Ind. Eng. Chem. Res. 2000, 39, 1095-1105
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Thermodynamic Modeling of Molecular Adsorption Using Parameters Derived from Binary Surface Excess Data Socrates Ioannidis* and Andrzej Anderko OLI Systems, Inc., 108 American Road, Morris Plains, New Jersey 07950
A methodology for modeling molecular adsorption in multicomponent systems has been developed. The adsorption process is described using the concept of exchange reactions at the solid-liquid interface. The nonideality of the adsorbed phase is represented using the Wilson solution model. For the computation of the equilibrium bulk phase activity coefficients, the nonrandom twoliquid (NRTL) model is used. The model allows for the existence of multiple adsorbed layers at the interface. Binary surface excess data are used to derive equilibrium constants for each exchange reaction. The internal consistency of the methodology is verified by using two separate consistency tests, i.e., the triangular rule for equilibrium constants in ternary systems and the invariance of equilibrium constants with concentration. The model has been applied to a variety of concentrated or dilute systems, including mixtures with liquid-phase immiscibility. In all examined cases, accurate representation of experimental data has been obtained. Introduction Molecular adsorption processes at the solid-liquid interface are important in the separation and purification of liquid streams in the chemical industry1 as well as in the prediction of the fate of organic molecules released in the environment.2 In the solid-liquid molecular adsorption literature, surface excess graphs are generally used to report and analyze adsorption in wide composition ranges whereas the more traditional Langmuir- and Freundlich-type adsorption isotherm plots are used for dilute aqueous mixtures.3 In the latter case, a thermodynamic framework such as the ideal adsorbed solution theory4 is required in order to utilize the isotherms to make predictions for multicomponent mixtures. In the former case, the interpretation of surface excess data requires a thermodynamic model that is valid for both adsorbed and equilibrium bulk phases over wide concentration ranges. The objective of our work is to construct such a model so that binary surface excess graphs can be accurately reproduced and the resulting thermodynamic parameters can be used with confidence for multicomponent systems. Solid-liquid adsorption systems are characterized by a variety of lateral interactions that occur between species in either the equilibrium bulk or the adsorbed phase as well as by vertical interactions between these species and the solid surface.5 For the representation of lateral interactions in the bulk phase, well-known activity coefficient models such as the Redlich-Kister6 or the Margules equation7 have been previously used. For the adsorbed phase, models based on the regular solution8 or lattice9 theories have been employed. In this work, we use the NRTL expression10 for the bulk phase nonideality because it has been proven to be accurate for multicomponent liquid mixtures and is applicable to systems with two liquid phases. To represent the adsorbed phase nonideality, we use the Wilson equa* Corresponding author. E-mail:
[email protected]. Tel: (973)-539-4996. Fax: (973)-539-5922.
tion,11 which has been proven to be accurate for ionexchange systems.12 The molecular adsorption model proposed in this work is based on the concept of exchange between the bulk and adsorbed phase species. The basic assumption in this approach is that the solid material is an inert species with a constant surface phase capacity, which is accessible to both species that participate in the elementary exchange reaction, i.e., molecular sieve effects are excluded. For quantitative modeling, it is convenient to couple the exchange approach with the assumption of multiple adsorbed layers.13 The methodology we present relies on the calculation of the selectivity coefficient from the liquid-phase activities and the adsorbed phase mole fractions. This methodology was also used for ion-exchange systems.12 The model proposed here is designed to ensure the internal consistency of the results. In particular, the obtained equilibrium constants should be invariant with respect to composition. Another consistency requirement is that the binary equilibrium parameters in a ternary system should satisfy the triangular rule.14 These consistency tests do not require experimental data in the whole concentration range, but they rely on the features of the model. The internal consistency of the model is additionally reinforced by using Rusanov’s theorem,15 which sets a minimum value for the number of layers allowed in the model.16 The model is applied to analyze several binary systems for which experimental data are available. These systems include dilute organic-aqueous mixtures and systems with immiscibility in the liquid phase. Model Development The adsorption exchange reaction between a bulk species A and a surface species B at the solid-liquid interface is written as
rAl + Bs ) rAs + Bl
10.1021/ie990719j CCC: $19.00 © 2000 American Chemical Society Published on Web 03/08/2000
(1)
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000
r is the ratio of the adsorption capacities of species A to species B and the superscripts l and s denote the bulk and adsorbed phases, respectively. The chemical potential for species i in the equilibrium bulk phase is customarily written as l l µli ) µol l + RT ln(γixi)
+ RT ln(γsi xsi ) + πvsi µsi ) µos i
(3)
where γsi denotes an activity coefficient in the adsorbed phase, which is characteristic for a solid medium, vsi is the partial molar volume of species i in the adsorbed phase, and π is the spreading pressure which is related to the surface tension of the mixture.4 For nonporous adsorbents we replace in eq 3 the partial molar volume with the partial molar area.14 The symmetric convention is used for the activity coefficients in both the adsorbed and equilibrium bulk phases. From the equality of eqs 2 and 3 at a given equilibrium point, we get
( ) γsi xsi γlixli
+ πvsi ) 0
(4)
ol where ∆µoi denotes the difference µos i - µi . By writing eq 4 for a specific neutral species j, we obtain the osmotic pressure:17
π)-
[
1 ∆µoj + RT ln s vj
( )] γsj xsj γljxlj
(5)
From eqs 4 and 5 and for the i-j exchange, we get
∆µoj + RT ln
( ) [ γsi xsi γlixli
-
vsi
vsj
∆µoj + RT ln
]
γsj xsj γljxlj
) 0 (6)
Next, we assume that the ratio of partial molar volumes of species i to species j is given by the molar ratio of the adsorption capacities of species j to species i.16 Then, eq 6 can be rewritten for the exchange of species A and B as
[
(xsAγsA)r xlBγlB xsBγsB
]
∆µoB - r∆µoA ) exp ) KAB RT (xl γl )r A A
(7)
Equation 7 introduces the equilibrium constant, which can be expressed in terms of the selectivity coefficient Kc:
KAB )
(γsA)r Kc γsB
(8)
Assuming that multiple adsorbed layers exist and for an energetically homogeneous surface, the reduced surface excess of component A is16
neA )
t xsB xsA + mA mB
(xsA - xlA)
xsA )
(2)
where xli denotes the mole fraction of species i and γli is its activity coefficient. Similarly, for the adsorbed species i, we write
∆µoi + RT ln
where t is the number of adsorption layers on the solid surface and mi denotes the surface phase capacity of the solid for species i. Equation 9 can be solved for the adsorbed phase mole fraction
(9)
[
xlA + neA/tmB 1 1 mB mA neA 1+ t
]
(10)
The adsorbed phase mole fraction as predicted by eq 10 should be less than 1, and should also satisfy Rusanov’s theorem:15
() ∂xsi ∂xli
g0
(11)
T,P
Substitution of eq 10 into eq 11 at constant temperature and pressure gives16
tmB g
(
e
∂nA 1 1 - 1 neA - l 1 + - 1 xlA r r ∂x
)
A
[ (
) ]
(12)
where r denotes the ratio mA/mB. For various classes of surface excess graphs,18 eq 12 gives the lower limit of the number of layers16 required in eq 9. As with the bulk phase, we can write a Gibbs-Duhem equation for the adsorbed phase8 by using eq 9:
xsA d ln γsA + xsB d ln γsB ) 0
(13)
Having defined the Gibbs-Duhem equation for the adsorbed phase, we can apply the formalism that was developed for ion exchange in a previous paper.12 Accordingly, the differential of eq 8, combined with eq 13, can be integrated from an arbitrarily selected point (Q1) to another point (Q2) on the adsorption isotherm surface. As shown in the previous paper, this results in12
ln
γsB(Q1) γsB(Q2)
+ ln
[ ] γsA(Q2)
γsA(Q1)
r
) ln Kc(Q1) - ln Kc(Q2) (14)
Equation 14 is consistent with eq 13. Another result of the derivation that leads to eq 14 is a relationship between KAB and Kc,19 i.e.,
ln KAB )
∫01ln Kc dxsA
(15)
Equation 15 can be used to estimate the equilibrium constant when experimental data cover the entire range of liquid-phase mole fractions. Additionally, we can derive theoretical values of the selectivity coefficient at the integration limits of eq 15 as
lim (Kc)1/r )
xlAf0
lim xAf0
and
[ ( )] xsA xlBγlB
xlAγlA xsB
1/r
)
xlA ∂xsA 1 1 lim ) lim (16) x f0 xl x f0 ∂xl γ∞l γ∞l A A A A A A
Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 1097
lim Kc )
xlAf1
lim xAf1
[( ) ] r
xsA
xlBγlB
xlAγlA
xsB
ln(ΛrABΛBA) - (1 + r - ΛAB - rΛBA) ) )
γ∞l B
lim xAf1
xlB xsB
)
[ ( ( ) )] [ ( ( ) )]
1 lim s l (17) xAf1 ∂x /∂x A A
γ∞l B
ln
( )
(18)
( )
(19)
∂neA ∂xlA xl f0 A
)1+
1 m Bt
)1+
e 1 ∂nA mAt ∂xl A
and
lim
xAf1
∂xsA ∂xlA
xlAf1
The combination of eqs 16 and 18 gives
lim (Kc)1/r )
xlAf0
( ( ) )
e 1 1 ∂nA 1 + mBt ∂xl γ∞A A
(20)
xlAf0
Equations 17 and 19 give
lim Kc )
xlAf1
γ∞A
( ( ) ) e 1 ∂nA 1+ mAt ∂xl A
(21)
xlAf1
Equations 15, 20, and 21 define a possible way to obtain the equilibrium constant through numerical integration. Specifically, a cubic spline may be used to link the experimental selectivity points and those defined by eqs 20 and 21. Integration of this spline curve makes it possible to estimate the equilibrium constant based on eq 15. For the computation of the parameters of the adsorbed solution model, eq 14 is applied. The Wilson model is used to calculate the adsorbed phase activity coefficients. Thus, the activity coefficient of species k in the adsorbed phase is given by11
ln γsk ) -ln(
∑j xsj Λkj) + 1 - ∑i
xsi Λik
∑j
(22)
ln γ∞s A ) -ln(ΛAB) + (1 - ΛBA)
ln
γ∞s B
τBA ) bBA + aBAT
(27)
where T is the temperature and bAB, bBA, aAB, and aBA are parameters regressed from experimental VLE data. The G functions in NRTL are
(
τAB T
(24)
The solution of eq 14 at the points of infinite dilution for both species A and B, combined with eqs 20, 21, 23, and 24, gives the following equation:
)
(28)
(
τBA T
)
(29)
and
where R is the nonrandomness parameter for the mixture. The activity coefficient in the equilibrium bulk phase is10
ln γli ) ) -ln(ΛBA) + (1 - ΛAB)
(26)
and
(23)
and
(25)
xlAf1
τAB ) bAB + aABT
GBA ) exp -R
where Λij and Λji are the model parameters for a binary i-j pair of molecules. The activity coefficients of the A and B species in the binary mixture at infinite dilution are
e 1 ∂nA mAt ∂xl A
Equation 25 indicates that the values of the surface excess at the composition limits establish a relation between the adsorbed phase parameters. While eq 25 imposes a constraint on the variation of the parameters in a two-parameter solution model, it can uniquely specify the parameter of a single-parameter solution model. Thus, it can be used to derive model parameters in conjunction with eq 8 for computing the equilibrium constant. However, the experimental uncertainty at the composition limits is high and makes the determination of the slopes in eq 25 inherently uncertain. Therefore, in this work, we utilize a computational method that minimizes the effect of experimental error. In this method, we obtain the adsorbed solution model parameters from eq 14. We usually start from the maximum point on the surface excess graph, and we utilize the sections where the experimental isotherm is smooth. Subsequently, the equilibrium constant is statistically derived by averaging values obtained from eq 8. The liquid-phase activity coefficients are obtained from the NRTL model.10 The NRTL parameters are regressed from vapor-liquid equilibrium (VLE) data over a range of temperatures. For the temperaturedependent interaction parameters between species A and B (i.e., τAB and τBA), we have
GAB ) exp -R
xsj Λij
-
xlAf0
ln γ∞s B / 1 +
To estimate the limits of eqs 16 and 17, we use eq 10 and we note that neA approaches zero as xlA approaches either zero or 1. Thus, we get
∂xsA lim l xAf0 ∂x A
e 1 ∂nA 1 1 + mBt ∂xl γ∞s A A
∑j τjiGjixlj Glixll
+
∑j
xljGij
∑j
Gljxll
( ) τij -
∑k xlkτkjGkj ∑l
(30)
Gljxll
The infinite dilution activity coefficients of species A and B in the binary mixture are
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000
ln γ∞l A ) bAB + aABT bBA + aBAT bAB + aABT + exp -R (31) T T T
(
)
and
ln γ∞l B ) bBA + aBAT bAB + aABT bBA + aBAT (32) + exp -R T T T
(
)
Multicomponent Mixtures Although the thermodynamic parameters are usually regressed from data for binary systems, the framework can be used for the prediction of adsorption in multicomponent mixtures. For an N-component mixture, the equilibrium constant of the i-j pair exchange is
Kij )
[ ][ ] xsi γsi
ri
xliγli
xljγlj
rj
xsj γsj
(33)
It should be noted that each species i has a unique stoichiometric coefficient ri for a full set of exchange reactions in a multicomponent system. The activity coefficients are estimated from eqs 22 and 30. From the N(N - 1)/2 nonlinear equations of the binary equilibrium constants of all mixture pairs, only N - 1 can be considered independent because of the triangular rule.14 In addition to the N - 1 equilibrium relations, we have the mole fraction balance constraint for the adsorbed phase, and thus, at a given vector of equilibrium compositions in the bulk phase, we can numerically solve for the N unknown adsorbed phase mole fractions. Then, the reduced surface excess of species i in the mixture can be computed from its definition:
nei
)t
xsi - xli N
xsi
∑ l)1 m
(34)
l
Model Application The first ternary system considered here is the adsorption of the benzene-cyclohexane-ethyl acetate system on activated carbon. The capacities of the solid for benzene, cyclohexane, and ethyl acetate are reported as 5.48, 4.3, and 4.95 mmol/g, respectively.6 According to eq 12, the lower limit for the number of layers t is less than 1 for all three binary systems. Therefore, the value of 1 is adopted. The sources of the original VLE data and the regressed NRTL parameters for the benzene-cyclohexane and ethyl acetate-cyclohexane
Figure 1. Surface excess calculations and experimental data for the adsorption of the binary systems ethyl acetate (1)-cyclohexane (2), benzene (1)-ethyl acetate (2), and benzene (1)-cyclohexane (2) on activated carbon.
systems are shown in Table 1. The Redlich-Kister expansion was used for the liquid activity coefficients of the benzene-ethyl acetate system.6 The surface excess predictions are shown in Figure 1. The values of the model parameters are collected in Table 2. The equilibrium constant values are close to the ones obtained by direct application of equations 15, 20, and 21, which are equal to 1.352, 0.565, and 0.350 for the benzene-cyclohexane, benzene-ethyl acetate, and ethyl acetate-cyclohexane binaries, respectively. For comparison, application of eq 12 with the integration limits defined by the range of experimental data (i.e., close to the mole fractions of 1 and zero) resulted in values for the equilibrium constants of 1.282, 0.548, and 0.332 for the same systems. The internal consistency of the model can be evaluated by using eqs 18 and 19. For example, from the experimental data for the system benzene (A)-cyclohexane (B), we calculate the values limxl f0 (∂xsA/∂xlA) ) 0.613 and limxl f0 (∂xsA/∂xlA) ) 8.648 A A while the values of 1 + (1/mAt)(∂nA/∂xlA)xl f1 and 1 + A (1/mBt)(∂nA/∂xlA)xl f0 as calculated from the surface exA cess graphs are 0.627 and 8.947, respectively. This shows the validity of eqs 20 and 21 for these systems. In Figure 2, the activity coefficients in both the equilibrium bulk and adsorbed phases are plotted. We note that the adsorbed phase excess Gibbs energy is a wellbehaved function. Figure 2 shows that the activity coefficients of the two benzene-containing binaries in the adsorbed phase are lower than 1. Therefore, these
Table 1. NRTL Parameters and VLE Data References for the Binary Pairs Studied system (1-2)
T range (K)
R
a12
a21
b12
b21
ref
benzene-cyclohexane ethyl acetate-cyclohexane propanol-water chlorobenzene-benzene chlorobenzene-n-heptane nitrobenzene-benzene nitrobenzene-n-hexane benzene-n-heptane methanol-water methyl acetate-water methanol-methyl acetate
281-333 293-328 273-323 298-348 323-343 293-343 294 293-328 273-328 273-376 293-313
0.20 0.20 0.45 0.20 0.20 0.20 0.50 0.20 0.20 0.20 0.20
-1.28 -1.04 -1.53 3.03 -1.65 11.2 0.0 1.65 -2.35 -3.43 -1.27
0.37 -1.11 5.28 -4.43 1.06 -14.2 0.0 -2.11 3.84 4.69 0.67
694 436 675 -1159 834 -3509 437 -81 631 1076 566
-237 616 -874 1673 -406 4550 697 468 -877 -473 -16
29-34 35, 36 37-40 41-44 45 46-48 49 50-55 56-59 60-63 64
Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 1099 Table 2. Thermodynamic Parameters and Adsorption Data References for the Binary Pairs Studied system (1-2)
r ) m1/m2
Λ12
Λ21
ln K12
t
ref
carbon: benzene-cyclohexane carbon: benzene-ethyl acetate carbon: ethyl acetate-cyclohexane A: water-propanol B: water-propanol C: water-propanol D: water-propanol E: propanol-water F: propanol-water G: propanol-water silica: chlorobenzene-n-heptane silica: benzene-chlorobenzene silica: nitrobenzene-benzene silica: nitrobenzene-n-heptane silica: benzene-n-heptane NaX: water-methanol NaX: water-methyl acetate NaX: methanol-methyl acetate coarse silt: pyrene-water EPA4: pyrene-water EPA5: pyrene-water EPA6: pyrene-water Overton: benzene-water Hasting: benzene-water Ca-clay: benzene-water Al-clay: benzene-water
1.274 1.107 1.151 4.167 4.167 4.167 4.167 0.240 0.240 0.240 0.844 1.123 0.876 0.830 1.055 1.735 2.717 1.566 0.234 0.234 0.234 0.234 0.345 0.345 0.345 0.345
0.197 0.369 1.051 1.242 0.585 1.870 1.878 0.174 1.281 2.917 1.953 2.511 1.882 0.314 0.072 0.275 6.120 1.934 0.238 0.204 0.188 0.246 0.366 0.504 0.359 0.141
5.042 2.707 0.485 0.065 0.340 0.001 0.001 0.523 0.550 0.607 0.144 0.153 0.525 0.252 4.199 3.631 8.528 0.002 0.001 0.227 0.632 0.001 0.001 0.001 0.001 0.978
1.290 0.670 0.381 0.851 0.370 1.487 0.890 0.092 0.513 0.655 1.052 0.621 0.961 0.480 1.590 1.320 3.159 0.992 0.846 0.218 0.141 0.259 2.888 2.854 3.155 3.034
1 1 1 2 2 1 2 2 1 1 2 1 1 4 2 2 2 2 1 1 1 1 1 1 1 1
6 6 6 20 20 20 20 20 20 20 21 21 21 21 21 7 7 7 25 27 27 27 28 28 28 28
Figure 2. Activity coefficients for the adsorption of the binary systems ethyl acetate (1)-cyclohexane (2), benzene (1)-ethyl acetate (2), and benzene (1)-cyclohexane (2) on activated carbon in both the equilibrium bulk phase (points) and the adsorbed phase (lines).
binaries exhibit stronger vertical than lateral interactions, compared to those in the ethyl acetate-cyclohexane binary.9 In Figure 3, we plot the values of the equilibrium constant as derived from eq 8. From a linear regression of the data plotted in Figure 3, we observe very low values of the slopes of the ln K versus xs1 curves. A near-zero slope signifies the invariance of the equilibrium constant over the entire composition range. Application of the triangular rule to these three binary systems requires that the ratio 1.151 (Kbenzene-ethyl acetateKethyl acetate-cyclohexane/Kbenzene-cyclohexane) be equal to 1. From the values reported in Table 2 we get a value of 0.871 for this ratio. In Figure 4, we plot our predictions for the system benzene-ethyl acetatecyclohexane.6 A satisfactory agreement with experimental data has been obtained.
Figure 3. Calculated equilibrium constants for the adsorption of the binary systems ethyl acetate (1)-cyclohexane (2), benzene (1)-ethyl acetate (2), and benzene (1)-cyclohexane (2) on activated carbon.
Experimental data for the adsorption of the waterpropanol system on several adsorbents (A, Serdolit CS-2; B, Amberlite CG-120; C, Serdolit CW-18; D, Amberlyst A-21; E, Amberlite XAD-8; F, Amberlite XAD-2; G, Amberlite XAD-4)20 reveal a variety of surface excess graphs. For the A, B, D, and E systems, we have assumed two layers, and for the systems C, F, and G we have assumed one layer. For the first four systems, water is reported in Table 2 as the most favorably adsorbed species (i.e., component 1), while for the last three systems, propanol becomes component 1. The references to the VLE data used to regress the liquid-phase model parameters are reported in Table 1, while the obtained parameters are shown in Table 2. The surface excess calculations using the thermodynamic parameters from Table 2 are shown in Figure 5 for all seven systems. The activity coefficients in both phases are plotted in Figure 6. From Figure 6, we note
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000
Figure 4. Prediction of adsorption of the ternary system benzeneethyl acetate-cyclohexane on activated carbon.
Figure 6. Activity coefficients for the adsorption of the binary system propanol (1)-water (2) on various adsorbents (see legend of Figure 5) in both the equilibrium bulk (points) and the adsorbed phase (lines).
Figure 5. Surface excess calculations and experimental data for the adsorption of the binary system propanol (1)-water (2) on various adsorbents (A, Serdolit CS-2; B, Amberlite CG-120; C, Serdolit CW-18. D, Amberlyst A-21; E:Amberlite XAD-8; F, Amberlite XAD-2; G, Amberlite XAD-4).
Figure 7. Variation of ln K1/ln Kt (eqs 12, 17, and 18) for the nitrobenzene-benzene and benzene-cyclohexane systems with the number of layers t.
that, as the nature of the sorbent becomes more hydrophobic (i.e., systems E-G), the vertical interactions increase. This is in contrast to the systems where the solid surface is more hydrophilic and the lateral interactions dominate. The values of the equilibrium constant are reported in Table 2. Calculations have also been performed for systems that include the binaries of chlorobenzene and nitrobenzene with benzene and n-heptane.21 The VLE data references and liquid-phase activity coefficient parameters are shown in Table 1. For the nitrobenzenen-heptane system for which adsorption data are reported21 over the whole composition range, no VLE data are available in the open literature. Therefore, the NRTL parameters for this system were set equal to those for the nitrobenzene-n-hexane system. The estimated numbers of the adsorption layers for all four systems are shown in Table 2. Also, Table 2 contains the model parameters for all four systems. The equi-
librium constant for the benzene-n-heptane system can be derived from either the benzene-chlorobenzene and chlorobenzene-n-heptane systems or the nitrobenzenebenzene and nitrobenzene-n-heptane systems. The problem that arises here is that, for each of these combinations, a different number of layers is obtained from eq 12. Because of the nonlinear nature of eqs 10 and 22, there is no obvious way to derive a formula for the variation of the equilibrium constant with respect to the number of adsorption layers. To circumvent this problem, we have assumed various numbers of layers for the two systems nitrobenzene-benzene and benzenechlorobenzene and we have derived the value of the equilibrium constant by using eqs 15, 20, and 21 for each number of layers. The results for both binary systems are shown in Figure 7. It can be observed that the ratio ln Ki/ln Kt, where t and i denote the numbers of layers with t > i, can be well regressed for both systems with straight lines. To further investigate the
Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 1101
observed constancy of the (ln Ki/t - ln Kt/i) term with respect to the number of layers, we consider eq 15:
ln Kt ) r
∫01 ln xs1 dxs1 + ∫01 ln xs2 dxs2 +
( ) xl γl
∫01 ln (xl2γl2)r
dxs1 )
1 1
( ) xl γ l
∫01 ln (xl2γl2)r
dxs1 (35)
1 1
The first two integrals on the right-hand side of eq 35 are equal to zero. Therefore, by substituting xsi from eq 10 into eq 35, we obtain
ln Kt ) I1 + I2 )
( ) xl γl
∫01ln (xl2γl2)r 1 1
∫0
1
ln
1 dxl1 + ne1 1 1 1+ t m2 m1
∂ne1 ∂xl1
( )[ xl2γl2
r (xl1γl1)
(
[
)
xl1 1 1 + t m2 m1 m2
1+
xl1
)]
(
ne1
1 1 t m2 m1
]
2
dxl1 (36)
On the basis of eq 36, we will now examine the term ln Kt1/t2 - ln Kt2/t1. For equal-size molecules, the first integral of eq 36 becomes zero. For molecules of different sizes, we get for the first integral (I1) on the right-hand side of eq 36
1+
(
ne1
∫
1
( )
( )
)
t2
(
)
e 2 1 1 t1 + t2 (n1) 1 1 + m2 m1 t1t2 t1t2 m2 m1
dxl1
(37)
2
When the third quotient on the right-hand side of eq 37 is close to unity, eq 37 becomes
I1(t1) I1(t2) t1 - t2 ≈ t2 t1 t1t2
( ) xl2γl2
∫0 ln xl γl 1
r
dxl1
(38)
1 1
and, because both the integrals ∫10ln(xl2/(xl1)r) dxl1 and ∫10ln(γl2/γl1) dxl1 are zero,22 we get
I1(t1) I1(t2) t1 - t2 ≈ (1 - r) t2 t1 t1t2
∫01ln γl1 dxl1
(39)
For example, for the nitrobenzene-benzene system the integral ∫10ln γl1 dxl1 is approximately 0.119 and r is 0.876. Hence, the right-hand side of eq 39 is nearly zero. For the integral I2, we obtain for equal-sized molecules
I2(t1) I2(t2) )0 t2 t1
straight lines in Figure 7. When this is the case, eq 40 is satisfied to a good approximation. Then, we can write the final approximation for the equilibrium constants, i.e.,
ln Kt1
xl2γl2
t1 - t2 ln × 0 l l r t1t2 (x1γ1) 1 1 t 1 + t2 1 + ne1 m2 m1 t1t2
I1(t1) I1(t2) ) t2 t1
Figure 8. Surface excess calculations and experimental data for the adsorption of the binary systems benzene (1)-chlorobenzene (2), benzene (1)-n-heptane (2), chlorobenzene-(1)-n-heptane (2), nitrobenzene (1)-n-heptane (2), and nitrobenzene (1)-benzene (2) on silica gel.
(40)
For molecules of different sizes, the denominator of the second integrand can be computed numerically. For example, such calculations have been performed for the two systems shown in Figure 7 to verify that the denominator is close to unity. This is illustrated by the
-
ln Kt2 t1
≈0
(41)
We apply eq 41 to the benzene-chlorobenzene system in order to go from one to two layers. Then, from the triangular rule and the equilibrium constant for the system chlorobenzene-n-heptane, we get the equilibrium constant for the benzene-n-heptane system. In Figure 8, we show the surface excess calculations for all four systems and the model predictions for the benzene-n-heptane system for which no experimental data were reported. To derive the adsorbed solution model parameters for the benzene-n-heptane system, the ternary data of the benzene-chlorobenzene-nheptane system were reduced by assuming two layers. For the data reduction, we have used eq 8 applied to the benzene-n-heptane pair. Trial calculations with different numbers of layers for the benzene-chlorobenzene system showed that the parameters of the adsorbed solution model do not vary significantly. The invariance of the parameters is particularly pronounced as the number of layers increases. The values of the Wilson parameters for the benzene-n-heptane system are reported in Table 2. The model has also been applied to the analysis of the adsorption behavior of the immiscible system watermethyl acetate on NaX zeolite, along with the completely miscible binaries water-methanol and methanolmethyl acetate.7 For the miscible systems, we have assumed two adsorption layers. The same number of layers is also used for the immiscible system in order to reduce the experimental data for the water-lean region. With two layers, we satisfy Rusanov’s theorem,15 and also the adsorbed phase mole fractions calculated from eq 10 satisfy the obvious condition of being less than 1. In Tables 1 and 2, the VLE data references
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Figure 9. Surface excess calculations and experimental data for the adsorption of the binary systems water-methanol, methanolmethyl acetate, and water-methyl acetate on NaX zeolite.
are reported for all three binaries along with the model parameters. The saturation adsorption capacity for water is 14.4 mmol/g,23 while the adsorption capacities for methanol and methyl acetate derived from Gurvitsch’s rule6 are 8.3 and 5.3 mmol/g, respectively. While the predictions of eq 9 for the water-lean region were good, negative values for the surface excess were obtained in the water-rich region. To obtain a reasonable surface excess value, it would be necessary to assume a high number of layers (∼15) in the water-rich region. However, it is important to maintain the same number of layers for all binaries. Therefore, to use two layers for each system, we have assumed the following empirical compositional transformation for the bulk phase water mole fraction (x′l1) in the water-rich region: lR l lβ lβ x′l1 ) xlR 1 + x2 (x1 - x1 )/x2
(42)
lβ In eq 42, xlR 1 ) 0.282 and x1 ) 0.929 are the values of the water mole fractions at the limits of immiscibility. Equation 42 ensures the continuity of the surface excess predictions at the mole fractions that correspond to the miscibility limits. Also, x′l1 ) 1 when xll ) 1. Good agreement with experimental data is obtained when eq 42 is substituted into eq 9. The surface excess calculations for all three binaries are shown in Figure 9 for two layers and with the Wilson parameters obtained from the water-lean region for the watermethyl acetate system. With the values reported in 1.566 Kmethyl acetate-methanol/ Table 2, the quantity Kwater-methanol Kwater-methyl acetate is equal to 0.905. Finally, we consider applications to systems that involve organic molecules under dilute aqueous conditions. Traditionally, the modeling of such systems involves the use of either the Langmuir or Freundlich models.24 In this work, we translate the reported concentrations of dilute solutions in the bulk or adsorbed phase into surface excess plots. For this purpose, we divide the reported liquid concentrations (moles/per kilogram) by the number of moles of water in 1 kg (∼55.5 M) and we define the surface excess as
ne1 ) ns[xs1 - xl1] ≈ nsxl1 ) ns1
(43)
Figure 10. Surface excess calculations and experimental data for the adsorption of the binary system pyrene-water on various adsorbents.
where ns denotes the number of moles in the adsorbed phase and ns1 is the number of moles of the solute in the adsorbed phase, which is reported in a Langmuir or Freundlich plot. First, the model is applied to the pyrene-water adsorption on coarse silt.25 The infinite dilution activity coefficient of pyrene in water is26 10 638 606. This value is assumed for the activity coefficient of pyrene for finite concentrations of pyrene in dilute solutions. For water, the activity coefficient is assumed to be equal to 1. The monolayer surface capacity of the solid for pyrene is estimated as 0.000 96 mmol/g. The monolayer surface capacity of the adsorbent for water is estimated from Gurvitsch’s rule.6 Experimental data for this binary are also reported for the adsorption of pyrene on various sediments27 (EPA4, EPA5, and EPA6). The model parameters are shown in Table 2, and the surface excess predictions for all four systems are shown in Figure 10. When the values of the parameters are examined, several linear correlations are identified for the three EPA samples. These correlations include a relationship between the Λ21 parameter and the reported cationexchange capacity ln(ln(CEC)), a correlation between Λ12 and ln(CEC), and a correlation of the adsorbed phase infinite dilution activity coefficient of pyrene with ln(CEC). For the first correlation, we note that the lateral interactions, which dominate in the adsorbed phase, remain unchanged for all three systems and, therefore, CEC can be identified as a measure of the magnitude of the vertical interactions. The other two correlations follow by inspection of eq 23. A relationship of ln K with ln(CEC) follows as well, based on eqs 8, 18, and 23. Figure 11 shows these correlations and the regression parameters (i.e., the intercept b0, the slope b1, and the degree of correlation r2). Also, the adsorption of the benzene-water system has been studied on two soils (Overton and Hasting) and a montmorillonite clay saturated with either Ca or Al (clay-Ca, clay-Al).28 Because experimental values of exchange capacities are not available for these systems, it was necessary to estimate them in order to apply the model. For practical purposes, the adsorption capacities were assumed to be somewhat larger than the surface
Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 1103
Figure 11. Correlations of the pyrene-water thermodynamic parameters with the cation-exchange capacity of three sediments.
written for the adsorbed phase. The model allows for the existence of multiple layers of adsorbed molecules in the adsorbed phase. The bulk phase nonideality is also considered as a contributor to the lateral interactions in the adsorbed phase. This effect is modeled using the NRTL model for the bulk phase. At the same time, the adsorbed phase nonideality is represented using the Wilson model. The analysis of experimental results shows that the model conveniently decouples the computation of equilibrium constants and adsorbed phase activity coefficient parameters. This gives significant flexibility for the analysis of data that do not extend to the composition limits of the surface excess graphs. The model has been successfully applied to both concentrated and dilute systems. A variety of types of surface excess plots18 has been satisfactorily reproduced. Two separate consistency tests have been introduced along with a test that defines the minimum number of layers.16 In particular, the triangular rule was shown to be satisfied for the binary pairs of selected ternary systems. Also, the equilibrium constant was shown to be invariant with composition. Additionally, a heuristic approach has been developed to relate the equilibrium constant to the number of adsorbed layers. A technique has been developed to reduce the number of model parameters, which may be useful when experimental data are not available. In summary, our model can be used to regress thermodynamic adsorption parameters, in an internally consistent way, from experimental data given in binary surface excess plots. These parameters can then be used to predict surface excess in multicomponent systems. Acknowledgment This work was supported by the Gas Research Institute under Contract 5094-250-2863. Discussions with Dr. Rajat Ghosh are gratefully acknowledged. Notation
Figure 12. Surface excess calculations and experimental data for the adsorption of the binary system benzene-water on various adsorbents.
excess reported for the highest adsorbate mole fraction in the aqueous phase. Gurvitsch’s rule was applied to estimate the water saturation adsorption capacity. The benzene activity coefficient was set equal to its infinite dilution value (i.e., 2416),26 while the water bulk activity coefficient in eq 4 was set equal to 1. The parameters obtained from the data reduction for all four systems are shown in Table 2. The lateral interactions in the adsorbed phase dominate for all four sorbents. The surface excess predictions for all four systems are presented in Figure 12. Conclusions A molecular adsorption framework has been developed for modeling multicomponent adsorption using parameters derived from binary surface excess data. The framework is based on the exchange concept, which was previously used to model ion-exchange systems.12 The model is consistent with the Gibbs-Duhem equation
a ) molar area, NRTL parameter A,B ) species b ) NRTL parameter G ) NRTL energy function I ) integral defined in eq 36 K ) equilibrium constant Kc ) selectivity coefficient m ) adsorption saturation capacity N ) number of species n ) number of moles P ) pressure Q ) point on the isotherm surface R ) gas constant r ) molar capacity ratio T ) temperature t ) number of layers v ) partial molar volume x ) mole fraction Greek Symbols R ) nonrandomness parameter γ ) activity coefficient ∆ ) difference Λ ) Wilson parameter µ ) chemical potential π ) osmotic pressure τ ) NRTL parameter
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Subscripts 1, 2 ) species, integrals in eqs 36-40 A, B ) species i, j, k ) species Superscripts ′ ) transformed mole fraction (eq 40) R, β ) immiscibility limits (eq 40) A, B ) species E, e ) equilibrium bulk phase ° ) standard state s ) adsorbed phase ∞ ) infinite dilution state
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Received for review September 29, 1999 Revised manuscript received January 20, 2000 Accepted January 26, 2000 IE990719J
