Ind. Eng. Chem. Res. 1998, 37, 1159-1166
1159
Isosteric Heats of Adsorption on Carbon Predicted by Density Functional Theory Huanhua Pan, James A. Ritter, and Perla B. Balbuena* Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Columbia, South Carolina 29208
Isosteric heats of adsorption (qst) of propane and butane adsorbed on carbon were studied by numerical differentiation of nonlocal density functional theory (NDFT) isotherms. The qst values of both adsorbates in slit-shaped pores were a weak function of temperature, and decreased with increasing pore width. Using the calculated pore size distribution (PSD) of an activated carbon, it was further determined that the qst for both propane and butane decreased with increasing loading, consistent with a heterogeneous adsorbent. The NDFT results utilizing the PSD were also in fair agreement with those obtained from the classical approach using experimental isotherms fitted to a model and applied to the Clausius-Clapeyron-type equation; both models predicted qst of butane to be ∼10 kJ/mol higher than that of propane at the same loading. On a model homogeneous carbon, the qst of both adsorbates increased with reduced surface coverage up to ∼0.5, then they dropped rapidly. The reduced surface coverages corresponding to monolayer completion were 0.61 for propane and 0.65 for butane, in agreement with published experimental results. Introduction The heat of adsorption yields valuable insights into the mechanism of adsorption. It is also a critical thermodynamic property for the design of adsorption processes because it provides a base to estimate the heat released (or consumed) during the adsorption (or desorption) process. Values of the heat of adsorption can be determined either from the variation of adsorption with temperature, or by direct calorimetric measurements; however, the amount of work involved in finding the heat of adsorption at many different loadings and different temperatures by experiment is usually prohibitive (Beebe et al., 1947a,b; Dunne et al., 1996). Some empirical adsorption isotherm equations can be used to determine the isosteric heat of adsorption (qst), but experimental adsorption data are required to determine the parameters in these models; thus, this method is not predictive. Furthermore, different isotherm equations usually lead to different qst results. Recently, there has been increasing interest in using molecular simulation tools to determine the qst for various adsorbate-adsorbent systems, especially Monte Carlo techniques (Vernov and Steele, 1991; Razmus and Hall, 1991; Karavias and Myers, 1991; Matranga et al., Bottani et al., 1994; Myers et al., 1997); however, far less work has been done using density functional theory (Jiang et al., 1992; Balbuena and Gubbins, 1993). In this work, density functional theory (DFT), a statistical mechanics model, was used to predict the qst of propane and butane adsorbed on carbon materials. DFT methods are based on the idea that the grand free energy of an inhomogeneous fluid can be expressed as a functional of F(r), the density profile in the pore. The free energy functional is modeled as a sum of two contributions: a hard sphere nonlocal core and a mean * Author to whom correspondence should be addressed. Phone: 803-777-8022. Fax: 803-777-8265. E-mail: Perla@ sun.che.sc.edu.
field attractive tail; and F(r) can be obtained by minimization of this functional. Once F(r) is known, all of the relevant thermodynamic functions can be calculated. The most elaborated form of this theory, the nonlocal density functional theory (NDFT), has been successfully used to describe fluids confined in narrow pores (Evans and Tarazona, 1984; Peterson et al., 1986; Balbuena and Gubbins, 1993). Lastoskie et al. (1993) also used NDFT to analyze the pore size distribution (PSD) of a microporous carbon, and recently, Ravikovitch et al. (1997) also used NDFT to characterize some siliceous catalyst supports and catalysts. In this work, the PSD of BAX-activated carbon was determined following the procedure of Lastoskie et al. (1993). Then, this PSD and NDFT were used to calculate the qst of propane and butane on the BAXactivated carbon by a numerical differentiation technique. The results were compared with those obtained from a classical approach using experimental isotherm data fitted to a model and applied to a ClausiusClapeyron-type equation (Sircar, 1985; Myers and Valenzuela, 1989). The effect of temperature and pore width on the qst in slit-shaped pores, along with the qst for a model homogeneous carbon were also investigated using NDFT. Theory It is assumed that the solid consists of slit-like pores represented by two semi-infinite parallel walls separated by a distance H. The fluid-fluid interaction potential µ(r) is described by a cut and shifted LennardJones 12-6 pair potential as follows (Weeks, et al., 1971):
uff(r) ) φff(r) - φff(rc) )0
if r > rc
with
S0888-5885(97)00586-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/07/1998
if r < rc (1)
1160 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998
[( ) ( ) ] σff r
φff(r) ) 4ff
12
σff r
-
6
(2)
where r is the separation distance between a pair of molecules, and ff and σff are parameters representing interaction energy and molecular size, respectively. The cutoff radius rc is chosen as 2.5 σff. For solid-fluid interaction potential φsf, Steele’s 10-4-3 potential is used (Steele, 1974), which is given by:
[( ) ( )
2 σsf φsf(z) ) 2πsfFsσ2sf∆ 5 z
10
σsf z
4
σ4sf
]
3
3∆(z + 0.61∆)
(3)
(4)
The potential parameters used in this work are shown in Table 1; parameters for solid-gas interactions are estimated by Lorentz-Berthelot mixing rules (Steele, 1974) and given by eqs 5a and 5b:
σij ) (σii + σjj)/2
(5a)
ij ) (iijj)0.5
(5b)
For a fluid in an external field Vext(r), and at a fixed temperature T, and chemical potential µ, the grand potential functional Ω is written as follows (Evans and Tarazona, 1984):
Ω[FL(r)] ) F[FL(r)] -
∫dr FL(r)[µ - Vext(r)]
(6)
F[FL(r)] is the Helmholtz free energy, which is expanded in a perturbative fashion about a reference system of hard spheres of diameter d to yield:
F[FL(r)] ) FH(F,d) +
1 2
∫dr dr′ F(2) L (r,r′) φatt(|r - r′|)
(7)
where FL(2)(r,r′) is the pair distribution function. The equivalent hard sphere diameter d is calculated as a function of temperature from eq 8 (Barker and Henderson, 1967):
η1kT/ff + η2 d ) σff η3kT/ff + η4
(8)
1 2
σ (Å)
/k (K)
3.4 3.681 5.061 4.997
28.0 91.5 254 410
Viscosity data.
FH[FL(r),d] ) Fid[FL(r)] + Fex[FL(r),d]
(10)
∫
Fid[FL(r)] ) kT dr FL(r)[ln(Λ3FL(r)) - 1] (11) where Λ ) h/x2πMkT is the thermal deBroglie wavelength. The excess term is a functional of the local density FL(r) and the excess Helmholtz free energy per molecule fex, as follows:
∫
Fex[FL(r),d] ) kT dr FL(r) fex[Fj(r),d]
∫dr dr′ FL(r) F(r′) φatt(|r - r′|)
(9)
Equation 9 neglects correlations due to attractive forces. The hard sphere term FH is further divided into an ideal gas component Fid and an excess component Fex:
(12)
where fex is given by the contribution of the local chemical potential µH, pressure PH, and ideal gas term for a hard sphere fluid of smoothed density Fj(r) and diameter d:
fex[Fj(r),d] ) µH[Fj(r),d] -
PH[Fj(r),d]
Fj(r) kT[ln(Λ3Fj(r)) - 1] (13)
The smoothed density is defined as follows:
Fj(r) )
∫dr′ FL(r′) W[|r - r′|,Fj(r)]
(14)
where W[|r - r′|, Fj(r)] is a weighting function, which takes into account nonlocal effects. Here, Tarazona’s (1985) model for the weighting function is used, and the hard sphere excess free energy is calculated (eq 13) with the Carnahan-Starling equation of state (Carnahan and Starling, 1969). To solve for the equilibrium density profile, the grand potential functional Ω of eq 6 is minimized with respect to density. The density profile at a certain H and T is obtained by numeric iteration. For convenience, the pore width and mean fluid density are scaled with respect to the fluid-fluid molecular diameter, respectively, as H* ) H/σff and F* ) Fσff3. After solving for the density profile, the pore size distribution f(H) is obtained by solving the adsorption integral equation (Szombathely et al., 1992; Jagiello, 1994), which is written as follows:
N(P) )
with η1 ) 0.3837, η2 ) 1.035, η3 ) 0.4249, and η4 ) 1 (Lastoskie, et al., 1993). The attractive part is treated in the mean field approximation as follows:
F[FL(r)] ) FH(F,d) +
a
interaction CsC N2sN2 C3H8sC3H8 n-C4H10sn-C4H10
The ideal contribution is given by eq 11:
where Fs is the solid density and ∆ is the separation between carbon layers. Values of Fs ) 0.114 Å-3 and ∆ ) 3.35 Å are used here (Steele, 1973). For a slit pore, the fluid molecule interacts with both walls, so the full external potential Vext(z) is written as follows:
Vext(z) ) φsf(z) + φsf(H - z)
Table 1. Lennard-Jones Parameters Used in the NDFT Calculations (Hirschfelder et al., 1954)a
∫HH
max
min
F(P,H) f(H) dH
(15)
The isosteric heat of adsorption is calculated by eq 16 (Sircar, 1985; Myers and Valenzuela, 1989):
ln P [∂ ∂T ]
qst ) RT2
N
(16)
When the isotherm is in the form of N d N(P,T), eq 16 is changed to the following form by the chain rule of calculus:
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1161
RT2 ∂N ∂N qst ) / P ∂T P ∂P T
( )( )
(17)
The derivatives in eqs 16 and 17 are evaluated analytically if explicit expressions are available, otherwise they are evaluated numerically. Experimental Section Adsorption isotherms for propane and butane adsorbed on BAX-activated carbon were measured gravimetrically using a VTI Integrated Microbalance System. The propane adsorption isotherms were obtained at six different temperatures ranging from 297 to 333 K. At each temperature, 22 points were obtained covering the pressure range from 6 to 333 kPa. The butane adsorption isotherms were obtained at nine different temperatures ranging from 297 to 367 K. At each temperature, 22 points were obtained covering the pressure range from 6 to 200 kPa. Prior to each isotherm measurement, the activated carbon was regenerated at 523 K for 2 h under reduced pressure of 4. At zero coverage, there are no interactions among the adsorbate molecules, so qsto is totally determined by the adsorbate-adsorbent interaction (Vext). When H* is small, both walls have significant effects on the adsorbate molecules. From eqs 3 and 4, Vext decreases with increasing H*; thus, qsto decreases when H* increases. If H* is large enough, φsf(H* - z) becomes very close to zero; then Vext(z) is controlled by φsf(z) (eq 4), the interaction between the adsorbate and the closer wall. At zero coverage, z is small and almost constant because the fluid molecules are adsorbed only in the first layer; therefore, qsto is nearly constant at large H*. Isosteric Heats of Adsorption from Adsorption Isotherm Correlations. The propane-BAX-activated
1162 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998
Figure 4. Adsorption isotherms of propane adsorbed on BAXactivated carbon. The points are the experimental data and the lines are the correlation with the TPLM, eq 19.
Figure 2. (A) Isosteric heats of adsorption of n-butane on carbon at different slit widths and temperatures, and (B) the corresponding adsorption isotherms calculated from the NDFT model.
Figure 5. Adsorption isotherms of n-butane adsorbed on BAXactivated carbon. The points are the experimental data and the lines are the correlation with the TPLM, eq 19.
( )
bi ) b0i exp
Bi T
(19b)
was used to correlate the experimental data. TPLM was fitted simultaneously to all of the isotherm data points in each system to determine the nine parameters using nonlinear regression. The average relative error (ARE), defined as follows:
ARE% )
Figure 3. Isosteric heats of adsorption of propane and butane at zero coverage calculated from an analytical expression (eq 18); (Steele, 1974).
carbon and butane-BAX-activated carbon isotherms are shown in Figures 4 and 5, respectively. The Three Process Langmuir Model (TPLM) (Drago et al., 1996), which is given by: 3
N)
qmibiP
i ) 1, 2, 3 ∑ i)1 1 + b P i
and
(19a)
100 Np
Np
∑ i)1
|(
)|
Nexp - Ncal Nexp
(20)
i
was used to indicate the goodness of the fit. The TPLM parameters and the corresponding ARE for each system are given in Table 2; the TPLM correlation for propane and butane are plotted against the experimental data in Figures 4 and 5, respectively. It is seen that the TPLM correlates quite well with both the propane and butane-BAX-activated carbon systems, with some minor deviations apparent in the butane correlation at low pressures and temperatures. The isosteric heats of adsorption were calculated from the TPLM by combining eqs 17 and 19. The derivatives in eq 17 were evaluated analytically using Maple III. The results are shown in Figures 6a and 7a, respectively, for propane and butane. It was reassuring to see
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1163 Table 2. TPLM Parameters and Goodness of Fita for the Propane and n-Butane-BAX Carbon Systems parameter
propane
n-butane
qm1 (mol/kg) b01 (1/kPa) B1 (K) qm2 (mol/kg) b02 (1/kPa) B2 (K) qm3 (mol/kg) b03 (1/kPa) B3 (K) ARE (%)a
6.007 2.56 × 10-7 2880 2.6047 8.78 × 10-7 3354.63 0.6008 2.21 × 10-7 4625.18 0.90
7.9862 8.79 × 10-8 3305.384 2.0174 4.43 × 10-7 3923.893 1.544895 3.0 × 10-9 6310.47 1.5
a
Defined by eq 20.
Figure 7. (A) Isosteric heats of adsorption of butane on BAXactivated carbon at 313 K predicted from the TPLM and the NDFT model, and (B) the corresponding adsorption isotherms.
Figure 6. (A) Isosteric heats of adsorption of propane on BAXactivated carbon at 313 K predicted from the TPLM and the NDFT model, and (B) the corresponding adsorption isotherms.
that for both systems, the TPLM was capable of predicting the characteristic decrease in qst with loading that is associated with a heterogeneous adsorbent. Pore Size Distribution. Following the procedure of Lastoskie et al. (1993), the density profile of N2 on carbon at 77 K was generated for a wide range of slitlike pores of diameter H*. Then, using the experimental adsorption isotherm for N2 on BAX-activated carbon at 77 K and eq 15, the PSD of the BAX-activated carbon was solved for using a regularization method with nonnegative constraints (Szombathely et al., 1992; Jagiello, 1994). The results are shown in Figure 8. It is seen that this carbon material has a wide range of pore sizes, with two peaks in the micropore range, one in the small mesopore range, and a broad distribution of pores up to 100 Å in diameter.
Figure 8. Pore size distribution of the BAX-activated carbon, determined from the N2 experimental adsorption isotherm at 77 K and a NDFT procedure outlined by Lastoskie et al. (1993), utilizing eq 15.
NDFT Isosteric Heats of Adsorption of Propane and Butane on BAX-Activated Carbon. The NDFT isosteric heat of adsorption is obtained from
∫HH
max
qst(N) )
min
qst(H,P) F(H,P) dVp(H)
∫HH
max
min
F(H,P) dVp(H)
(21)
once qst(H,P) and the mean density at a certain pressure
1164 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998
and temperature (where the equilibrium loading is N) at different pore sizes is determined. The accumulative pore volume Vp(H) is calculated from the PSD function f(H) () dV/dH) of the material, which is given by:
Vp(H) )
∫HH
min
f(H′) dH′
(22)
In this work, qst values for propane and butane on carbon at 313 K were calculated for different slit widths ranging from 1.6 to 20, with truncation at H* ) 20 because the contribution to adsorption by pores with H* > 20 was very small in the pressure range investigated. Then, using eqs 21 and 22 and the PSD result obtained earlier (Figure 8), the qst values for propane and butane on the BAX-activated carbon at 313 K were calculated. The results are shown in Figures 6a and 7a, along with the experimental results obtained from the TPLM correlations. The corresponding adsorption isotherms are shown in Figures 6b and 7b. There have been some discussions about the difference between the absolute properties derived from simulation studies and the excess properties obtained from experiment (Sircar, 1985; Myers et al., 1997). Myers et al. (1997) suggest that the difference between absolute and excess adsorption may be ignored when the pore volume of the adsorbent is negligible compared with the adsorption second virial coefficient (B1s). The TPLM adsorption second virial coefficients of propane and butane on the BAX-activated carbon at 313 K are 1216.6 and 7592.9 cm3/g, respectively; the pore volume of the BAX-activated carbon is 0.915 cm3/g (Jagiello, 1997). Therefore, the difference between absolute and excess adsorption was considered negligible and the NDFT results are compared directly to the TPLM (experimental) results in Figures 6 and 7. The qst values and the adsorption isotherms obtained from the TPLM (experiment) are in fair agreement with those from NDFT. For propane, the adsorption isotherms obtained from the two models agree very well in the low pressure range, but begin to deviate when P exceeds 30 kPa; in contrast, butane exhibits deviation in the low pressure range and fairly good agreement at higher pressures. It is also seen that the qst for propane predicted from the NDFT agrees fairly well with the TPLM results over most of the loading range, except at very low loadings, whereas qst for butane calculated from the two models agree well only at low loadings. These differences may have been caused by the inability of the TPLM to represent the full heterogeneity of the systems or to correlate the experimental data precisely in the entire pressure range (Figure 5). Oversimplification of the structure of the adsorbent and the mean field approximation for the attractive potential in NDFT may have also contributed to this discrepancy between the models. The drop of qst with loading for both propane and butane is typical of heterogeneous adsorbents (Rudzinski and Everett, 1992); and it is reassuring to see that both NDFT and TPLM capture this characteristic. This BAX-activated carbon has a wide range of pore sizes (Figure 8): 12% of its pore volume has H
