Langmuir 1997, 13, 1205-1210
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Prediction of Multicomponent Adsorption Equilibrium Using a New Model of Adsorbed Phase Nonuniformity† Craig R. C. Jensen and Nigel A. Seaton* Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K.
Vladimir Gusev and James A. O’Brien Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286 Received December 4, 1995. In Final Form: July 29, 1996X We discuss the origin of adsorbed-phase nonuniformity and its implications for the prediction of a multicomponent adsorption equilibrium in microporous adsorbents. Current treatments of adsorbed phase nonuniformity are reviewed and a new model, the multispace adsorption model (MSAM), is presented. The MSAM multicomponent predictions have as inputs the pure-species adsorption isotherms and a single parameter characteristic of the adsorbent. We extend the original formulation of the model to include the temperature dependence of the parameters. Our model’s performance is compared with that of the other thermodynamic models for multicomponent adsorption using our own experimental results and the literature data.
Introduction The design of adsorptive separation processes requires the prediction of multicomponent adsorption equilibrium over a range of temperatures, pressures, and compositions based on a modest amount of experimental data. The ideal adsorbed solution theory (IAST) of Myers and Prausnitz1 may be regarded as an industry standard adsorption thermodynamic model. It is thermodynamically rigorous and is based on the elegant concept of creating the ideal multicomponent adsorbed phase by mixing pure adsorbed phases at a constant spreading pressure and temperature. The prediction of multicomponent equilibrium relies on the assumption of a uniform adsorbed phase, in the sense that each adsorbed molecule experiences the same interaction with the adsorbent, regardless of location. This assumption of uniformity (which is standard in bulk thermodynamics) is unlikely to be realized in practice, since the interactions with the adsorbent may be quite complex. The IAST does not account for the structural and chemical heterogeneity present in the industrial adsorbents, and indeed does not explicitly distinguish between different adsorbents. It simply assumes that all of the information regarding a particular adsorbent is intrinsically included in the calculation by means of the pure-component isotherms. This very convenient assumption allows for the prediction of multicomponent behavior using only pure-species adsorption data. Valenzuela and Myers2 have carried out a systematic evaluation of the IAST using many different pure and binary gas adsorption data. The average error in the predicted selectivity, defined by
S1,2 )
x1/y1 x2/y2
(1)
* Author to whom correspondence should be addressed. Telephone: (+44) 1223 334786. Fax: (+44) 1223 334796. E-mail:
[email protected]. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysts on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, February 15, 1997. (1) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121.
S0743-7463(95)01510-1 CCC: $14.00
was found to be approximately 40%. (Here x and y refer to the adsorbed and bulk phase mole fractions, respectively.) The first attempt to account for the deviations of the IAST from experimental results made use of activity coefficients (analogous to the treatment of nonideality in bulk liquid mixtures).1 As with the IAST, this approach treats the adsorbed solution as though it were uniform with deviations between the IAST and experimental results being ascribed exclusively to nonidealities in the adsorbate-adsorbate interactions. Typically, the values of the activity coefficients have been found to be less than unity, i.e., showing a negative deviation from Raoult’s law.3,4 In many cases, mixtures which are almost ideal in the bulk display large deviations from ideality as an adsorbed phase. Subsequently, Myers5 presented an alternative explanation of this experimental observation. He showed that the adsorption of a gas mixture on a heterogeneous surface has a behavior similar to adsorption on a homogeneous surface with negative deviations from Raoult’s law. It is now widely accepted that the main reason for the errors in the IAST predictions lies in the treatment of the adsorbate as a uniform phase and not in the nonideality of the adsorbate-adsorbate interactions.6-8 The conventional thermodynamic treatment of adsorption views the adsorbate as a uniform two-dimensional phase, neglecting composition and density variations both parallel and perpendicular to the adsorbent surface. We consider two possible sources of deviations from this ideal picture. One source of nonuniformity is an energetically heterogeneous adsorbent surface introduced, for example, by surface structure or chemical impurities. Energetic heterogeneity leads to local variations in the extent of adsorption and adsorbate composition over the adsorbent surface. A second source of adsorbed-phase nonuniformity is a consequence of the adsorption process itself. For a (2) Valenzuela, D.; Myers, A. L. Sep. Purif. Methods 1973, 19, 453. (3) Minka, C.; Myers, A. L. AIChE J. 1973, 19, 453. (4) Costa, E.; Sotelo, L. J.; Calleja, G.; Marron, C. AIChE J. 1981, 27, 5. (5) Myers, A. L. AIChE J. 1983, 29, 691. (6) Myers, A. L. Proceedings of the International Conference on the Fundamentals of Adsorption; Engineering Foundation: New York, 1984; p 365. (7) O’Brien, J. A.; Myers, A. L. Proceedings of the International Conference on the Fundamentals of Adsorption; Engineering Foundation: New York, 1987; p 451. (8) Sircar, S. AIChE J. 1995, 41, 1135.
© 1997 American Chemical Society
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Figure 1. Schematic drawing of a pore cross section highlighting the inherent nonuniformity of the adsorption process in microporous adsorbents.
homogeneous surface in the Henry’s law limit, each molecule experiences the same surface interaction. However at higher loadings, adsorption occurs both on the surface itself and on the already-adsorbed molecules with molecules experiencing a different local environment depending on their location. If adsorption occurs on a planar surface, the nonuniformity is a consequence of multilayer adsorption, in a BET sense, with the first layer having different properties from the subsequent layers. For more complex surface structures (especially those of microporous adsorbents), the degree of nonuniformity is greatly increased and the structure of the adsorbed phase is no longer adequately described in terms of layers. Thus, at all but the lowest pressures, adsorbed-phase nonuniformity is inherent to the adsorption process. Figure 1 depicts what we term the “inherent” nonuniformity of the adsorbed phase in microporous solids. Here, the picture of adsorption is essentially three-dimensional with complex composition and density variations as a function of position within the pores. We emphasize that this nonuniformity arises regardless of the energetic heterogeneity of the surface. In practice, both the heterogeneous surface and the inherent contributions to the adsorbedphase nonuniformity are likely to be present and their relative magnitude is an open question. Attempts to account for nonuniformity as a result of energetically heterogeneous surfaces have led to the development of the heterogeneous ideal adsorbed solution theory (HIAST).9-13 In the HIAST, the complex physical and chemical structure of porous adsorbents is accounted for in terms of an adsorption-site energy distribution. The adsorbent is considered to be composed of a set of patches, each large enough to be essentially independent of the next, yet small enough for all of the sites within each patch to have the same energy. The IAST is applied independently to each patch. The overall adsorbed phase composition is calculated by summing the amount of each species adsorbed on each of the patches. In the HIAST, the distribution of the site energies for each species is estimated from their pure-species adsorption isotherms. Although this approach does not give the detailed shape of the energy distribution, if a particular form for the energy distribution is assumed (e.g., binomial or uniform), then the mean and variance of the distribution may be calculated. In order to calculate mixture adsorption, it is necessary to assume a relationship between the adsorption energies of the various species on a single site, to perform “site matching”. The most straightforward procedure9,11,13 is to assume that the lowest energy site for one species is also the lowest energy site for the other species, similarly for the second-lowest site and so on up the energy (9) Myers, A. L. Proceedings of the International Conference on the Fundamentals of Adsorption; Engineering Foundation: New York, 1987; p 3. (10) Nakahara, T. Chem. Eng. Sci. 1986, 41, 2093. (11) Valenzuela, D.; Myers, A. L.; Talu, O.; Zwiebel, I. AIChE J. 1988, 34, 397. (12) Moon, H.; Tien, C. Chem. Eng. Sci. 1988, 43, 2967. (13) Hu, X.; Do, D. AIChE J. 1995, 41, 1585.
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Figure 2. Schematic drawing a pore cross section showing the notional division of the pore volume into spaces I and II.
distribution; this is “cumulative site matching”. More sophisticated procedures, involving fitting parameters to binary data, have also been used.12 The quality of agreement with the experimental results depends critically on the site matching procedure adopted.12 Unfortunately, there are no general guidelines as to which energy distribution or site matching procedure is most appropriate to a particular application, and this has limited the use of the HIAST in practical applications. In this paper we describe a new theory of multicomponent adsorption equilibrium, the multispace adsorption model (MSAM).14 This theory pursues the idea that the inherent nonuniformity illustrated in Figure 1 makes the dominant contribution to the overall nonuniformity of the adsorbed phase, while ignoring the contribution of the energetic heterogeneity of the surface. Firstly, we summarize the model presented by Gusev et al.14 We then extend it by proposing a temperature dependence for the model parameters and investigate its performance in predicting both the equilibrium selectivity and the absolute amounts adsorbed of each species. We go on to compare the performance of the MSAM with that of other adsorption thermodynamic models: the IAST, the HIAST, and the vacancy solution model (VSM) of Danner and co-workers.15,16 The Multispace Adsorption Model The MSAM accounts for the inherent nonuniformity of the adsorbed phase in microporous adsorbents by conceptually dividing the pore volume into two “spaces”, shown schematically in Figure 2. In space I, close to the adsorbent surface, the adsorbate molecules interact strongly with the adsorbent. On the other hand, in space II, further from the adsorbent surface, the molecules interact with the molecules in space I and not directly with the surface. Effectively, the molecules in space I constitute a “pseudoadsorbant” on which the molecules in space II adsorb. Thus, adsorption in space II depends on the occupancy of space I. (A space in the MSAM is not identified with a layer, e.g., in the BET sense, although of course the latter is a case of the former.) On the basis of this model, the total adsorption of pure species i, ni, is given by:14
ni ) Ti,I(P) [R + (1 - R)Ti,II(P)] ai
(2)
Adsorption in Spaces I and II is described by isotherm equations, denoted by Ti,I(P) and Ti,II(P), respectively. R is the fraction of the total pore volume occupied by space I, ai is a measure of the total adsorption capacity, and P is the pressure. Adsorption equilibrium predictions often require extrapolation of the pure component isotherms to pressures significantly above the maximum experimental pressure. (14) Gusev, V.; O’Brien, J. A.; Jensen, C. R. C.; Seaton, N. A. AIChE J., in press. (15) Suwanayuen, S.; Danner, R. P. AIChE J. 1980, 26, 76. (16) Cochran, T. W.; Kabel, R. L.; Danner, R. P. AIChE J. 1985, 31, 268.
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Since it is important to accurately describe the highpressure regions, we represent adsorption in each space with an isotherm equation that accounts for the compressibility of the adsorbate in the pores by having a nonzero limiting slope at high pressure:17
reflect the fact that mixture adsorption in space II takes place on the mixture occupying space I, rather than on pure species i. We use the following quadratic mixing rules to calculate the effective Henry’s constant for the adsorption of species i in space II, K ˜ i,II:
ni,I Ki,IP(1 + κiP) ) Ti,I(P) ) Rai Rai + (Ki,I + Raiκi)P
Kij,II ) Kji,II ) xKi,II Kj,II, Kii,II ) Ki,II
ni,II (1 - R)ai
(3)
K ˜ i,II )
) Ti,II(P) )
(11)
Equation 4 rewritten for mixture adsorption becomes
Ki,IIP(1 + κiP) (1 - R)ai + (Ki,II + (1 - R)aiκi)P
(4)
ni,I and ni,II are the number of moles of species i adsorbed in spaces I and II, respectively. Ki,I and Ki,II are the Henry’s law constants for the two spaces. κi is the compressibility of the adsorbate, and ai is the zero-pressure intercept of the linear high-pressure asymptote. The product κiai is the limiting slope of the isotherm at high pressure. The MSAM predicts multicomponent adsorption equilibrium by applying the IAST separately to each space.14 This implies treating the spaces as distinct, uniform phases, each independently in equilibrium with the bulk fluid, although of course in reality there is no phase boundary between spaces I and II. The spreading pressure, composition, and total amount adsorbed are calculated for each space at the specified bulk conditions. The mathematical framework for the IAST calculation in each space is as follows. The integrated Gibbs isotherm
πA ) RT
ni(t) dt t
∫0P ° i
(5)
relates the spreading pressure, π, to the bulk pressure, Pi° (this pressure is used as a standard-state pressure in the mixture calculation), using the pure-species isotherms, ni(P). Here A is the adsorbent surface area, R is the universal gas constant, T is the temperature, and t is a dummy variable of integration. The assumption of ideal adsorbed solution behavior, given by1
Pi ≡ Pyi ) Pi°(π) xi
(6)
gives the mixture composition. The total amount adsorbed for the mixture, n˜ t, is calculated by
1 ) n˜ t
∑
xi ni°
(7)
ni° is the number of moles of species i that would be adsorbed from the pure gas at Pi°(π). The number of moles of each species adsorbed from the mixture follows directly:
n˜ i ) xin˜ t
(8)
Equation 5 is interpreted differently for the two spaces. For space I, the adsorption behavior is independent of space II and eq 5 is written simply as
πIA ) RT
∑j Kij,IIxj,I
(10)
Pi,I°Ti,I(t)Rai
∫0
t
dt
(9)
In contrast, adsorption in space II cannot be determined independently of the space I calculation but rather depends on both the composition and the amount of adsorption in space I. Firstly, the Henry’s constants for space II must (17) Jensen, C. R. C.; Seaton, N. A. Langmuir 1996, 12, 2866.
T ˜ i,II(P) )
K ˜ i,IIP(1 + κiP) (1 - R)ai + (K ˜ i,II + (1 - R)aiκi)P
(12)
Secondly, the amount adsorbed in space II is scaled by the occupancy of space I, θ, which is calculated by:
θ)
n˜ i,I
∑ Rai
(13)
Equation 5 applied to adsorption in space II is written as
πIIA ) RT
∫0P
i,II°
T ˜ i,II(t) (1 - R)ai t
θ(t)
(14)
Since space I evolves with pressure, θ appears within the integral. Thus, it is necessary to solve the space I problem completely before tackling space II. Finally, summing over the two spaces gives the aggregate number of moles of each species adsorbed
n˜ i ) n˜ i,I + n˜ i,II
(15)
and the total number of moles of all species adsorbed.
n˜ t ) n˜ t,I + n˜ t,II
(16)
The overall composition of the adsorbed phase follows directly from eqs 15 and 16. We propose the following temperature dependence for the parameters in eq 2. Ki,I(T) and Ki,II(T) are represented by the usual Henry’s constant temperature dependence 0 exp(-i,I/RT) Ki,I(T) ) Ki,I
(17)
0 Ki,II(T) ) Ki,II exp(-i,II/RT)
(18)
where i,I and i,II represent adsorption energies. The temperature dependence of ai is based on a Taylor series expansion, truncated at the linear term:
ai(T) ) ci + diT
(19)
The limiting slope of the pure-species isotherms at high pressures, κiai, varies slowly with temperature. Assuming κiai to be constant gives the temperature dependence for κi.
κi(T) ai(T) ) bi ) constant
(20)
Since ai(T) decreases with increasing temperature, the compressibility, κi(T), increases with temperature, as expected. Results In this section we apply the MSAM to experimental measurements of pure species and binary adsorption of methane and ethane on BPL activated carbon (Calgon
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Figure 3. Figure showing the MSAM fits, with R ) 0.2 (lines), to the pure-species adsorption data (points) for methane on BPL activated carbon at three different temperatures. Fitted 0 ) -0.002 821 03 mmol parameter values are as follows: Ki,I 0 -1 -1 -1 g bar , i,I ) 14 578 J mol , Ki,II ) -0.012 763 25 mmol g-1 bar-1, i,II ) 1830.7 J mol-1, ci ) 53.634 55 mmol g-1, di ) -0.110 767 mmol g-1 K-1, bi ) 0.042 960 4 mmol g-1 bar-1.
Figure 4. Figure showing the MSAM fits, with R ) 0.2 (lines), to the pure-species adsorption data (points) for ethane on BPL activated carbon at three different temperatures. Fitted 0 ) -0.000 079 60 mmol g-1 parameter values are as follows: Ki,I 0 -1 -1 bar , i,I ) 32 871 J mol , Ki,II ) -0.000 697 53 mmol g-1 bar-1, i,II ) 22 609 J mol-1, ci ) 14.7145 mmol g-1, di ) -0.027 962 mmol g-1 K-1, bi ) 0.009 442 4 mmol g-1 bar-1.
Carbon Corporation). Details of the experimental apparatus and the materials used are given elsewhere.14 A static volumetric technique was used for the pure-species adsorption measurements, and a flow-through technique was used for the mixture measurements. The flowthrough technique enabled the bulk mole fraction to be held constant over a range of temperatures and pressures (we used gas-phase methane mole fractions of 48% and 75%). Measurements were carried out at 308.15, 333.15, and 373.15 K and at pressures from several Torr to 25 000 Torr (about 33 atm). The input parameters for the MSAM multicomponent calculations are obtained by correlating the pure-species data using the MSAM pure-species isotherm, eq 2, including the temperature dependence of the parameters, defined by eqs 17 to 20. This involves fitting seven parameters, namely Koi,I, i,I, Koi,II, i,II, ci, di, and bi, simultaneously to the pure-species adsorption isotherm over a range of temperatures. In Figures 3 and 4 the methane and ethane pure-species data at 308.15, 333.15, and 373.15 K are correlated in this way. The fitted parameter values are given in the figure captions. The adsorbent-specific parameter R, which appears in the MSAM pure-species isotherm, is obtained by fitting it to limited mixture data. (In principle, one suitably chosen binary data point is sufficient.) Gusev et al.14 set R ) 0.2 for BPL carbon. (It is worth emphasizing here that R is in no sense a binary interaction parameter of the kind
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Figure 5. Experimental selectivities and MSAM predictions, with R ) 0.2, for the adsorption of a 52% ethane, 48% methane mixture on BPL at three different temperatures.
Figure 6. Experimental selectivities and MSAM predictions, with R ) 0.2, for the adsorption of a 25% ethane, 75% methane mixture on BPL at three different temperatures.
Figure 7. Comparison of MSAM, with R ) 0.2, IAST, and VSM predictions for the adsorption of a 52% ethane, 48% methane mixture on BPL at T ) 308.15 K.
found in activity coefficient models and equations of state. Although data for a particular binary pair at some suitable temperature and pressure are required to set R, this value is used to generate the MSAM predictions over a range of temperatures, pressures, and compositions and for different mixtures.) Figures 5 and 6 show the MSAM predictions of the experimental selectivities at different temperatures and bulk mole fractions over a range of pressures, using R ) 0.2. These predictions, based on the temperature-dependent form of the MSAM, are virtually identical to the calculations based on applying the MSAM separately at each temperature.14 We now compare the MSAM predictions with those of other theories. In Figures 7 and 8 we reproduce two of the sets of the experimental and MSAM results from Figures 5 and 6 and also show the predictions of the IAST
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Figure 8. Comparison of MSAM, with R ) 0.2, IAST, and VSM predictions for the adsorption of a 25% ethane, 75% methane mixture on BPL at T ) 373.15 K.
Figure 10. Comparison of MSAM predictions, with R ) 0.2, and HIAST predictions for the adsorption of a 25% ethane, 75% methane mixture on BPL at T ) 308.15 K and T ) 373.15 K.
Figure 9. Comparison of MSAM predictions, with R ) 0.2, and HIAST predictions for the adsorption of a 52% ethane, 48% methane mixture on BPL at T ) 308.15 K and T ) 373.15 K.
Figure 11. Experimental adsorption uptakes (points), MSAM predictions, with R ) 0.2, IAST predictions, VSM predictions, and HIAST predictions for the adsorption of a 25% ethane, 75% methane mixture on BPL at T ) 308.15 K.
and the VSM. The IAST calculations are based on fitting the MSAM pure-species isotherm, eq 2, to the purecomponent data. (The IAST is not dependent on the specific form of the isotherm equation, and thus there are many plausible ways to integrate the pure-species data for an IAST calculation. The results from using other isotherm equations as well as a cubic spline fit to the data are similar to the curves shown.) The VSM calculations were carried out using the method of Cochran et al.16 which uses Flory-Huggins activity coefficients to describe the vacancy solution. The MSAM predictions are clearly superior to those of both the IAST and the VSM which fail to capture correctly the decline in the experimental selectivity with pressure. It is worth pointing out here that the MSAM is identical with the IAST in two limits: in the Henry’s law region where only space I adsorption is significant and in the limit of R ) 1, where space II disappears. The comparison between the various theories is qualitatively similar for the other experimental temperatures shown in Figures 5 and 6, omitted for clarity from Figures 7 and 8. Figures 9 and 10 compare the MSAM and HIAST predictions using the data of Figures 5 and 6 but omitting the middle temperature for clarity. The HIAST calculations were carried out using a uniform energy distribution and cumulative site matching as described by Hu and Do.13 Both of these models, which account for adsorbedphase nonuniformity in different ways, perform substantially better than the IAST. At T ) 308.15 K and low pressure, the performance of these two approaches is very similar for both of the bulk mole fractions, whereas at high temperatures and pressures the predictions are increasingly different. (We will comment on this observation in the summary and discussion section.) Over all of these conditions, the performance of the MSAM is superior to that of the HIAST.
Figure 12. Experimental selectivities (points) of Reich et al.,18 MSAM predictions, with R ) 0.2, IAST predictions, VSM predictions, and HIAST predictions for the adsorption of a 46.4% ethylene, 53.6% methane mixture on BPL at T ) 301.4 K.
Accurate prediction of the absolute amounts adsorbed, as well as the composition of the adsorbed phase, is important in the design of industrial separation processes. Figure 11 compares the MSAM, IAST, VSM, and HIAST predictions of the amounts adsorbed for mixture conditions at 308.15 K and a bulk methane mole fraction of 48%. The MSAM and HIAST perform much better than the IAST and VSM, which consistently underpredict the adsorption of methane and overpredict that of ethane. Figures 12 and 13 demonstrate the ability of the MSAM to describe adsorption over a range of adsorbate mixtures. The experimental data were measured on BPL carbon by Reich et al.18 In principle, this is the same adsorbent as we used in our experiments (although the batches were presumably produced 15 or so years apart) so the same value of R, 0.2, applies. Figure 12 compares MSAM, IAST,
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Figure 13. Ternary adsorption data (points) of Reich et al.,18 IAST predictions and MSAM predictions, with R ) 0.2 for the adsorption of a 62.4% methane, 17.4% ethane, and 20.2% ethylene mixture on BPL at T ) 301.4 K.
VSM, and HIAST predictions with binary adsorption data for a methane-ethylene mixture. The MSAM predictions are in good agreement with the experimental data and are superior to the other models. Figure 13 compares the MSAM and the IAST for the ternary mixture methaneethane-ethylene. As might be expected, agreement is not as good for the ternary mixture as for the constituent binaries. Nevertheless, the MSAM predictions are significantly better than those of the IAST, confirming the importance of taking into account the nonuniformity of the adsorbed phase. Discussion In this paper we have described a new theory for multicomponent adsorption equilibrium, the multispace adsorption model, and have extended it to include temperature-dependent parameters. We have demonstrated the MSAM by applying it to multicomponent adsorption on a particular adsorbent, BPL carbon, using data obtained in two laboratories. Using as inputs pure-species isotherms and a single structural parameter, fitted to limited mixture data, the MSAM was shown to give a good description of the adsorption of two binary mixtures and one ternary mixture. Both the MSAM and the HIAST were shown to yield substantial improvements over the industry standard IAST, as well as the VSM, over a range of conditions. This suggests that for these systems (light hydrocarbons on BPL carbon) the adsorbed phase is significantly nonuniform. It is useful at this point to summarize the similarities and differences between the MSAM and HIAST. Both the MSAM and the HIAST account for adsorption separately in the different regions of the pore space: patches or spaces. However, the HIAST assumes that adsorption occurs independently on the various energy sites, while in the MSAM, the two spaces interact. In particular, in the MSAM, adsorption in space II depends on the composition and amount adsorbed in space I. The superiority of the MSAM over the HIAST demonstrated in this paper suggests that, for BPL carbon, inherent nonuniformity of the adsorbed phase is more important than surface heterogeneity. In making the comparison between the MSAM and HIAST, it should be pointed out that more sophisticated versions of the HIAST may perform better than the version of Hu and Do,13 which we used in the above comparison. For example, the version of Moon and Tien12 gives highquality agreement for some of the ternary data of Reich et al.18 for adsorption on BPL. However, their calculation (18) Reich, R.; Ziegler,W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 336. (19) Szepesy, L.; Illes, V. Acta Chim. Hung. 1963, 35, 245.
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of the multicomponent equilibrium involves three adjustable binary parameters (one for each binary pair) at each temperature. These parameters, as well as the choice type of site matching, are determined by fitting to binary data. In contrast, the MSAM has only a single structural parameter, independent of adsorbate mixture and temperature. Thus, it seems that the physical basis of the MSAM is more sound in the case of this adsorbent. Although we do not expect the importance of inherent nonuniformity of the adsorbed phase (as described by the MSAM) to be specific to BPL carbon, it is clearly not universal. Multicomponent adsorption equilibrium on some carbons can be predicted adequately using models that do not account for any form of adsorbed-phase nonuniformity. For example, the binary data of Szepesy and Illes19 for the adsorption of mixtures of light hydrocarbons on Nuxit-AL activated carbon is well predicted by the IAST and the VSM, as well as by the MSAM.14,16 We pointed out in the previous section that the MSAM reduced to the IAST in two limits. There is also a limit in which the MSAM and the HIAST become very similar, although not identical. If the Henry’s constants for space I are much larger than those for space II, for all species, space I becomes essentially full at very low pressure. Adsorption in space II then takes place on a fully-formed pseudoadsorbant consisting of the molecules in space I. Adsorption in space II depends on adsorption in space I only to the extent that the effective Henry’s constants for space II, K ˜ i,II, depend on the composition of space I. If, in addition, R ) 0.5, the MSAM is approximately equivalent to the HIAST with the two adsorption sites having widely different energies. As the difference between the space I and space II Henry’s constants is largest at low temperature, this is where the MSAM and HIAST predictions are the most similar, at least at low pressure. The deviation between the models at higher pressures, even at low temperature, is partly a consequence of the evolution of the composition of space I with pressure, but is also related to the choice of the pure-species isotherm equation for each model. The HIAST is based on the assumption that adsorption is Langmuirian on each adsorption site,13 whereas the MSAM isotherm equation (eq 2) accounts for an adsorbed phase compressibility. In both of these cases, the IAST calculation, whether for a site or a space, requires an extrapolation to pressures well above those experimentally measured; Gusev et al.14 have demonstrated the improvement to the multicomponent predictions at high pressure that result from taking into account the compressibility of the adsorbed phase. The aim behind the development of the MSAM was to provide an improved engineering model of adsorption suitable for use in routine design calculations. For this purpose, in addition to accuracy, the speed and simplicity of the calculation are important. The isotherm equations for each space, eqs 3 and 4, are analytically integrable, so the calculation of the spreading pressure, eq 5, in each space is fast. The parameters for the model can be obtained from isotherm data at a single temperature. The inclusion of the temperature dependence of the parameters, presented in this paper, gives the MSAM greater flexibility by enabling prediction of multicomponent adsorption equilibrium at temperatures other than those where pure-species data are available. Acknowledgment. C.R.C.J. acknowledges a scholarship from Trinity College, Cambridge. This work is supported in part by the U.S. National Science Foundation through Grant CTS-9215604. LA951510N
