Langmuir 2006, 22, 9623-9631
9623
Application of the Vacancy Solution Theory to Describe the Enthalpic Effects Accompanying Mixed-Gas Adsorption K. Nieszporek* Department of Theoretical Chemistry, Faculty of Chemistry, MCS UniVersity, M.C. Skłodowska sq. 3, 20-031 Lublin, Poland ReceiVed June 27, 2006. In Final Form: August 24, 2006 The possibility of utilizing vacancy solution theory (VST) to study the enthalpic effects accompanying mixed-gas adsorption equilibria is presented. Besides heterogeneity, the interaction effects by using the regular adsorbed solution, Flory-Huggins, and Wilson models of nonideality in the adsorbed phase are taken into account. To predict adsorption phase diagrams and calorimetric effects in the mixed-gas adsorption system, only a knowledge of the single-gas adsorption isotherms and accompanying calorimetric effects is required. The possibility of simplification of the obtained theoretical expressions is shown. The obtained agreement between theory and experiment is very satisfactory.
1. Introduction Attempts to obtain a better separation of gases in industrial processes has led to the growing use of adsorption phenomena. The application of this modern technology is also accompanied by less energy consumption. Such processes are controlled by computer programs in which subroutines calculating gas adsorption equilibria play a substantial role. The accuracy and speed of calculations result in a higher degree of purity for the separated gases. Because the adsorption of gases is accompanied by enthalpic effects, the possibility of taking these into account may have an unquestionable influence on the process productivity. Although the adsorption phenomena for gas separation processes were theoretically well described, the calorimetric effects accompanying the mixed-gas adsorption are still not well investigated. Only a few papers devoted to calorimetric effects accompanying mixed-gas adsorption can be found in the literature. For example, the thermodynamic analysis of the isosteric heats of multicomponent adsorption was performed by Sircar,1 and the molecular simulations of the heat effects were performed by Karavias and Myers2 and Vuong and Monson3 as well as Nicholson4 and Steele et al.27 The theoretical expressions for isosteric heats of simultaneously adsorbed components obtained using the integral equation approach were obtained by Jaroniec.5 Talu et al.6 used vacancy solution theory to describe isosteric heats of adsorption of single gases. Theoretical studies of the heat effects of a two- and three-component gas mixtures were made by Sundaram and Yang.7 Dekany et al.8,9 studied the * E-mail:
[email protected]. (1) Sircar, S. Ind. Eng. Chem. Res. 1992, 31, 1813. (2) Karavias, F.; Myers, A. L. Langmuir 1991, 7, 3118. (3) Vuong, T.; Monson, P. A. Langmuir 1996, 12, 5425. (4) Nicholson, D. Langmuir 1999, 15, 2508. (5) Jaroniec, M. J. Colloid Interface Sci. 1977, 59, 371. (6) Talu, O.; Kabel, R. L. AIChE J. 1987, 33, 510. (7) Sundaram, N.; Yang, R. T. J. Colloid Interface Sci. 1998, 198, 378. (8) Dekany, I. Pure Appl. Chem. 1992, 64, 1499. (9) Dekany, I. Pure Appl. Chem. 1993, 65, 901. (10) Nieszporek, K.; Rudzinski, W. Colloids Surf., A 2002, 196, 51. (11) Nieszporek, K. Langmuir 2002, 18, 9334. (12) Nieszporek, K. Appl. Surf. Sci. 2003, 207 1-4, 208. (13) Nieszporek, K. Chem. Eng. Sci. 2005, 60, 2763-2769. (14) Dubinin, M. M. New Results in Investigations of Equilibria and Kinetics of Adsorption of Gases on Zeolites, presented at the Fourth International Conference on Molecular Sieves, April 18-22, 1977, University of Chicago. (15) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1972, 41, 156. (16) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1973, 42, 554. (17) Lucassen-Reynders, E. H. Prog. Surf. Membr. Sci. 1976, 10, 253. (18) Suwanayuen, S.; Danner, R. P. AIChE J. 1980, 26, 77.
adsorption of binary liquid mixtures and the accompanying heat effects at the solid/liquid interface. In our recent paper, we have investigated the theoretical methods of predicting heat effects accompanying mixed-gas adsorption by using the integral equation (IE) approach,10 ideal adsorbed solution (IAS) theory,11,12 and potential theory (PT).13 A basic principle in theoretical investigations proposed by us is the possibility of calculating the mixed-gas adsorption equilibria and accompanying enthalpic effects by using information that can be determined only from the theoretical analysis of singlegas systems. In this article, we examine vacancy solution theory (VST).
2. Theory 2.1. Adsorption Isotherms. Dubinin14 first proposed the vacancy formalism. He treated the single-component adsorption as an equilibrium between two “vacancy solutions” having different compositions. Lucassen-Reynders15-17 used a similar idea to describe the surfactant adsorption, and its consideration can be extended to predict the adsorption equilibrium of a mixed surfactant system. On the basis of the achievements of Dubinin and LucassenReynders, Suwanayuen and Danner18 proposed the approach known as vacancy solution theory. Briefly speaking, the model of adsorption equilibrium assumes that the gas and adsorbed phases are the solutions of adsorbates in a hypothetical solvent called a “vacancy”. A vacancy (i.e., an empty site) can be filled by adsorbate molecules. Rudzinski et al.19 applied VST to the case of mixed-gas adsorption equilibria on heterogeneous solid surfaces. Using different models of nonideality of the adsorbed phase, the authors19 proposed the following expressions for the case of single-gas adsorption equilibria: (19) Rudzin˜ski, W.; Nieszporek, K.; Moon, H.; Rhee, H.-K. Heterog. Chem. ReV. 1994, 1, 275. (20) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969; p 230. (21) Cochran, T. W.; Kabel, R. L.; Danner, R. P. AIChE J. 1985, 31, 268. (22) Dunne, J. A.; Mariwala, R.; Rao, M.; Sircar, S.; Gorte, R. J.; Myers, A. L. Langmuir 1996, 12, 5888. (23) Dunne, J. A.; Rao, M.; Sircar, S.; Gorte, R. J.; Myers, A. L. Langmuir 1996, 12, 5896. (24) Dunne, J. A.; Rao, M.; Sircar, S.; Gorte, R. J.; Myers, A. L. Langmuir 1997, 13, 4333. (25) Siperstein, F.; Gorte, R. J.; Myers, A. L. Langmuir 1999, 15, 1570. (26) Nieszporek, K. Appl. Surf. Sci. 2004, 227, 205. (27) Steele, W. A.; Bojan, M. J. Pure Appl. Chem. 1989, 61, 11.
10.1021/la061847x CCC: $33.50 © 2006 American Chemical Society Published on Web 09/27/2006
9624 Langmuir, Vol. 22, No. 23, 2006
Nieszporek
the Appendix), xsV is the mole fraction of the vacancy (third component), and NΣ and MΣ are defined as follows
1. Regular adsorbed solution
{} ( )
0i θit piKi exp ) kT 1 - θit
ci/kT
{ } ωiiθit kT
exp -
2. Flory-Huggins activity
{} ( ) {
0i θit piKi exp ) kT 1 - θit
ci/kT
}
(aiV)2θit exp 1 + aiVθit
(2)
where θit is the surface coverage, ci/kT is the heterogeneity parameter, Ki exp{0i /kT} is the Langmuir constant (0i is the most probable value of the adsorption energy), and ωii and aiV are the interaction parameters. 3. Wilson activity
{} ( ) [ {(
0i θit ) piKi exp kT 1 - θit exp -
1 - (1 - ΛVi)θit
ci/kT
]
ΛiV ΛiV + (1 - ΛiV)θit
ΛVi(1 - ΛVi)θit
1 - (1 - ΛVi)θit
+
(1 - ΛiV)θit
)}
ΛiV + (1 - ΛiV)θit
{
}
υj λii - λij exp υi kT
}
(3)
(5)
{}
λii ΛiV ) exp and ΛVi ) 1 kT
{[
ci Mi - MΣ exp - 1 ln γsV + ln xsV - Πmi NΣ kT
(10)
lim
) (aiV + 1)-1 exp{aiV}
(11)
) ΛiV exp{ΛVi - 1} ) ΛiV
(12)
All remaining symbols in eq 7 can be determined from the singlegas adsorption isotherms by applying eqs 1-3. 2.2. Enthalpic Effects. To calculate the isosteric heats of adsorption, we use the following well-known equations
[ ] [ ] ∂ ln pi
qsti ) -k Qsti ) -k
∂(1/T)
∂lnpi
∂(1/T)
(13)
θit
{θit}
i ) 1, 2
(14)
where qsti and Qsti are the isosteric heats for single- and mixed-gas adsorption systems, respectively. In the case of pure gas adsorption, a combination of eq 13 with eqs 1-3 leads to the following expressions: 1. Regular adsorbed solution (RAS)
qsti ) qst0 i - ci ln
θit + ωiiθit 1 - θit
(15)
where qst0 i ) k(d ln Ki′/d(1/T)) is frequently called the nonconfigurational isosteric heats of adsorption and Ki′ ) K0i exp{0i /kT}. ωii is the same interaction parameter as in eq 1, so the simultaneous analysis of the isotherms and isosteric heats of single-gas adsorption gives the possibility of increasing the accuracy of determination of ωii. The value of the parameter qst0 i shifts the heat curve only on the y axis. 2. Flory-Huggins (FH) activity
θit 1 - θit 2aiVθit(1 + aiVθit) - (aiVθit)2 (16) ξiV (1 - aiVθit)2
(7)
where Xit is the mole fraction of component i in the adsorbed phase, Πmi is the parameter whose value depends on the definition of the limits of adsorption energies (when infinite limits are assumed, Πmi ) 0), γsi represents the activity coefficients (see
1
NΣf0γs i
st - ci ln qsti ) qi0
]}
1
3. Wilson activity (W)
(6)
Therefore, to adjust eq 3 for the single-gas adsorption isotherms we have four best-fit parameters: monolayer capacity Mi, heterogeneity parameter ci/kT, Langmuir constant Ki exp{0i /kT}, and interaction parameter λii. On the basis of the single-gas adsorption isotherms, the mixedgas adsorption equilibria can be calculated using the following equation19
[ ] ][
{ }
WiV 1 ) exp s NΣf0γ kT i lim
lim
Equation 5 makes it possible to minimize the number of best-fit parameters used during the adjustment of eq 3 to single-gas isotherms. Because the interaction energy between the adsorbed molecule and the vacancy is equal to zero, we can write
{}
(9)
NΣf0γs i
(4)
Mi λii - λij Λij ) exp Mj kT
0i Mi 1 ) Xitγsi NΣ lim piKi exp kT MΣ NΣf0 γs i
MΣ ) X1tM1 + X2tM2
where Nit is the adsorbed amount of component i and Mi is the maximum loading. The definition of the limit limNΣ f0(1/γsi ) depends on the model of nonideality used: 1. Regular adsorbed solution (RAS)
where υj/υi is the size ratio of the mixture components and λij is the potential energy of interaction between species i and j. For the case of gas adsorption, eq 4 becomes
{
(8)
2. Flory-Huggins activity (FH)
where ΛiV and ΛVi are the Wilson coefficients. Generally, they can be treated as two additional best-fit parameters, but these parameters can also be estimated from an equation analogous to that introduced for a bulk liquid mixture20
Λij )
NΣ ) N1t + N2t
(1)
where
ξiV ) k
( ) ∂aiV
∂(1/T)
{Xi}
(17)
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Langmuir, Vol. 22, No. 23, 2006 9625
The calculation of the derivative ξiV ) k(∂aiV/∂(1/T))Nit is a separate problem. Generally, there is no theoretical calculation of this value.21 In this case, a semitheoretical method can be proposed, which consists of drawing the graph of aiV versus 1/T for a few isotherms measured at different temperatures. Then, assuming that this dependence is linear, the value of ξiV is equal to the tangent of the slope of the straight line. The other method is to treat this value as an additional best-fit parameter that is helpful in the adjustment of the theoretical isosteric heat of adsorption (eq 16) to the experimental data. 3. Wilson (W) coefficients
θit qsti ) qst0 + i - ci ln 1 - θit
[
θit
-
ΛiV(ΛiV + θit - ΛiVθit)
2
]
ΛVi(2 + (ΛVi - 1)θit)ζVi (1 + (ΛVi - 1)θit)
2
(18)
ζij ) k
( ) ∂Λij
∂(1/T)
{Xi}
Wij ) ωii + ωjj - 2ωij ) 0
(20)
where Wij is called the interchange energy and the above case refers to ideal solutions, where Wij vanishes. Therefore, this assumption means that ωij is simply the arithmetic average of ωii and ωjj. Using the FH model of nonideality, the calculation of the cross-interaction parameter aij from single-gas adsorption data can be made as proposed by Cochran et al.21
aiV + 1 -1 ajV + 1
(21)
Then, differentiating eq 21 with respect to 1/T, we obtain
( ) ∂(1/T)
{} ( ) [
piKi exp
0i θit ) kT 1 - θit
ci/kT
)
ξiV(ajV + 1) - ξjV(aiV + 1)
{Xi}
(ajV + 1)
]
ΛiV
ΛiV + (1 - ΛiV)θit
{
(1 - ΛiV)θit
}
ΛiV + (1 - ΛiV)θit
(26)
and for the single-gas isosteric heats of adsorption
[
θit (ΛiV(θit - 2) - θit)ζiV + θit 1 - θit ΛiV(ΛiV + θit - ΛiVθit)2
]
(27)
(19)
and ΛVi ) 1. The problem of calculating ξij is the same as in eq 17. In the case of mixed-gas adsorption, cross-interaction parameters ωij, aij, Λij, ζij and ξij represent the interaction effects between molecules of mixture components. Below we consider the possibility of calculating these parameters. In the case of RAS, cross-interaction parameter ωij is defined by the following equation
aij )
(25)
While summarizing the case of the Wilson coefficients, the use of eq 5 leads to a simpler form of the single-gas isotherm equation
qsti ) qst0 i - ci ln
where
ξij ) k
λij ) (λiiλjj)1/2
exp -
(ΛiV(θit - 2) - θit)ζiV
∂aij
coefficients λii and λjj:
2
Before considering the case of a mixed-gas adsorption system, we want to emphasize the main idea of the presented model of calculations. It consists of the possibility of calculating the isosteric heats of adsorption in the mixed system by analyzing only the single-gas adsorption data. In other words, the theoretical expressions can include best-fit parameters that can be determined from isotherms and isosteric heats of single-gas adsorption. To determine the equation for the isosteric heat of adsorption of a given component in the presence of another component, we use eqs 7 and 14. Differentiation leads to the following expression:
Qsti ) qst0 i -k
∂ ln γsi ∂(1/T)
(
ζij ) Λij(λii - λij)
(23)
ζiV ) ΛiVλii
(24)
and then
because in the single-gas adsorption system λij ) 0 and ξVi ) 0. Finally, in the gas mixture the cross-interaction coefficient λij can be assumed to be a geometric mean of the pure-component
( )]
∂ 1 ln lim ∂(1/T) NΣf0γsi
+
)
∂Πmi ∂ ln γsi Mi - MΣ -1 k (28) - ci ln xsV - k NΣ ∂(1/T) ∂(1/T)
The term ∂/∂(1/T)(limNΣ f0(ln γsi )) occurs in eq 28. To calculate the value of this expression, we can substitute the logarithm of the limit by the limit of the logarithm:
( )
ln lim
(22)
Wilson coefficients Λij and ζij, with i, j ) 1, 2, represent the interactions between neighboring adsorbed molecules. A large number of interaction parameters in the Wilson model of nonideality can be reduced by using eq 5. By substituting eq 5 into eq 19, ζij can be calculated by using the following equation
[ )(
-k
NΣf0
1 ) - lim (ln γsi ) s NΣf0 γi
( )
∂ ln γsi ∂ ( lim (ln γsi )) ) -k lim -k NΣf0 ∂(1/T) ∂(1/T) NΣf0
(29)
(30)
and then
-k
( )
∂ ln γsi ∂ 1 ln lim s ) k lim NΣf0 ∂(1/T) ∂(1/T) NΣf0 γi
(31)
Such mathematical transformations can be verified in the case of RAS when the expression ∂/∂(1/T)(limNΣ f0(ln γsi )) appearing in eqs 28 and 31 can be calculated directly from eq 10. In both cases, it leads to the same result. Therefore, the values of the derivatives (eq 31) are following:
9626 Langmuir, Vol. 22, No. 23, 2006
Nieszporek
1. Regular adsorbed solution (RAS)
k
∂ ( lim (ln γsi )) ) -WiV ∂(1/T) NΣf0
(32)
2. Flory-Huggins activity (FH)
k
ξiVaiV ∂ ( lim (ln γsi )) ) aiV + 1 ∂(1/T) NΣf0
(33)
3. Wilson coefficients (W)
k
ζiV ∂ + ζiV ( lim (ln γsi )) ) ΛiV ∂(1/T) NΣf0
the ideal solution case, and we assumed infinite values of adsorption energies. Such brief assumptions lead to important simplifications: Πmi ) 0 and k(∂Πmi /∂(1/T)) ) 0. While considering the RAS approach, we assumed that all interchange energies Wij are equal to 0, which leads to ln γsi ) 0, limNΣ f0(1/γsi ) ) 1, k(∂ ln γsi /∂(1/T)) ) 0, and k(∂/∂(1/T)) ln(limNΣ f0(1/γsi )) ) 0. Therefore, the theoretical phase diagram (eq 7) takes the following form
{}
piKi exp
0i ) kT
(34)
2.3. Summing Up of Mathematical Considerations. Because of a large number of mathematical expressions presented in the previous sections, for the reader’s convenience we summarize the theoretical considerations: RAS. Equation 1 can be used to analyze the experimental single-gas isotherms, and eq 15 can be used to fit the isosteric heats of single-gas adsorption. As a result, we determine the single-gas best-fit parameters. Then, while studying the mixedgas adsorptions, eq 7 should be used, whereas to calculate the isosteric heats of adsorption of a given component in a mixture eq 28 can be used. NΣ and MΣ appearing in eqs 7 and 28 are defined by eqs 8 and 9, and the activity coefficients and the mole fractions in the vacancy solution are presented in the Appendix. The cross-interaction parameters appearing in γsi can be calculated from eq 20 (assuming an ideal solution). The limits appearing in eqs 7 and 28 are defined by eqs 10 and 32, and the other parameters are determined from single-gas adsorption systems. Therefore, in the RAS we have only five best-fit parameters for each component (Mi, ci/kT, Ki′, ωii, and qst0 i ), which are necessary for theoretical calculations of adsorption phase diagrams and isosteric heats of adsorption of a given component in the mixture system. FH. The pure-gas adsorption systems can be analyzed by eqs 2 and 16. To use eqs 7 and 28, we must calculate the crossinteraction parameters aij and ξij defined by eqs 21 and 22. Finally, the limits appearing in eqs 7 and 28 are defined by eqs 11 and 33. The assumption of infinite values of adsorption energies (i.e., Πmi ) 0) reduces the number of best-fit parameters. While summarizing, in the FH model of nonideality we have six bestfit parameters for each mixture component (Mi, ci/kT, Ki′, aiV, ξiV, and qst0 i ) that are necessary for theoretical calculations of adsorption phase diagrams and isosteric heats of adsorption of a given component in the mixture system. W. The single-gas analysis can be performed by using eqs 26 and 27. On the basis of eqs 7 (adsorption phase diagrams) and 28 (isosteric heat of a given component in the presence of second component), we can calculate the cross-interaction parameters λij (appearing in the expressions for activity coefficients γsi ) by using eq 25. ζij is defined by eqs 23 and 24, and the limits occurring in eqs 7 and 28 are defined by eqs 12 and 34. Therefore, the theoretical prediction of adsorption phase diagrams and isosteric heats of components in the gas mixture requires five best-fit parameters for each mixture component: Mi, ci/kT, Ki′, ΛiV, and qst0 i .
3. Further Remarks The profound analysis of the presented model of calculations shows its unquestionable advantage. Although the presented equations are complicated, some of the simplifications that are applied lead to relatively simple calculations. Namely, we used
{(
)( )}
Mi Mi - MΣ ci XitNΣ ln xs exp -1 MΣ NΣ kT V
(35)
and the theoretical isosteric heat of adsorption (eq 28) simplifies to
(
Qsti ) qst0 i + ci
)
Mi - MΣ - 1 ln xsV NΣ
(36)
In eqs 35 and 36, the interaction parameter ωii does not occur. Thus, this best-fit parameter improves the agreement only between the single-gas isotherms and the single-gas isosteric heats. While considering FH and W models of nonideality, the assumptions of aiV ) 0 and ζiV ) 0 (leading to the elimination of the interaction parameters from eqs 7 and 28) also lead to eqs 35 and 36. The use of eqs 7 and 28 without any simplifications improves the agreement but produces additional best-fit parameters.
4. Calculations Now we examine the theoretical expressions describing singleand mixed-gas adsorption equilibria and heat effects accompanying them shown in previous sections. The theoretical analysis of single-gas isotherms and isosteric heats of single-gas adsorption provides enough knowledge to predict the behavior of a mixed adsorption system. To carry out the numerical exercises, we use the experimental data by Dunne et al.22-24 and Siperstein et al.:25 (C2H6 + CH4) on MFI at 23 °C, (CO2 + C2H6) on NaX (32 °C), and (SF6 + CH4) on MFI. Although our model of calculations requires the same temperature for all single- and mixed-gas experiments, because of small differences between measurement temperatures we neglect them. We used these experimental data in our previous studies,10-13 and one of our conclusions is that the adsorption of C2H6 and CH4 on MFI takes place on the homogeneous surface. The homogeneous character of adsorption processes in the case of MFI justifies the simplification of the theoretical isosteric heats of adsorption
qsti ) qst0 i + ωiiθit qsti
)
qst0 i
(37)
2aiVθit(1 + aiVθit) - (aiVθit)2
- ξiV
qsti ) qst0 i + θit
[
(1 - aiVθit)2 (ΛiV(θit - 2) - θit)ζiV
]
ΛiV(ΛiV + θit - ΛiVθit)2
(38)
(39)
where in comparison with eqs 15, 16, and 27 the logarithmic term with the heterogeneity parameter as a multiplier vanishes. Then, in the case of C2H6 and CH4 adsorption on MFI the isosteric heats were analyzed by eqs 37-39. In the case of the (CO2 + C2H6) adsorption system, our previous numerical studies suggest that strong interaction effects between the adsorbed molecules exist.
Enthalpic Effects Accompanying Mixed-Gas Adsorption
Langmuir, Vol. 22, No. 23, 2006 9627
Table 1. Values of the Parameters Obtained by Applying Equation 1 to the Experimental Isotherms of C2H6 and CH4 Adsorbed on MFI, CO2 and C2H6 Adsorbed on NaX,22-24 and SF6 and CH4 Adsorbed on MFI25 a adsorption system
T (°C)
Mi (mmol/g)
kT/ci
ωii/kT
Ki exp(0i /kT)
linear regression error
qst0 i (kJ/mol)
MFI
C2H6 CH4
23.31 23.07
1.97 1.96
0.991 0.987
0.404 0.304
1.5 × 10-2 6.2 × 10-4
2.2 × 10-5 2.5 × 10-4
31.05 21.17
NaX
CO2 C2H6
31.4 32.4
8.97 3.78
0.802 0.98
-4.488 1.225
5.1 × 10-2 3.5 × 10-3
4.2 × 10-2 1.7 × 10-3
38.98 28.44
MFI
SF6 CH4
1.98 2.35
0.751 0.978
1.265 0.204
1.4 × 10-2 5.0 × 10-4
4.2 × 10-3 3.1 × 10-5
36.00 21.50
∼24 ∼23
a The best-fit values obtained from adopting the theoretical equations for the heat of adsorption (15 and 37) were inserted in the last column. We also present the error values corresponding to fitting quality, defined as the residual sum of the squares.12
For clarity of presentation, we examine the models of nonideality separately. 4.1. Regular Adsorbed Solution. The results of the analysis of isotherms and isosteric heats of single-gas adsorption by eqs 1 and 15 were presented in our previous paper.12 (We studied the ideal adsorbed solution approach). The best-fit parameters obtained then are presented in Table 1. The best-fit parameters collected in Table 1 are sufficient for modeling a mixed-gas adsorption system. It is the master rule of the proposed model of calculations because the single-gas isotherms and calorimetric data are relatively easy to determine and provide complete information necessary to calculate adsorption phase diagrams and the isosteric heats accompanying mixedgas adsorption. While performing calculations of the mixed-gas adsorption equilibria, we must explain a certain property of the analyzed experimental data. Namely, probably because of technical complications, the adsorption phase diagrams and the enthalpic
Figure 1. Adsorption of the (C2H6 + CH4), (CO2 + C2H6), and (SF6 + CH4) gaseous mixtures reported by Dunne et al.22-24 and Siperstein et al.25 Comparison with the experiment (b) of the X-Y composition phase diagrams calculated by applying eq 7 (-) and regular adsorbed solution theory. The theoretical calculated points are hidden to increase the clarity. Y1 is the mole fraction in the gas phase whereas X1 relates to the adsorbed phase. The theoretical phase diagrams were calculated with the assumption of infinite values of adsorption energies (i.e., Πm ) 0).
effects accompanying them were determined at varied total pressure. Therefore, the calculated theoretical adsorption phase diagrams and isosteric heats of adsorption are the collection of points calculated for each mole fraction of the gas phase determined during the experiment. For readability, we connected all theoretical points by a line. It simplifies the comparison of the calculation results with the experiment. The calculations of the adsorption phase diagrams were performed by using eq 7 whereas the isosteric heats of adsorption were performed by using eq 28. For simplicity, we assumed infinite values of the adsorption energies (Πmi ) 0). The equations for the activity coefficients are collected in the Appendix (eqs A1-A5). The results are shown in Figures 1 and 2. Besides the case of isosteric heats of (CO2 + C2H6) adsorption, we can observe satisfactory agreement between the theoretically calculated curves and the experimental data. In all cases, the predicted isosteric heats of the components in the mixture are slightly overstated, which appear in the position of the heat curves on the Y axis. 4.2. Flory-Huggins Activity. As in the case of RAS, the single-gas isotherms and accompanying calorimetric effects were
Figure 2. Comparison of the individual theoretical isosteric heats of adsorption from the gaseous mixtures22-25 (black points) with those calculated theoretically by using eq 28 and regular adsorbed solution theory. Theoretical curves (-) were calculated by using the parameters collected in Table 1. To separate the heat effects of components, a thin, slightly broken line divides the panels.
9628 Langmuir, Vol. 22, No. 23, 2006
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Table 2. Values of the Parameters Obtained by Applying Equation 2 to the Experimental Isotherms of C2H6 and CH4 Adsorbed on MFI, CO2 and C2H6 Adsorbed on NaX,22-24 and SF6 and CH4 Adsorbed on MFI25 a adsorption system MFI NaX MFI
C2H6 CH4 CO2 C2H6 SF6 CH4
T (°C) 23.31 23.07 31.4 32.4 ∼24 ∼23
Mi (mmol/g) 2.29 2.65 6.45 5.59 1.95 2.69
kT/ci 1.000 0.987 0.678 0.999 0.967 0.981
aiV
Ki exp{0i /kT}
linear regression error
qst0 i (kJ/mol)
ξiV
0.2 0.3 1.3 0.2 0.2 0.1
1.3 × 10 4.6 × 10-4 3.9 × 10-2 2.8 × 10-3 2.8 × 10-2 4.4 × 10-4
3.9 × 10 2.4 × 10-5 1.1 × 10-2 9.4 × 10-3 6.3 × 10-4 3.4 × 10-6
30.85 20.93 38.54 26.25 34.86 21.29
-4.95 -2.15 0.00 -26.71 -10.22 -8.03
-2
-4
a The best-fit values obtained by applying the theoretical equations for the heat of adsorption (16 and 38) to the experimental data were inserted in the last two columns. We also present the error values corresponding to fitting quality, defined as the residual sum of the squares.
Figure 3. Adsorption of the (C2H6 + CH4), (CO2 + C2H6), and (SF6 + CH4) gaseous mixtures reported by Dunne et al.22-24 and Siperstein et al.25 Comparison with the experiment (b) of the X-Y composition phase diagrams calculated by applying eq 7 (-) and the FloryHuggins model of nonideality.
Figure 4. Comparison of the individual theoretical isosteric heats of adsorption from the gaseous mixtures22-25 (black points) with those calculated theoretically by using eq 28 and the Flory-Huggins model of nonideality. Theoretical curves (-) were calculated by using the parameters collected in Table 2.
examined in the previous paper26 by using eqs 2 and 16. The results of that numerical exercise are presented in Table 2. It is important to remember that all single-gas isosteric heats of adsorption except that for carbon dioxide were examined by a homogeneous version of eq 16 (i.e., by eq 38). This is due to the fact that only in the case of CO2 adsorption does heterogeneity parameter kT/ci differ distinctly from 1. Applying the values of the best-fit parameters collected in Table 2, we calculated the adsorption phase diagrams (Figure 3) and the isosteric heats of adsorption of the components adsorbed in a mixed system (Figure 4). While comparing Figures 1 and 3, we can find similar, poor agreement of the theoretical curves with experiment, but the use of the Flory-Huggins activity coefficients leads to a visibly better theoretical description of the heat effects. Particularly in the case of CO2 adsorption on NaX (in the presence of C2H6), the shape of the theoretically calculated isosteric heat of adsorption projects the experimental data. 4.3. Wilson Activity. First we examined the experimental data by using the single-gas isotherm (eq 26). In the case of adsorption of pure C2H6 and CH4 on MFI, the values of hetero-
geneity parameter kT/ci are almost 1. Therefore, in that case while examining the isosteric heats of single gases we use eq 39 (where in comparison to eq 27 the logarithmic term multiplied by the heterogeneity parameter vanishes). While adjusting the theoretical isosteric heat of adsorption (eq 27 or 39), there are two possibilities: one is to treat ζiV as the best-fit parameter and the other is to calculate ζiV by using eq 23 (i.e., by utilizing the ΛiV values). The second case decreases the number of best-fit parameters used in the proposed theoretical model. The common numerical technique used to adjust the experimental isotherms is to draw them in logarithmic coordinates. In the case of the Wilson model, the isotherms can be fitted by the following logarithmic form of eq 26:
ln
(
) [
]
Nit/M ΛiV + ln 1 - Nit/M ΛiV + (1 - ΛiV)Nit/M (1 - ΛiV)Nit/M ΛiV + (1 - ΛiV)Nit/M
)
(
)
0i kT kT ln pi + ln Ki + (40) ci ci kT
The fitting procedure of eq 40 to the experimental isotherms
Enthalpic Effects Accompanying Mixed-Gas Adsorption
Figure 5. Fitting procedure of eq 26 (-) to the experimental data (b, 2) reported by Dunne and co-workers22-24 for C2H6 and CH4 adsorbed on silicalite, for CO2 and C2H6 adsorbed on NaX, and for the data reported by Siperstein et al.25 (SF6 and CH4 adsorbed on MFI).
consists of suitable choices of ΛiV and Mi so that the plot of the left-hand side of eq 40 versus ln pi will be linear (the minimalization of the residual sum of squares). Then, from the linear regression parameters the values of kT/ci and Ki exp{i0/kT} can be calculated. While looking at eq 26, it can be seen that because of the complicated mathematical form the above-described procedure is inconvenient. Here we want to propose another way to determine the adsorption parameters. Namely, while looking at eq 26 it can be seen that in this mathematical form adsorbate pressure pi is a function of adsorbed amount pi ) pi(Nit). We propose to draw the experimental isotherms in the reverse coordinates (i.e., pi vs Nit) and adjust them directly using eq 26 and suitably choosing best-fit parameters. This procedure should lead to a precise determination of the best-fit parameters, especially in the case of maximum adsorbed amount Mi. (In the case of linear regression, the Mi values are frequently overstated.) Figure 5 shows the theoretical isotherms fitted to the experimental data in the way described here.
Langmuir, Vol. 22, No. 23, 2006 9629
Figure 6. Comparison of the theoretical isosteric heats of adsorption of pure components (eq 27) (-) with the analyzed experimental data.22-25 The dashed lines are the case when ζiV values are calculated by using eq 23.
Figure 6 shows the comparison between the theoretically calculated and the experimentally measured isosteric heats. Table 3 collects the set of best-fit parameters whose values of qst0 i in parentheses correspond to the case when ζiV values are calculated. Figures 5 and 6 show excellent agreement between the theoretical curves and the experimental data. Because we analyzed the experimental data reported by Dunne et al.22-24 and Siperstein et al.25 by various theoretical models describing single-gas adsorption equilibria and the accompanying enthalpic effects, we can state that it is one of the best theoretical descriptions of the isosteric heats of adsorption. Simultaneously, slightly worse agreement of the theoretical curves calculated when eq 23 was applied (dashed lines) can be seen in Figure 6. On the basis of the values of the best-fit parameters, we can perform the calculations for mixed systems. These results are shown in Figures 7 and 8. One can state that in the case of (CO2 + C2H6) the agreement between the theoretical curves and the
Table 3. Values of the Parameters Obtained by Applying Equation 26 to the Experimental Data22-25
a
Ki exp{0i /kT}
qst0 i (kJ/mol)
ζiV
MFI 0.821 0.932
1.439 × 10-2 5.727 × 10-4
30.78 (31.43) 20.93 (21.11)
-0.84 -0.54
0.516 0.872
NaX 0.733 0.374
9.473 × 10-3 1.920 × 10-3
40.78 (37.66) 19.79 (28.89)
2.01 -3.87
0.871 0.989
MFI 0.810 0.998
2.248 × 10-2 5.858 × 10-4
30.70 (38.97) 21.29 (21.54)
-5.53 0.687
adsorption system
T (°C)
Mi (mmol/g)
kT/ci
C2H6 CH4
23.31 23.07
1.987 2.17
0.988 0.990
CO2 C2H6
31.4 32.4
7.32 4.023
SF6 CH4
∼24 ∼23
1.96 2.25
ΛiV
a The best-fit values obtained by applying the theoretical equations for the heat of adsorption (27 or 39) to the experimental data were inserted in the last two columns. The values of qst0 i in parentheses correspond to the case when ζiV values were calculated by using eq 23.
9630 Langmuir, Vol. 22, No. 23, 2006
Nieszporek
experimental data fails completely. It can be seen in Figure 7 that the theoretical adsorption phase diagram of (CO2 + C2H6) is completely different from the experiments that can probably be attributed to the existence of strong interaction effects in the mixed system between CO2 and C2H6 molecules or the assumption of infinite limits of the adsorption energies. When ζiV values were calculated by using eq 23, a slightly better theoretical description was obtained. Meanwhile, for that time in the case of the other adsorption systems (i.e., (C2H6 + CH4) and (SF6 + CH4)), the obtained results deserve citation. When eq 23 was used to calculate ζiV (dashed lines), a better theoretical description of the isosteric heats of adsorbed components can be observed. Especially in the case of the (C2H6 + CH4) system the obtained results are excellent.
5. Conclusions
Figure 7. Adsorption of the (C2H6 + CH4), (CO2 + C2H6), and (SF6 + CH4) gaseous mixtures.22-25 Comparison with the experiment (b) of the X-Y composition phase diagrams calculated by applying eq 7 (-) and the Wilson model of nonideality. Theoretical curves (-) were calculated by using the parameters collected in Table 3.
While summarizing our considerations, we can point to relatively good results in the application of our theoretical expressions. Besides heterogeneity, the interaction effects are taken into account. By using some assumptions, the theoretical adsorption phase diagrams and isosteric heats of a given mixture component reduce to simple mathematical expressions. The theoretical studies performed by us previously11-13 when we studied other theories of mixed-gas adsorption equilibria (IE, IAS) indicate that the isosteric heat of adsorption in the mixed system always is the logarithmic function of the surface coverage. Another interesting fact is that all information necessary to calculate the adsorption phase diagrams and isosteric heats of adsorption in the mixed system originate from the analysis of pure-gas adsorption systems. The set of best-fit parameters determined by using, for example, eqs 1 and 15 can be utilized by various theories of mixed-gas adsorption equilibria. One of them besides the integral equation approach and the ideal adsorbed solution theory is the vacancy solution theory studied in this article.
Appendix Regular Adsorbed Solution (RAS):12
kT ln γs1 ) W12(xs2)2 + W1V(xsV)2 + (W12 + W1V - W2V)xs2 xsV (A1) kT ln γs2 ) W12(xs1)2 + W2V(xsV)2 + (W12 + W2V - W1V)xs1 xsV (A2) kT ln γsV ) W1V(xs1)2 + W2V(xs2)2 + (W1V + W2V - W12)xs1 xs2 (A3) After differentiation over (1/T),
k Figure 8. Comparison of the individual theoretical isosteric heats of adsorption from the gaseous mixtures22-25 (black points) with those calculated theoretically by using eq 28 and the Wilson model of nonideality. Theoretical curves were calculated by using the parameters collected in Table 3. Solid lines indicate the case when ζiV values are the best-fit parameters whereas dashed lines denote the case when ζiV values were calculated by using eq 23 (Table 3, values of qst0 i in parentheses).
k
( ) ∂ ln γs1 ∂(1/T)
{Xi}
( ) ∂ ln γs2 ∂(1/T)
{Xi}
) W12(θ2t)2 + W1V(1 - θ1t - θ2t)2 + (W12 + W1V - W2V)θ2t(1 - θ1t - θ2t) (A4) ) W12(θ1t)2 + W2V(1 - θ1t - θ2t)2 + (W12 + W2V - W1V)θ1t(1 - θ1t - θ2t) (A5)
Enthalpic Effects Accompanying Mixed-Gas Adsorption
Langmuir, Vol. 22, No. 23, 2006 9631
Flory-Huggins Model (FH):26
(
ln γs1 ) -ln xs1 +
xs2
+
a12 + 1
[ ( 1-
ln
γs2
(
) -ln
xs1
a21 + 1
+
+
[ ( 1-
(
ln γsV ) -ln
xs1
aV1 + 1
+
xsV a1V + 1
xs1
xs2
lnγs2 ) 1 - ln[xs1Λ21 + xs2 + xsVΛ2V] -
)
[
)]
a2V + 1
)
aV2 + 1
[ ( 1-
(A6)
[
aV1 + 1
+
x1sΛ1V
)]
) xs1(xs2ζ12 + xsVζ1V)
(xs1 + xs2Λ12 + xsVΛ1V)
(A8)
k
( ) ( [( ∂ ln
∂(1/T)
)
{Xi}
xs1 + xs2Λ12 + xsVΛ1V
xs2 xsV + + a12 + 1 a1V + 1
(
xs1
( ) ( [( ∂ ln γs2 ∂(1/T)
)
{Xi}
(xs2 + xs1Λ21 + xsVΛ2V)2
)
xsVζV1
-1
xsV + xs1ΛV1 + xs2ΛV2
-
)]
xsV xs2 + + a12 + 1 a1V + 1
-2
(A9)
∂(1/T) -
)
xs1ξ21
xs3ξ2V + (a21 + 1)2 (a2V + 1)2
xsV xs1 s + x2 + a21 + 1 a2V + 1
(
)
(A14)
)
{Xi}
xs1ζ12
+
xs1 + x2sΛ12 + xsVΛ1V
xs1Λ12(xs2ζ12 + xsVζ1V) (xs1 + xs2Λ12 + xsVΛ1V)2
+
-
-1
xs1ζ21 + xsVζ2V
-
)]
xs2 + xs1Λ21 + xsVΛ2V
-2
xsVζV2
(A10)
xsV + xs1ΛV1 + xs2ΛV2
+
-
xsVΛV2(xs1ζV1 + xs2ζV2) (xsV + xs1ΛV1 + xs2ΛV2)2
(A15)
The mole fractions in the vacancy solution formalism can be calculated from the following relations:
lnγs1 ) 1 - ln[xs1 + xs2Λ12 + xsVΛ1V] -
xs1 + xs2Λ12 + xsVΛ1V
(xsV + xs1ΛV1 + xs2ΛV2)2
(xs2 + xs1Λ21 + xsVΛ2V)2
Wilson Model (W):
xs1
-
xsVΛV1(xs1ζV1 + xs2ζV2)
+
xs2(xs1ζ21 + xsVζ2V)
xsV xs1 + xs2 + a21 + 1 a2V + 1
[
( ) ∂ ln γs2
-
+
xs2Λ21(xs1ζ21 + xsVζ2V)
)
xsVξ1V
+ (a12 + 1)2 (a1V + 1)2
xs1
k
xs2ζ12 + xsVζ1V
2
xs2ζ21
and
xs2ξ12
(A13)
{Xi}
xs2 + xs1Λ21 + xsVΛ2V γs1
]
and
-1
+
(A12)
+
x1sΛV1 + xs2ΛV2 + xsV
(A7)
∂(1/T)
aV2 + 1
xs1Λ21 + xs2 + xsVΛ2V
-1
( )
xsV
xs2Λ2V
+
xsV
∂ ln γs1
xs2
]
ln γsV ) 1 - ln[xs1ΛV1 + xs2ΛV2 + xsV] -
+ xsV +
xs1
xs1ΛV1 + xs2ΛV2 + xsV
+
)]
+
xsVΛV2
xs1 + xs2Λ12 + xsVΛ1V
)
xs1Λ21 + xs2 + xsVΛ2V
-1
xsV xs1 + xs2 + a21 + 1 a2V + 1
xs2
xs2
+
xs1 + xs2Λ12 + xsVΛ1V
+
xs2 xsV + + a12 + 1 a1V + 1
xsV
xs1Λ12
+
xsi ) Nit xs2Λ21
xs1Λ21 + xs2 + xsVΛ2V
[
]
N1t + N2t N1tM1 + N2tM2
i ) 1, 2 and xsV ) 1 - xs1 - xs2 (A16)
+
xsVΛV1 xs1ΛV1 + xs2ΛV2 + xsV
and
]
xs1 ) θ1t (A11) LA061847X
xs2 ) θ2t
xsV ) (1 - θ1t - θ2t)
