Langmuir 2009, 25, 2309-2312
2309
A New Isotherm for Multisite Occupancy Adsorption of Binary Gaseous Mixtures Saeid Azizian* and Hadis Bashiri Department of Physical Chemistry, Faculty of Chemistry, Bu-Ali Sina UniVersity, Hamedan, 65174, Iran ReceiVed NoVember 5, 2008. ReVised Manuscript ReceiVed NoVember 20, 2008 Adsorption is one of the most popular methods for reducing pollutants or separation of gases. Therefore it is important to introduce new and extended isotherm equations for binary systems. In the previous adsorption isotherms for binary systems, it is assumed that an adsorbate occupies one site in lattice sites. However, the adspecies may occupy more than one site in the lattice. Here we propose an adsorption isotherm for binary systems where the adsorbates occupy more than one site. The new isotherm equation for multisite occupancy of binary mixtures was derived based on statistical thermodynamics. The present new adsorption isotherm provides information about the structure of molecules in adsorbed form. Finally, the results of the present theoretical study were confirmed by analysis of two experimental systems.
Introduction Adsorption on solid surfaces is one of the most powerful techniques for reducing pollutants in industrial and natural systems. Since adsorption plays a key role for gas separation or water purification, it is important to investigate this process theoretically and experimentally. Both kinetics and equilibrium are important in adsorption studies. There are different and simple kinetic models for adsorption at interfaces.1-6 Recently, some of the kinetic models for adsorption were derived theoretically by Azizian based on the theory of activated adsorption/desorption (TAAD) approach7,8 and the statistical rate theory (SRT) approach.9,10 Rudzinski et al. also investigated the kinetics of adsorption based on SRT.11-14 The design of industrial adsorption equipment requires the availability of reliable equilibrium data and theoretical models for their accurate prediction. Until recently, equilibrium adsorption (thermodynamics) has been studied extensively, and there are various isotherm equations that relate the amount of adspecies to the pressure or concentration at equilibrium.15-20 Most recently, we proposed a new isotherm equation based on classical21 and
* To whom correspondence should be addressed. Fax: +98-811-8257404. E-mail:
[email protected]. (1) Lagergren, S. K. SVen. Vetenskapsakad. Handl. 1898, 24, 1. (2) Ho, Y. S.; Mckay, G. Water Res. 2000, 34, 735. (3) Blanchard, G.; Maunaye, M.; Martin, G. Water Res. 1984, 18, 1501. (4) Yang, X.; Al-Duri, B. J. Colloid Interface Sci. 2005, 287, 25. (5) Azizian, S.; Haerifar, M.; Basiri-parsa, J. Chemosphere 2007, 68, 2040. (6) Azizian, S.; Haerifar, M.; Bashiri, H. Chem. Eng. J. 2008, 146, 36. (7) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47. (8) Azizian, S. J. Colloid Intefrace. Sci. 2006, 302, 76. (9) Azizian, S.; Bashiri, H.; Iloukhani, H. J. Phys. Chem. C 2008, 112, 10251. (10) Azizian, S.; Bashiri, H. Langmuir 2008, 24, 11669. (11) Rudzinski, W.; Plazinski, W. J. Phys. Chem. B 2006, 110, 16514. (12) Rudzinski, W.; Plazinski, W. J. Phys. Chem. C 2007, 111, 15100. (13) Rudzinski, W.; Plazinski, W. Appl. Surf. Sci. 2007, 253, 5827. (14) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (15) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (16) Frumkin, A. N. Z. Phys. Chem. 1925, 116, 466. (17) Freundlich, H. Kapillarchemie; Akademische Verlagsgesellschaft: Leipzig, Germany, 1992. (18) Markin, V. S.; Volkova-Gugeshashvili, M. I.; Volkov, A. G. J. Phys. Chem. B 2006, 110, 110. (19) Koter, S.; Terzyk, A. P. J. Colloid Interface Sci. 2005, 282, 335. (20) Rudzinski, W.; Lee, S.-L.; Yan, C.-C. S.; Panczyk, T. J. Phys. Chem. B 2001, 105, 10847. (21) Azizian, S.; Volkov, A. G. Chem. Phys. Lett. 2008, 454, 409.
statistical22 thermodynamics, known as the Azizian-Volkov (AV) isotherm. This new isotherm equation accounts for two different states of an adsorbate.21,22 Since there are many industrial situations that contain several components, it is necessary to introduce isotherm equations for binary adsorption. In recent years, multicomponent equilibrium studies have been passed through many stages of development. There are several thermodynamic models to predict mixture adsorption equilibrium: the extended Langmuir equation23 and the Langmuir-Freundlich,24 Toth,25 P-factor,26 ideal adsorbed theory (IAS),27 and bi-Langmuir28 isotherms. Other binary isotherms have also been listed in the literature.29-32 Usually, it is assumed that an adsorbed molecule occupies one site in the lattice of adsorbent sites. But there may be other situations in which an adsorbate molecule occupies more than one site in the lattice. Therefore it seems that considering multisite occupancy in adsorption models is necessary. On the basis of statistical thermodynamics, the EA isotherm has been derived for the adsorption of linear k-mers in such a way that each molecule occupies k sites in the lattice.33 Also, statistical thermodynamics has been applied for the derivation of other adsorption isotherms.22,34-37 The purpose of the present work is to derive a new isotherm for binary adsorption, in such a way each molecule occupies more than one site in the lattice sites. (22) Azizian, S.; Bashiri, H.; Volkov, A. G. Colloids Surf. A 2008, DOI: 10.1016/j.cdsurfa.2008.10.030. (23) Butler, J. A. V.; Orckrent, C. J. Phys. Chem. 1930, 34, 2841. (24) Benedetti, M. F.; Milen, C. J.; Kinniburgh, D. G.; Riemsdijk, W. H. V.; Koopal, L. K. EnViron. Sci. Technol. 1995, 29, 446. (25) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Book; Prentice Hall: Englewood Cliffs, NJ, 1989. (26) Mckay, G.; Al-Duri, B. Chem. Eng. Sci. 1991, 46, 193. (27) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (28) Jain, J. S.; Snoeyink, V. K. J. Water Pollut. Control Fed. 1973, 45, 2463. (29) Schay, G. J.; Fejes, F. P.; Szethmary, J. Acta Chim. Acad. Sci. Hung. 1957, 12, 299. (30) Nieszporek, K.; Szabelski, P.; Drach, M. Langmuir 2005, 21, 7335. (31) Seidel, A.; Gelbin, D. Chem. Eng. Sci. 1988, 43, 79. (32) Lin, B.; Ma, Z.; Shirazi, S. G.; Guiochon, G. J. Chromatography A 1989, 475, 1. (33) Roma, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Langmuir 2003, 19, 6770. (34) Du, Z.; Dunne, L. J.; Manos, G.; Chaplin, M. F. Chem. Phys. Lett. 2000, 318, 319. (35) Kalies, G.; Bra¨uer, P.; Messow, U. J. Colloid Interface Sci. 2004, 275, 90. (36) Bae, J. H.; Lim, Y. R.; Sung, J. Langmuir 2008, 24, 2569. (37) Belmabkhout, Y.; Frere, M.; De Weireld, G. Mol. Simul. 2006, 32, 495.
10.1021/la803675h CCC: $40.75 2009 American Chemical Society Published on Web 01/06/2009
2310 Langmuir, Vol. 25, No. 4, 2009
Azizian and Bashiri
Table 1. The Constants of the New Adsorption Isotherm Equations (Eqs 18) for Experimental Binary Systemsa system
qm (mmol/g)
KA (1/bar)
KB (1/bar)
propane (A) and butane (B) methane (A) and carbondioxide (B)
1.16
9.30 × 10-2
9.78 × 10-2
0.86
0.06 × 10-1
0.42 × 10-1
a
The first system is propane and butane on Vycor glass, and the second one is methane and carbon dioxide on wet Tiffany coal.
Theory
[M - (k - 1)NA - (k ′ -1)NB]! NA NB Q(M,NA,NB,T) ) q q (1) NA ! NB ! [M - kNA - k ′ NB]! A,s B,s where, qA,s and qB,s are the molecular partition functions of adspecies A and B, respectively, and T is the absolute temperature. In the canonical ensemble, the chemical potential of adspecies i (µsi ) relates to Q(M,NA,NB,T) through
(
µis ) -kBT
∂ ln Q(M,Ni,T) ∂Ni
)
ln[(M - (k - 1)NA - (k ′ -1)NB] + ln NB - k ′ ln(M - kNA - k ′ NB) - ln qB,s] (4)
Rearrangement of the above equations yields
µBs ) kBT
ln
(M - (k - 1)NA - (k ′ - 1)NB)k-1NA (M - kNA - k ′NB)kqA,s
) )
(M - (k - 1)NA - (k ′ - 1)NB)k ′-1NB (M - kNA - k ′NB)k ′qB,s
(5)
(6)
By defining the fractional surface coverages of A and B molecules as θA ) (kNA)/(M) and θB ) (k′NB)/(M), eqs 5 and 6 can be rewritten as
( ((
k-1 k-1 k′-1 1θA θB θA k k′ µAs ) kBT ln k(1 - θA - θB)kqA,s
( (
)
(
) )
k ′-1 k-1 k′-1 1θA θB θB k k′ µBs ) kBTln k ′(1 - θA - θB)kqB,s
(
) (
) )
) )
(7)
xbB
k-1
A
k(1 - θA - θB)kqA,s
µBob + kBT ln xBb )
(
θA
B
k ′-1
k-1 k′-1 1-( θ -( θ ( k ) k′ ) ) k T ln B
A
θB
B
k ′(1 - θA - θB)k ′qB,s
) )
(13)
(14)
By defining KA and KB as constant parameters, ob µA
KA ) kqA,se kBT
(15)
µBob
KB ) k ′ qB,se kBT
(16)
Equations 13 and 14 can be simplified as
k-1 k′-1 1-( θ -( θ ( k ) k′ ) ) ) A
θA
B
(17-i)
(1 - θA - θB)k
k ′-1
k-1 k′-1 1-( θ -( θ ( k ) k′ ) ) K x ) A
b B B
θB
B
(17-ii)
(1 - θA - θB)k ′
Equations 17 relate the bulk concentration to the surface concentration at equilibrium, and they are the final forms of the new isotherm for binary adsorption of A and B molecules, in such a way that each A and B molecule occupies k and k′ lattice sites, respectively. For the gas phase, xAb and xbB can be replaced by the partial pressure of A and B species, respectively. The plot of the term on the right-hand side of eq 17-i versus xAb is a line with the slope of KA. Also the KB constant can be obtained by plotting the term on the right-hand side of eq 17-ii versus xbB. By using the definition of θA ) qA/qm and θB ) qB/qm (qA and qB are the amount of adspecies A and B, respectively, and qm is the maximum amount of adspecies), eqs 17 convert to
k-1 k′-1 q -( q -( q ( k ) k′ ) ) )
k-1
KAxAb
m
A
B
qA (18-i)
(qm - qA - qB)k
k ′-1
(8)
The chemical potentials of A and B molecules in the bulk phase (µAs, µsB)are obtained by
xAb
(
k-1 k′-1 1-( θ -( θ ( k ) k′ ) ) k T ln
KAxAb
T,M,Nj*i
µBs ) kBT[(k ′ -1)
( (
(12)
k-1
µAs ) kBT[(k - 1) ln[(M - (k - 1)NA - (k ′ -1)NB] + ln NA - k ln(M - kNA - k ′ NB) - ln qA,s] (3)
ln
µAob + kBT ln xAb )
(2)
where kB is the Boltzmann constant. By substitution of eq 1 into eq 2, and using Stirling approximation, one arrives at
µAs ) kBT
(11)
µBb ) µBs Substitution of eqs 7-10 into eqs 11 and 12 gives
B
Let us assume that there is a two-dimensional lattice of M sites for binary adsorption of A and B molecules. NA and NB molecules of A and B are adsorbed on the surface, respectively, in such a way that each A molecule occupies k lattice sites and each B molecule occupies k′ lattice sites. It is assumed that there is no interaction between adspecies. The canonical partition function Q(M,NA,NB,T) for this system is
µAb ) µAs
µAb ) µAob + kBT ln xAb
(9)
µBb ) µBob + kBT ln xBb
(10)
where and are the concentration or pressure of A and B molecules in the bulk phase, respectively. At equilibrium
k-1 k′-1 q -( q -( q ( ) k k′ ) ) K x ) b B B
m
A
(qm - qA - qB)k ′
B
qB (18-ii)
Therefore, the KA and KB constants can be obtained by plotting the terms on the right-hand side of eqs 18 versus xAb and xbB, respectively. In these linear plots, the qm parameter is considered an adjustable parameter for the best linear plots. It is also possible to derive more simple isotherm equations for binary adsorption when one of the components occupies one site in lattice sites (for example, k′ ) 1). When assuming that k′ ) 1, eqs 17 simplify to
Isotherm for Multisite Occupancy of Binary Mixtures
Langmuir, Vol. 25, No. 4, 2009 2311
Figure 1. Equilibrium of competitive adsorption of propane and butane on Vycor glass for (a) propane and (b) butane. Open circles are experimental data, and the solid lines are theoretical curves calculated from eqs 18 with the best fit parameters displayed in Table 1.
Figure 2. Equilibrium of competitive adsorption of methane and carbon dioxide on wet Tiffany coal for (a) methane and (b) carbon dioxide. Open circles are experimental data, and the solid lines are theoretical curves calculated from eqs 18 with the best fit parameters displayed in Table 1.
(1 - ( k -k 1 )θ )
k-1
KAxAb )
KBxBb )
A
θA (19-i)
(1 - θA - θB)k θB (1 - θA - θB)
(19-ii)
It should be noted that eq 19-ii is the extended Langmuir isotherm. Therefore, when all the adsorbates occupy one site in the lattice sites (k ) 1 and k′ ) 1), the new isotherm equation (eqs 17) converts to the Langmuir isotherm for binary mixtures. By combination of eqs 19 we obtain k-1 k-1 θA 1KA θBxAb k ) KB θ xb (1 - θA - θB)k-1 A B
( (
) )
(20)
Eq 20 can be transformed to linear form by taking the logarithm and then rearranging:
( )
ln
θBxAb
θAxBb
(
) (k - 1) ln
1-
( k -k 1 )θ
A
1 - θA - θB
)
- ln
KA KB
(21)
Therefore, it is expected that the plot of ln[(θBxAb)/(θAxbB)] versus ln[(1 -[(k - 1)/k]θA)/(1 - θA - θB)] is a line (for k′ ) 1 and k * 1). The tangent and intercept of this plot are (k - 1) and -ln(KA)/(KB), respectively.
Results and Discussion In this section, we want to analyze the applicability of the derived equations (eqs 18) for multisite occupancy adsorption of binary systems at equilibrium. Two different sets of experimental data have been selected from the literature38,39 to be analyzed by applying the newly obtained isotherms for binary systems (eqs 18). The first experimental system that was considered is the binary adsorption of propane and butane on Vycor glass.38 Different
values of k and k′ were tested until a reasonable reproduction of the experimental kinetic data was obtained. It is found that the best fitting can be obtained with k ) 3 and k′ ) 2. This means that each propane molecule occupies three sites (k ) 3), and each butane molecule occupies two sites (k′ ) 2). Of course, for understanding the exact structure of these molecules on the surface, spectroscopic studies are necessary. The terms on the right-hand side of eqs 18 were calculated by using the experimental data of the binary system, and then the obtained data for propane and butane were plotted versus their partial pressure (plots not shown here). The value of qm was adjusted until a reasonable reproduction of the experimental equilibrium data was obtained. The obtained value of qm is presented in Table 1. The coefficients of eqs 18 (Kpropane and Kbutane) were obtained from the tangent of these plots and are listed in Table 1. From eqs 18, and the obtained values of qm, Kpropane, and Kbutane, the equilibrium amounts of adsorbed propane and butane molecules at different partial pressures were calculated. Figure 1 shows the experimental (open circles) and calculated (solid lines) adsorption isotherm, i.e., the value of the equilibrium amount of adspecies as a function of partial pressure. Figure 1a and Figure 1b are for propane and butane molecules in a binary system, respectively. As shown in Figure 1, there is good agreement between the experimental and calculated values. This agreement indicates that the obtained isotherm equations for multisite occupancy of binary adsorption (eqs 18) are suitable expressions. The next competitive adsorption system that has been selected to be analyzed by eqs 18 is the binary adsorption of methane and carbon dioxide on wet Tiffany coal.39 The results of fitting of the equilibrium experimental data by eqs 18 are presented in Table 1. The best fitting was obtained when assuming that each (38) Cermakova, J. R.; Markovic, A.; Uchytil, P.; Seidel-Morgenstern, A. Chem. Eng. Sci. 2008, 63, 1568. (39) Fitzgerlad, J. E.; Pan, Z.; Sudibandriyo, M.; Robinson, R. L., Jr.; Gasem, K. A. M.; Reeves, S. Fuel 2005, 84, 2351.
2312 Langmuir, Vol. 25, No. 4, 2009
methane molecule occupies one site (k ) 1) and each carbon dioxide occupies two sites (k′ ) 2) in lattice sites. In Figure 2, the open circles are experimental data and solid lines are calculated values of the amount of adspecies by eqs 18. Figure 2a and Figure 2b show the adsorption isotherms of methane and carbon dioxide in binary mixtures, respectively. As shown in Figure 2 there is acceptable agreement between the calculated and experimental data, but the agreement is not perfect, especially for carbon dioxide. There are different reasons for this derivation, including surface heterogeneity and adsorbate-adsorbate interactions, which are not considered in eqs 18. In summary, the behavior of equilibrium experimental data of the systems studied here are in good agreement with our new isotherm equations. The k and k′ parameters in eqs 18 provide information about the structure of adsorbed species under equilibrium conditions. Thus, the obtained isotherm equations (eqs 18) not only fit to the experimental data, but also provide
Azizian and Bashiri
important information about adsorbates structures in binary systems and under equilibrium conditions.
Conclusion A new isotherm equation for multisite occupancy of binary mixtures was developed on the basis of statistical thermodynamics. Information about the structure of adsorbed species by using equilibrium adsorption data can be found with this new isotherm equation. A simpler isotherm equation for binary adsorption has been derived when one of the components occupies one site in the lattice sites. The obtained isotherm equation has been applied to experimental systems including binary adsorption of propane and butane on Vycor glass, and also adsorption of methane and carbon dioxide on wet Tiffany coal. In the both cases, the new obtained isotherm equations show good fitting to the equilibrium experimental data. LA803675H
