Langmuir 1998, 14, 6323-6327
Examination of the Approximations Used in Determining the Isosteric Heat of Adsorption from the Clausius-Clapeyron Equation
6323
area A from the bulk gas phase (G) to the adsorbed phase (a):7
qst )
Huanhua Pan, James A. Ritter, and Perla B. Balbuena* Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Columbia, South Carolina 29208
Theory The isosteric heat of adsorption qst is defined as the differential change in energy δQ that occurs when an infinitesimal number of molecules δN are transferred at constant pressure P, temperature T, and adsorbent surface * To whom correspondence should be addressed. Telephone: (803) 777-8022. Fax: (803) 777-8265. E-mail:
[email protected]. (1) Sircar, S. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527. (2) Myers, A. L.; Valenzuela, D. P. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (3) Din, F. Thermodynamic Functions of Gases; Butterworths: London, 1961; Vol. 3. (4) Din, F. Thermodynamic Functions of Gases; Butterworths: London, 1962; Vol. 2. (5) Gupta, D.; Eubank, P. T. J. Chem. Eng. Data 1997, 42, 961. (6) Yang, R. T. Gas Separation by Adsorption Process; Butterworths: Boston, MA, 1987.
∂Q ∂Na
(1)
P,T,A
A change in the integral heat of adsorption -dQ associated with an interfacial system is defined as
-dQ ) TdS
Received March 26, 1998. In Final Form: August 4, 1998
A method that is widely used to calculate the isosteric heat of adsorption involves the application of the Clausius-Clapeyron equation, which relates the adsorption heat effects to the temperature dependence of the adsorption isotherm.1,2 Two approximations are introduced in deriving the Clausius-Clapeyron equation: (1) the bulk gas phase is considered ideal and (2) the adsorbed phase volume is neglected. These two assumptions are reasonable at low pressures but may not be true at higher pressures, and as far as the authors are aware, no work has been done in validating these assumptions. Two questions are addressed in this work: Do these assumptions hold true at relatively high pressure, and if not, how large is the error caused by them? To answer these questions, isosteric heats of adsorption of ethane, propane, and butane on BAX activated were calculated in three different ways: (1) using the classic Clausius-Clapeyron-type equation, which assumes an ideal gas and neglects the adsorbed phase volume; (2) assuming ideal gas for the bulk gas phase, while taking into account the adsorbed phase volume; (3) relaxing both assumptions and predicting the adsorbed phase molar volume by nonlocal density functional theory (NDFT), and describing the bulk gas by the Carnahan-Starling equation of state plus a mean-field attraction (CSPMA). In addition, all of the adsorption isotherms were generated by NDFT. The isosteric heats of adsorption obtained for each adsorbate using the three different methods were compared with each other over a wide range of pressures and thus loadings to expose the relative importance of the two assumptions. The accuracy of CSPMA and the ideal gas law were also checked by comparing the PVT behavior of each gas predicted by these two models with experimental data from the literature.3-5 The results showed that the assumptions may cause significant errors in calculated isosteric heats adsorption, even at moderate pressures that are relevant to industrial adsorption processes.6
( )
(2)
where the total entropy S is the sum of the entropies of the different phases
S ) S G + S a + Ss
(3)
where SG, Sa, and Ss are the entropies of the gas, adsorbed and solid phases, respectively. From a mass balance and assuming an inert adsorbent
dNG ) -dNa
(4)
Based on eqs 2-4, eq 1 is rewritten in the following form:
( )
qst ) -T
∂S ∂Na
[( ) ( ) ] ∂SG ∂NG
) -T
P,T,A
∂Sa ∂Na
-
P,T
(5)
P,T,A
The total derivative of the chemical potential µ for the adsorbed phase is expressed as a function of P, T, and A:
dµa )
( ) ∂µa ∂T
dT +
P,A
( ) ∂µa ∂P
T,A
dP +
( ) ∂µa ∂A
dA
P,T
(6)
From the Maxwell relations and noting that1
( ) ∂µa ∂T
( )
)P,A
∂Sa ∂Na
(7)
P,T,A
eq 6 is rewritten as
( )
dµa ) -
∂Sa ∂Na
dT + Va dP +
P,T,A
( ) ∂µa ∂A
dA
(8)
P,T
where Va is the adsorbed phase molar volume. For a bulk gas phase in equilibrium with the adsorbed phase
( )
dµG ) -
∂SG ∂NG
dT + VG dP
(9)
P,T
where VG is the gas phase molar volume. At a fixed adsorbed phase loading Na and for a constant adsorbent surface area A, equating eqs 8 and 9 leads to
dP (dT )
N,A
)
(∂SG/∂NG) - (∂Sa/∂Na) V G - Va
(10)
Substituting eq 10 into eq 5 gives a general relation for qst as
qst ) T(VG - Va)
dP (dT )
N,A
(11)
(7) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: New York, 1982.
S0743-7463(98)00337-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/24/1998
6324 Langmuir, Vol. 14, No. 21, 1998
Notes
By neglecting the adsorbed phase molar volume and assuming an ideal gas for the bulk gas phase, eq 11 reduces to
(
)
d(ln P) qst ) RT dT 2
N
(12)
which is the familiar form of the Clausius-Clapeyron equation that is widely used in adsorption studies. When the isotherm is in the form of N ) N(P,T), Equation 11 is changed to the following form by the chain rule of calculus:
qst ) -TV
∂N / (∂N ∂T ) ( ∂P ) P
T
(13)
where V ) VG - Va and is calculated in different ways according to the different assumptions outlined below. In method 1, the ideal gas law is assumed and the adsorbed phase molar volume is neglected; thus
V)
RT P
(14)
where R is the universal gas constant. In method 2, the ideal gas law is assumed, but the adsorbed phase molar volume is taken into account; thus
V)
RT 1 RT - Va ) P P Fa
(15)
where Fa is the mean density of the adsorbed phase, which results from averaging the local density obtained by minimizing the grand potential functional with respect to density in a slit-shaped pore in NDFT.8 In method 3, both assumptions are relaxed, and VG and Va are predicted using the values of the bulk density Fb and adsorbed phase density Fa obtained from NDFT calculations; thus
V)
1 1 Fb F a
∫
dr FL(r)[µ - Vext(r)] (17)
F[FL(r)] is the Helmholtz free energy functional, 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
interaction
σff (Å)
ff/k (K)
C-C C2H6-C2H6 C3H8-C3H8 n-C4H10-n-C4H10
3.4 4.418 5.061 4.997
28.0 230 254 410
of the fluid-fluid interaction. The equivalent hard sphere diameter d is calculated as a function of temperature from9
η1kT/ff + η2 d ) σff η3kT/ff + η4
∫dr dr′ F2L(r,r′)φatt(|r - r′|)
(18)
where FH(F,d) is the free energy functional of an inhomogeneous hard sphere fluid, FL(2)(r,r′) is the pair distribution function, and φatt(|r - r′|) is the attractive part (8) Pan, H; Ritter, J. A.; Balbuena, P. B. Ind. Eng. Chem. Res. 1998, 37, 1159.
(19)
where σff and ff are the Lennard-Jones parameters, and η1 ) 0.3837, η2 ) 1.035, η3 ) 0.4249, and η4 ) 1.10 The attractive part in eq 18 is treated in the mean field approximation as follows:
F[FL(r)] ) FH(F,d) +
1 2
∫dr dr′FL(r)F(r′)φatt(|r - r′|)
(20)
The fluid-fluid and fluid-solid intermolecular interactions are calculated from the Lennard-Jones and Steele’s 10-4-3 potentials,11 respectively. The potential parameters used for each pure gas are given in Table 1,12 and the parameters for gas-solid interactions are estimated by Lorentz-Berthelot mixing rules,11 which are given by eq 21a, b.
σij ) (σii + σjj)/2
(21a)
ij ) (iijj)1/2
(21b)
The bulk phase behavior is described by a CarnahanStarling equation of state13 plus a mean-field attraction term (CSPMA). The CS equation of state for a hard-sphere (hs) is given by
(16)
NDFT methods are based on the idea that the grand free energy (Ω) of an inhomogeneous fluid can be expressed as a functional of F(r), the density profile in the pore. For a fluid in an external field Vext(r), and at a fixed temperature T and chemical potential µ, Ω is written as
Ω[FL(r)] ) F[FL(r)] -
Table 1. Lennard-Jones Parameters used in NDFT Calculations12
Phs[Fb] 1 + y + y2 - y3 ) FbkT (1 - y)3
(22)
π y ) Fbd3 6
(23)
where
k is the Boltzmann constant and d is the equivalent hard sphere diameter, which is given by eq 19. According to Tarazona’s NDFT model,14 FH(F,d) in eq 20 is further divided into an ideal gas component and an excess component. The ideal term is exactly a functional of the local density F(r), and the excess part is considered as a functional of a smoothed density. (More detailed descriptions of the model are given elsewhere.8,10,14,15) The equilibrium density profile of the fluid confined in the (9) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 4714. (10) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (11) Steel, W. A. The Interaction of Gases with Solid Surface; Pergamon: Oxford, England, 1974. (12) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954. (13) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (14) Tarazona, P. Phys. Rev. A 1985, 31, 2672. (15) Balbuena, P. B.; Gubbins, K. E. Langmuir 1993, 9, 1801.
Notes
Langmuir, Vol. 14, No. 21, 1998 6325
Figure 1. (A) The isosteric heat of adsorption of ethane on BAX carbon at 300 K calculated from the three different methods. (B) The corresponding adsorption isotherm of ethane on BAX carbon predicted by NDFT. (C) P-V curves of ethane at 300 K predicted from the ideal gas law and CSPMA compared with experimental data.3
pore is obtained by minimizing the grand potential functional Ω of eq 17 with respect to density. Results and Discussion The isosteric heats of adsorption of ethane and propane in slit-shaped pores at 300 K and butane in slit-shaped pores at 350 K were calculated for a slit width H* ()H/σff) ranging from 1.6 to 20, according to eq 13. The derivatives in eq 13 were evaluated numerically based on NDFT
Figure 2. (A) The isosteric heat of adsorption of propane on BAX carbon at 300 K calculated from the three different methods. (B) The corresponding adsorption isotherm of propane on BAX carbon predicted by NDFT. (C) P-V curves of propane at 300 K predicted from the ideal gas law and CSPMA compared with experimental data.4
calculations of adsorption isotherms at different temperatures.8 Three different V’s, given by eqs 14-16, were used for each adsorbate. Then on the basis of the pore size distribution obtained in previous work by the authors for BAX activated carbon,8 qst values for each adsorbate were obtained as a function of loading on the BAX carbon. The results are shown in Figures 1a, 2a, and 3a, for ethane, propane, and butane, respectively. The corresponding adsorption isotherms obtained from NDFT are shown in Figures 1b, 2b, and 3b. Note that the pressure ranges studied in each case differed based on the saturation vapor
6326 Langmuir, Vol. 14, No. 21, 1998
Figure 3. (A) The isosteric heat of adsorption of butane on BAX carbon at 350 K calculated from the three different methods. (B) The corresponding adsorption isotherm of butane on BAX carbon predicted by NDFT. (C) P-V curves of butane at 350 K predicted from the ideal gas law and CSPMA compared with experimental data.5
pressures of the adsorbates, which are Psat(300 K) ) 43.18 atm, Psat(300 K) ) 9.87 atm, and Psat(350 K) ) 9.37 atm for ethane, propane, and butane, respectively. To check the accuracy of CSPMA used in the NDFT calculations, the P-V curves for each adsorbate predicted by CSPMA and the ideal gas law are compared with experimental data taken from the literature in Figures 1c, 2c, and 3c, respectively. Figure 1a shows that the qst’s calculated from the three different methods for ethane were almost the same in the low pressure region (N < 1.5 mol/kg, or P < 0.3 atm). This
Notes
is reasonable because at low pressure the adsorbed phase molar volume is negligible compared with the gas phase molar volume, and the ideal gas law is more valid at low pressures as shown in Figure 1c. As the loading increases, however, the qst’s from the three methods began to deviate from each other. At loadings higher than about 4 mol/kg (corresponding to P > 4.6 atm), the difference became considerable: for example, the qst’s calculated from method 1 (classic Clausius-Clapeyron equation) began to increase with loading in this range, the qst’s calculated from method 2 (ideal gas but accounting for Va) began to level off, and the qst’s calculated from method 3 (relaxing both assumptions) kept on decreasing with increasing loading. The difference became even more marked at higher pressures. At N = 6 mol/kg (corresponding to P = 23 atm), the qst from method 1 was 42.5% higher than that obtained from method 3, and the qst from method 2 was 27.3% higher than that obtained from method 3. Clearly, at these higher pressures, the approximations used in the ClausiusClapeyron equation are not valid and may lead to erroneous values for qst; because of the deviations caused by the ideal gas law, little improvement is obtained by accounting for the adsorbed phase volume. Actually, the ideal gas assumption caused a larger error than the error caused by neglecting the adsorbed phase volume. The P-V curves of ethane at 300 K predicted from the ideal gas law and CSPMA are compared to experimental data3 in Figure 1c. The CSPMA curve agreed very well with the experimental data over the whole pressure range, while the ideal gas law only showed good agreement at low pressures. At higher pressures, the ideal gas law overpredicted the gas phase molar volume at the same pressure. The three P-V curves were quite close, however, for P < 10 atm; nevertheless, the qst’s in Figure 1a deviated from each other even for P < 10 atm. This demonstrates that even small differences in V may make considerable differences in the qst’s. What is quite disturbing about these results is that the higher pressure range is still within the operational pressures of industrial adsorption processes designed for light hydrocarbon and other light gas separations.6 Figure 2a shows that the qst’s of propane calculated from the three different methods were close to each other when the loading was lower than about 3 mol/kg (corresponding to P < 2 atm). Above 3 mol/kg, however, the deviation became considerable and increased with loading. At P ) 8.4 atm, the qst obtained by method 1 was about 22% higher than that obtained by method 3, and the qst obtained by method 2 was 15.5% higher than that obtained by method 3. Moreover, the qst from method 1 was generally very close to that obtained from method 2, suggesting that neglecting the adsorbed phase volume had very little effect in this pressure range and that differences in the qst’s at high pressures were caused mainly by the inability of the ideal gas law to describe the bulk gas behavior. Figure 2c shows that the P-V curves of propane at 300 K predicted from the ideal gas law and CSPMA were quite close to the experimental data.4 However, the ideal gas model began to overpredict the gas phase molar volume at higher pressures. This slight overprediction in the P-V curve apparently had a significant effect on the qst curve for propane, again at pressures which are relevant to industrial adsorption processes. Figure 3a shows that the qst’s obtained by the different methods for butane were close to each other at low loadings. The qst’s from method 3, however, began to deviate from the other two methods at loadings higher than 3 mol/kg, (corresponding to P > 0.3 atm), with the
Notes
Langmuir, Vol. 14, No. 21, 1998 6327
difference becoming larger with increasing pressure. At P ) 6.75 atm, the qst obtained from method 1 was about 18% higher than that obtained from method 3, and the qst from method 2 was about 15% higher than that obtained from method 3. Figure 3a also shows that the qst’s obtained from method 1 were very close to those and followed the same trend as those obtained from method 2 over the whole pressure range, even closer than that for the case of propane. This demonstrates that the adsorbed phase molar volume is indeed negligible for relatively strong adsorbates at least in the pressure ranges studied. The P-V curve of butane at 350 K predicted by CSPMA agreed very well with that from the experimental data,5 while the ideal gas law curve showed some deviation at higher pressures (Figure 3c). Again the correction for nonideality in PVT seemed small, but it certainly caused a marked difference in the qst’s (Figure 3a). This effect of small corrections in the gas phase molar volume also appeared to be more significant for stronger adsorbates having lower saturation vapor pressures. Moreover, these effects on butane were very much in the range of the operating conditions of commercial adsorption processes, which suggests that the ideal gas assumption must be used with caution in calculating qst for many adsorbates. Conclusions The two critical assumptions used in deriving the Clausius-Clapeyron equation, i.e., an ideal bulk gas phase and a negligible adsorbed phase molar volume, have a significant effect on the calculated isosteric heats of adsorption (qst), especially at high relative pressures for heavy adsorbates. A study was carried out based on NDFT calculations for ethane, propane, and butane adsorbed on the heterogeneous BAX activated carbon at 300, 300, and 350 K, respectively and over pressure ranges of commercial interest. The largest effect came from the ideal gas
assumption, which was valid for all of these adsorbates up to around 3 mol/kg. Above this loading, however, deviations in the qst’s were quite significant. In contrast, the negligible adsorbed phase volume assumption affected the calculated qst’s only at very high pressures approaching the saturation vapor pressure of the adsorbate. However, for lighter adsorbates, the effects become more pronounced at lower relative pressures. Therefore, depending on the adsorbate and the operating pressure (loading) range of interest, both assumptions can cause significant errors in the calculated qst’s. Nevertheless, it is fortuitous that the assumption with the largest effect is the easiest to correct. The molar volume of the bulk gas phase is relatively easy to predict with high accuracy. As an example, this study showed that the Carnahan-Starling equation of state coupled with a mean field attraction term agrees very well with experimental PVT data from the literature. Correcting for the adsorbed phase molar volume is not so easy, however, especially based on experimental data, and NDFT calculations like those used here may not be accurate enough for calibrating the Clausius-Clapeyron equation. Note that the hydrocarbons in this work were assumed to be spherical and described by the LennardJones model, but because they are small, these approximations should be reasonable. Nevertheless, the use of eq 11 (or eq 13) with a suitable equation of state for real gases should give reasonable values for qst even at relatively high loadings, at least significantly better than those obtained from the Clausius-Clapeyron equation. Acknowledgment. J.A.R. acknowledges financial support from the National Science Foundation under Grant CTS-9410630 and from the Westvaco Charleston Research Center. LA9803373
