Langmuir 1985,1,97-100
97
Correlation of Adsorption Equilibrium Data Using a Modified Antoine Equation: A New Approach for Pore-Filling Models John J. Hacskaylo and M. Douglas LeVan* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22901 Received June 26,1984 The Antoine equation for the pure component vapor pressure of saturated liquid is modified to correlate adsorption equilibrium data for porous adsorbents by letting the constants depend on loading. Simple forms are proposed that have a Henry’s law limit at low loading and are consistent with vapor-liquid equilibrium at complete filling of the pore volume. Data for light hydrocarbons adsorbed on activated carbon are analyzed. In most cases a simple two-parameter form correlates data with lower variance,than the three-parameter Dubinin-Astakhov equation.
Introduction The number of isotherm equations that are suitable for describing the adsorption of vapors on porous adsorbenta over the full concentration range, from trace values to saturation, and over a wide temperature range is small. For condensables, equations containing a monolayer capacity that is reached only in the limit of an infinite pressure are inappropriate. The current pore-filling models are based on the Polanyi potential theory with 8 given by an explicit function of 6 , where
e w/wo
(1)
R T In (p,/p)
(2)
6 3
8 is the fractional filling of the pore volume, W the volume of adsorbate per unit mass of adsorbent, W, the pore volume per unit mass of adsorbent, E the adsorption potential, R the gas constant, T the absolute temperature, pathe pure component vapor pressure, and p the pressure of the adsorbed species. Popular forms are the DubininRadushkevich (D-R) equation’
0 = ex~[-(~/(BE))~l
(3)
which contains two parameters ( Wo,BE), and the more general Dubinin-Astakhov (D-A) equation2J
e = exp[--(~/(BE))”l
(4)
which contains three parameters (Wo,BE, n). PE is the product of a scaling factor 0 and an energy of adsorption E; n is related to the pore-size distribution. These relations are well-behaved near €3 = 1, where the pressure of the vapor over the adsorbent becomes equal to the pure component vapor pressure. The principal deficiency of these equations is their behavior in the limit of zero loading. A finite Henry’s law slope is not reached and the heat of adsorption becomes unbounded. Similarities between vapor-adsorbate and vapor-liquid equilibrium have been noted previ~usly.~J’As for purecomponent vapor pressure curves, adsorption isosteres when plotted as In p vs. T’ are approximately straight (1) Dubinin, M.M.;Radushkevich, L. V. Dokl. Akad. Nauk SSSR 1947, 55, 327. (2) Dubinin, M. M.; Astakhov, V. A. Izv. Akad. Nauk SSSR, Ser. Khim. 1971, 1, 5. (3) Dubinin, M. M. Carbon 1979, 17, 505. (4) Brunauer, S. ”The Adsorption of Gases and Vapors”; Princeton Univ. Press: Princeton, NJ, 1945; Vol. 1. (5) deBoer, J. H. “The Dynamical Character of Adsorption”; Interscience: New York, 1964.
0743-7463/85/2401-0097$01.50/0
lines. Friday and LeVan6 have taken advantage of this behavior to develop a correlation, using a modified Antoine equation, for water vapor adsorbed on 4A zeolite. The coefficients in the Antoine equation were allowed to depend on loading. At complete saturation of the adsorbent the correlation becomes the traditional Antoine equation for the pure component vapor pressure of water. This paper examines the suitability of correlating adsorption equilibrium data over wide ranges of temperature and pressure for light hydrocarbons adsorbed on activated carbon using a modified Antoine equation. Simple forms of such an equation are proposed based on the behavior of adsorption equilibria in the two limits, low loading and complete filling of the pore volume, and on the variation of heat of adsorption with loading. The forms are used to analyze data of Szepesy and Il16s7 and Reich et a1.8 Results are compared with correlations based on the Dubinin-Radushkevich and Dubinin-Astakhov equations.
Theory We consider a pore that can be filled with adsorbate to different levels as shown in Figure 1. When the pore is completely filled with adsorbate, corresponding to 8 = 1, the pressure of the adsorbed component in the vapor space above the pore is the pure component vapor pressure of the adsorbate as liquid at the adsorption temperature. This vapor pressure can be calculated as a function of temperature from any of several relations such as the traditional Antoine equation B lnp,=A-(5) C+T where A , B, and C are constants for a particular species. For a partially filled pore, the pressure over the adsorbent will be a function of both 8 and T. We will assume that the pressure can be described by the functional form of the vapor pressure equation with the constants allowed to take on different values for each 8. Thus, for the Antoine equation we write D’
u
l n p = A’- C’+ T where A’, B’, and C’depend on 0. Because adsorption isotherms are continuous functions relating 0 and p for (6) Friday, D.K.;LeVan, M. D. AIChE J. 1982,28, 86. (7) Szepesy, L.; Ill&, V. Acta Chim. Hung. 1963,35,37, 53. (8) Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 336.
0 1985 American Chemical Society
98 Langmuir, Vol. 1, No. 1, 1985
Hacskaylo and LeVan Table I. Physical Properties of Adsorbed Species
vapor pressuren methane ethylene ethane propylene propane butane “quation
A
B
C
6.49447 6.60633 6.80196 6.68443 6.80329 6.75547
933.5 1369.6 1528.3 1774.9 1872.5 2154.9
-5.37 -16.99 -16.47 -27.64 -25.11 -34.42
5, p in MPa, T in K. b p = do
liquid densityb d , X lo3 d, X lo5 -14.00 10.42 3.683 -3.579 -1.619 0.455 0.763 -2.565 0.327 -1.518 -1.235 0.188
dn X 10 10.80 5.658 7.710 9.573 8.460 8.771
dn X lo8 -28.54 8.333 -1.287 -1.378 -0.722 -0.390
+ d,T + d 2 P + d,P; p in g/cm3, T i n K.
[
p = PAT)=exp A -
Dependences of heat of adsorption on loading can now be proposed and used to construct adsorption equilibrium relations. For a heat of adsorption that is independent of loading we can write simply
CYT]
A’=A+ln8
B’=B
C’=C
(10)
This is the linear isotherm 8 = p / p , and contains one parameter ( Wo),If we allow for a linear variation of the heat of adsorption with loading we obtain A’=A+ln8
Figure 1. Depiction of
pore-filling
= -RP-
a
model.
kplq
C’=C
(11)
which contains two parameters (Wo,b). A similar form with a more complicated dependence for the heat of adsorption on loading is
particular temperatures, we expect A’, B’, and C’ to be continuous functions of 8 for 8 I1 and at 8 = 1 to be equal to the corresponding constants in eq 5. An important assumption has been made here, namely, that pressures at constant 8 for vapor-adsorbate equilibrium follow approximately the same empirical temperature dependence as pure component vapor pressures for vapor-liquid equilibrium. The isosteric heat of adsorption is given with little approximation by the Clausius-Clapeyron equation
x
B’=B+b(l-8)
(7)
where q, quantity adsorbed, is held constant in taking the derivative. We follow Ross and Olivierg and replace the derivative taken at constant q in eq 7 by a derivative at constant 8. This is a reasonable assumption because, compared to the pressure over an adsorbent, the adsorbate density is only a weak function of temperature.’O Then, from eq 6, we obtain
Thus, it follows from the use of the vapor pressure equation that isosteric heats of adsorption for vapor-adsorbate equilibrium have approximately the same empirical temperature dependence as heats of condensation for vaporliquid equilibrium. In the limit of zero loading, no competition exists among molecules for adsorption sites. An isotherm equation must approach 0 = Kp or lnp=ln0-lnK (9) where K = K ( T ) is the Henry’s law slope. Comparing eq 6 and 9 indicates that A’ must contain the In 8 term. Furthermore, B’and C’must be bounded as 8 approaches zero in order for the heat of adsorption to behave properly. (9) Ross, S.;Olivier, J. P. “On Physical Adsorption”; Interscience: London, 1964. (10) Suwanayuen, S.;Danner, R. P. AlChE J. 1980, 26,68.
A‘= A
+ In 8 B’= B + b ( 1 - 8 ) c’ = c + c ( i - e)
(12)
which contains three parameters (Wo,b, c). More complicated dependences of A’, B‘, and C’on 8 are of course possible involving expansions in higher order polynomials of 1 - 0 and nonlinear dependences.
Results We examine here the data of Szepesy and Ill&’ for methane, ethylene, ethane, propylene, propane, and butane adsorbed on Nuxit-A1 activated carbon and the data of Reich et a1.8 for methane, ethylene, and ethane adsorbed on type BPL activated carbon. Various schemes, all subject to uncertainty, have been proposed to evaluate the density of the adsorbed phase for pore-filling models. Because some of the data that we analyze are for adsorption temperatures that exceed the vapor-liquid critical temperature of the adsorbed species, we have adopted the method of Lewis et al.,” which was also used by Szepesy et a1.12 to analyze their data. The density of the adsorbed phase is assumed to be equal to the density of the adsorbed species as saturated liquid evaluated at the temperature at which the pressure of the adsorbed species would be its pure component vapor pressure. Thus for a given pressure, eq 5 is solved for the temperature at which to evaluate the density of the adsorbed phase. Data for liquid densities for appropriate temperature ranges (93-189 K for methane, 153-293 K for butane) were obtained13 and fit to a cubic polynomial. Values for the coefficients are given in Table I. Also shown in Table I are the Antoine coefficients, taken from a standard source14and converted to SI units, that were used to obtain our correlations. (11) Lewis, W.K.;Gilliland, E. R.; Chertow, B.; Cardigan, W. P. Znd. Eng. Chem. 1960,42, 1326. (12) Szepesy, L.; Ill&, V.; Benedek, P. Acta Chim.Hung. 1963,35,433. (13) American Petroleum Institute Research Project 44, “Selected Values of Properties of Hydrocarbons and Related Compounds”,Texas A&M University, College Station, TX. (14) Dean, J. A., Ed. “Lange’s Handbook of Chemistry”, 12th ed.; McGraw-Hill: New York, 1979.
Langmuir, Vol. 1 , No. 1, 1985 99
Correlation of Adsorption Equilibrium Data
6 I
I
5 0
40
4.0
2
r cn
-2 3O \
\
5 3.0 E w
u
20
293.15'K 313.15'K 333.15'K 363.15'K
0 0 0
IO
0
0 0 0
0
0'1
03
02
04
0'5
293.15OK 313.15'K 3 3 3 15°K 363 15°K
0'6
017
0.8
p (MPa)
Figure 4. Data of Szepesy and 1116s for propane adsorbed on Nuxit-Al activated carbon. Solid lines are plots of eq 11 with Wo = 445 cm3/kg and b = 1633.8.
1
1.
5.0
7.5
I
0
I
I
I
02
04
06
I
I
00
t
1 1
10
I
0
I
I
0.5
1.0
1.5
2.0
p (MPa)
Values of W,,the pore volume, were obtained by plotting characteristic curves and noting the value of W at saturation. We obtained 445 cm3/kg for the data of Szepesy and 1116s and 435 cm3/kg for the data of Reich et al. Nonlinear regressions based on a least-squares criterion were then performed on all of the data for each adsorbate-adsorbent pair to obtain best fits for the parameters appearing in eq 11,12,3 (D-R), and 4 (D-A). The following variance was defined 1 N
-C(A In p ) 2
Ni
I
I
I
3.0
3.5
4.0
p (MPa)
Figure 3. Data of Szepesy and Ill& for propylene adsorbed on NuxibAl activated carbon. Solid lines are plots of eq 11 with W, = 445 cm3/kg and b = 1565.9.
var
I
2.5
Figure 5. Data of Raich et al. for methane adsorbed on typeBPL activated carbon. Solid lines are plots of eq 11 with W , = 435 cm3/kg and b = 883.5.
7.51/*
6.0
(13)
where N is the number of data points for a particular system and A In p is the difference between a measured and a calculated value of In p . The use of logarithms in the definition of variance assures that percentage differences in p will be roughly constant over the entire range of partial pressures. Newton's method was used to minimize the variance. The results of the nonlinear regressions are shown in Table 11. Note the quality of the fits as measured by the variances. Comparing the two-panmeter equations, eq 11 is found to give variances lower than those for the Dubinin-Radushkevich equation in all cases by roughly an order of magnitude. In fact, eq f l gives variances lower than those for the three-parameter Dubinin-Astakhov equation in six of the nine cases considered. Comparing the three-parameter equations, eq 12 gives variances lower
* , 0.4
I
I
saturation
0.8
1 I
1
1.2
I.6
2.0
p (MPa)
Figure 6. Data of Reich et al. for ethane adsorbed on type BPL activated carbon. Solid lines are plots of eq 11 with Wo = 435 cm3/kg and b = 1214.6. than those for the Dubinin-Astakhov equation for all but one case, the methane data of Reich et al. General trends in the behavior of the parameter b can be noted. The ratio bf B is roughly constant for the various adsorbate-adsorbent pairs indicating a relatively uniform variation with 8 of the ratios of heat of adsorption to heat of condensation.
100 Langmuir, Vol. 1, No. 1, 1985
Hacskaylo and LeVan Table 11. Variances and Correlation Parameters eq ll/eq 12
methane" ethylene" ethanea propylene" propanea butanea methaneb ethyleneb ethaneb
var 0.011 0.0054 0.024 0.0052 0.0054 0.0036 0.024 0.014 0.020 0.018 0.13 0.13 0.038 0.038 0.072 0.035 0.016 0.015
b 867.0 578.2 1147.8 910.0 1223.3 1031.9 1565.9 1184.8 1633.8 1446.6 1825.5 1833.6 883.5 893.2 1129.7 688.7 1214.6 1150.1
eq 3 (D-R)/eq 4 (D-A) C
-51.5 -49.3 -23.8 -40.2 -18.8 0.7 1.5 -45.2 -6.3
var 0.16 0.025 0.15 0.0060 0.14 0.0070 0.27 0.037 0.26 0.032 0.49 0.15 0.13 0.035 0.49 0.071 0.31 0.052
RE, kJ/mol 8.732 6.786 9.678 8.724 10.019 9.201 11.352 11.198 11.683 11.421 12.927 12.697 7.076 6.783 7.893 7.967 8.441 8.581
n 1.387 1.448 1.498 1.443 1.490 1.435 1.506 1.219 1.354
OData of Szepesy and 1116s; Wo = 445 cm3/kg. bData of Reich et al.; Wo = 435 cm3/kg.
Our fits for the ethane, propylene, and propane data of Szepesy and 1116s and the methane and ethane data of Reich et al. using the two-constant eq 11 are shown in Figures 2-6. Predicted values of q are calculated from q = W,/(MW) where p is adsorbate density and MW is molecular weight. Points at which a saturated vapor phase is formed occur in Figures 3 and 6 where they are marked with an asterisk. Since W, is a property only of the adsorbent, only one parameter specific to an adsorbate-adsorbent pair is contained in the fitting equation. Nevertheless, the data are predicted with reasonable accuracy over the full ranges of temperatures.
Discussion We have shown that a traditional vapor pressure equation can be modified to treat vapor-adsorbate equilibrium by letting the constants depend on loading. Proper limits can be imposed on the allowable variations of the constants. Implicit in our approach is the assumption that temperature dependences for vapor-adsorbate equilibria are similar to those for vapor-liquid equilibria. Our approach can be implemented to correlate data over wide ranges of temperature and pressure using very few fitted parameters. We have considered here parameters that affect the heat of adsorption. Our fits are generally better in a statistical sense than those obtained from the Dubinin-Radushkevich and Dubinin-Astakhov equations. Effects of interfacial tension have been neglected throughout our approach as is generally the case for pore-filling models. The principal contribution to the total pore volume of activated carbons designed for vapor-phase applications occurs in pores about 2 nm in diameter. The Kelvin equation has been found not to apply to pores smaller than 8 nm in diameter.15 According to Gregg and Sing,lGthe plateau of a type I isotherm may cut the p = p 8 vertical line sharply or may (15) Fisher, L. R.; Israelachvili, J. N. J. Colloid Interface Sci. 1981, 80, 528. (16) Gregg, S. J.; Sing, K. S. W. 'Adsorption, Surface Area, and Porosity", 2nd ed.; Academic Press: New York, 1982.
show a tail as the saturation pressure is approached, depending on the pore structure of the adsorbent. The data here show no evidence of a tail. However, it should still be possible to correlate data that do show a tail by using a modified Antoine equation, although more complicated dependences of A'and B'on 8 would be required. Compared to the plateau, the tail requires a larger change in 8 to produce an equivalent change in p , indicating that near 8 = 1the primed variables will be only weak functions of 8. Furthermore, if the isotherm joins the p = p s line smoothly with dp/d8 = 0 at 8 = 1,then it is easily shown by differentiating eq 6 that derivatives of A', B', and C' with respect to 8 must be zero at 8 = 1. Two points should be noted concerning eq 10-12. First, A'affects the linearity of isotherms but not the isosteric heat of adsorption. For example, adding a(1- 8 ) to A in eq 10 gives nonlinear isotherms along which the isosteric heats of adsorption are constants. Second, even an equation as simple as eq 11, with a proper Henry's law slope and linear variation of heat of adsorption with loading, cannot be rearranged to the explicit closed form 8 = 8(p,T),the traditional representation of an isotherm equation. The prediction of the density of the adsorbed phase remains a problem for pore-filling models. Several methods have been proposed but none is clearly superior. We have analyzed all of the data of Szepesy and 1116s' and Reich et a1.8 except for data on acetylene7 and carbon di~xide.~B These components undergo liquid-solid phase transitions within the temperature ranges needed to apply the method of Lewis et al." Applying the method to these data would give unrealistic discontinuities in the densitites of the adsorbed phases. With the exception of this discontinuity, however, which implies a sudden rather than a gradual change in the nature of the adsorbed phase, the method of Lewis et a1.l1 gives trends that seen to be qualitatively correct. Furthermore, the method avoids the difficulty, common to other methods, of estimating the density of the adsorbed phase at temperatures exceeding the vapor-liquid critical temperature of the adsorbed component.
