Langmuir 2001, 17, 5503-5507
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A Simple Isotherm Equation for Modeling the Adsorption Equilibria on Porous Solids over Wide Temperature Ranges† Li Zhou* and Junshe Zhang Hydrogen Energy Research Center, Tianjin University, Tianjin 300072, China
Yaping Zhou Department of Chemistry, Tianjin University, Tainjin 300072, China Received January 3, 2001. In Final Form: April 10, 2001 A new isotherm equation is derived from the distribution of energetic heterogeneity of adsorbent surface. It is simple in form and is physically compatible with the mechanism of monolayer adsorption. The proposed equation is used to model the experimental adsorption equilibrium of methane on activated carbon for the range of 158-333 K. It successfully represents the whole set of data for both sides of the critical temperature. Compared with the Langmuir-Freundlich equation, it yields a parameter of saturation adsorption, which is in agreement with the fictitious limiting adsorption defined by the intersection of linear isotherms.
Introduction A mathematical model is usually proposed to describe the isotherms in order to understand the nature of the adsorption equilibrium between gases and solids. Following the pioneering works of Polanyi1 and Langmuir,2 a wide variety of isotherm equations have been developed.3 However, because of the complexity of the interactions between gas molecules and solid surfaces, studies on pertinent models are still being actively pursued.4-8 Some isotherm equations are semiempirical; others are deducted from fundamental theoretical reasoning. The physical meaning of any parameter in a theoretical equation is definite, which is advantageous over the semiempirical ones. However, a theoretical isotherm sometimes loses the simplicity in form and fails to explain highly complicated practical cases because of the assumptions made under its deduction. It is more difficult to represent adsorption equilibria over a wide range of temperatures by a single isotherm equation. The mechanism assumed for subcritical conditions may not be valid for supercritical temperatures, as is shown by the adsorption of nitrogen on mesoporous silica gel.9 The experimental isotherms on * To whom correspondence should be addressed. Tel: 86 22 87891466. Fax: 86 22 87891466. E-mail:
[email protected]. † This work was accomplished in the High Pressure Adsorption Laboratory, Hydrogen Energy Research Center, Tianjin University, Tianjin 300072, P R China. (1) Polanyi, M. Verh. Dtsch. Phys. Ges. 1914, 16, 1012; 1916, 18, 55. (2) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (3) Adamson, A. W. Physical Chemistry of Surfaces; John Wiley & Sons: New York, 1976. (4) Jovanovich, D. S.; Kolloid, Z. Z. Polymer 1969, 235, 1203. (5) Dubinin, M. M.; Astakhov, V. A. Development of the Concepts of Volume Filling of Micropores in the Adsorption of Gases and Vapors by Microporous Adsorbents. Translated from Izv. Akad. Nauk SSSR, Ser. Khim. 1971, 1, 5-11. (6) Jaroniec, M. Colloid Polym. Sci. 1976, 254, 601. (7) Subramanian, R.; Pyada H.; Lira, C. T. An engineering model for adsorption of gases onto flat surfaces and clustering in supercritical fluids. Ind. Eng. Chem. Res. 1995, 34, 3830. (8) Do, D. D.; Wang, K. Predictions of Adsorption Equilibria of Nonpolar Hydrocarbons onto Activated Carbon. Langmuir 1998, 14, 7271-7277. (9) Yang B. An Experimental Study on the Adsorption Behavior of Gases Crossing Critical Temperature. M.S. Thesis, Tianjin University, China, January 2000.
microporous adsorbents share the same shapes below or above the critical temperature. Both of them belong to type I, though a maximum may appear at high pressures for supercritical isotherms. The mechanism underlying the type I isotherms could be assumed to be a process of monolayer surface coverage, on the basis of which a new isotherm equation is proposed. The new isotherm equation bears merits of simplicity in form and applicability for all type I isotherms on both sides of the critical temperature. The experimental adsorption equilibrium data of methane on activated carbon at 158-333 K are used to verify the proposed isotherm equation. Critical temperature (190.6 K) is included in the range. A comparison with the Langmuir-Freundlich (L-F) equation is made. Deduction of the Isotherm Equation. Porous solids such as activated carbon typically have irregular pores making it practically impossible to account for all the details of the pore structure. Half width, r, of a pore is usually used to identify pore size in the literature. Although there are other factors affecting adsorption, the decisive one is the intensity of the interaction between the adsorbate molecules and the surface atoms, which is basically determined by the pore size for a given adsorbate species. It is known from the Lennard-Jones potential function that the process of surface coverage must begin with the smallest pores accessible to the adsorbate molecules and end up with the largest ones, i.e., the flat surface. Consequently, there should be a critical pore size, rc, for a given equilibrium condition (T, p) under which all pores with r e rc have been covered, but those with r > rc have not been covered. Therefore, the fractional surface coverage under a given equilibrium condition could be determined by the pore size distribution (PSD) of the adsorbent. Any PSD information for a microporous adsorbent is acquired from adsorption data. Because of the compounded effects of the surface topology with the adsorption potentials, the profile of which is shown in Figure 1a, it is indeed a distribution of surface energetic heterogeneity rather than a pure geometric property of the adsorbent. Therefore, the effective PSD for adsorption could not be symmetric (see Figure 1c), even for an adsorbent with a geometrically symmetric PSD (see Figure
10.1021/la010005p CCC: $20.00 © 2001 American Chemical Society Published on Web 08/10/2001
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assumption is (rc - ra) ∝ p, eq 3 is thus transformed to
θ ) 1 - exp(-bpq)
(4)
where b is a parameter related to adsorption energy change. If the amount adsorbed at θ is n, then n becomes n0, the saturated capacity, at θ ) 1. Therefore,
n ) n0[1 - exp(-bpq)]
(5)
Equation 5 is physically reasonable because n ) n0 at high pressures, and
n ≈ n0[1 - (1 - bpq)] ) Kpq
(6)
when p approaches zero. It does not reduce to Henry’s law, but seems more reasonable because n ∝ p is valid for homogeneous surfaces; however, n ∝ pq (0 < q < 1) is rational for heterogeneous surfaces. Equation 6 appears to be the same as the Freundlich equation,3 but only for a region of low pressure. Equation 5 looks similar to the Jovanovich equation, but is not exactly the same. As summarized by Misra,13 the Jovanovich equation, the Langmuir equation, and Henry’s law can be generated from the same differential equation that follows Figure 1. (a) Lennard-Jones potential, (b) distribution of geometric pore sizes, and (c) an example of the detected PSD.
1b). This argument is proven by the experimental PSD data reported.10,11 The differential distribution function of r is more likely to be steeper on the side of smaller pore sizes and skewed slowly on the other side toward larger pores after the peak as shown in Figure 1c. We use the Weibull function12 for such a nonasymmetric distribution. This function is simple in mathematical expression, yet has found many practical applications.
(
)
dθ ) c(1 - θ)k dp
where c and k are constants and k g 1 to satisfy all the requirements of physical compatibility. The integral of eq 7 yields the Jovanovich isotherm
θ ) 1 - exp(- cp) for k ) 1
(8)
the Langmuir isotherm
θ ) cp/(1 + cp) for k ) 2
q
(r - ra) q f(r) ) (r - ra)q-1 exp ; (r g ra) R R
(7)
(9)
If temperature is constant, then rc is determined only by pressure for a given adsorbent/adsorbate system. A simple
and Henry’s law for k ) 0. Equations 8 and 9 reduce to Henry’s law when the pressure approaches zero. However, both the isotherms do not contain a parameter that accounts for the effect of heterogeneity of the adsorbent surface and, therefore, are not sufficient to represent the experimental isotherms derived from porous solid studies. Experimental Isotherms Spanning the Critical Temperature. Adsorption equilibrium measurements of methane on activated carbon were carried out twice in our laboratory. Li’s measurement14 covered a temperature range of 233-333 K, the details of which are published elsewhere.15 The temperature range was extended down to 158.15 K spanning the critical temperature (190.6 K) in Yang’s work.9 The sample of activated carbon with specific surface area of about 3000 m2/g and micropore volume of about 1.3 mL/g was served as the adsorbent in both measurements. The measurement setup used in the experiments is based on volumetric principle. Eight isotherms at 158.15-298.15 K with 20 K increments were obtained in Yang’s measurement, to which two isotherms at 313 and 333 K obtained in Li’s measurement are added. All the experimental excess adsorption data are presented by “dots” in Figure 2. The curves represent the predicted isotherms by the subsequently proposed model. Appar-
(10) Nguyen, C.; Do, D. D. A New Method for the Characterization of Porous Materials. Langmuir 1999, 15, 3608-3615. (11) Shethna, H. K.; Bhatia, S. K. Interpretation of Adsorption Isotherms at Above-Critical Temperatures Using a Modified Micropore Filling Model. Langmuir 1994, 10, 870-876. (12) Weibull, W. Fatigue Testing and the Analysis of Results; The Macmillan Company: New York, 1961.
(13) Misra, D. N. Monomolecular Adsorption Isotherms. J. Colloid Interface Sci. 1980, 77(2), 543-547. (14) Li, M. High-Pressure Adsorption of Methane on Activated Carbon AX-21. M.S. Thesis, Tianjin University, China, January 1998. (15) Zhou, L.; Li, M.; Zhou, Y. Measurement and Theoretical Analysis of the Adsorption of Supercritical Methane on Superactivated Carbon. Sci. China, Ser. B 2000, 43(2), 143-153.
f(r) ) 0; (r < ra)
(1)
where ra is the radius of an adsorbate molecule. As mentioned above, there is a definite critical pore size, rc, for a given condition of (T, p) at equilibrium. The fractional surface coverage under the condition could be calculated simply by
∫rr f(r)dr r )∫ θ) ∞ ∫r f(r)dr r c
a
c
a
f(r)dr
(2)
a
The substitution of eq 1 into eq 2 yields
θ)-
∫r
rc a
(
)
(
)
(r - ra)q (rc - ra)q d exp ) 1 - exp R R (3)
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Langmuir, Vol. 17, No. 18, 2001 5505
Figure 2. Adsorption isotherms of CH4 on activated carbon spanning critical temperature.
Figure 3. Linear isotherms of methane for formulating the absolute adsorption.
ently, all methane isotherms on activated carbon belong to type I, although on some of them a maximum is shown. The proposed equation is shown to model all isotherms with type I features including those with maxima as illustrated by the set of methane isotherms. A comparison with the L-F equation is provided with respect to modeling. Modeling the Experimental Isotherms by the Proposed Equation. As disclosed in Figure 2, a maximum appears on some isotherms; therefore, the difference between the excess adsorption and the absolute adsorption has to be accounted for. The Gibbs definition of adsorption is thus used as a framework in modeling
atures, they all can be modeled by the following linear equation
n ) nt - VaFg
(10)
where n is the measured or the excess adsorption, nt is the so-called absolute adsorption, Va is the volume of the adsorbate adsorbed, and Fg is the density of the ambient gas. Determination of Va is critical in the modeling. A formulation method was proposed for the determination of absolute adsorption16,17 and was satisfactorily applied for several modeling studies.15,18 The principle of the method is quite simple in that nt = n in the region of low concentration of adsorbate on the surface. Therefore, we used the experimental data in this region to formulate absolute adsorption. For the sake of reliability of the formulation, the equilibrium data in the part of low surface concentration is linearized. So, a plot of ln ln(n) vs 1/ln p of the data was suggested as shown in Figure 3. To avoid taking logarithms of negative numbers, pressure p is expressed in kPa and n is enlarged 10-fold as shown in the ordinate of Figure 3. All the isotherms have some points on linear plots. Although low-temperature isotherms do not merge at the same point as those of higher temper(16) Zhou, L.; Zhou, Y. A. Mathematical Method for the Determination of Absolute Adsorption from Experimental Isotherms of Supercritical Gases. Chin. J. Chem. Eng. 2001, 9(1), 110-115. (17) Zhou, L.; Zhou, Y. Linearization of Adsorption Isotherms for High-Pressure Applications. Chem. Eng. Sci. 1998, 53(14), 2531-2536. (18) Zhou, L.; Zhou, Y.; Bai, Sh.; Lu¨, Ch.; Yang, B. Determination of the Adsorbed Phase Volume and Its Application in Isotherm Modeling for the Adsorption of Supercritical Nitrogen on Activated Carbon. J. Colloid Interface Sci. 2001, 239, 33-38.
ln ln(δnt) ) R +
β ln p
(11)
from which
1 β nt ) exp exp R + δ ln p
[ (
)]
(11a)
is derived. The parameter δ ) 10 for the methane data and R and β are determined only by temperature. For example, in the range of 158.15-333.15 K
R ) 2.198 046 196 7 - 0.005 440 279 878 5T + 0.000 020 890 773 15T2 (11b) β ) - 2.202 034 502 7 + 0.036 343 241 7T 0.000 178 030 920 91T2 (11c) The volume of the adsorbed phase can then be evaluated using the formulation
Va )
nt - n Fg
(12)
where n is the experimental data, while the corresponding nt is determined using eqs 11a-c. The parameter Fg is evaluated from an appropriate state equation of real gases, the third virial equation in the present paper, and is correlated with pressure at each temperature for the convenience of computation
Fg )
cipi ∑ i)1
(13)
The coefficients of eq 13 at the experimental temperatures are presented in Table 1. The evaluated Va is shown in Figure 4. Va increases with pressure and generally expands with the increasing temperature. Equation 10 suggests that the difference (nt - n) is trivial at a very low density of gas, Fg. Reliable values of Va could not be expected under
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Table 1. Coefficients in Correlating the Density (mmol/mL) of Gaseous Methane T (K) 333.15 313.15 298.15 278.15 258.15 238.15 218.15 198.15 178.15 158.15
c1
c2
c3
c4
c5
1.9005 × 10-5 9.4601 × 10-4 2.6320 × 10-3 -2.8137 × 10-3 -0.114 08 -0.022 828 0.046 424
-3.3939 × 10-5 -6.0115 × 10-5 8.6821 × 10-4 1.4297 × 10-2 0.012 669
10-3
3.2275 × 4.9239 × 10-3 7.1491 × 10-3 0.010 983 0.011 583 0.011 890 0.037 697 0.314 86 0.091 646 0.073 280
0.364 34 0.387 11 0.403 51 0.429 56 0.467 17 0.511 99 0.543 33 0.430 87 0.660 79 0.764 83
5.5691 × 10-4
Table 2. Coefficients in Correlating the Volume (mL/g) of the Adsorbed Methane T (K)
c0
c1
c2
c3
c4
333.15 313.15 298.15 278.15 258.15 238.15 218.15 198.15 178.15 158.15
-0.979 21 -1.0522 -1.3899 -1.3794 -2.0291 -1.4999 -0.827 40 3.7789 2.3657 -
0.740 32 0.750 25 0.922 57 1.0446 1.7876 1.5031 0.431 04 -6.8392 -6.2603 -
-0.079 751 -0.083 163 -0.117 30 -0.182 15 -0.480 19 -0.469 92 -0.027 966 4.1785 3.8742 -
3.2295 × 10-3 33 836 × 10-3 5.4298 × 10-3 0.015 513 0.067 144 0.080 532 7.4067 × 10-4 -1.1600 -0.680 53 -
-5.0848 × 10-4 -4.4231 × 10-3 -6.7836 × 10-3
Figure 4. Adsorbed phase volume of methane on activated carbon.
such conditions. All reliable Va values could be correlated with pressure at each temperature by another polynomial function for the sake of convenience of modeling
Va )
cjpj ∑ j)0
(14)
The coefficient values of eq 14 are given in Table 2. The working model for the experimental excess isotherms is established by the substitution of eqs 5, 13, and 14 into eq 10, where the isotherm equation is used to describe the absolute adsorption as it should be
n ) nt0[1 - exp(-bpq)] - (
cipi) ‚ (∑cjpj) ∑ i)1 j)0
(15)
The three parameters of the model nt0, b, and q are evaluated on fitting the model to the experimental isotherms by nonlinear regression analysis. The results
0.157 34
c5
1.0249 × 10-4 2.2111 × 10-4 -8.4023 × 10-3
Figure 5. Variation of parameter nt0 with temperature.
are shown in Figures 5-7, respectively. In data fitting, Va was set to zero for the isotherm at 158.15 K because no reliable values could be used for it. The proposed equation gives two constant values for the parameter nt0, the higher value for subcritical temperatures and the lower for supercritical temperatures. It seems quite reasonable because volume filling is more likely to happen in micropores at subcritical temperatures. The parameters b and q yielded by the proposed equation vary continuously with temperature for the whole temperature range tested. It seems physically reasonable that parameter b, which is associated with energy change in adsorption, increases as temperature decreases. Noticeable is the variation of q with temperature, that it spans only the range of 0-1. Therefore, the exponent parameter q of the proposed isotherm serves as an appropriate index of surface heterogeneity. It always increases with temperature at above-critical temperatures; however, a minimum is shown near critical temperatures. The minimum value of q is possibly caused by the special compressibility of fluids
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Langmuir, Vol. 17, No. 18, 2001 5507
nexp is the experimental amount, Nm is the number of data at a temperature, and NT is the number of isotherms. A Comparison with the L-F Equation. Although potential theory is very important and practical for the adsorption at subcritical temperatures, it may not be a pertinent model for supercritical adsorption. The first trouble in applying the theory to supercritical isotherms lies in the definition of the adsorption potential. More important is the fact that the tendency of condensation or coagulation of adsorbate molecules cannot be comparable to that at subcritical temperatures, therefore, only short distance interaction is important for adsorption at supercritical temperatures. In other words, instead of volume filling, a monolayer coverage mechanism seems more practical. However, the classical monolayer model, the Langmuir equation, is applicable only to homogeneous surfaces and, therefore, is not sufficient to describe the isotherms obtained on heterogeneous surfaces. The L-F equation that follows is a successful modification to the Langmuir equation, in that one more parameter is introduced to account for the effect of heterogeneity of surface.
(
n ) nt0
Figure 6. Variation of parameter b with temperature.
)
bpq 1 + bpq
(17)
Applying eq 17 for the absolute adsorption we have another model similar to eq 15
( )
n ) nt0
Figure 7. Variation of parameter q with temperature.
near the critical point because q reflects the topology of the adsorbate adsorbed rather than the topology of adsorbent surface. The total disagreement of the model with the experimental isotherms is defined by eq 16. It is about 1.5% for the whole range of temperatures tested
d(%) )
1 NT
NT
(
1
Nm
∑ ∑ j)1 N i)1 m
)
100 × ABS(ncal - nexp) nexp
(16)
where ncal is the amount adsorbed calculated by model,
bpq
q
1 + bp
-(
cipi)(∑cjpj) ∑ i)1 j)0
(18)
Fitting the model to the experimental isotherms of methane, we obtained a total disagreement of about 1.5%, which is as fit to the experimental isotherms as the proposed model. The parameter values shown in Figures 5-7 are also quite similar to those of the proposed model. Parameters b and q given by the two models have similar trends of variation with temperature, as shown in Figures 6 and 7. However, the L-F equation gives parameter nt0 a constant value of 35.25 mmol/g in both sub- and supercritical regions. On the other hand, the proposed equation yields two different values of nt0 for the sub- and the supercritical regions, and the one for the supercritical region is 27.5 mmol/g, which is in agreement with the fictitious limiting adsorption, nlim, given by the intersection point of linear isotherms shown in Figure 3. The limiting adsorption value is 0.1 × exp[exp(1.73)] ) 28.2 mmol/g which is quite close to 27.5 mmol/g. It is also seen in Figure 3 that the linear isotherms for subcritical temperatures tend to intersect at a higher point giving a larger quantity of limiting adsorption, which was also predicted by the proposed equation. Additional verifications and investigations into the new isotherm equation by other authors with more data are expected. Acknowledgment. This work is subsidized by the Special Funds of Major State Basic Research Projects (G2000026404) and supported by the National Natural Science Foundation of China (No. 29936100). LA010005P
