Potential Theory of Adsorption for Associating Mixtures: Possibilities

Article pubs.acs.org/IECR

Potential Theory of Adsorption for Associating Mixtures: Possibilities and Limitations Martin G. Bjørner,* Alexander A. Shapiro, and Georgios M. Kontogeorgis Center for Energy Resources Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark ABSTRACT: The applicability of the multicomponent potential theory of adsorption (MPTA) for the prediction of the adsorption equilibria of several associating binary mixtures on different industrial adsorbents was investigated. In the MPTA, the adsorbates are considered to be distributed fluids subject to an external potential field emitted by the adsorbent. In this work, the theory was extended to include the cubic-plus-association (CPA) equation of state (EoS) for the description of the fluid−fluid interactions of associating mixtures. The Dubinin−Radushkevich−Astakhov (DRA) potential function was utilized to describe the solid−fluid interactions. The potential was extended to include adsorbate−absorbent-specific capacities rather than an adsorbent-specific capacity. Correlations of pure-component isotherms were generally found to be excellent with individual capacities, although adsorption on silicas at different temperatures still poses a challenge. The quality of the correlations was usually found to be independent of the applied EoS. Predictions for binary mixtures indicate that MPTA + SRK is superior when adsorption occurs on nonpolar or slightly polar adsorbents, whereas MPTA + CPA performs better for polar adsorbents or when the binary mixtures contain only associating compounds. Predictions were typically improved by about 3% when individual capacities were employed, but improvements in some cases were as large as 45%. When individual capacities and the bestperforming EoS were used, average absolute deviations of the selectivity were as low as 7−12%. Predictions of the selectivity were generally found to be superior to predictions of the adsorbed amounts. The sensitivity of the model was also tested, and it was concluded that the predictions are very sensitive to the adsorption energies.

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INTRODUCTION Adsorption of gases and liquids on porous and microporous materials is of importance in many natural and industrial processes. Knowledge of the adsorption equilibrium is essential in the proper design of equipment for several of these processes, such as in the separation of pollutants and impurities from fluid mixtures and in the design of heterogeneous chemical reactors.1 It is therefore of considerable interest to have robust models that are capable of predicting multicomponent adsorption equilibria. Several models for describing microporous adsorption equilibria have been proposed, ranging from simple explicit equations such as the Langmuir and Toth isotherms2 to theories based on statistical mechanics and molecular simulations.3,4 One of the most well-known and used models for the prediction of multicomponent adsorption equilibria is the ideal adsorbed solution theory (IAST) suggested by Myers and Prausnitz,1 in which both the adsorbed solution and the bulk phase are assumed to be ideal, in analogy with Raoult’s law for vapor− liquid equilibria. Most industrial cases, however, involve complex multicomponent mixtures at high pressures or temperatures, for which the assumption of ideality is unlikely to be true. Unfortunately, most approaches to nonideal mixtures, such as the real adsorbed solution theory (RAST),5,6 are nonpredictive, as they are usually fitted to binary data by use of a bulk-phase activity coefficient model that might be inappropriate for describing the behavior of the adsorbed phase.7 An engineering model that has potential for describing nonideal adsorption on microporous media in a predictive way is the multicomponent potential theory of adsorption (MPTA) (Shapiro and Stenby8), based on the original potential theory of adsorption (PTA) originally formulated by Polanyi.9,10 The © 2013 American Chemical Society

theory uses a bulk-phase equation of state (EoS) to determine the chemical potentials of the components in the fluid phase and a separate equation to describe the fluid−solid interactions. In principle, this approach makes it possible to model strongly nonideally behaving mixtures. Previous works have indicated that the MPTA is successful in the description of adsorption equilibria in instances where other engineering models fail. For example, Monsalvo and Shapiro used the MPTA to successfully describe both supercritical adsorption and strongly non-Langmuirian behavior.11,12 The theory is completely predictive for multicomponent mixtures, as it uses only pure-compound parameters regressed from single-component isotherms. To date, the thermodynamic model for the fluid phase used with the MPTA has mostly been the Soave−Redlich−Kwong (SRK) EoS,13 although both the Soave−Benedict−Webb− Rubin EoS14 and the simplified perturbed-chain (PC) statistical associating fluid theory (SAFT)15 have been investigated on relatively simple mixtures.11,12 The SRK EoS is not capable of predicting the phase behavior of associating (hydrogen-bonding) mixtures and is not expected to be accurate for complex adsorption equilibria either. The cubic-plus-association (CPA) EoS16 combines the SRK EoS with an association (perturbation) theory.17−20 This approach has previously allowed accurate predictions and correlations of fluid phase equilibria for several associating multicomponent mixtures.21 Received: Revised: Accepted: Published: 2672

August 9, 2012 January 14, 2013 January 24, 2013 January 24, 2013 dx.doi.org/10.1021/ie302144t | Ind. Eng. Chem. Res. 2013, 52, 2672−2684

Industrial & Engineering Chemistry Research

Article

parameters for existing equations of state, resulting in significant computational simplifications. This also illustrates a limitation of the theory: it will require modifications in cases where molecular interactions make a dominant contribution to the adsorption.11 The system of N equations given by eq 1 can be transformed into expressions that are more convenient for calculations by introducing the fugacity coefficient (φ) and rearranging8 to give

In this work, we extend the MPTA by applying the CPA EoS as a mixture model and test its predictive capabilities on several sets of complex binary mixtures containing one or more associating compound(s) adsorbed on industrially relevant microporous adsorbents. The obtained results are compared with results based on application of the SRK EoS. We limit our investigation to gasphase adsorption, although the MPTA is readily extendable to liquid mixtures.22,23 The Dubinin−Radushkevich−Astakhov (DRA) expression was also applied and extended to include the possibility of different adsorbate−adsorbent capacities. The emphasis in the present work was not to repeat the previous successes with the MPTA, but rather to investigate and illustrate limitations of the model “as is” and its applicability when complex associating mixtures are considered.

φi[P(z)] P(z)

(2)

Summing eq 2 over all N components gives N

■

0=

∑

φi bxibP b exp[εi(z)/RT ]

i

MULTICOMPONENT POTENTIAL THEORY OF ADSORPTION (MPTA) The MPTA developed by Shapiro and Stenby8 originates from the concept of the potential theory of adsorption (PTA) introduced by Polanyi.9,10 In this theory, the adsorbate is considered to be a distributed fluid subjected to an external attractive potential field, ε(z), emitted by the adsorbent, the magnitude of which is determined solely by the nature of the adsorbate and its spatial position, z, relative to the adsorbent. The emitted potential is thus assumed to be independent of other molecules within the field, including those already adsorbed.24 A central assumption of the theory is that the distributed fluid, subjected to the external field, behaves in accordance with an appropriate bulk-phase EoS,24 that is, the description of the fluid−fluid interaction is not modified by the proximity to the adsorbent, except by variation of intensive variables. Consider the isothermal adsorption equilibrium of a mixture containing N components. The bulk composition is characterized by xb = {xb1, ..., xbN}. Close to the surface, the ith component is affected by an adsorption potential, εi(z). The potential can vary between the different components, as a result of different interactions with the solid. This leads to a rise in the effective pressure and, for mixtures, an additional segregation with respect to the composition close to the surface. Thermodynamic equilibrium for the ith component in the segregated mixture is reached when the potential emitted from the solid equals the difference between the chemical potential, μ, of the bulk phase (b) and the chemical potential at position z8 μi [P(z), x(z)] − εi(z) = μi b (P b , x b)

φi bxibP b exp[εi(z)/RT ]

xi(z) =

φi[P(z)] P(z)

−1 (3)

Equations 2 and 3 uniquely determine the equilibrium distribution of pressure and components, if the bulk pressure and composition is known.8 Robust and effective algorithms for solving the system of eqs 2 and 3 with respect to pressure and mole fractions were presented by Shapiro et al.8 At each value of the potential, eq 3 is solved with regard to pressure at fixed values of xi(z) using common Newton−Raphson iterations. Then, new estimates of xi(z) are obtained from eq 2. The procedure is repeated until convergence.8 The bulk-phase pressure and composition are used in the first iteration, and for each new value of z, the initial approximations to the solution are taken from the previous iteration. As the effective pressure rises, the stability of the (presumed) vapor phase is tested at each value of the potential, εi(z). At the phase transition, the fugacity coefficient and mole fraction of the gas phase is substituted by the mole fraction and fugacity coefficient of the liquid as determined by the stability analysis.26 The surface excess of component i in a mixture is defined as the difference between the amount of component i in the mixture and the amount that would be present in the bulk phase in the absence of adsorption.27 In terms of the MPTA and the TVFM, the surface excess can be calculated as8 Γi =

∫0

∞

[xi(z) ρ(z) − xibρ b ] dz

(4)

where ρ is the molar density and z has the meaning of volume in accordance with the TVFM. The total surface excess is calculated from N

(1)

Γ=

where i = {1, ..., N} and x(z) is the composition vector of the mixture at position z. In the theory of volume filling of micropores (TVFM), which we use in the following sections, variable z has the meaning of a volume rather than a distance. It follows from eq 1 that the total free energy in the potential adsorption field consists of a contribution from the fluid−fluid interactions (obtained from a bulk-phase EoS) and the solid− liquid interactions (the adsorption potential). A model that relies on similar ideas is the density functional theory (DFT). This theory also contains fluid−fluid and fluid−solid terms and equilibriates these with a bulk EoS.4,25 A proper description of the fluid−fluid parameters, however, seems to be problematic. Compared to DFT, the description of fluid parameters in the MPTA is much less complicated because it is assumed that the fluid−fluid interactions in a mixture close to an adsorbent can be expressed by an unmodified bulk-phase EoS. This nonstatistical approach makes it possible to take advantage of databases of

∑ Γi = ∫ i=1

0

∞

[ρ(z) − ρb ] dz

(5)

The mole fraction of the ith component in the adsorbate (a) is given by xia =

Γi Γ

(6)

The presented equations (eqs 1−3) are quite general and can, in principle, be used with any appropriate EoS and potential function. In this work, we limit our investigation to the classical SRK EoS13 and the CPA EoS.16 These two equations of state are combined with the DRA potential. Potential Function. The DRA (Dubinin−Radushkevich− Astakhov) expression is possibly the most extensively studied potential function.8,28−30 This semiempirical expression was originally proposed for the adsorption of gases on microporous activated carbon,28 but it was later extended to a wide variety of 2673

dx.doi.org/10.1021/ie302144t | Ind. Eng. Chem. Res. 2013, 52, 2672−2684


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materials.29 In the DRA expression, the variable z has the meaning of a volume rather than a distance in accordance with the TVFM.28,29 The DRA expression can be expressed in a generalized form as a relationship between the porous volume z and the potential εi2,8 ⎛ ε ⎞β z(εi) = z exp⎜ − i0 ⎟ ⎝ εi ⎠

P=

α (T ) RT 1 RT ⎛ 1 ∂ ln g ⎞ − − ⎜1 + ⎟ V−b V (V + b) 2 V ⎝ V ∂(1/V ) ⎠

∑ xi ∑ (1 − X A ) i

i

(8)

Ai

where b is the covolume, α(T) is the energy parameter of the SRK EoS, V the molar volume, xi is the mole fraction of component i, g is the radial distribution function (RDF), and finally XAi is the mole fraction of hydrogen-bonding sites A on molecule i that are not bound at other sites. This fraction can be defined as37

0

(7)

Here, z0 is interpreted as the total volume or capacity of micropores accessible to the adsorbate; ε0i is a characteristic adsorption energy representing the strength of the solid−fluid interactions of component i; and β is a parameter related to the heterogeneity of the porous space, which can vary from 2 to 7 depending on the microporous solid.28−30 Often, β = 2 for activated carbons, and β = 3 for silica gel and molecular sieves.11 Because the porous volume, z0, is not well-defined, it is treated as an adjustable parameter. In the context of the MPTA, both ε0i and z0 are thus adjusted to single-component isotherms. In earlier works, it was assumed, quite successfully, that z0 is the same for all components in a mixture.8,11,12,22,23 Thus, adsorption of an N-component mixture on a solid adsorbent uses only N + 1 adjustable pure-compound parameters. These parameters are adjusted on the basis of single-component data and used without further modification in mixtures. Ideally, the total volume of the micropores is thus accessible to all adsorbates, and z0 is the same for all components and proportional to the true adsorbent volume. It is likely that this method will be insufficient when adsorbates of widely different polarities and associative natures are considered. There might, for example, be parts of the porous volume more accessible to an associating species than to a nonassociating species or vice versa.31 For this reason, the accessible porous volume is also treated as an adsorbate−adsorbent-specific parameter, and z0 is replaced by z0i in eq 7. This procedure is pragmatic; it does not account for the possibility that the accessible volumes might change upon mixing, nor does it take into account the fact that the nature of the solid−fluid interactions might be different for associating species. With this procedure, an additional N − 1 parameters are introduced in the model. A disadvantage of the DRA approach is that the DRA potential is essentially empirical and does not account explicitly for specific structures of the porous space. Several other potentials have been suggested for the PTA.2 To the best of our knowledge, the only potential aside from the DRA potential that has been used with the MPTA is the celebrated Steele 10−4−3 potential.32,33 Application of this more rigorous potential improves correlations slightly, but predictions in the vapor phase are more ambivalent, sometimes being improved (e.g., for methane/nitrogen) but also sometimes being worsened (e.g., for methane/carbon dioxide).12 Equation of State. The CPA EoS (Kontogeorgis et al.16) is an association EoS that combines the simplicity of the SRK EoS, as the physical term, with the association term from Wertheim’s theory17−20 used, for instance, in the more elaborate SAFT EoS.34−36 This pragmatic approach allows the CPA EoS to accurately model many complex associating systems, whereas it reduces to the SRK EoS for nonassociating systems. The CPA EoS for mixtures can be expressed, in terms of pressure, as the SRK EoS added to Wertheim’s association term16

X Ai =

1 1 + (1/V ) ∑j ∑B xjX BjΔA iBj

(9)

j

where Δ is the association strength between two unlike sites (A and B) on different molecules (i and j respectively)16 AiBj

⎤ ⎡ ⎛ ε A iBj ⎞ ΔA iBj = g (V )⎢exp⎜ ⎟ − 1⎥bijβ A iBj ⎦ ⎣ ⎝ RT ⎠

(10)

where ε is the association energy and β is the association volume between unlike sites on different molecules. bij is the covolume of a mixture. The CPA EoS originally employed the Carnahan−Starling (CS) hard-sphere expression for the RDF but was later modified to use the simplified hard-sphere expression38 AiBj

g (V ) =

AiBj

1 1 − 1.9η

(11)

where η is the reduced density (b/4V). It has been shown that similar results are obtained with the two expressions.39 In this work, the simplified hard-sphere expression (eq 11) was employed. Finally, the energy parameter, α(T), follows a Soave-type temperature dependency13 α(T ) = a0[1 + c1(1 −

Tr )]2

(12)

where a0 and c1 are adjustable pure-compound parameters and Tr (= T/Tc) is the reduced temperature. All pure-compound parameters are usually fitted to bulk-phase vapor pressures and liquid densities. It is clear that the CPA EoS has five adjustable pure-compound parameters: three in the physical SRK term (b, a0, and c1) and two additional parameters (εAiBj and βAiBj) in the association term. The extension of the CPA EoS to mixtures requires the use of mixing and combining rules in the physical term. The classical van der Waals one-fluid (vdW1f) mixing rules and the geometric and arithmetic combining rules were applied in this work.13,21 A temperature-independent interaction parameter can often successfully be used in the CPA EoS, giving it predictive capabilities between different temperatures.40 No mixing rules are required in the association term, but if more than one associating compound is present, combining rules are needed. One common combining rule is CR-141,42 ε A iBj =

ε A iBi + ε A jBj 2

(13a)

β A iBj =

β A iBi β A jBj

(13b)

This combining rule was also utilized here. For solvating mixtures (such as benzene/methanol), eq 13b cannot be used. Instead, the 2674



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Table 1. Correlated Adsorption Energies (ε0i ) and Common Adsorption Capacities (z0) for MPTA + SRK and MPTA + CPA on Different Adsorbents Including the Absolute Average Deviations of the Total Amounts Adsorbed (AADn) MPTA + SRK adsorbate a

n-hexane benzenea acetonea methanola n-hexane benzenea acetone methanola n-hexanea benzene acetone methanola 1-butanol p-xylene cyclohexane benzene ethanol 2-propanol water average a b

adsorbent

z0 (m3/kg) −4

MS-13X

2.17 × 10

MS-5A

1.91 × 10−4

AC G-2X

4.59 × 10−4

Z-Yb

2.17 × 10−4

S

9.99 × 10−5

SGCC

2.22 × 10−4

ε0i (kJ/mol)

MPTA + CPA AADn (%)

29.20 27.32 43.36 26.93 29.89 24.97 26.47 11.49 31.09 16.96 20.83 8.94 17.41 16.56 3.39 5.29 10.70 8.33 3.50

3.5 6.0 2.5 2.1 3.0 3.3 9.5 3.8 1.4 13.6 17.5 17.5 2.4 3.2 30.0 17.6 12.2 15.9 26.5 10.1

z0 (m3/kg) −4

1.60 × 10

1.46 × 10−4

3.51 × 10−4

1.88 × 10−4 8.64 × 10−5

1.72 × 10−4

ε0i (kJ/mol)

AADn (%)

ref

36.46 34.88 47.15 25.76 43.78 30.53 26.81 11.37 38.40 18.03 21.48 8.98 16.82 16.86 3.50 5.39 10.33 9.02 4.01

7.5 4.7 4.1 3.3 7.1 5.9 9.5 6.8 8.8 24.8 17.6 22.2 2.3 3.1 29.0 17.5 13.5 14.8 21.8 11.8

47

47

47

48 49

50

Data for fitting were extrapolated from pure-compound parameters belonging to the two-dimensional fluid model presented in Konno et al.47 Where β = 3.

Table 2. Correlated Adsorption Energies (ε0i ) and Common Adsorption Capacities (z0) for MPTA + SRK and MPTA + CPA on Cab-O-Sil (S) at T = 303.15 K Including the Absolute Average Deviations (AADn) MPTA + SRK adsorbate cyclohexane benzene ethanol average

adsorbent S

z0 (m3/m2) −10

3.52 × 10

ε0i (kJ/mol) 5.03 6.56 11.19

MPTA + CPA AADn (%) 19.6 6.9 6.9 12.0

z0 (m3/m2) −10

3.05 × 10

ε0i (kJ/mol)

AADn (%)

ref

5.17 6.75 10.76

16.9 6.8 5.0 10.0

51

obtained from critical properties and the acentric factor, and the CPA EoS was employed with appropriate association schemes and pure parameters from the literature21,44,45 correlated to purecompound bulk-phase data. The value for the heterogenity parameter, β, in the DRA expression was adjusted to be either 2 or 3 depending on the quality of the correlations. Unless otherwise indicated, the parameters obtained with β = 2 gave the best correlations. This was even the case for molecular sieves (MS) and silica gels (SG), for which other values of β have often been employed.8,11 Thus, the only adjustable parameters were the characteristic potential, εi, and the capacity, z0 (or z0i ), of the adsorbent. An important assumption in the MPTA is that the potential field is independent of temperature.8,11,12 Thus, if single-component experimental data were available at several temperatures, the parameter regression routine used the complete temperature interval. In a recent study, Dundar et al.46 modified the MPTA by including a simple temperature dependence in the potential, for modeling of supercritical adsorption over a wide temperature range. The temperature ranges considered in this work, however, were small in comparison, and it was assumed that such temperature effects were negligible. The adjusted pure-compound parameters are presented in Tables 1 and 2 for the case in which the capacity is treated as a characteristic adsorbent-specific parameter (N + 1 pure-

modified CR-1 combining rule is utilized. Here, the crossassociation volume is fitted to experimental bulk-phase data, whereas eq 13a remains unchanged. Prior to using the CPA EoS on associating compounds, the association scheme, that is, the number and type of association sites, needs to be determined in order to calculate XAi. The terminology of Huang and Radosz is commonly used43 for classification of an association scheme. The suggested association schemes and pure-compound parameters for the compounds used in this work can be found in the literature.21,44,45

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RESULTS The correlative and predictive performances of the MPTA with the SRK EoS (MPTA + SRK) and the CPA EoS (MPTA + CPA) on seven different microporous adsorbents were investigated. The adsorption of six different associating compounds (methanol, 1-butanol, ethanol, 2-propanol, water, and acetone) and four nonassociating compounds (n-hexane, benzene, pxylene, and cyclohexane) was investigated and compared to experimental data. We note that, whereas acetone is polar and not associating, it was treated here as an associating compound abiding by the 2B scheme (Huang and Radosz 43 ) in correspondence with the approach of Folas et al.44 Pure-Compound Correlations. When correlating purecompound parameters, the fluid parameters in the SRK EoS were 2675



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Table 3. Correlated Adsorption Energies (ε0i ) and Individual Adsorption Capacities (z0i ) for MPTA + SRK and MPTA + CPA on Different Adsorbents Including the Absolute Average Deviations MPTA + SRK adsorbate a

n-hexane benzenea acetonea methanola n-hexane benzenea acetone methanola n-hexanea benzene acetone methanola 1-butanol p-xylene cyclohexane benzene ethanol 2-propanol water 2-propanol water average a b

adsorbent MS-13X

MS-5A

AC G-2X

Z-Yb S

SG-CC SG

z0i (m3/kg)

ε0i (kJ/mol)

−4

2.06 × 10 1.93 × 10−4 2.27 × 10−4 2.20 × 10−4 1.83 × 10−4 1.79 × 10−4 2.17 × 10−4 1.82 × 10−4 4.61 × 10−4 4.75 × 10−4 6.14 × 10−4 3.79 × 10−4 2.33 × 10−4 1.96 × 10−4 9.87 × 10−5 1.00 × 10−5 9.36 × 10−5 2.14 × 10−4 2.22 × 10−4 1.54 × 10−4 7.20 × 10−5

MPTA + CPA AADn (%)

30.64 30.94 40.62 26.67 32.58 26.71 23.04 11.70 30.55 21.65 15.73 10.04 15.79 20.35 3.51 5.29 11.08 8.76 3.50 14.03 5.09

2.2 1.3 0.9 1.9 1.4 1.6 3.2 2.6 1.5 1.8 4.4 4.7 1.4 1.6 29.6 17.6 9.9 15.2 26.5 3.8 15.0 7.1

z0i (m3/kg) −4

1.84 × 10 1.75 × 10−4 1.74 × 10−4 1.53 × 10−4 1.63 × 10−4 1.63 × 10−4 1.67 × 10−4 1.30 × 10−4 4.09 × 10−4 4.30 × 10−4 4.72 × 10−4 2.71 × 10−4 1.99 × 10−4 1.70 × 10−4 9.06 × 10−5 8.64 × 10−4 7.78 × 10−5 1.72 × 10−4 1.72 × 10−4 1.28 × 10−4 5.57 × 10−5

ε0i (kJ/mol)

AADn (%)

ref

30.68 30.96 40.71 27.02 32.59 26.68 23.09 11.80 30.90 21.61 15.88 9.82 15.41 20.65 3.50 5.39 11.27 9.02 3.97 14.59 5.88

2.2 1.5 0.9 1.9 1.4 1.6 3.4 3.1 1.5 1.8 4.4 5.8 1.4 1.6 29.0 17.5 9.8 14.8 21.6 3.9 14.9 6.9

47

47

47

48 49

50 50

Data for fitting were extrapolated from pure-compound parameters belonging to the two-dimensional fluid model presented in Konno et al.47 Where β = 3.

Table 4. Correlated Adsorption Energies (ε0i ) and Individual Adsorption Capacities (z0i ) for MPTA + SRK and MPTA + CPA on Cab-O-Sil at T = 303.15 K Including the Absolute Average Deviations MPTA + SRK adsorbate cyclohexane benzene ethanol average

adsorbent S

0

3

2

z (m /m ) −10

2.57 × 10 2.99 × 10−10 3.99 × 10−10

ε0i

(kJ/mol)

MPTA + CPA AADn (%)

5.65 7.30 9.66

6.5 6.3 3.2 5.3

0

3

2

z (m /m ) −10

2.37 × 10 2.74 × 10−10 3.36 × 10−10

ε0i (kJ/mol)

AADn (%)

ref

5.65 7.18 9.78

6.5 6.5 3.3 5.4

51

particularly well-illustrated for MPTA + CPA used on molecular sieves, where the capacities of n-hexane, benzene, and acetone became almost identical whereas the capacity of methanol deviated from these. This might suggest that the original assumption of a common capacity8 is, in fact, an approximation that is valid only for similar molecules. This is substantiated in Figures 1 and 2, where correlations on MS-13X and MS-5A using both individual and common capacities with MPTA + CPA are shown. Although the correlations with a common capacity are acceptable, adsorption was overestimated for methanol whereas it was underestimated for the other components. From the individual parameters in Table 3, it can be seen that the capacities of all components except for methanol are similar. Remarkably, acetone (as a 2B molecule) with MPTA + CPA was found to have roughly the same capacity as the nonassociating molecules, whereas this is not the case when it is considered to be inert. It can be seen especially from Figure 2 that the behavior of the methanol isotherm is clearly very different from that of the other isotherms, which are much more similar to each other. These results seem to warrant the use of individual capacities, at least for different types of molecules. It must be mentioned, however, that it was necessary to extrapolate most of the isothermal adsorption data from Konno et al.47 The data were extrapolated from pure-

compound parameters). In general, the correlations were unsatisfactory even though the two molecular sieves47 and the zeolite48 (Z) seemed to be correlated with acceptable accuracy. The characteristic energies on MS-13X and MS-5A, however, were quite large compared to previous results.11,12 Unfortunately, the correlations on both activated carbon (AC) and Cabo-Sil (S) were quite poor. The correlations with individual capacities (2N pure-compound parameters) are shown in Tables 3 and 4. Not surprisingly, the correlations were improved by the added flexibility of almost twice the number of adjustable parameters. Correlations of the adsorption on AC, in particular, were significantly improved. Correlations to the data from Perfetti and Wightman49 on Cab-o-Sil and the data from Wolf and Schlünder50 on two types of SG showed only small improvements, however. From Tables 1−4, we note that the capacities were always found to be smaller when MPTA + CPA was used; the individual characteristic potentials often differed by several kilojoules per mole; and, finally, the quality of the correlations seemed to be almost independent of the employed EoS. In fact, MPTA + SRK performed better in several cases. There seems to be a systematic difference between the various capacities of the nonassociating and associating compounds on the same adsorbent. This is 2676



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were not presented. Although this model is considered to correlate the data to within an AADn of no more than 2.5% for all components (see Appendix A), the adjusted parameters based on these extrapolations must be viewed with some caution. Although the best correlations were obtained with β = 2 on MS-13X, deviations with β = 3 were of the same order of magnitude (increased AADn by 1% or less). Because all experimental data on MS-13X were extrapolated, it might well be that the true experimental data would be better fit with β = 3. The parameters from these correlations are listed in Tables 5 and 6. For the mixture of water and 2-propanol on SG, convergence to a common capacity was not obtained within the tolerance and characteristic potential search interval (0−150 kJ/mol). Table 3 shows that the individual capacities differ significantly from each other, which might explain the convergence issues. Although the correlations suggest that the associating and nonassociating components have different accessible volumes, the apparent differences in capacity and generally improved correlations might simply be the result of an extra compoundspecific adjustable parameter and not an actual difference in capacity. For this reason, binary predictions were performed using both the parameters obtained with a common capacity and those obtained with individual capacities. Predictions for Binary Mixtures. The adsorption behavior of binary mixtures was predicted without binary interaction parameters and with the classical vdW1f mixing rules. The CR-1 or modified CR-1 mixing rule was used when MPTA + CPA was applied. All mixtures, except cyclohexane/benzene, contained at least one associating compound. The different combinations of binary adsorption pairs, ranging from strongly associating to nonassociating, and their deviations from the experimental data are presented in Table 7 in the case of a common capacity. It is clear from Table 7 that the predictions in many cases are unsatisfactory, irrespective of the EoS employed. As was the case with the correlations, predictions with MPTA + CPA were often worse than those with MPTA + SRK, even for mixtures containing associating molecules such as methanol or ethanol. Predictions were particularly poor for adsorption on MS-5A and silica gel. On the other hand, several predictions were quite satisfactory, for instance, for the mixtures with acetone on MS13X. The results of predictions with individual capacities are reported in Table 8. Predictions were generally either similar or improved compared to those with common capacities, although there are exceptions. Predictions with MPTA + SRK were still better in many cases. It is encouraging, however, that all predictions that involved only associating species were more accurate with MPTA + CPA (see Figure 3). Irrespective of the EoS employed or whether individual or common capacities were

Figure 1. Experimental and fitted pure-compound isotherms of nhexane (squares), benzene (triangles), acetone (circles), and methanol (diamonds) on MS-13X at T = 303.15 K. Solid lines are individual correlations with MPTA + CPA, whereas dashed lines are correlations to a common capacity (with β = 2). Experimental data from Konno et al.47

Figure 2. Experimental and fitted pure-compound isotherms of nhexane (squares), benzene (triangles), acetone (circles), and methanol (diamonds) on MS-5A at T = 303.15 K. Solid lines are individual correlations with MPTA + CPA, whereas dashed lines are fits to a common capacity (with β = 2). Experimental data from Konno et al.47

compound parameters using the two-dimensional fluid model presented by Konno et al.,47 because the actual experimental data

Table 5. Correlated Adsorption Energies (ε0i ) and Common Adsorption Capacities (z0) for MPTA + SRK and MPTA + CPA on MS-13X with β = 3 Including Absolute Average Deviations MPTA + SRK adsorbate a

n-hexane benzenea acetonea methanola average a

adsorbent MS-13X

z0 (m3/kg) −4

2.06 × 10

ε0i (kJ/mol) 27.77 26.21 38.66 25.93

MPTA + CPA AADn (%) 3.0 6.2 3.1 2.3 3.6

z0 (m3/kg) −4

1.52 × 10

ε0i (kJ/mol)

AADn (%)

ref

32.66 31.30 41.51 25.36

7.7 5.2 5.0 4.2 5.5

47

Data for fitting were extrapolated from pure-compound parameters belonging to the two-dimensional fluid model presented in Konno et al.47 2677



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Table 6. Correlated Adsorption Energies (ε0i ) and Individual Adsorption Capacities (z0i ) for MPTA + SRK and MPTA + CPA on MS-13X with β = 3 Including Absolute Average Deviations MPTA + SRK adsorbate a

n-hexane benzenea acetonea methanola average a

−4

1.96 × 10 1.87 × 10−4 2.20 × 10−4 2.06 × 10−4

MS-13X

MPTA + CPA

ε0i (kJ/mol)

z0i (m3/kg)

adsorbent

28.82 28.23 35.64 25.93

1.3 1.7 1.1 2.3 1.6

−4

1.75 × 10 1.72 × 10−4 1.67 × 10−4 1.44 × 10−4

ε0i (kJ/mol)

AADn (%)

ref

28.70 28.03 36.12 26.23

1.3 2.1 0.9 2.1 1.6

47

Data for fitting were extrapolated from pure-compound parameters belonging to the two-dimensional fluid model presented in Konno et al.47

Table 7. AADx and AADn Values of the Binary Predictions Obtained Using Both MPTA + SRK and MPTA + CPA with a Common Capacity (z0) MPTA + SRK adsorbate methanol + acetone methanol + benzene methanol + nhexane acetone + nhexane methanol + acetone acetone + benzene acetone + nhexane methanol + benzene 1-butanol + pxylene methanol + benzene methanol + acetone methanol + nhexane acetone + nhexane acetone + benzene ethanol + benzene ethanol + cyclohexane cyclohexane + Benzene ethanol + benzene ethanol + cyclohexane benzene + cyclohexane water +2propanol total/average a

z0i (m3/kg)

AADn (%)

MPTA + CPA

DP

AADx (%)

AADn (%)

AADx (%)

AADn (%)

10

22.9

18.0

19.0

15.4

14

8.0

5.5

23.2

23.4

8

30.0

23.8

41.4

37.4

9

33.2

13.6

17.8

12.5

8

55.5

30.6

62.7

34.2

5

4.8

11.9

1.1

15.6

6

4.6

10.3

3.9

12.9

6

19.4

21.5

19.5

24.4

zeolite Y

24

22.7

−

27.6

−

AC G-2X

8

27.1

6.0

17.7

28.4

7

23.7

27.5

18.6

23.8

3

0.9

3.6

6.5

18.3

6

15.4

9.8

8.8

3.2

8

14.1

4.7

12.4

2.1

10

11.4

15.1

13.6

11.8

8

10.2

71.6

12.2

66.9

9

3.1

17.8

7.0

19.4

9

4.5

24.3

6.5

24.3

8

2.8

26.2

3.3

23.8

8

5.6

19.1

5.9

21.9

63

48.1

51.2

20.4

30.6

237

17.5

20.6

16.6

22.5

adsorbent MS-5A

MS-13X

S

S

SGCCa

Table 8. AADx and AADn Values of the Binary Predictions Obtained Using Both MPTA + SRK and MPTA + CPA with Individual Capacities (zi) MPTA + SRK adsorbate methanol + acetone methanol + benzene methanol + nhexane acetone + nhexane methanol + acetone acetone + benzene acetone + nhexane methanol + benzene 1-butanol + pxylene methanol + benzene methanol + acetone methanol + nhexane acetone + nhexane acetone + benzene ethanol + benzene ethanol + cyclohexane cyclohexane + Benzene ethanol + benzene ethanol + cyclohexane benzene + cyclohexane water +2propanol water +2propanol total/average

Average value based on the AADs at different temperatures.

a

used, the AADn was overall found to be larger than the AADx (see Appendix A), in agreement with previous works.11 The adsorption of acetone/hexane on AC G-2X is shown in Figure 4. Here, MPTA + SRK with individual capacities performed better at low to moderate mole fractions of acetone,

MPTA + CPA

DP

AADx (%)

AADn (%)

AADx (%)

AADn (%)

10

13.0

7.8

7.7

5.6

14

10.0

10.0

19.7

11.4

8

31.1

24.6

31.5

23.0

9

8.0

5.9

22.1

14.9

8

43.7

24.2

28.0

18.1

5

15.2

12.0

4.9

10.1

6

5.7

8.8

4.4

8.1

6

23.1

26.2

11.2

17.8

zeolite Y

24

11.1

−

21.6

−

AC G-2X

8

7.0

4.0

15.5

5.0

7

13.8

13.9

14.6

11.7

3

2.9

2.9

28.0

25.0

6

6.3

7.4

14.8

12.7

8

6.9

2.1

14.2

5.6

10

7.6

13.6

10.5

11.8

8

9.1

71.3

11.5

68.4

9

6.3

26.8

6.0

27.0

9

4.6

20.0

7.1

17.2

8

2.7

25.2

3.4

22.8

8

5.3

17.0

5.4

17.2

SGCCa

63

49.0

52.0

21.2

27.8

SGa

33

34.7

43.5

22.2

28.1

270

14.4

20.0

14.8

18.5

adsorbent MS-5A

MS-13X

S

S

Average value based on the AADs at different temperatures.

as was often observed in the cases where MPTA + SRK performed better than MPTA + CPA. Figure 5 shows the adsorption of ethanol/benzene on Cab-o-Sil at 293.15 K. Just as 2678



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Figure 3. Predicted composition of the mixture methanol (1)/acetone (2) on MS-5A at 303.15 K and 4 kPa. Experimental data are from Konno et al.47 Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Bold lines are with individual capacities.

Figure 5. Predicted composition of the mixture ethanol (1)/benzene (2) on Cab-o-Sil at 293.15 K and 4 kPa. Experimental data are from Perfetti and Wightman.49 Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Bold lines are with individual capacities.

Figure 4. Predicted composition of the mixture acetone (1)/hexane (2) on AC G-2X at 303.15 K and 4 kPa. Experimental data are from Konno et al.47 Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Bold lines are with individual capacities.

Figure 6. Predicted composition of the mixture acetone (1)/n-hexane (2) on MS-5A at 303.15 K and 4 kPa. Experimental data are from Konno et al.47 Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Bold lines are with individual capacities.

on MS-13X, both models capture the trend of the data, but in this case, MPTA + SRK performs better than MPTA + CPA. The predicted composition diagram of the mixture acetone/nhexane on MS-5A is shown in Figure 6. It can be seen that the predictions with MPTA + SRK and individual capacities match the experimental data very well, unlike in the case where common capacities were employed. The predictions with MPTA + CPA lie on either side of the experimental data. This might suggest that the reason for the unsatisfactory match is the sensitivity of the model. An excellent prediction of methanol/ benzene on MS-13X is shown in Figure 7. Here, MPTA + CPA with individual capacities performed better, although all models captured the behavior qualitatively well. Predictions on MS-13X with the parameters from Table 6 where β = 3 are reported in Table 9. Excellent predictions were obtained with these parameters. In particular, the predictions of the mixtures methanol/acetone and methanol/benzene were improved. The two other mixtures were predicted slightly worse than with individual capacities and β = 2, but the deviations were

still within experimental error. Regressions with β = 2 and β = 3 for adsorption of the acetone/methanol mixture on MS-13X are compared in Figure 8.

■

DISCUSSION Correlations of the individual adsorption isotherms were, in general, found to be of similar quality irrespective of the EoS employed. This was not the case for the predictions of adsorption of binary mixtures, where results from MPTA + CPA and MPTA + SRK often deviated substantially from each other. Nevertheless, the average deviations from experimental data with the two models were very similar with respect to both composition and total amounts adsorbed. The average deviation was reduced by about 2−3% when individual capacities were used, irrespective of the EoS employed. However, even in the bestcase scenario, the overall average error in the composition was about 14%, and that in the total amounts adsorbed was about 20%. This is almost 10% higher than previously published results 2679



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was found to perform better than MPTA + CPA on all adsorbents except MS-13X. On the other hand, application of MPTA + CPA definitely improved predictions of mixtures containing two associating species. Both MPTA + CPA and MPTA + SRK generally captured the trends of the experimental data, with respect to both composition and amounts adsorbed. The most noteworthy exceptions to this were on MS-5A, where both models had problems predicting the composition of adsorbed mixtures, in particular, with n-hexanecontaining mixtures (see, for example, Figure 6). A possible trend can be found by considering the polarity of the adsorbents: MS-5A and AC G-2X are both nonpolar,47 MS13X and SG/SGCC are polar,47,50 and Cab-o-Sil exhibits an intermediate polarity.49,51 If the predictions are scrutinized within these polarity groups, then the results suggest that MPTA + SRK is superior for (associating/nonassociating) mixtures on nonpolar and slightly polar adsorbents and that MPTA + CPA is superior for adsorption on polar adsorbents. Table 10

Figure 7. Predicted composition of the mixture methanol (1)/benzene (2) on MS-13X at 303.15 K and 4 kPa. Experimental data are from Konno et al.47 Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Bold lines are with individual capacities.

Table 10. Overall Average Deviations of Associating/ Nonassociating Mixtures Divided between the Three Different Adsorbent Types: Nonpolar (MS-5A, AC G-2X), Intermediate (Cab-o-Sil, Zeolite Y), and Polar (MS-13X, SG, SGCC) Obtained Using the MPTA with Individual Capacities (z0i )

Table 9. Binary Predictions for MS-13X Obtained Using Both MPTA + SRK and MPTA + CPA with Individual Capacities (zi) and β = 3 MPTA + SRK

MPTA + CPA

adsorbate

adsorbent

DP

AADx (%)

AADn (%)

AADx (%)

AADn (%)

methanol + acetone methanol + benzene acetone + nhexane acetone + benzene total/average

MS-13X

10

28.0

20.1

17.5

16.2

14

18.4

25.9

4.9

17.2

8

5.6

10.9

3.6

6.2

9

22.5

15.1

4.6

9.9

41

18.6

18.0

7.7

12.4

MPTA + SRK

MPTA + CPA

adsorbent type

AADx (%)

AADn (%)

AADx (%)

AADn (%)

nonpolar intermediate polar

11.0 6.7 26.4

8.7 29.0 27.9

18.7 9.4 12.3

12.7 27.4 17.6

summarizes the results with individual capacities in terms of polarity, where the predictions with β = 3 were used for MS-13X (Table 9) and each mixture was given the same weight. These results suggest that, if predictions are performed with the EoS that performs best depending on the polarity of the adsorbent, then average deviations in composition can be decreased to 7− 12%, depending on the mixture type. This error is of the same order of magnitude as determined for simple mixtures in previous works.12,22 The predictions were (almost) always improved by individual capacities when only the “best-performing” EoS is considered. It is noted that the mole fraction of adsorbed associating species on nonpolar adsorbents was generally underpredicted at low mole fractions of associating species when MPTA + CPA was applied to associating/ nonassociating mixtures. No such trends were observed for MPTA + SRK. The fact that predictions were improved with MPTA + CPA for mixtures containing only associating species suggests that the choice of phase equilibrium model is of importance. It is surprising, however, that MPTA + SRK performs better than MPTA + CPA for associating/nonassociating mixtures on several adsorbents. There might be several reasons for this apparent discrepancy between our findings and what was expected, and we attempt to address some of these in the following paragraphs. It is quite possible that the DRA potential is not capable of correctly taking the polar interactions into account, because the DRA potential is a semiempirical equation and its development was based on observations on nonassociating adsorbates, such as nitrogen on activated carbon. This means that a more sophisticated, theoretically justified potential might be needed.

Figure 8. Predicted composition of the mixture methanol (1)/acetone (2) on MS-13X at 304.15 K and 4 kPa. Experimental data are from Konno et al.47 Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Bold lines are with β = 3.

with simple mixtures such as ethene/ethane.11 These deviations are similar to those in predictions of supercritical mixtures.12 For mixtures containing only one associating species, MPTA + SRK 2680



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This is partly justified by the work of Chen and Yang.52 The authors derived an isotherm equation based on statistical thermodynamics and a consideration of dispersion forces only. They showed that the DRA expression is an approximate form of this “dispersion” isotherm. An alternative could be a potential that is capable of taking polar interactions into account. This might, for instance, be the potential suggested by Zhao and Johnson,53 derived in a similar fashion as the Steele 10−4−3 potential but including the effect of dipoles and quadrupoles. This potential has a built-in temperature dependence for dipole− quadrupole and quadrupole−quadrupole interactions, which could further explain the large deviations observed for simultaneous correlations on silica at several temperatures with the DRA potential. Another possibility is that the adsorption capacity for silica varies at higher temperatures, which could be modeled similarly to the temperature dependence suggested by Dundar et al.46 Another reason for the discrepancy might be that a large part of the microporous adsorption occurs in extremely small pores and cavities, with each cavity holding only a few molecules. Molecules within such a cavity are effectively shielded from interactions with species outside the cavity. At low mole fractions of an associating species, most of these cavities will hold either a single associating molecule or none at all. On nonpolar adsorbents, which do not “like” polar species, this would effectively diminish the effect of self-association, thus making the CPA EoS less appropriate than the SRK EoS. On polar adsorbents, however, the associating species are attracted, and the effect of association is significant. This would also explain why the CPA EoS is more appropriate for mixtures with two associating components, where the full effect of association is retained because of the presence of both cross- and self-association. Moreover, Wertheim’s association term, used in the CPA EoS, assumes a continuous distribution of hydrogen-bonding compound in the fluid. In the confined spaces within pores, such a distribution might not be possible, particularly for nonpolar adsorbents with very few associating molecules in the cavities, requiring a modification of the expression or its parameters. This modification would, however, lose some of the advantage of the MPTA, namely, that an unmodified EoS can be used for both the bulk and adsorbed phases. One problem, common to all adsorption models, is the fact that most binary predictions are based on parameter regression of relatively few experimental pure-compound data points. Furthermore, these data often show signs of capillary condensation and/or hysteresis effects,2 for which the MPTA does not apply. As experimental multicomponent adsorption data for mixtures containing associating and polar species seem to be rare, it might be beneficial in future works to compare results with the MPTA to Monte Carlo simulation results. The lack of sufficient amounts of reliable experimental data might, for instance, be a problem if a model is sensitive to the value of the regressed parameters. To test this possibility, the mixture acetone/n-hexane was investigated. The characteristic energy of acetone was varied between 25 and 29 kJ/mol in steps of 2 kJ/ mol, and all other parameters were kept constant. The predictions are shown in Figure 9 using both MPTA + CPA and MPTA + SRK. Apparently, the predictions are very sensitive to the value of the characteristic potential, particularly with MPTA + CPA. The same was observed upon variation of the characteristic n-hexane potential (not shown), although in this case the models seemed to be equally sensitive. High sensitivity

Figure 9. Investigation of the sensitivity of the mixture acetone (1)/nhexane (2) at different values of the characteristic potential of acetone. Other parameters are constant. Solid and dashed lines correspond to MPTA + CPA and MPTA + SRK, respectively. Arrows point in the direction of increasing ε01.

to the parameters might require improvement of the correlation procedure.

■

CONCLUSIONS In this work, we have extended the MPTA to include both the CPA EoS and an extension to the DRA potential, which allows individual capacities. The correlative and predictive capabilities were tested for associating species. The results were compared with those obtained using MPTA + SRK with and without individual capacities. Correlations with a common capacity were found to be unsatisfactory in many cases, whereas the use of individual capacities significantly improved the correlations. The choice of EoS did not seem to affect the correlative power of the MPTA. The MPTA has previously been shown to successfully predict the adsorption equilibria of relatively simple mixtures of hydrocarbons and other components such as hydrogen sulfide, carbon dioxide, and nitrogen on silica gels, molecular sieves, and activated carbons.8,11,12 In the present work, several predictive calculations of the adsorption of various associating mixtures on similar adsorbents were presented. Surprisingly, the predictions show that MPTA + SRK is superior to MPTA + CPA in many cases, although the overall average deviations are similar. The results indicate that MPTA + SRK is the best choice for nonpolar and slightly polar adsorbents, as long as the binary mixture consists of an associating and a nonassociating compound. On the other hand, MPTA + CPA is superior for polar adsorbents and for mixtures consisting of two associating compounds. These indications should be treated with care, because the amounts of available experimental data are insufficient for far-reaching conclusions. Overall, predictions were improved by about 3% with individual capacities. In particular, when the “bestperforming” EoS was used (i.e., SRK for nonpolar adsorbents and CPA for polar adsorbents), the relative error of the model could be reduced to an acceptable value of 7−12%. The predictions with individual capacities were, in many cases, as good as those obtained in previous works on simpler mixtures (with a common capacity). This suggests that it might be 2681



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Greek Letters

necessary to use individual capacities for more complex adsorbates on the same adsorbent. MPTA + SRK with individual capacities was found to be, at least partly, successful in predicting the adsorption equilibria of mixtures containing only one associating species, particularly if one considers only nonpolar adsorbents. For strongly associating mixtures, however, MPTA + SRK completely fails, whereas better predictions are found with MPTA + CPA. In these cases, it might be recommended to apply a more theoretically justified interaction potential than the DRA potential. Further investigation is required, for example, with respect to determining an optimal correlation procedure; modified mixture models; and in particular, different adsorption potentials and their possible temperature dependency.

α(T) = energy parameter of the SRK/CPA EoS αij = cross energetic parameter between components i and j β = empirical parameter related to the heterogeneity of porous space βAiBj = association volume between two unlike sites, A and B, on different molecules i and j Γ = surface excess ΔAiBj = association strength between two unlike sites, A and B, on different molecules i and j εAiBj = association energy between two unlike sites, A and B, on different molecules i and j ε(z) = adsorption potential field (as a function of distance z) ε0i = characteristic adsorption energy of component i η = reduced density (= b/4V) μ = chemical potential ρ = molar density φ = fugacity coefficient

■

APPENDIX A Two quantifiable measures were used to estimate the deviations from experimental data in this work: the absolute average deviation with respect to the amounts adsorbed (AADn) AADn (%) =

100 DP

DP

∑ i

|Nicalc ,ex

−

Superscripts

Ai = Ath hydrogen-bonding site on molecule i b = bulk phase Bj = Bth hydrogen-bonding site on molecule j or i N = number of specific components in a mixture 0 = characteristic pure-component parameter exp = experimental data calc = calculated data

Niexp ,ex |

Ni ,ex

(14)

and the absolute average deviation with respect to the mole fraction of adsorbed species (AADx) AADx (%) =

100 DP

DP

∑ |xia,calc − xia,exp| i

Subscripts

(15)

i = ith component in a mixture ex = excess amount adsorbed

where DP is the number of data points; Nex is the excess amount adsorbed; and superscripts calc and exp denote the calculated and experimental values, respectively.

■

List of Abbreviations

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge Statoil (Norway) for supporting this project. The authors are grateful to Ph.D. student Bjørn Maribo-Mogensen and M.Sc. Sofie Bartholdy for valuable discussions and suggestions.

■

LIST OF SYMBOLS a0 = pure-compound energy term b = covolume bij = cross covolume between components i and j c1 = pure-compound energy term g(V) = radial distribution function n = number of moles P = pressure R = ideal gas constant [J/(mol K)] T = temperature (K) V = molar volume x = mole fraction x = composition vector of mole fractions X = mole fraction of molecules not bound at other sites z = spatial position variable z0 = total volume of micropores accessible to the adsorbates z0i = total volume of micropores accessible to adsorbate i

■

AADn = absolute average deviation in terms of the total amounts adsorbed AADx = absolute average deviation in terms of the composition AC = activated carbon CC = coated ceramic CPA = cubic-plus-association CR-1 = combining rule 1 DP = number of data points DRA = Dubinin−Radushkevich−Astakhov EoS = equation of state MPTA = multicomponent potential theory of adsorption MS = molecular sieve PC = perturbed-chain PTA = potential theory of adsorption RDF = radial distribution function S = silica (Cab-o-Sil) SAFT = statistical associating fluid theory SG = silica gel SRK = Soave−Redlich−Kwong TVFM = theory of volume filling of micropores vdW = van der Waals vdW1f = van der Waals one-fluid mixing rule

REFERENCES

(1) Myers, A. L.; Prausnitz, J. M. Thermodynamics of mixed-gas adsorption. AIChE J. 1965, 11, 121−127. (2) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; John Wiley & Sons: New York, 1997. (3) Henderson, D. Fundamentals of Inhomogeneous Fluids; Marcel Dekker: New York, 1992.

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Potential Theory of Adsorption for Associating Mixtures: Possibilities

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