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Adsorption of Chain Molecules
Journal of Colloid and Interface Science 213, 457– 464 (1999) Article ID jcis.1999.6160, available online at http://www.idealibrary.com on
Adsorption of Chain Molecules G. L. Aranovich and M. D. Donohue 1 Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218 Received September 21, 1998; accepted February 17, 1999
molecule contains r A segments, and the B molecule has r B segments. Each A molecule on the surface is attracted to the surface with an energy r Ae AS. For the B molecule, this energy is r Be BS. Figure 1 illustrates chain-like molecules on the surface.
We analyze the influence of chain length on the adsorption isotherm using the framework of lattice theory. Each molecule is represented as a chain of segments occupying separate sites in the lattice. Adsorption equilibria (particularly adsorption isotherms) are analyzed for one-component and two-component mixtures of chain molecules. © 1999 Academic Press Key Words: chain molecules; phase equilibria; lattice theory; adsorption isotherm for gas mixture.
EQUATIONS OF EQUILIBRIUM
Consider the exchange of an A molecule on the surface with r A vacancies (chain of vacancies) in the bulk:
INTRODUCTION
Mixtures of chain molecules often are separated by adsorption (1). For example, there are many industrial processes where mixtures of normal alkanes are separated by adsorbents, such as alumina, silica, carbon, or composites of these (2). Here we analyze the influence of chain length on the adsorption isotherm using the framework of lattice theory. Each chain molecule is represented as a string of segments occupying adjacent sites of the lattice. We assume that all segments of a molecule have the same interaction energies; for n-alkanes this implies that methane, methyl, and methylene groups behave similarly. This allows the effect of chain length to be predicted using simple, analytical equations. These assumptions are analogous to those used by Russell and LeVan in their group-contribution theory (3). MODEL
Consider a ternary mixture of molecules and holes on a lattice with a boundary. Each site of the lattice can contain a segment of an A molecule, a segment of a B molecule, or a hole. There are interactions between nearest neighbors with e AA, e AB, and e BB being the energy of adsorbate–adsorbate (segment–segment) interactions and e AS and e BS being the energy for adsorbate–surface (segment–surface) interactions. We assume that the lattice fluid is in contact with a flat surface at the plane of i 5 0 and that the first layer of adsorbed molecules is in the plane of i 5 1. To simplify the mathematics, we assume that if a molecule is on the surface, all segments touch the surface. The A 1
A s 1 r AV b 3 A b 1 r AV s.
[1]
If this transfer occurs at equilibrium, then DH 2 TDS 5 0,
[2]
where DH and DS are the enthalpy and entropy changes (for a lattice system, the enthalpy is equal to the configurational energy). We have shown previously (4) that the equations of equilibrium for the exchange of molecules shown in Eq. [1] are equivalent to the calculation of the partition function for the lattice system and minimization of free energy as proposed by Ono and Kondo (5). The value of DS can be represented in the form DS 5 k Bln W 1 2 k Bln W 2 ,
[3]
where W 1 is the number of configurations where an A molecule is on the surface and r A vacancies are in the bulk, W 2 is the number of configurations where an A molecule is in the bulk and there is a “chain” of r A vacancies on the surface. Let W 0 be the overall number of configurations for the system. Then: W 1 x AS 5 ~1 2 x A 2 x B! r A W 0 2r A
To whom correspondence should be addressed. 457
[4]
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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ARANOVICH AND DONOHUE
Equation [1] can be considered also for B molecules: B s 1 r BV b 3 B b 1 r BV s.
[9]
Equations [6] and [7] then take the following form: DS 5 k Bln FIG. 1. A and B molecules on the surface.
1 ~3r B 1 2!~ e BBx B 1 e ABx A!. W2 xA 5 ~1 2 x AS 2 x BS! r A, W 0 3r A
~3x AS/ 2!~1 2 x A 2 x B! r A . x A~1 2 x AS 2 x BS! r A
Equations [2], [10], and [11] give:
[6]
The change in enthalpy can be calculated in the mean-field approximation as: DH 5 2@r Ae AS 1 ~2r A 1 2!~ e AAx AS 1 e ABx BS! 1 r A~ e AAx A 1 e ABx B!# 1 @~4r A 1 2!~ e AAx A 1 e ABx B!#.
[7]
In Eq. [7], the first term is the energy for a molecule on the surface, and the second term is the energy for a molecule in the bulk. From Eqs. [2], [6], and [7] it follows that ~3x AS/ 2!~1 2 x A 2 x B! r A r Ae AS 1 x A~1 2 x AS 2 x BS! r A k BT 1
~2r A 1 2!~ e AAx AS 1 e ABx BS! k BT 2
~3r A 1 2!~ e AAx A 1 e ABx B! 5 0. k BT
[11]
[5]
where x AS is the probability of the surface site being occupied with a segment of an A molecule, x BS is the probability of the surface site being occupied with a segment of a B molecule, x A is the probability that the site infinitely distant from the surface is occupied by a segment of an A molecule, and x B is the probability that the site infinitely distant from the surface is occupied by a segment of a B molecule. The value of 2r A in Eq. [4] is the number of configurations of the chain on the surface where a certain site is occupied with a segment of the chain, and the value of 3r A in Eq. [5] is this number for the bulk. The values of (1 2 x A 2 x B) r A and (1 2 x AS 2 x BS ) r A are probabilities of having a chain of vacancies in the bulk and on the surface, respectively. Substituting Eqs. [4] and [5] into Eq. [3] we obtain: DS 5 k Bln
[10]
DH 5 2r Be BS 2 ~2r B 1 2!~ e BBx BS 1 e ABx AS!
and
ln
~3x BS/ 2!~1 2 x A 2 x B! r B , x B~1 2 x AS 2 x BS! r B
[8]
ln
~3x BS/ 2!~1 2 x A 2 x B! r B r Be BS 1 x B~1 2 x AS 2 x BS! r B k BT 1
~2r B 1 2!~ e BBx BS 1 e ABx AS! k BT 2
~3r B 1 2!~ e BBx B 1 e ABx A! 5 0. k BT
[12]
Equations [8] and [12] determine the values of x AS and x BS as functions of x A, x B, e AA, e AB, e BB, e AS, e BS, r A, r B, and T. Adsorption isotherms for component A can be calculated as a function of x A at constant x B and T. Note also that the ratio (1 2 x A 2 x B)/(1 2 x AS 2 x BS) in the logarithms of Eqs. [8] and [12] goes to a finite value when 1 2 x A 2 x B goes to zero. This can be shown by expanding x AS 1 x BS in powers of 1 2 x A 2 x B. A noteworthy feature of Eqs. [8] and [12] is that they do not turn into equations for monomers when r A 3 1 and r B 3 1. This can be explained as follows. The chain molecules are one-dimensional (1D), the surface is two-dimensional (2D), and the bulk is three-dimensional (3D); therefore, we have a (1D,2D,3D) problem. In case of monomers, we have a (0D,2D,3D) problem. However, the solution of the (1D,2D,3D) problem does not necessarily reduce to the solution for (0D,2D,3D) problem as r A goes to unity. The difference between the solution for (0D,2D,3D) problem and for (1D,2D,3D) problem at r A 5 1 comes from the entropy term. In this term, for the (1D,2D,3D) problem there are 3r A configurations in the bulk and 2r A configurations on the surface. The ratio of these values, s, is 23, and it does not depend on r A. For the (0D,2D,3D) problem, both numbers of configurations are unity, and their ratio is unity. Of course, for rigid molecules, the value of s depends on the shape of molecules. rA rA In general, s 5 ¥ i51 (r ib 1 1)/¥ i51 (r is 1 1), where r ib and r is are the numbers of possible distinguishable rotations for the molecule in the bulk and on the surface where the center of rotation is the ith segment. The values of r ib and r is depend on the dimensionality and symmetry of the molecule. For mono-
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ADSORPTION OF CHAIN MOLECULES
mers, r ib 5 r is 5 0 and s 5 1. For linear molecules, r ib 5 2, r is 5 1, and s 5 23. When r A goes to unity, the symmetry of the molecule changes. Therefore, the value of s does not change continuously. SPECIAL CASES AND COMPARISON WITH GROUP-CONTRIBUTION THEORY
For this case, the group-contribution theory of Russell and LeVan (3) gives the following equations: x AK A~T! 5
x AS ~1 2 x AS 2 x BS! r A
[19]
x BK B~T! 5
x BS . ~1 2 x AS 2 x BS! r B
[20]
and
In the Henry’s law range (limit of small x A), Eq. [8] gives: 2x A~1 2 x BS! x AS 5 3~1 2 x B! r A
rA
F
G
2r Ae AS 2 ~2r A 1 2! e ABx BS 1 ~3r A 1 2! e ABx B 3 exp . k BT [13] From Eq. [12], for small x B we have: x BS 5
2x B~1 2 x AS! r B 3~1 2 x A! r B
F
3 exp
G
2r Be BS 2 ~2r B 1 2! e ABx AS 1 ~3r B 1 2! e ABx A . k BT [14]
The assumptions in our model are very close to those of the group-contribution theory of Russell and LeVan (3). However, there are differences which we consider here. The nature of these differences can be shown by comparisons of particular cases. For e AA 5 e AB 5 e BB 5 0, Eqs. [8] and [12] yield Langmuir-like equations for a binary mixture which can be represented in the form: x A f A~T! 5
x AS ~1 2 x AS 2 x BS! r A
[15]
Here K A(T) 5 K *AP A/x A and K B(T) 5 K *BP B/x B are adjustable, temperature-dependent parameters, P A and P B are partial pressures, and K *A and K *B are temperature-dependent functions. Equations [15] and [16] are identical in form to Eqs. [19] and [20], respectively. The only difference is that the coefficients f A(T) and f B(T) in our equations are derived explicitly but the K A(T) and K B(T) of group-contribution theory are adjustable parameters. As seen from Eqs. [17] and [18], for small x A and x B (more exactly x A 1 x B ! 1/r A and x A 1 x B ! 1/r B) the coefficients f A(T) and f B(T) become functions only of temperature and do not depend on x A or x B. Next, consider the case where r A 5 r B 5 1 and e AA, e AB, and e BB are not zero. In this case, the group-contribution theory of Russel and LeVan (3) reduces to Frumkin’s isotherm for binary system (6) (Russell and LeVan refer to this as the multicomponent Fowler–Guggenheim isotherms (7)): x AK A~T! 5 @ x AS/~1 2 x AS 2 x BS!# 3 exp@4~ e AAx AS 1 e ABx BS!/k BT#
[21]
and x BK B~T! 5 @ x BS/~1 2 x AS 2 x BS!# 3 exp@4~ e BBx BS 1 e ABx AS!/k BT#.
[22]
and For this case Eqs. [8] and [12] give: x BS x B f B~T! 5 , ~1 2 x AS 2 x BS! r B
[16]
x A f A~T! 5 @ x AS/~1 2 x AS 2 x BS!#exp@4~ e AAx AS 1 e ABx BS!/k BT 2 5~ e AAx A 1 e ABx B!/k BT#
where fA 5
2 exp~2e AS/k BT! ~1 2 x A 2 x B! r A
and [17] x B f B~T! 5 @ x BS/~1 2 x AS 2 x BS!#exp@4~ e BBx BS 1 e ABx AS!/k BT 2 5~ e BBx B 1 e ABx A!/k BT#.
and fB 5
2 exp~2e BS/k BT! . ~1 2 x A 2 x B! r B
[23]
[18]
[24]
Comparison of Eqs. [21] and [22] with Eqs. [23] and [24] shows that the group-contribution theory (3) does not take into
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ARANOVICH AND DONOHUE
FIG. 2. Dependence of the surface density, x AS, on the bulk density, x A, for one-component dimers at e AS/k BT 5 20.5 and e AA/k BT 5 0. Also shown are the dependence for monomer at e AS/k BT 5 21.0 and e AA/k BT 5 0 and Monte Carlo simulations.
account interactions of adsorbate molecules with the bulk. This is a reasonable approximation when x A ! x AS
ln
3x AS~1 2 x A! r A r Ae AS ~2r A 1 2! e AAx AS 1 1 2x A~1 2 x AS! r A k BT k BT
[25] 2
and x B ! x BS.
FIG. 4. Dependence of the surface density, x AS, on the bulk density, x A, for one-component chains (tetramers) at e AS/k BT 5 20.5 and e AA/k BT 5 0. Also shown are this dependence for monomer at e AS/k BT 5 22.0 and e AA/k BT 5 0 and Monte Carlo simulations. The dotted line shows the prediction of Eq. [27] for ^r A& 5 2.78.
[26]
However, Eqs. [25] and [26] might be not met if adsorption of one of the components is small or if the bulk concentration becomes large. For a one-component chain molecule (A molecules and holes), Eq. [8] can be written in the following form:
FIG. 3. Dependence of the surface density, x AS, on the bulk density, x A, for one-component dimers at e AS/k BT 5 20.5 and e AA/k BT 5 20.4. Also shown are the dependence for monomer at e AS/k BT 5 21.0 and e AA/k BT 5 20.4 and Monte Carlo simulations.
~3r A 1 2! e AAx A 5 0. k BT
[27]
COMPARISON WITH SIMULATIONS AND EXPERIMENTS
Equations [8], [12], and [27] are nonlinear with respect to x AS and x BS and cannot be solved analytically. However, they easily can be solved numerically. Figures 2–5 show adsorption isotherms for one-component systems predicted by Eq. [27].
FIG. 5. Dependence of the surface density, x AS, on the bulk density, x A, for one-component chains (tetramers) at e AS/k BT 5 20.5 and e AA/k BT 5 20.4. Also shown are this dependence for monomer at e AS/k BT 5 22.0 and e AA/k BT 5 20.4 and Monte Carlo simulations. The dotted line shows the prediction of Eq. [27] for ^r A& 5 2.78.
ADSORPTION OF CHAIN MOLECULES
461
FIG. 6. Surface densities of components A and B, x AS and x BS, as functions of the bulk density of component A, x A, for x B 5 0.2, e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.3 and different e AS/k BT, e BS/k BT, r A, and r B: (a) e BS/k BT 5 20.4, r A 5 3, r B 5 2, and different e AS/k BT: 20.4 (1) and 20.8 (2); (b) e AS/k BT 5 e BS/k BT 5 20.4, r B 5 2, and different r A: 3 (1) and 7 (2); (c) e AS/k BT 5 e BS/k BT 5 20.4, r A 5 3, and different r B: 12 (1) and 2 (2); (d) e AS/k BT 5 20.4, r A 5 3, r B 5 2, and different e BS/k BT: 20.8 (1) and 20.4 (2).
Also shown in these figures are calculations for monomers with the same molecule/surface and segment/segment interaction energies. Figures 2 and 3 illustrate isotherms for dimers with segment–surface and segment–segment interaction energies of e AS/k BT 5 20.5, e AA/k BT 5 0, and e AS/k BT 5 20.5, e AA/k BT 5 20.4, respectively. As seen from Figs. 2 and 3, Eq. [27] predicts results which are reasonably close to Monte Carlo simulation data. The slight under-prediction of the surface density for dimers is because the theory does not account for molecules adsorbed perpendicular to the surface. In a previous paper (8), we derived more exact equations for dimers which take the effect of these perpendicular dimers into account.
Figures 4 and 5 compare isotherms for tetramers with Monte Carlo simulations and with monomers at e AS/k BT 5 20.5, e AA/k BT 5 0, and e AS/k BT 5 20.5, e AA/k BT 5 20.4, respectively. Figures 4 and 5 show that the differences between predictions of Eq. [27] and Monte Carlo simulations are greater for tetramers than for dimers. At least part of this difference is due to the fact that the Monte Carlo simulations were performed by taking into account all possible configurations, but Eq. [27] was derived by considering only configurations where all segments touch the surface if the molecule is on the surface. However, the average number of segments touching the surface, ^r A&, is less than r A. Equation [27] can be corrected
462
ARANOVICH AND DONOHUE
FIG. 7. Adsorption isotherm for butane from gas mixture with propane on activated carbon at T 5 293 K. Here r A 5 4, r B 5 3, e AS/k BT 5 e BS/k BT 5 22.8, and e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.2. Experimental data from Ref. (10).
without complication by using an adjusted value of ^r A& instead of r A. The dotted lines in Figs. 4 and 5 show predictions of Eq. [27] where ^r A& was used instead of r A. In Figs. 4 and 5, ^r A& 5 2.78 which was calculated in the limit of low density (see Eq. [33] in Appendix). As seen from Figs. 4 and 5, this results in closer agreement with the Monte Carlo simulations. As seen from Figs. 2–5, the predictions of the monomer model do not represent the behavior of the chain molecules. The surface density of monomers is always greater than for chain molecules for the same molecule–surface interactions and for monomer–monomer interaction equal to the segment– segment interaction. This suggests that adsorption models derived for monomers should not be very accurate when used for chain molecules; this should not be surprising as the analogous conclusion for bulk solutions was made by Hildebrand (9) more than sixty years ago. Figure 6 shows predictions of Eqs. [8] and [12] for twocomponent mixtures. Figures 6a and 6b illustrate that increasing the A molecule length, r A, has an effect that qualitatively is similar to increasing the energy between A molecules and surface, e AS/k BT. Figure 6c shows that increasing the B molecule length, r B, increases the surface concentration of B molecules, x BS, but decreases the surface concentration of A molecules, x AS. Figure 6d shows the influence of e BS/k BT on x AS and x BS. Adsorption isotherms for two-component mixtures can be represented in different ways. One way is to keep x B constant and to vary x A. In this case, the dependences of x AS and x BS on x A show the isotherm at constant x B. This is useful in an analysis of the mechanisms of adsorption. However, most experiments are performed by varying x A while keeping the sum of x A 1 x B constant (10). Experimentally, this is done by varying the partial pressure of one of the components while keeping the total pressure constant. We have applied the theory presented above to the analysis of adsorption of hydrocarbon mixtures on activated carbon at T 5 293 K (10). We solved Eqs. [8] and [12] numerically
under the constraint x A 1 x B 5 constant. We then calculated mole fractions of the first component in the adsorbed layer as a function of mole fraction of this component in the gas phase. To determine e AS/k BT and e BS/k BT, we used experimental values for the differential heats of adsorption at small filling for the pure components on graphite. These values are about 6.7 kcal/mol for butane and 5.0 kcal/mol for propane (11). The segment–surface interaction energy is then e AS ' 6.7/4 5 1.675 kcal/mol for butane, and e BS ' 5.0/3 ' 1.67 kcal/mol for propane. Therefore, the reduced hydrocarbon segment–surface interactions energies are e AS/k BT ' e BS/k BT ' 22.8 at T 5 293 K. The energy of the segment–segment interactions can be evaluated from the heat of vaporization, H v. At T 5 293 K, H v ' 5 kcal/mol for butane, and H v ' 3.6 kcal/mol for propane (12). Therefore, H v/kT ' 8 for butane, and H v/kT ' 6 for propane. The value of H v/kT can be represented as 2( z/ 2) e /kT where z is the average coordination number in liquid phase, and e is the energy of interactions between molecules of the pure components. The coordination number determined from the partition function for the square-well fluid is 18.9 (13). Assuming z 5 20, we have e */kT ' 20.8 for butane and e */kT ' 20.6 for propane where e* is the molecule– molecule energy of interaction. Then, for the segment–segment interactions we have e AA/k BT ' e AB/k BT ' e BB/k BT ' 20.2. Figure 7 shows adsorption isotherms for butane from a mixture with propane on activated carbon at T 5 293 K. For the theoretical curve, we used e AA/k BT 5 e AB/k BT 5 e BB/ k BT 5 20.2, e AS/k BT 5 e BS/k BT 5 22.8, r A 5 4, and r B 5 3. Experimental data are from Ref. (10) (p. 224). As seen from Fig. 7, the theoretical prediction is quite good given that the parameters all are estimated (not fitted to the adsorption isotherms). Figures 8 and 9 show adsorption isotherms for ethane from mixtures with methane and for propane from mixtures with ethane on the same activated carbon as in Fig. 7. The theoretical predictions are calculated with the same segmental ener-
FIG. 8. Adsorption isotherm for ethane from gas mixture with methane on activated carbon at T 5 293 K. Here r A 5 2, r B 5 1, e AS/k BT 5 e BS/k BT 5 22.8, and e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.2. Experimental data from Ref. (10).
463
ADSORPTION OF CHAIN MOLECULES
getic parameters. The only difference is that, for ethane–methane mixture, r A 5 2 and r B 5 1, and, for propane– ethane mixture, r A 5 3 and r B 5 2. As seen from Figs. 8 and 9, theoretical predictions are quite good. To compare our model with the group-contribution theory of Russell and LeVan (3), we calculated adsorption isotherms from Eqs. [21]–[24]. Figure 10 presents the total density in the adsorbed layer, x AS 1 x BS , as a function of the bulk density of component A, x A , for x B 5 0.2, r A 5 3, r B 5 2, e AA /k B T 5 e AB /k B T 5 e BB /k B T 5 20.3, and e AS / k B T 5 e BS /k B T 5 20.4. Also shown in Fig. 10 are the group-contribution theory predictions and Monte Carlo simulations. As seen from Fig. 10, the theory presented above is in agreement with Monte Carlo simulation data and the group-contribution theory is not. Figure 10 shows that our model and the groupcontribution theory agree at small densities. However, at large densities, these two theories predict quite different compositions of the adsorbed layer, and they give very different limits for x A 3 0.8 (which is equivalent to x A 1 x B 3 1). In this limit, ( x AS 1 x BS ) must go to unity as ( x A 1 x B ) goes to unity. Figure 10 illustrates that our theory predicts this limit correctly but the group-contribution theory does not. As discussed previously, this is because the group-contribution theory does not take into account interactions of adsorbate molecules with the bulk. CONCLUSION
A new model for adsorption of chain-like molecules on solid surfaces is presented for one- and two-component mixtures in the framework of the Ono and Kondo theory. This model shows the influence of energetic parameters and chain length on the composition in an adsorbed layer and on adsorption isotherm for each component. Comparison with experimental data for adsorption of butane–propane, pro-
FIG. 10. Total density in the adsorbed layer, x AS 1 x BS, as a function of the bulk density of component A, x A, at x B 5 0.2, r A 5 3, r B 5 2, e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.3, and e AS/k BT 5 e BS/k BT 5 20.4.
pane– ethane, and ethane–methane gas mixtures on activated carbon demonstrates the ability of this model to predict the composition of adsorbed layer as the function of the bulk composition with only two parameters: segment–segment and segment–surface energies of interactions which can be evaluated from differential heats of adsorption and heats of vaporization. APPENDIX
Rigorously speaking, the value of ^r A& is a function of the density and temperature. However, we can estimate ^r A& in the small density limit. In this limit, the number of possible configurations of the chain molecule on the surface, n i , is a function of the number of chain contacts, i, with the surface. The energy of the chain molecule is i e AS. From Eq. [27], it follows that in the small density limit: H5
S
D
x AS 2 r Ae AS 5 exp 2 , xA 3 k BT
[28]
where H is Henry’s coefficient. Applying formula [28] for each configuration and taking into account numbers of configurations, one obtains: Hi 5
S
D
x ASi 2 i e AS 5 v i exp 2 . xA 3 k BT
[29]
Since FIG. 9. Adsorption isotherm for propane from gas mixture with ethane on activated carbon at T 5 293 K. Here r A 5 3, r B 5 2, e AS/k BT 5 e BS/k BT 5 22.8, and e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.2. Experimental data from Ref. (10).
Ox rA
x AS 5
i51
ASi
,
[30]
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ARANOVICH AND DONOHUE
ACKNOWLEDGMENT
then
x AS 2 5 xA 3
O v expS 2 ike T D . rA
AS
[31]
i
B
i51
REFERENCES
Defining ^r A& by the following equation,
S
O
D
x AS 2 A ^r A& e AS 5 ~ v i !exp 2 , xA 3 i51 k BT r
[32]
we obtain from Eqs. [31] and [32]:
^r A& 5 2
F S DG
k BT ln rA e AS ¥ i51 vi
O v exp 2 ike T rA
AS
i
i51
M.D. acknowledges support by the Division of Chemical Sciences of the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-FG02-87ER13777.
[33]
B
For tetrameric molecules, numbers of different configurations are v 1 5 21, v 2 5 28, v 3 5 12, and v 4 5 24. With these numbers and with e AS/k BT 5 20.5, Eq. [33] gives ^r A& ' 2.78.
1. Masel, R., “Principles of Adsorption and Reaction on Solid Surfaces.” John Wiley, New York, 1996; Russell, B. P., and LeVan, M. D., Ind. Eng. Chem. Res. 36, 2380 (1997). 2. Perry, R. H., and Chilton, C. H., “Chemical Engineers’ Handbook,” Chap. 16. McGraw-Hill Book Co., New York, 1973. 3. Russell, B. P., and LeVan, M. D., Chem. Eng. Sci. 51, 4025 (1996). 4. Aranovich, G. L., and Donohue, M. D., J. Colloid Interface Sci. 200, 273 (1998). 5. Ono, S., and Kondo, S., in “Molecular Theory of Surface Tension in Liquids” (S. Flu¨gge, Ed.), Encyclopedia of Physics, Vol. 10, p. 134. Springer, Berlin, 1960. 6. Frumkin, A., Z. Phys. Chem. (Leipzig) 116, 466 (1925). 7. Fowler, R. H., and Guggenheim, E. A., “Statistical Thermodynamics.” Cambridge Univ. Press, Cambridge, UK, 1949. 8. Wu, D.-W., Aranovich, G. L., and Donohue, M. D., J. Colloid Interface Sci. 1998, in press. 9. Hildebrand, J. H., J. Am. Chem. Soc. 51, 66 (1929); Hildebrand, J. H., “Solubility of Non-Electrolytes.” Reinhold Publishing Corp., New York, 1936. 10. Valenzuela, D. P., and Myers, A. L., “Adsorption Equilibrium Data Handbook.” Prentice Hall, Englewood Cliffs, NJ, 1989. 11. Kiselev, A. V., and Yashin, Y. I., “Gas Adsorption Chromatography,” p. 131. Nauka, Moscow, 1967. 12. Perry, R. H., and Chilton, C. H., “Chemical Engineers’ Handbook,” Chap. 3, p. 115. McGraw-Hill Book Co., New York, 1973. 13. Lee, K.-H., Lombardo, M., and Sandler, S. I., Fluid Phase Equilibria 21, 177 (1985).
Adsorption of Chain Molecules G. L. Aranovich and M. D. Donohue 1 Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218 Received September 21, 1998; accepted February 17, 1999
molecule contains r A segments, and the B molecule has r B segments. Each A molecule on the surface is attracted to the surface with an energy r Ae AS. For the B molecule, this energy is r Be BS. Figure 1 illustrates chain-like molecules on the surface.
We analyze the influence of chain length on the adsorption isotherm using the framework of lattice theory. Each molecule is represented as a chain of segments occupying separate sites in the lattice. Adsorption equilibria (particularly adsorption isotherms) are analyzed for one-component and two-component mixtures of chain molecules. © 1999 Academic Press Key Words: chain molecules; phase equilibria; lattice theory; adsorption isotherm for gas mixture.
EQUATIONS OF EQUILIBRIUM
Consider the exchange of an A molecule on the surface with r A vacancies (chain of vacancies) in the bulk:
INTRODUCTION
Mixtures of chain molecules often are separated by adsorption (1). For example, there are many industrial processes where mixtures of normal alkanes are separated by adsorbents, such as alumina, silica, carbon, or composites of these (2). Here we analyze the influence of chain length on the adsorption isotherm using the framework of lattice theory. Each chain molecule is represented as a string of segments occupying adjacent sites of the lattice. We assume that all segments of a molecule have the same interaction energies; for n-alkanes this implies that methane, methyl, and methylene groups behave similarly. This allows the effect of chain length to be predicted using simple, analytical equations. These assumptions are analogous to those used by Russell and LeVan in their group-contribution theory (3). MODEL
Consider a ternary mixture of molecules and holes on a lattice with a boundary. Each site of the lattice can contain a segment of an A molecule, a segment of a B molecule, or a hole. There are interactions between nearest neighbors with e AA, e AB, and e BB being the energy of adsorbate–adsorbate (segment–segment) interactions and e AS and e BS being the energy for adsorbate–surface (segment–surface) interactions. We assume that the lattice fluid is in contact with a flat surface at the plane of i 5 0 and that the first layer of adsorbed molecules is in the plane of i 5 1. To simplify the mathematics, we assume that if a molecule is on the surface, all segments touch the surface. The A 1
A s 1 r AV b 3 A b 1 r AV s.
[1]
If this transfer occurs at equilibrium, then DH 2 TDS 5 0,
[2]
where DH and DS are the enthalpy and entropy changes (for a lattice system, the enthalpy is equal to the configurational energy). We have shown previously (4) that the equations of equilibrium for the exchange of molecules shown in Eq. [1] are equivalent to the calculation of the partition function for the lattice system and minimization of free energy as proposed by Ono and Kondo (5). The value of DS can be represented in the form DS 5 k Bln W 1 2 k Bln W 2 ,
[3]
where W 1 is the number of configurations where an A molecule is on the surface and r A vacancies are in the bulk, W 2 is the number of configurations where an A molecule is in the bulk and there is a “chain” of r A vacancies on the surface. Let W 0 be the overall number of configurations for the system. Then: W 1 x AS 5 ~1 2 x A 2 x B! r A W 0 2r A
To whom correspondence should be addressed. 457
[4]
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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ARANOVICH AND DONOHUE
Equation [1] can be considered also for B molecules: B s 1 r BV b 3 B b 1 r BV s.
[9]
Equations [6] and [7] then take the following form: DS 5 k Bln FIG. 1. A and B molecules on the surface.
1 ~3r B 1 2!~ e BBx B 1 e ABx A!. W2 xA 5 ~1 2 x AS 2 x BS! r A, W 0 3r A
~3x AS/ 2!~1 2 x A 2 x B! r A . x A~1 2 x AS 2 x BS! r A
Equations [2], [10], and [11] give:
[6]
The change in enthalpy can be calculated in the mean-field approximation as: DH 5 2@r Ae AS 1 ~2r A 1 2!~ e AAx AS 1 e ABx BS! 1 r A~ e AAx A 1 e ABx B!# 1 @~4r A 1 2!~ e AAx A 1 e ABx B!#.
[7]
In Eq. [7], the first term is the energy for a molecule on the surface, and the second term is the energy for a molecule in the bulk. From Eqs. [2], [6], and [7] it follows that ~3x AS/ 2!~1 2 x A 2 x B! r A r Ae AS 1 x A~1 2 x AS 2 x BS! r A k BT 1
~2r A 1 2!~ e AAx AS 1 e ABx BS! k BT 2
~3r A 1 2!~ e AAx A 1 e ABx B! 5 0. k BT
[11]
[5]
where x AS is the probability of the surface site being occupied with a segment of an A molecule, x BS is the probability of the surface site being occupied with a segment of a B molecule, x A is the probability that the site infinitely distant from the surface is occupied by a segment of an A molecule, and x B is the probability that the site infinitely distant from the surface is occupied by a segment of a B molecule. The value of 2r A in Eq. [4] is the number of configurations of the chain on the surface where a certain site is occupied with a segment of the chain, and the value of 3r A in Eq. [5] is this number for the bulk. The values of (1 2 x A 2 x B) r A and (1 2 x AS 2 x BS ) r A are probabilities of having a chain of vacancies in the bulk and on the surface, respectively. Substituting Eqs. [4] and [5] into Eq. [3] we obtain: DS 5 k Bln
[10]
DH 5 2r Be BS 2 ~2r B 1 2!~ e BBx BS 1 e ABx AS!
and
ln
~3x BS/ 2!~1 2 x A 2 x B! r B , x B~1 2 x AS 2 x BS! r B
[8]
ln
~3x BS/ 2!~1 2 x A 2 x B! r B r Be BS 1 x B~1 2 x AS 2 x BS! r B k BT 1
~2r B 1 2!~ e BBx BS 1 e ABx AS! k BT 2
~3r B 1 2!~ e BBx B 1 e ABx A! 5 0. k BT
[12]
Equations [8] and [12] determine the values of x AS and x BS as functions of x A, x B, e AA, e AB, e BB, e AS, e BS, r A, r B, and T. Adsorption isotherms for component A can be calculated as a function of x A at constant x B and T. Note also that the ratio (1 2 x A 2 x B)/(1 2 x AS 2 x BS) in the logarithms of Eqs. [8] and [12] goes to a finite value when 1 2 x A 2 x B goes to zero. This can be shown by expanding x AS 1 x BS in powers of 1 2 x A 2 x B. A noteworthy feature of Eqs. [8] and [12] is that they do not turn into equations for monomers when r A 3 1 and r B 3 1. This can be explained as follows. The chain molecules are one-dimensional (1D), the surface is two-dimensional (2D), and the bulk is three-dimensional (3D); therefore, we have a (1D,2D,3D) problem. In case of monomers, we have a (0D,2D,3D) problem. However, the solution of the (1D,2D,3D) problem does not necessarily reduce to the solution for (0D,2D,3D) problem as r A goes to unity. The difference between the solution for (0D,2D,3D) problem and for (1D,2D,3D) problem at r A 5 1 comes from the entropy term. In this term, for the (1D,2D,3D) problem there are 3r A configurations in the bulk and 2r A configurations on the surface. The ratio of these values, s, is 23, and it does not depend on r A. For the (0D,2D,3D) problem, both numbers of configurations are unity, and their ratio is unity. Of course, for rigid molecules, the value of s depends on the shape of molecules. rA rA In general, s 5 ¥ i51 (r ib 1 1)/¥ i51 (r is 1 1), where r ib and r is are the numbers of possible distinguishable rotations for the molecule in the bulk and on the surface where the center of rotation is the ith segment. The values of r ib and r is depend on the dimensionality and symmetry of the molecule. For mono-
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ADSORPTION OF CHAIN MOLECULES
mers, r ib 5 r is 5 0 and s 5 1. For linear molecules, r ib 5 2, r is 5 1, and s 5 23. When r A goes to unity, the symmetry of the molecule changes. Therefore, the value of s does not change continuously. SPECIAL CASES AND COMPARISON WITH GROUP-CONTRIBUTION THEORY
For this case, the group-contribution theory of Russell and LeVan (3) gives the following equations: x AK A~T! 5
x AS ~1 2 x AS 2 x BS! r A
[19]
x BK B~T! 5
x BS . ~1 2 x AS 2 x BS! r B
[20]
and
In the Henry’s law range (limit of small x A), Eq. [8] gives: 2x A~1 2 x BS! x AS 5 3~1 2 x B! r A
rA
F
G
2r Ae AS 2 ~2r A 1 2! e ABx BS 1 ~3r A 1 2! e ABx B 3 exp . k BT [13] From Eq. [12], for small x B we have: x BS 5
2x B~1 2 x AS! r B 3~1 2 x A! r B
F
3 exp
G
2r Be BS 2 ~2r B 1 2! e ABx AS 1 ~3r B 1 2! e ABx A . k BT [14]
The assumptions in our model are very close to those of the group-contribution theory of Russell and LeVan (3). However, there are differences which we consider here. The nature of these differences can be shown by comparisons of particular cases. For e AA 5 e AB 5 e BB 5 0, Eqs. [8] and [12] yield Langmuir-like equations for a binary mixture which can be represented in the form: x A f A~T! 5
x AS ~1 2 x AS 2 x BS! r A
[15]
Here K A(T) 5 K *AP A/x A and K B(T) 5 K *BP B/x B are adjustable, temperature-dependent parameters, P A and P B are partial pressures, and K *A and K *B are temperature-dependent functions. Equations [15] and [16] are identical in form to Eqs. [19] and [20], respectively. The only difference is that the coefficients f A(T) and f B(T) in our equations are derived explicitly but the K A(T) and K B(T) of group-contribution theory are adjustable parameters. As seen from Eqs. [17] and [18], for small x A and x B (more exactly x A 1 x B ! 1/r A and x A 1 x B ! 1/r B) the coefficients f A(T) and f B(T) become functions only of temperature and do not depend on x A or x B. Next, consider the case where r A 5 r B 5 1 and e AA, e AB, and e BB are not zero. In this case, the group-contribution theory of Russel and LeVan (3) reduces to Frumkin’s isotherm for binary system (6) (Russell and LeVan refer to this as the multicomponent Fowler–Guggenheim isotherms (7)): x AK A~T! 5 @ x AS/~1 2 x AS 2 x BS!# 3 exp@4~ e AAx AS 1 e ABx BS!/k BT#
[21]
and x BK B~T! 5 @ x BS/~1 2 x AS 2 x BS!# 3 exp@4~ e BBx BS 1 e ABx AS!/k BT#.
[22]
and For this case Eqs. [8] and [12] give: x BS x B f B~T! 5 , ~1 2 x AS 2 x BS! r B
[16]
x A f A~T! 5 @ x AS/~1 2 x AS 2 x BS!#exp@4~ e AAx AS 1 e ABx BS!/k BT 2 5~ e AAx A 1 e ABx B!/k BT#
where fA 5
2 exp~2e AS/k BT! ~1 2 x A 2 x B! r A
and [17] x B f B~T! 5 @ x BS/~1 2 x AS 2 x BS!#exp@4~ e BBx BS 1 e ABx AS!/k BT 2 5~ e BBx B 1 e ABx A!/k BT#.
and fB 5
2 exp~2e BS/k BT! . ~1 2 x A 2 x B! r B
[23]
[18]
[24]
Comparison of Eqs. [21] and [22] with Eqs. [23] and [24] shows that the group-contribution theory (3) does not take into
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ARANOVICH AND DONOHUE
FIG. 2. Dependence of the surface density, x AS, on the bulk density, x A, for one-component dimers at e AS/k BT 5 20.5 and e AA/k BT 5 0. Also shown are the dependence for monomer at e AS/k BT 5 21.0 and e AA/k BT 5 0 and Monte Carlo simulations.
account interactions of adsorbate molecules with the bulk. This is a reasonable approximation when x A ! x AS
ln
3x AS~1 2 x A! r A r Ae AS ~2r A 1 2! e AAx AS 1 1 2x A~1 2 x AS! r A k BT k BT
[25] 2
and x B ! x BS.
FIG. 4. Dependence of the surface density, x AS, on the bulk density, x A, for one-component chains (tetramers) at e AS/k BT 5 20.5 and e AA/k BT 5 0. Also shown are this dependence for monomer at e AS/k BT 5 22.0 and e AA/k BT 5 0 and Monte Carlo simulations. The dotted line shows the prediction of Eq. [27] for ^r A& 5 2.78.
[26]
However, Eqs. [25] and [26] might be not met if adsorption of one of the components is small or if the bulk concentration becomes large. For a one-component chain molecule (A molecules and holes), Eq. [8] can be written in the following form:
FIG. 3. Dependence of the surface density, x AS, on the bulk density, x A, for one-component dimers at e AS/k BT 5 20.5 and e AA/k BT 5 20.4. Also shown are the dependence for monomer at e AS/k BT 5 21.0 and e AA/k BT 5 20.4 and Monte Carlo simulations.
~3r A 1 2! e AAx A 5 0. k BT
[27]
COMPARISON WITH SIMULATIONS AND EXPERIMENTS
Equations [8], [12], and [27] are nonlinear with respect to x AS and x BS and cannot be solved analytically. However, they easily can be solved numerically. Figures 2–5 show adsorption isotherms for one-component systems predicted by Eq. [27].
FIG. 5. Dependence of the surface density, x AS, on the bulk density, x A, for one-component chains (tetramers) at e AS/k BT 5 20.5 and e AA/k BT 5 20.4. Also shown are this dependence for monomer at e AS/k BT 5 22.0 and e AA/k BT 5 20.4 and Monte Carlo simulations. The dotted line shows the prediction of Eq. [27] for ^r A& 5 2.78.
ADSORPTION OF CHAIN MOLECULES
461
FIG. 6. Surface densities of components A and B, x AS and x BS, as functions of the bulk density of component A, x A, for x B 5 0.2, e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.3 and different e AS/k BT, e BS/k BT, r A, and r B: (a) e BS/k BT 5 20.4, r A 5 3, r B 5 2, and different e AS/k BT: 20.4 (1) and 20.8 (2); (b) e AS/k BT 5 e BS/k BT 5 20.4, r B 5 2, and different r A: 3 (1) and 7 (2); (c) e AS/k BT 5 e BS/k BT 5 20.4, r A 5 3, and different r B: 12 (1) and 2 (2); (d) e AS/k BT 5 20.4, r A 5 3, r B 5 2, and different e BS/k BT: 20.8 (1) and 20.4 (2).
Also shown in these figures are calculations for monomers with the same molecule/surface and segment/segment interaction energies. Figures 2 and 3 illustrate isotherms for dimers with segment–surface and segment–segment interaction energies of e AS/k BT 5 20.5, e AA/k BT 5 0, and e AS/k BT 5 20.5, e AA/k BT 5 20.4, respectively. As seen from Figs. 2 and 3, Eq. [27] predicts results which are reasonably close to Monte Carlo simulation data. The slight under-prediction of the surface density for dimers is because the theory does not account for molecules adsorbed perpendicular to the surface. In a previous paper (8), we derived more exact equations for dimers which take the effect of these perpendicular dimers into account.
Figures 4 and 5 compare isotherms for tetramers with Monte Carlo simulations and with monomers at e AS/k BT 5 20.5, e AA/k BT 5 0, and e AS/k BT 5 20.5, e AA/k BT 5 20.4, respectively. Figures 4 and 5 show that the differences between predictions of Eq. [27] and Monte Carlo simulations are greater for tetramers than for dimers. At least part of this difference is due to the fact that the Monte Carlo simulations were performed by taking into account all possible configurations, but Eq. [27] was derived by considering only configurations where all segments touch the surface if the molecule is on the surface. However, the average number of segments touching the surface, ^r A&, is less than r A. Equation [27] can be corrected
462
ARANOVICH AND DONOHUE
FIG. 7. Adsorption isotherm for butane from gas mixture with propane on activated carbon at T 5 293 K. Here r A 5 4, r B 5 3, e AS/k BT 5 e BS/k BT 5 22.8, and e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.2. Experimental data from Ref. (10).
without complication by using an adjusted value of ^r A& instead of r A. The dotted lines in Figs. 4 and 5 show predictions of Eq. [27] where ^r A& was used instead of r A. In Figs. 4 and 5, ^r A& 5 2.78 which was calculated in the limit of low density (see Eq. [33] in Appendix). As seen from Figs. 4 and 5, this results in closer agreement with the Monte Carlo simulations. As seen from Figs. 2–5, the predictions of the monomer model do not represent the behavior of the chain molecules. The surface density of monomers is always greater than for chain molecules for the same molecule–surface interactions and for monomer–monomer interaction equal to the segment– segment interaction. This suggests that adsorption models derived for monomers should not be very accurate when used for chain molecules; this should not be surprising as the analogous conclusion for bulk solutions was made by Hildebrand (9) more than sixty years ago. Figure 6 shows predictions of Eqs. [8] and [12] for twocomponent mixtures. Figures 6a and 6b illustrate that increasing the A molecule length, r A, has an effect that qualitatively is similar to increasing the energy between A molecules and surface, e AS/k BT. Figure 6c shows that increasing the B molecule length, r B, increases the surface concentration of B molecules, x BS, but decreases the surface concentration of A molecules, x AS. Figure 6d shows the influence of e BS/k BT on x AS and x BS. Adsorption isotherms for two-component mixtures can be represented in different ways. One way is to keep x B constant and to vary x A. In this case, the dependences of x AS and x BS on x A show the isotherm at constant x B. This is useful in an analysis of the mechanisms of adsorption. However, most experiments are performed by varying x A while keeping the sum of x A 1 x B constant (10). Experimentally, this is done by varying the partial pressure of one of the components while keeping the total pressure constant. We have applied the theory presented above to the analysis of adsorption of hydrocarbon mixtures on activated carbon at T 5 293 K (10). We solved Eqs. [8] and [12] numerically
under the constraint x A 1 x B 5 constant. We then calculated mole fractions of the first component in the adsorbed layer as a function of mole fraction of this component in the gas phase. To determine e AS/k BT and e BS/k BT, we used experimental values for the differential heats of adsorption at small filling for the pure components on graphite. These values are about 6.7 kcal/mol for butane and 5.0 kcal/mol for propane (11). The segment–surface interaction energy is then e AS ' 6.7/4 5 1.675 kcal/mol for butane, and e BS ' 5.0/3 ' 1.67 kcal/mol for propane. Therefore, the reduced hydrocarbon segment–surface interactions energies are e AS/k BT ' e BS/k BT ' 22.8 at T 5 293 K. The energy of the segment–segment interactions can be evaluated from the heat of vaporization, H v. At T 5 293 K, H v ' 5 kcal/mol for butane, and H v ' 3.6 kcal/mol for propane (12). Therefore, H v/kT ' 8 for butane, and H v/kT ' 6 for propane. The value of H v/kT can be represented as 2( z/ 2) e /kT where z is the average coordination number in liquid phase, and e is the energy of interactions between molecules of the pure components. The coordination number determined from the partition function for the square-well fluid is 18.9 (13). Assuming z 5 20, we have e */kT ' 20.8 for butane and e */kT ' 20.6 for propane where e* is the molecule– molecule energy of interaction. Then, for the segment–segment interactions we have e AA/k BT ' e AB/k BT ' e BB/k BT ' 20.2. Figure 7 shows adsorption isotherms for butane from a mixture with propane on activated carbon at T 5 293 K. For the theoretical curve, we used e AA/k BT 5 e AB/k BT 5 e BB/ k BT 5 20.2, e AS/k BT 5 e BS/k BT 5 22.8, r A 5 4, and r B 5 3. Experimental data are from Ref. (10) (p. 224). As seen from Fig. 7, the theoretical prediction is quite good given that the parameters all are estimated (not fitted to the adsorption isotherms). Figures 8 and 9 show adsorption isotherms for ethane from mixtures with methane and for propane from mixtures with ethane on the same activated carbon as in Fig. 7. The theoretical predictions are calculated with the same segmental ener-
FIG. 8. Adsorption isotherm for ethane from gas mixture with methane on activated carbon at T 5 293 K. Here r A 5 2, r B 5 1, e AS/k BT 5 e BS/k BT 5 22.8, and e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.2. Experimental data from Ref. (10).
463
ADSORPTION OF CHAIN MOLECULES
getic parameters. The only difference is that, for ethane–methane mixture, r A 5 2 and r B 5 1, and, for propane– ethane mixture, r A 5 3 and r B 5 2. As seen from Figs. 8 and 9, theoretical predictions are quite good. To compare our model with the group-contribution theory of Russell and LeVan (3), we calculated adsorption isotherms from Eqs. [21]–[24]. Figure 10 presents the total density in the adsorbed layer, x AS 1 x BS , as a function of the bulk density of component A, x A , for x B 5 0.2, r A 5 3, r B 5 2, e AA /k B T 5 e AB /k B T 5 e BB /k B T 5 20.3, and e AS / k B T 5 e BS /k B T 5 20.4. Also shown in Fig. 10 are the group-contribution theory predictions and Monte Carlo simulations. As seen from Fig. 10, the theory presented above is in agreement with Monte Carlo simulation data and the group-contribution theory is not. Figure 10 shows that our model and the groupcontribution theory agree at small densities. However, at large densities, these two theories predict quite different compositions of the adsorbed layer, and they give very different limits for x A 3 0.8 (which is equivalent to x A 1 x B 3 1). In this limit, ( x AS 1 x BS ) must go to unity as ( x A 1 x B ) goes to unity. Figure 10 illustrates that our theory predicts this limit correctly but the group-contribution theory does not. As discussed previously, this is because the group-contribution theory does not take into account interactions of adsorbate molecules with the bulk. CONCLUSION
A new model for adsorption of chain-like molecules on solid surfaces is presented for one- and two-component mixtures in the framework of the Ono and Kondo theory. This model shows the influence of energetic parameters and chain length on the composition in an adsorbed layer and on adsorption isotherm for each component. Comparison with experimental data for adsorption of butane–propane, pro-
FIG. 10. Total density in the adsorbed layer, x AS 1 x BS, as a function of the bulk density of component A, x A, at x B 5 0.2, r A 5 3, r B 5 2, e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.3, and e AS/k BT 5 e BS/k BT 5 20.4.
pane– ethane, and ethane–methane gas mixtures on activated carbon demonstrates the ability of this model to predict the composition of adsorbed layer as the function of the bulk composition with only two parameters: segment–segment and segment–surface energies of interactions which can be evaluated from differential heats of adsorption and heats of vaporization. APPENDIX
Rigorously speaking, the value of ^r A& is a function of the density and temperature. However, we can estimate ^r A& in the small density limit. In this limit, the number of possible configurations of the chain molecule on the surface, n i , is a function of the number of chain contacts, i, with the surface. The energy of the chain molecule is i e AS. From Eq. [27], it follows that in the small density limit: H5
S
D
x AS 2 r Ae AS 5 exp 2 , xA 3 k BT
[28]
where H is Henry’s coefficient. Applying formula [28] for each configuration and taking into account numbers of configurations, one obtains: Hi 5
S
D
x ASi 2 i e AS 5 v i exp 2 . xA 3 k BT
[29]
Since FIG. 9. Adsorption isotherm for propane from gas mixture with ethane on activated carbon at T 5 293 K. Here r A 5 3, r B 5 2, e AS/k BT 5 e BS/k BT 5 22.8, and e AA/k BT 5 e AB/k BT 5 e BB/k BT 5 20.2. Experimental data from Ref. (10).
Ox rA
x AS 5
i51
ASi
,
[30]
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ARANOVICH AND DONOHUE
ACKNOWLEDGMENT
then
x AS 2 5 xA 3
O v expS 2 ike T D . rA
AS
[31]
i
B
i51
REFERENCES
Defining ^r A& by the following equation,
S
O
D
x AS 2 A ^r A& e AS 5 ~ v i !exp 2 , xA 3 i51 k BT r
[32]
we obtain from Eqs. [31] and [32]:
^r A& 5 2
F S DG
k BT ln rA e AS ¥ i51 vi
O v exp 2 ike T rA
AS
i
i51
M.D. acknowledges support by the Division of Chemical Sciences of the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-FG02-87ER13777.
[33]
B
For tetrameric molecules, numbers of different configurations are v 1 5 21, v 2 5 28, v 3 5 12, and v 4 5 24. With these numbers and with e AS/k BT 5 20.5, Eq. [33] gives ^r A& ' 2.78.
1. Masel, R., “Principles of Adsorption and Reaction on Solid Surfaces.” John Wiley, New York, 1996; Russell, B. P., and LeVan, M. D., Ind. Eng. Chem. Res. 36, 2380 (1997). 2. Perry, R. H., and Chilton, C. H., “Chemical Engineers’ Handbook,” Chap. 16. McGraw-Hill Book Co., New York, 1973. 3. Russell, B. P., and LeVan, M. D., Chem. Eng. Sci. 51, 4025 (1996). 4. Aranovich, G. L., and Donohue, M. D., J. Colloid Interface Sci. 200, 273 (1998). 5. Ono, S., and Kondo, S., in “Molecular Theory of Surface Tension in Liquids” (S. Flu¨gge, Ed.), Encyclopedia of Physics, Vol. 10, p. 134. Springer, Berlin, 1960. 6. Frumkin, A., Z. Phys. Chem. (Leipzig) 116, 466 (1925). 7. Fowler, R. H., and Guggenheim, E. A., “Statistical Thermodynamics.” Cambridge Univ. Press, Cambridge, UK, 1949. 8. Wu, D.-W., Aranovich, G. L., and Donohue, M. D., J. Colloid Interface Sci. 1998, in press. 9. Hildebrand, J. H., J. Am. Chem. Soc. 51, 66 (1929); Hildebrand, J. H., “Solubility of Non-Electrolytes.” Reinhold Publishing Corp., New York, 1936. 10. Valenzuela, D. P., and Myers, A. L., “Adsorption Equilibrium Data Handbook.” Prentice Hall, Englewood Cliffs, NJ, 1989. 11. Kiselev, A. V., and Yashin, Y. I., “Gas Adsorption Chromatography,” p. 131. Nauka, Moscow, 1967. 12. Perry, R. H., and Chilton, C. H., “Chemical Engineers’ Handbook,” Chap. 3, p. 115. McGraw-Hill Book Co., New York, 1973. 13. Lee, K.-H., Lombardo, M., and Sandler, S. I., Fluid Phase Equilibria 21, 177 (1985).