated values of separation factors and experimental data is shown in Table 1. The classic studies of Lewis and co-workers were treated by the least squares method to determine the D-A theory parameters, E and β. When the adsorption temperature exceeded the critical temperature, Tc, we followed the Dubinin-Nicolaev recommendation36 and substituted for P0 the value Pc(T/Tc)2, in which Pc is the critical pressure and Tc is the critical temperature. When gases were employed, the parameter β for activated carbon was calculated with eq 31 and indicated by an asterisk (*); in the case of vapors, parameter β was calculated from experimental data. The data in Table 1 are related to activated carbon, except for one case in which silica gel was used, indicated by (SG). All calculations were (28) Stoeckli, F. J. Colloid Interface Sci. 1977, 59, 184-185. (29) Stoeckli, F. In ref 9, pp 3-17. (30) Dubinin, M. M. In ref 4, pp 186-192. (31) Eltekova, N. A.; Eltekov, Yu. A. In ref 9, pp 203-211. (32) Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Calogan, W. P. Ind. Eng. Chem. 1950, 42, 1326. (33) Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Hoffman, W. H. J. Am. Chem. Soc. 1950, 72, 1153. (34) Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Calogan, W. P. Ind. Eng. Chem. 1950, 42, 1319. (35) Keltzev, N. V. In ref 3, p 254. (36) Nicolaev, K. M.; Dubinin, M. M. Izv. Akad. Nauk. SSSR, Ser. Khim. 1958, 1165.
performed with a set of functions from Microsoft Excell.37 All experimental data were obtained at a temperature of 25 °C, with equal partial pressures in the binary gas phase and a total pressure 1 atm, except where indicated. The correlation is good both for vapor mixture adsorption and for adsorption of vapor-gas combination such as the ethylene-acetylene system. In the case of the gas system, the adsorption temperature exceeds the critical temperature and we cannot expect good agreement, because of poor prediction of the value of θ by the D-A theory. In these cases the value of θ is relatively small, and the separation factor may be very responsive to the small inaccuracy in θ (Figure 1). Special consideration should be given to the fact that in compliance with theory, the separation factor for the ethylene-propane system decreases as the total pressure increases. This fact shows that there is real dependence of adsorption phase composition on the occupied volume fraction, θ and its behavior cannot be explained in terms of the solution properties, because the relative composition of the gas phase was invariable. It seemed to be confirmation of the assumption that the adsorption phase composition is mainly determined by the adsorbent-adsorbate interaction and not by the normal volume solution properties. The results of calculation of the separation factors and adsorption amounts in the case of activated carbon with bimodal structure are summarized in Table 2. Measurements were made by Keltzev38 for the propane-butane system with Russian activated carbon, AG2. The selectivity coefficient was measured at 25 °C and p1 ) p2 ) 0.5 atm. Propane and butane saturated pressures at this temperature are 9.335 and 2.398 atm, respectively. The similarity coefficients and structural characteristics39 of AG2 were given with respect to benzene as a standard substance. Parachors of benzene, propane, and butane, evaluated by summation of the additive contributions40 are 205.4, 150.8, and 190.8, respectively. Thus, the similarity coefficient for propane, β, is 0.734 and that for butane is 0.929. The data listed in Table 2 demonstrate that there is good agreement between predicted and experimental values of the separation factor. Comparison between the contributions of the two substructures to the total carbon properties shows the important role of the characteristic adsorption energy and the clearly defined growth of a separation factor with the increasing energy E. The results of the comparison listed in Tables 1 and (37) Microsoft Excel, Function Reference. Doc. Number AB262980592, 1992. (38) Keltzev, N. V. In ref 5, p 30. (39) Kolyshkin, D. A.; Mikhaylova, K. K. Active Carbon; Khimia: Leningrad, 1972 (Russian). (40) Chemical Engineers’ Handbook; Perry, J. H., Ed.; McGraw-Hill: New York, 1963; pp 3-214.
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2 demonstrate that the developed correlation can be applied to the description of the binary mixture adsorption behavior. Isotherm Equation for Multicomponent Adsorption in Micropores In the case of multicomponent adsorption, the relationship eqs 20-27 must be satisfied for each pair of components i and j. Let ni and nj be the mole numbers of the components in the adsorbed phase. Taking into account the equality n
1
nj) ) ∑ j)1
χi ) ni/(
n
(
(42)
nj)/ni ∑ j)1
and the relation xjji ) nj/ni, we obtain for the mole fractions, χi of component i in the adsorbed phase
χi )
1
(43)
n
xjji ∑ j)1 where xjji is determined by the system of n equations analogous to eq 27. The series of n values of χi gives the mole fractions of all components in the adsorbed phase. Just as for binary adsorption, the n values of χip at the characteristic point θ ) 1/e may be evaluated by n equations analogous to eq 36, each of which corresponds to a pair ji. As a result, in the case of multicomponent adsorption, eq 35 must be modified as follows: n
βm )
∑ χip βi
(44)
I)1
And finally the equation for total adsorption takes the form n
n
∑ γI RT ln Poi/Pi)/I)1 ∑ χipβiE]2} I)1
θm ) exp{-[(
(45)
This equation contains only individual isotherm parameters and prescribes the surface of the overall occupied adsorption volume in (n + 1)-dimensional space. Summary, Discussion, and Conclusions Let us summarize the main assumptions underlying the correlation under discussion: 1. Adsorption of the individual components is described by the D-A theory. 2. The adsorption energy of each component depends only on the adsorption site and not on the presence of another component. 3. Interaction between components in the adsorbed phase is neglected. 4. The activated complex of desorption is equal to the second adsorption state, existing in the extramicropore volume. The theoretical basis for the formerly empirical D-A equation was recently given by Chen and Yang.41 In the present work, the process of micropore volume filling is assumed to comply with Polanyi’s theory and Polanyi’s (41) Chen, S. G.; Yang, R. T. Langmuir 1994, 10, 4244.
insight of adsorption potential.42 The D-A equation is considered to be based on the Weibull distribution of micropore volume over the adsorption potential, as originally postulated by Astakhov et al.7 The proportions of the components in the adsorbed phase are determined by subsequent statistical distribution based on Frenkel’s mechanism. The correlation under discussion was obtained for the case when the parameter n (eq 2) is equal to 2. For fine microporous carbonaceous adsorbents, the exponent n is taken to be 3. In the case of zeolites, different values of the exponent are used for different adsorption centers and can be as high as 5-6.7,35 When n > 2, the above-mentioned solutions are not valid and should be replaced by numerical solutions. The model used in the present study to represent the adsorbed phase is one in which adsorbate-adsorbent interactions predominate over lateral interactions between adsorbed molecules. On the basis of the assumption that the physical forces within the micropore are stronger than those acting in the solution, the adsorption energy is considered to be determined by the adsorption sites and to be independent of the presence of the other component. The comparison of interchange energy in solution with the adsorption energy in a micropore shows that, at least for nonpolar components, the contribution of interchange energy may be considered to be negligible. Therefore, owing to the predominant micropore influence, the composition of the adsorbed phase in the micropore is also determined by the adsorbent-adsorbate interaction. Such a simplification seems to correspond to the micropore concept better than the hypothesis of a normal liquid solution in the micropore space.13,14 The fundamental question is the choice of the activated complex of desorption. To the knowledge of the author, the molecular mechanism of desorption from micropores has not yet been studied. Even in the special monographs there are no molecular details of desorption from microporous adsorbents.43,44 The great energy barrier between the molecule adsorbed in micropore and that in the gas phase makes it likely that desorption involves passage through an intermediate state with a lower barrier. In light of the polydispersion nature of such adsorbents as activated carbons and silica gels, and the existence vacant energy states in meso- and macropores, the hypothesis that an intermediate state exists in the extramicropore volume seems to be likely. The good agreement between results predicted on the base of this hypothesis and the experimental data seems to justify it. It is worth mentioning here that the same equations may be obtained without this hypothesis, if we consider the adsorption phase as comprising a compressed but not condensed component. However, such an approach contradicts Dubinin’s theory in the case of vapor adsorption. To make the calculations of the separation factor, the experimental values of the component affinity coefficients and the adsorbent characteristic energy are needed. In addition to these parameters, the value of the limiting adsorption volume, W0, is necessary for the mixture isotherm prediction. From an examination of experimental data for model testing, it is easy to understand the following statement, which was written by Yang in 1987, “Compared to the number of models that have been proposed, experimental data for mixture adsorption are indeed scarce.” The every set of experimental data is (42) Flood, E. A. In The Solid-Gas Interface; Flood, E. A., Ed.; Mir: Moscow, 1970; pp 62-75. (43) Lukin, V. D.; Ancipovich, I. S. Adsorbent Recovery; Khimia: Leningrad, 1983; pp 5-110 (Russian). (44) Lukin, V. D.; Novoselski, A. B. Cyclic Adsorption Processes; Khimia: Leningrad, 1989; pp 81-106 (Russian).
Adsorption of Multicomponent Mixtures
incomplete. Thus, the calculation of W0 from Lewis’ experiments with gases and low-boiling liquids32-34 is troublesome in view of the problem of adsorbate density evaluation. The methods recommended for this purpose36,44-45 lead to the different results, and the limiting adsorption volume is usually calculated from benzene adsorption.29 Further studies based on the adsorbents with the well-known porous structure are desirable before the general applicability of this model can be determined. Glossary A ai b d0 E ∆E f h kB k m n ni Pi P0, Pc Pr, Prst q Q#, Qa r jr R s S T, Tc Vm, Vi W W0 xj xjp
differential molar work of adsorption; adsorption potential adsorbed amount (mol/g) constant, eq 24 equilibrium bond length characteristic adsorption energy energy barrier that must be overcome for desorption to occur Mee potential parameter Planck’s constant Boltzmann’s constant constant of the desorption rate molecule mass Weibull distribution parameter number of molecules; the mole number of the components in the adsorbed phase partial pressure in the gas phase saturated pressure of the liquid adsorbate; critical pressure parachors of the component and the standard gas net differential adsorption heat partition functions of the activated complex and the adsorbed molecule ratio of the average residence times in an elementary volume dW average value r for a finite occupied volume W gas constant Mee potential parameter separation factor; interface between volume W and the gas phase absolute temperature, critical temperature molar volumes of the adsorbed phase and the pure component occupied adsorption volume limiting adsorption volume component molar ratio in the occupied volume W component molar ratio at the characteristic point θ ) 1/e
Greek Symbols R
thermal coefficient of limiting adsorption
(45) Cook, W. H.; Basmadjian, D. Can. J. Chem. Eng. 1964, 4, 146. (46) Astakhov, V. A.; Kozhuschko, V. V.; Novoselski, A. B. Izv. Akad. Nauk. Bel. SSR, Ser. Khim. 1987, 6, 48.
Langmuir, Vol. 12, No. 4, 1996 993 affinity (similarity) coefficient vibration frequency of the adsorbate-surface bond molar fraction of the component in the gas phase desorption heat molecule residence time frequency factor in eq 6 degree of micropore filling (fractional adsorption) molar fraction in the adsorbed phase
β ν γ λ τ τ0 θ χ
Subscripts I,j m 0 1,2 1,2 Σ p
components i,j mixture pure component number of components of a binary mixture number of a substructure (eqs 37-39) total structure of activated carbon value at characteristic point θ ) 1/e
Appendix A The integral (eq 22)
∫A∞A exp[-(A/E)2 + ∆β A/RT] dA
I ) 2/E2
(A1)
may be evaluated proceeding from the tabulated integral22
∫xn exp(-a2x2 + cx) dx ) 1/an+1 × n
exp(c2/4a2)
(nk)(c/2a)n-k ∫ tk exp(-t2) dt ∑ k)0
(A2)
where t ) ax - c/2a. In this case
x ) A; n ) 1; a ) 1/E; c ) ∆β/RT
(A3)
and the sum in eq A2 transforms to the sum of two integrals
∫t∞ exp(-t2) dt + ∫t∞ t exp(-t2) dt )
J ) c/2a
c/2a π0.5/2 [1 - erf(t)] + 1/2 exp(-t2) (A4) where t ) A/E - ∆β E/2RT. Taking account of the notation 24 and equality A/E ) (ln 1/θ)0.5 following from eq 23, one obtains
t ) (ln 1/θ)0.5 - b
(A5)
From (A1), (A2), and (A4)
I ) 2/E2 × 1/a2 exp(c2/4a2)J
(A6)
Substituting (A3), (A4), and (A5) in eq A6, yields
I ) exp(b2){exp{-[(ln 1/θ)0.5 - b]2} + π0.5b{1 - erf[(ln 1/θ)0.5 - b]}} (A7) LA9505327
