Langmuir 2005, 21, 9555-9561
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Estimation of Adsorption Parameters From Temperature-Programed-Desorption Thermograms: Application to the Adsorption of Carbon Dioxide onto Naand H-Mordenite Jose´ A. Delgado* and Jose´ M. Go´mez Department of Chemical Engineering, Universidad Complutense Madrid, 28040, Madrid, Spain Received April 11, 2005. In Final Form: June 8, 2005 In this work, a model is proposed for the estimation of the adsorption parameters from TPD thermograms when the adsorption cell can be modeled as a well-mixed reactor, evaluating the adsorption and desorption rate constants from statistical thermodynamics. The estimation procedure consists of fitting the model to the experimental TPD thermograms using numerical methods. The study of the effect of readsorption in this system reveals that this effect must be taken into account in most cases. Only with high activation energies of adsorption may this effect be negligible. The model is used to estimate the adsorption parameters of the systems CO2-Na-mordenite and CO2-H-mordenite, including an analysis about the degrees of freedom of the adsorbed phase. The estimated values of the adsorption enthalpy have been compared with the ones obtained from adsorption equilibrium data.
Introduction Temperature-programmed desorption (TPD) is an experimental technique which is often used to characterize the adsorption properties of zeolites, for example, the acidity and basicity. Normally, the peak maximum temperature obtained from TPD thermograms is used to measure the strength of adsorption sites. Nevertheless, as the position of this maximum can depend on phenomena such as diffusional resistance and readsorption and on parameters such as the weight of catalyst and the flow of carrier,1,2 the comparison of the results obtained in different conditions may lead to wrong conclusions. A better parameter to estimate the adsorption strength is the enthalpy of adsorption (∆H) extracted from the TPD data,1,3 which requires the use of a physical model considering the effect of the system parameters on the TPD thermogram. In this estimation procedure, it is always necessary to estimate another parameter, in addition to the adsorption enthalpy, related to the adsorption entropy. This parameter depends on the mathematical expression used for the preexponential factor of the adsorption and desorption rate constants, which affects the estimated value of the adsorption enthalpy, as some correlation between both parameters cannot be avoided. Furthermore, the preexponential factor gives valuable information about the degrees of freedom of the adsorbate in the adsorbed phase. The precise physical significance of the preexponential factors can be obtained from the statistical treatment of the adsorption process.4,5 According to this treatment, the preexponential factors depend on temperature in a complex manner. Most * Corresponding author. Phone: +34 91 3944119. Fax: +34 91 3944114. E-mail:
[email protected]. (1) Kapustin, G. I.; Brueva, T. R.; Klyachko, A. L.; Beran, S.; Wichterlova, B. Appl. Catal. 1988, 42, 239. (2) Sawa, M.; Niwa, M.; Murakami, Y. Zeolites 1990, 10, 307. (3) Joly, J. P.; Perrard, A. Langmuir 2001, 17, 1538. (4) Laidler, K. J. In Catalysis, Fundamental Principles (Part 1); Emmett, P. H., Ed.; Reinhold Publishing Corporation: New York, 1954; Vol. 1. (5) Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press: London, 1978.
methods proposed in the literature for extracting the adsorption enthalpy from TPD data focus on the determination of the adsorption enthalpy only, and they are based on the assumption that the preexponential factors are constant.1,2,6,7 Other methods to estimate adsorption parameters also based on temperature programmed adsorption techniques do not consider the temperature dependence of the preexponential factor based on statistical thermodynamics.8 Recently, Joly and Perrard3 have proposed a model for estimating both the adsorption enthalpy and the degrees of freedom of the adsorbate from TPD measurements, using an expression for the adsorption equilibrium constant based on statistical thermodynamics. However, this model is limited to the case when the temperature is significantly lower than the characteristic vibration temperature of the adsorbate, which is not valid for physisorption on zeolites.5 In all of the models commented previously, several TPD experiments are necessary to estimate the adsorption parameters of the sorbate-sorbent system. In a previous work,9 a theoretical model was proposed for the quantitative analysis of TPD thermograms when the adsorption cell can be modeled as a well-mixed reactor, similar to the one proposed by Gorte,10 which can be used for the analysis of TPD data obtained from zeolites. The model allows the adsorption enthalpy to be estimated from one TPD experiment, taking into account the possible effects of accumulation in the gas cell, diffusional resistance, and readsorption. This model was validated with the system CO2-Na-mordenite. However, in this model, the adsorption and desorption rate constants were modeled in a classical manner (with constant preexponential factors). This approach has several disadvantages: (i) the effect of readsorption cannot be evaluated adequately, as the absolute value of the adsorption rate constant cannot (6) Cvetanovic, R. J.; Amenomiya, Y. Adv. Catal. 1967, 17, 103. (7) Palermo, A.; Aldao, C. M. Thermochim. Acta 1998, 319, 177. (8) Mugge, J. Adsorption Isotherms from Temperature Programmed Physisorption-Equilibrium and Kinetics. Ph.D. Thesis, University of Twente, 2000. (9) Delgado, J. A.; Go´mez, J. M. Langmuir 2005, 21, 3503. (10) Gorte, R. J. J. Catal. 1982, 75, 164.
10.1021/la050966u CCC: $30.25 © 2005 American Chemical Society Published on Web 09/02/2005
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be estimated from fundamental relations, (ii) the effect of temperature on the preexponential factors is not considered, so the mobility of the adsorbate in the adsorbed phase cannot be discussed, and (iii) the estimated value of the adsorption enthalpy has some error, because of the correlation between the preexponential factors and this parameter in the estimation procedure. In this work, a solution to these deficiencies is proposed by evaluating the adsorption and desorption rate constants from statistical thermodynamics. The new model is used to estimate again the adsorption parameters of the system CO2-Na-mordenite, including an analysis about the degrees of freedom of the adsorbed phase, and it is also applied to the system CO2-H-mordenite. The estimated values of the adsorption enthalpy have been compared with the ones obtained from adsorption equilibrium data.
θ is the surface coverage (q/qsat), T0 is the initial temperature, and β is the heating rate (T ) T0 + βt). The differential mass balance in the adsorbed phase for the surface diffusion model is given by
Experimental Section
where ka (Pa-1 s-1) and kd (s-1) are the adsorption and desorption rate constants, respectively. The initial conditions for the differential equations are p ) 0 and θ ) θ0. The normalized signal predicted by the model is given by
Na-mordenite was used as the adsorbent (20% binder, Si/Al ) 6, crystals of 1 µm), supplied by CU Chemie Uetikon AG. The BET surface of this material was 350 m2 g-1, with a pore volume of 0.13 cm3 g-1, as determined from the nitrogen adsorption isotherm at 77 K. The H form was obtained by ionic exchange of Na+ by NH4+ followed by calcination under air at 773 K. After this process, the concentration of sodium in the zeolite decreased from 4.6% to 0.5% (w/w), as measured with XRF. TPD experiments were carried out with a commercial apparatus (Micromeritics TPD/TPR 2900). Helium was used as the carrier gas (35 cm3 min-1). In a typical TPD experiment, 169 mg of crushed sample were put into the adsorption cell, dispersed in quartz wool (free volume ) 1.8 cm3). The resulting sample had a wide particle size distribution, the maximum particle size being about 0.3 mm. The sample previously outgassed overnight at 523 K was saturated with pure CO2 at 1 atm at room temperature for 1 h, purged with helium for 30 min, and then heated under helium flow at 10 K min-1. Adsorption equilibrium data were obtained with a volumetric installation. The adsorbent was outgassed at 523 K at least for 12 h prior to the adsorption experiments. Equilibrium data, for the determination of the adsorption enthalpy, were obtained at 279, 293, and 308 K.
Theory The model used to predict the TPD signal is based on the assumption that the sample cell is a well-stirred tank of volume V being pumped at a volumetric flow rate Q. The mass balance of sorbate in this volume is
dc mol net desorption rate ) Qc + V s dt
( )
(1)
where c is the concentration in the adsorption cell. The net desorption rate is
net desorption rate )
-3WD ∂q rp2 ∂x
( )
x)1
(2)
p
QT p p+ β x)1 VT0 T
( )
p
(3)
where p denotes the sorbate pressure in the adsorption cell, qsat is the maximum adsorption capacity of the sorbent,
( )
(4)
The boundary conditions for eq 4 are given by
x)0 x)1
∂θ )0 ∂x
3 ∂θ D ) kap(1 - θ) - kdθ rp2 ∂x
N.S. )
(5)
Q(p/RT0) Wqsatθ0
(6)
The absolute values of the adsorption and desorption rate constants may be evaluated from statistical thermodynamics.4 Taking into account that the adsorption equilibrium constant is equal to the ratio between the adsorption and desorption constants, their absolute values may be evaluated if the adsorption equilibrium constant and the adsorption rate constant are known. In the case of equilibrium between the gaseous phase and the adsorbed phase, the thermodynamic potentials of both phases must be equal. Denoting the number of molecules in the gaseous phase and the adsorbed phase respectively by Ng and Na, and the corresponding partition functions by fg and fa, then the condition of equilibrium may be written as
Na fa E ) exp Ng fg RT
( )
(7)
where E is the energy decrease per mol of adsorbate at the absolute zero due to adsorption. This parameter is the difference between the activation energies of desorption and adsorption. Furthermore, each partition function is made up of translational, rotational, and vibrational contributions; consequently eq 7 may be written as
(
)
vib frot Ea,des - Ea,ads Na ftrans a a fa ) trans rot vib exp Ng f RT f f g
where W is the sample weight, D is the intracrystalline diffusivity, rp is the particle radius (spherical particles), q is the adsorbed concentration (mol kg-1), and x is the dimensionless radial coordinate. Taking into account the ideal law of gases, and the variation of temperature with time in the mass balances
dp -3WRTDqsat ∂θ dt ∂x r 2V
∂θ D 1 ∂ 2∂θ ) x ∂t r 2 x2 ∂x ∂x
g
g
(8)
It is assumed that the intramolecular vibrations are not significantly changed upon adsorption, so that the internal entropy is the same in the gas phase and after sorption, which is reasonable for physisorption.5 For an ideal gas, ftrans is given by g
) ftrans g
(
)
(2πmkT)1/2 3NgkT h p
(9)
where m is the mass of a molecule of adsorbate, k is the Boltzmann’s constant, and h the Planck’s constant. For is given by the adsorbed phase, ftrans a
Estimation of Adsorption Parameters
ftrans ) a
(
)
(2πmkT)1/2 h
Langmuir, Vol. 21, No. 21, 2005 9557
ta
Vta/3 pore
(10)
where ta are the translational degrees of freedom in the adsorbed phase (from zero to 3) and Vpore is the accessible intracrystalline pore space. In this equation, the zeolitic cavities are idealized as boxes or tubes of rectangular crosssection, with the three sides of similar length, a simplification that has been previously proposed in the literature.5 Although this idealization is very simple for the mordenite pores, it reduces the complexity of the translational partition function significantly. For ta ) 0 (localized adsorption), the number of uncovered adsorption sites must be included in the equilibrium expression4 vib frot Na a fa ) trans rot vib NsNg f f f f g
g
g
(
sites
)
Ea,des - Ea,ads RT
exp
(11)
where fsites is the partition function of the adsorption sites (equal to 1). The rotational partition function is given by
frot )
( )
1 T σ Θr
rdf/2
(12)
(
i
( )) -hνi kT
-1
(13)
where νi is the vibration frequency in s-1, the product being taken over all modes of vibration. For linear molecules, there are 3N - 5 modes of vibration, where N is the number of atoms in the molecule, and for nonlinear ones, 3N - 6. If the internal vibrations of the molecules in the gaseous phase are preserved in the adsorption process, but a certain number of translational degrees of freedom are lost, these degrees of freedom are converted into modes of vibration in the adsorbed phase. If rotational degrees of freedom are also lost, they can also be converted into additional vibrations, especially for localized sorption.5 On this basis, the ratio between vibrational partition functions becomes
fvib a fvib g
[
(
) 1 - exp -
hν kT
])
-(t+r)
(14)
where ν is a mean value for the vibrations of the guest molecule relative to its intracrystalline environment, t are the translational degrees of freedom lost upon adsorption, and r are the rotational ones. Substituting eqs 9, 10, 12, and 13 into eq 8, the following equation is obtained:
Na
[
] ( ) ( ( ))
1/2 ta-3
(2πmkT) ) ta/3 h V
(
exp -
hν kT
T σΘr
-(t+r)
[
]( ) ( ( ) ))
(2πmkT)1/2 h
-r/2
1-
)
Ea,des - Ea,ads p (15) exp RT kT
-t
-(r/2)
T 1σΘr -(t+r) Ea,des - Ea,ads p hν (16) exp exp kT RT kT
ca ) N1-(t/3) ‚θ ) sat
(
where Nsat is the number of adsorption sites per unit of intracrystalline volume. This parameter can be estimated as qsatFpNA/p, where Fp is the particle density, p is the particle porosity, and NA is Avogadro’s number. Note that the expression is correct for t ) 3, because ca is replaced by Na/Ns, according to eq 11. From eq 16, it is deduced that
[
]( ) ( (
ka (2πmkT)1/2 ) kd h
-t
T σΘr
exp
where Θr is the characteristic rotational temperature, which depends on the existing moments of inertia in the molecule, σ is a symmetry number, and rdf are the rotational degrees of freedom. For linear molecules in the gaseous phase, rdf is 2, and for nonlinear molecules, it is 3. For the adsorbed phase, rdf may range from zero to the maximum corresponding value. The vibrational partition function is given by
fvib ) Π 1 - exp
Considering low pressures, so the number of available adsorption sites is near the maximum number, noting that Na/Vta/3 is the adsorbed concentration in units consistent with the translational degrees of freedom in the adsorbed phase (ca), and that t ) 3 - ta, the following equation is obtained from eq 15:
-(r/2)
(
1 - exp -
)
))
hν kT
-(t+r)
Ea,des - Ea,ads 1 (17) RT N1-(t/3) kT sat
For evaluating the value of ka, the activated complex theory4 states that adsorption takes place when an activated complex, in equilibrium with the gaseous phase, overcomes an energy barrier, which is the activation energy of adsorption. For simplicity’s sake, it will be assumed that the activated complex loses the same translational and rotational degrees of freedom as the adsorbate. The assumption of equilibrium between the activated complex and the gaseous adsorptive gives rise to the equation
(
)
vib frot -Ea,ads Nac ftrans ac ac fac ) trans rot vib exp Ng f RT f f g
g
g
(18)
where Nac are the number of activated complexes and Ea,ads is the energy of the complexes with reference to the gaseous adsorptives, at the absolute zero, and is therefore the activation energy at that temperature. For the case of localized activated complexes, the number of sites must be included in eq 18, like in eq 11. For the activated complex, one of the modes of vibration is of a very different nature from the rest, since it corresponds to a very loose vibration which allows the complex to dissociate into the products of reaction, i.e., into the adsorbed state in the present case of adsorption. For this degree of freedom, one may therefore employ, in place of the usual factor (1 - exp(-hνac/(kT)))-1, the value of this function in the limit at which ν tends to zero, kT/(hνac). The factors for the rest of modes of vibration are close to unity at ordinary temperatures, which is also applicable to the factors corresponding to the gaseous adsorptive. For the activated complex, ftrans is given by ac
) ftrans ac
(
)
(2πmkT)1/2 h
ta
Ata/2
(19)
where A is the available area for adsorption, which for zeolites is the available external area of the crystals. For ta ) 2, this expression for the translational partition function of the activated complex is equal to the one
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proposed by De Boer11 for a two-dimensional gas. The rate of adsorption is the product of the concentration of activated complexes by the frequency of their transformation, which may be calculated as
]( ) (
[
Nac (2πmkT)1/2 νac ta/2 ) νacN′satθ ) h A
-t
-(r/2)
T σΘr
)
Ea,ads p (20) RT h
exp -
where N′sat is the number of sites per unit of external area (for t ) 1). For zeolites, this parameter may be calculated as (Nsatrp/3)3/2-t/2. The expression for ka resulting from eq 20 is
ka )
[
]( )
(2πmkT)1/2 h
-t
T σΘr
-(r/2)
1 (Nsatrp/3)3/2-t/2h Ea,ads (21) exp RT
(
)
For the case of localized activated complex (t ) 3, r ) 0), this equation is equal to the one commonly used in the literature.4,12 For t ) 1 and r ) 0, this equation leads to the well-known expression for the adsorption rate constant derived from the Langmuir approach,13 replacing the term Nsatrp/3 by the saturation capacity in molecules m-2. The expression for kd is obtained by substituting eq 21 into eq 17, leading to
kd )
()
kT (t/6)-(1/2) rp N h sat 3
(
(t/2)-(3/2)
(
1 - exp -
))
hν kT
(
(t+r)
)
Ea,des exp (22) RT The complete model was solved numerically using the PDECOL program,14 which uses orthogonal collocation on finite elements technique. Further information about the calculation and fitting procedures used in this work can be found elsewhere.15 Results and Discussion Effect of Readsorption. The deviation from equilibrium at the fluid-solid interface in an adsorption process is not typically considered in gaseous systems, although it may be important in the adsorption of proteins.16,17 In a previous work,9 it was established that readsorption is negligible in a TPD apparatus (with an adsorption cell that can be modeled as a well-stirred tank) when the following condition is fulfilled:
kavWqsatRT0/Q < 0.1
(23)
where kav is an average adsorption rate constant calculated at T ) 0.5(T0 + 773 K). For kavWqsatRT0/Q > 20, free readsorption can be assumed. The statistical thermody(11) de Boer, J. H. The Dynamical Character of Adsorption, 2nd ed.; Oxford University Press: London, 1968. (12) Derrouiche, S.; Bianchi, D. Langmuir 2004, 20, 4489. (13) Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: Singapore, 1997. (14) Madsen, N. K.; Sincovec, R. F. ACM Trans. Math. Software 1979, 5, 326. (15) Delgado, J. A.; Nijhuis, T. A.; Kapteijn, F.; Moulijn, J. A. Chem. Eng. Sci. 2002, 57, 1835. (16) Rodrigues, A. E.; Ramos, A. M. D.; Loureiro, J. M.; Diaz, M.; Lu, Z. P. Chem. Eng. Sci. 1992, 47, 4405. (17) Azevedo, D. C. S.; Pais, L. S.; Rodrigues, A. E. J. Chromatogr. A 1999, 865, 187.
Figure 1. Simulated TPD signals for various cases: (a) localized adsorption without rotation (t ) 3, r ) 3, Ea,ads ) 0, solid line), (b) localized and activated adsorption without rotation (t ) 3, r ) 3, Ea,ads ) 50 kJ mol-1, dashed line), (c) localized and activated adsorption without readsorption (t ) 3, r ) 3, ka ) 0, dotted line). Other model parameters are given in the text.
namics approach allows the value of kav to be evaluated separately, so it can be known when the readsorption effect is important. A typical case considered in the literature3,12 is a nonactivated adsorption step (Ea,ads ) 0), with localized adsorption (t ) 3), whereas the adsorbate retains the rotational degrees of freedom (r ) 0). For this case, the adsorption rate constant is given by
ka )
h2 (2πmkT)3/2
(24)
For a commercial apparatus (as the one studied in this work), where a zeolite sorbent is studied, reasonable values for the rest of the parameters in eq 23 are W ) 150 mg, qsat ) 1 mol kg-1, T0 ) 298 K, and Q ) 35 mL min-1. This gives a value of the dimensionless number of 1.4 × 106, which clearly indicates that readsorption cannot be neglected and that free readsorption can be assumed. To avoid the effect of readsorption, very low W/Q ratios should be employed, making it difficult to detect the desorbed compound. Moreover, the lower this ratio is, the more important the effect of diffusional resistance is,9 which could affect the TPD signal. Therefore, when high carrier flow rates or high pumping rates are used to reduce the partial pressure of the adsorbate, the absence of diffusional resistance must be verified. According to eq 21, the case with a lower readsorption effect is when t and r have maximum values (localized adsorption without rotation), as the translation and rotation factors are higher than 1, assuming nonactivated adsorption. This case may occur in the desorption of ammonia from acid zeolites.3 For ammonia, σΘr ) 15 K and m ) 17 × 10-3/NA kg. Assuming t ) 3 and r ) 3 leads to kavWqsatRT0/Q ) 2.8 × 104, which is still very high to neglect readsorption. The remaining possibility to get negligible readsorption is that the activation energy of adsorption is significant, as it occurs in the chemisorption of gases as H2 and O2 on some metals and metal oxides.4 The activation energy of adsorption in these cases ranges between 40 and 80 kJ mol-1. Assuming Ea,ads ) 50 kJ mol-1 (other parameters are kept constant for comparison’s sake), the value of kavWqsatRT0/Q is about 0.36, indicating that readsorption begins to be significant. Figure 1 shows a comparison between the TPD signals obtained with and without activation energy of adsorption [for Ea,des ) 130 kJ mol-1, hν/kT . 1 (all of the adsorbate molecules in the fundamental level of energy), β ) 0.166 K s-1, V ) 1.8 × 10-6 m3, θ0 ) 0.8], with the no-readsorption case (ka ) 0). It was checked that free readsorption was
Estimation of Adsorption Parameters
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Figure 3. TPD thermogram of CO2 on H-mordenite. Experimental conditions are given in Table 1. Dashed line is calculated with the model, using the parameters given in Tables 1 and 2, for b ) 0 (eq 26). Dotted line is calculated with bθ0 ) 5.0 kJ mol-1. Table 1. Parameters Used for Fitting the Experimental TPD Data
Figure 2. TPD thermograms of CO2 on Na-mordenite. Experimental conditions for panel a are given in Table 1. The same conditions apply to panel b except for W ) 268 mg, T0 ) 300 K, q0 peak 1 ) 0.34 mol kg-1 and q0 peak 2 ) 0.70 mol kg-1. Dashed lines are calculated with the model, using the same model parameters for peaks 1 and 2 in both figures (Tables 1 and 2), except for the indicated ones.
applicable to the case with Ea,ads ) 0. The effect of activated adsorption is clear, resulting in a signal similar to the one obtained without readsorption, although with some difference, which confirms the prediction of the criterion. The diffusivity was set to a very high value to avoid diffusional control in this simulation. Therefore, as the activation energy of adsorption is low in many cases, it is deduced that readsorption must be taken into account when working with commercial TPD devices. Only with high activation energies of adsorption this effect may be negligible. Estimation of the Adsorption Parameters. The proposed model was tested in the analysis of the TPD of CO2 on Na-mordenite and H-mordenite. Desorption of CO2 from Na-mordenite gives two different peaks (Figure 2), which indicates that CO2 adsorbs on two types of sites, whereas one peak is obtained with H-mordenite (Figure 3), with a peak maximum temperature intermediate between the ones corresponding to Na-mordenite. It seems that the double peak obtained with Na-mordenite is due to the presence of sodium cations inside the channels. The two peaks observed for Na-mordenite were deconvoluted using a Gaussian fitting routine (Origin 5.1) and normalized. The model was fitted considering various states for the adsorbed phase, from a two-dimensional gas (t ) 1, r ) 0), to completely localized adsorbate, that is, t ) 3 and r ) 2. Other model parameters are shown in Table 1. The sorbent saturation capacity qsat was estimated from the adsorption equilibrium data of the CO2-Na-mordenite and CO2-H-mordenite systems. As the interaction of carbon dioxide with both Na- and H-mordenite comprises the interaction of its quadrupole moment with the field gradients created by sodium cations in Na-mordenite, and silanol groups in H-mordenite,
parameter
apeak 1, Na-MOR
peak 2, Na-MOR
H-MOR
Q (m3 s-1) V (m3) W (kg) Fp (kg m-3) p T0 (K) β (K s-1) rp (m) D0 (m2 s-1) Ea,diff (kJ mol-1) qsat (mol kg-1) q0 (mol kg-1) m (kg) σΘr (K) Ea,ads (kJ mol-1)
5.83 × 10-7 1.8 × 10-6 1.69 × 10-4 1260 0.26 296 0.166 0.5 × 10-6 9 × 10-10 23 5.47 0.26 7.3 × 10-26 1.15 0
5.83 × 10-7 1.8 × 10-6 1.69 × 10-4 1260 0.26 296 0.166 0.5 × 10-6 9 × 10-10 23 5.47 0.68 7.3 × 10-26 1.15 0
5.83 × 10-7 1.8 × 10-6 1.66 × 10-4 1260 0.26 300 0.166 0.5 × 10-6 9 × 10-10 23 3.9 0.77 7.3 × 10-26 1.15 0
a
This peak is the one with lower peak maximum temperature.
together with dispersion and polarization interactions,5 it is reasonable to assume that chemisorption does not occur, so the activation energy of adsorption is low. On this basis, this parameter was set to zero. With regards to the diffusion parameters, as diffusivity data of carbon dioxide in Na-mordenite were not found in the literature, the diffusivity parameters were assumed to be equal to the ones corresponding to a similar zeolite with smaller pore size, for which these data are available (CO2-4A,18 D (m2 s-1) ) 9 × 10-10 exp(-2768/T)). It was observed that diffusivity was not a controlling step using these diffusion parameters, so it is reasonable to assume that the same applies to the systems studied in this work. As the sample consisted of crushed pellets, it was also checked that the diffusional resistance in the macropore network present in the particles formed by zeolite crystals and binder was negligible even for the largest particles (about 0.3 mm), using a criterion proposed in a previous work for zeolite pellets.9 The adjustable parameters were the average vibration frequency in the adsorbed phase (ν), and the activation energy of desorption (Ea,des). Fitting results are shown in Table 2. A comparison between the calculated and the experimental TPD thermograms is shown in Figures 2 and 3. It is observed that the model reproduces adequately the experimental peaks, especially for Hmordenite. All of the states give a very similar calculated curve, so the calculated curves correspond to any of the (18) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites; John Wiley and Sons: New York, 1992.
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Table 2. Fitting Results for Different States in the Adsorbed Phase and Calculated Adsorption Enthalpies at 293 K peak
t
r
(-∆H)293 K (kJ mol-1)
Ea,des (kJ mol-1)
ν (s-1)
r2
Na-MOR, peak 1
1
0
38.2
34.6
1.47 × 1014
0.9740
2 3 3 1
0 0 2 0
46.0 46.5 46.9 46.2
42.8 45.7 48.8 44.8
1.15 × 1013 3.65 × 1012 1.86 × 1012 8.09 × 1011
0.9904 0.9906 0.9910 0.9917
2 3 3 1 2 3 3
0 0 2 0 0 0 2
47.0 47.9 49.6 39.2 39.8 40.4 41.5
46.8 48.7 52.5 37.9 39.6 41.3 44.6
6.41 × 1011 5.93 × 1011 6.75 × 1011 7.07 × 1011 5.46 × 1011 5.01 × 1011 5.93 × 1011
0.9920 0.9924 0.9929 0.9963 0.9964 0.9966 0.9968
Na-MOR, peak 2
H-MOR
assumed states, except for the state (t ) 1, r ) 0) for peak 1 in Na-mordenite, which gave poorer results. In the previous section, it was deduced that the assumption of free readsorption is probably valid if the activation energy of adsorption is low. To verify this point, the model was tested with an experiment performed with a different weight of sample (268 mg instead of 169 mg), with the rest of the conditions being the same. If free readsorption is applicable, the peak position must change with the ratio W/Q, being displaced to higher temperatures as this ratio increases. Figure 2b shows that this is the case. The peak shift was also predicted adequately by the model, as it is shown in this figure. To determine the state of the adsorbed CO2, the quality of fitting is not very helpful, as it is similar for all of the states. However, this criterion allows excluding the state (t ) 1, r ) 0) for the first peak in Na-mordenite, which is quite reasonable, as the presence of sodium cations in the channels impedes the movements of a two-dimensional gas inside them. According to Barrer,5 the states with t ) 1 are improbable in porous crystals, because the energy contour across the cages or channels is not flat, so that translations must be replaced by soft vibrations for transverse motions. Therefore, this state may be excluded for the other peaks. Other approach to the degrees of freedom of adsorbed CO2 can be made by considering the value of the vibration frequency. The same author5 proposes that, for soft vibrations of physically sorbed molecules one expects the vibrational energy quanta (hν) to be small compared with kT. This criterion applied to the adsorption of CO2 in zeolites suggests that the vibration frequency of this adsorbate should be not much higher than 1012 s-1 (less than about 2 × 1012 s-1), considering the binding energy of this adsorbate. This limit may be used to check the reasonableness of the states assumed for the adsorbed phase. Based on this criterion, only the state (t ) 3, r ) 2) is reasonable for peak 1 in Na-mordenite, whereas the rest may be excluded. For the other peaks, this criterion does not allow excluding any state, as all of the estimated frequencies are clearly below the limit. Another criterion is the value of the adsorption enthalpy resulting from each state, which can be calculated by application of van’t Hoff’s equation to eq 17
-∆H )
(t +2 r + 1)RT + (E
a,des
- Ea,ads) -
hν (t + r)R k
hν kT (25) hν 1 - exp kT
( )
(
exp -
(
)
)
Figure 4. Adsorption enthalpy of CO2 adsorbed in Na- and H-mordenite for different adsorbate loadings. The solid line corresponds to the values estimated from adsorption equilibrium data for Na-mordenite, and the dashed one, for H-mordenite. Squares correspond to the values estimated from the TPD thermogram for peak 1 in Na-mordenite, circles for peak 2 in Na-mordenite (covering squares partially), and triangles for H-mordenite.
The value of the adsorption enthalpy at 293 K calculated with eq 25 is given in Table 2 for each state. This enthalpy can been compared with the adsorption enthalpy obtained from adsorption equilibrium data, to see if both methods give similar results and to see which states give adsorption enthalpies more similar to the values obtained from equilibrium data. In Figure 4, these adsorption enthalpies are compared for the same adsorbed concentration, where the state (t ) 1, r ) 0) has been excluded. For Namordenite, it was not possible to obtain reliable equilibrium data at the same loading as the TPD experiment because the equilibrium gas pressures for this loading were very low, in the precision limit of the pressure meter. It is observed that both methods give similar results for both sorbents, which proves that the proposed model is valid. As far as the reasonableness of the different states is concerned, it seems that the most localized state (t ) 3, r ) 2) is the most probable one for Na-mordenite, as it gives the highest adsorption enthalpies for both peaks, nearer to the one estimated from the adsorption equilibrium data. The loss of rotational degrees of freedom can be attributed to the orientation of CO2 molecules in the channels because of the interaction of its permanent quadrupole moment with the electrostatic field gradient caused by sodium cations.5 The high value of the estimated frequency for peak 1 (1.9 × 1012 s-1) with respect to peak 2 (7 × 1011 s-1) can be explained in terms of a lower entropy of the adsorbate in the sites associated to peak 1, since the vibrational entropy decreases as the vibration frequency increases. The lower entropy for peak 1 may be due to the restrictions imposed to CO2 molecules adsorbed in the associated sites. These sites may be the sodium cations located in the side-pockets along the wide channels of mordenite,5 which offer a more limited space for adsorption than the channels. For H-mordenite, the criterion based on the adsorption enthalpy seems to indicate that the state (t ) 3, r ) 0) is the most probable (Figure 4), although the states (t ) 2, r ) 0) and (t ) 3, r ) 2) cannot be ruled out. Considering that in H-mordenite the strong field gradients due to sodium cations are not present, the preservation of the rotational degrees of freedom in the adsorbed phase seems reasonable. Validity of the Assumption of Constant Activation Energy of Desorption. The model used in this work to estimate the adsorption parameters from TPD measurements is based on the assumption that the activation
Estimation of Adsorption Parameters
Langmuir, Vol. 21, No. 21, 2005 9561
energy of desorption does not change with adsorbate loading; that is, the sorbent is energetically homogeneous. A previous study19 has shown that this assumption may result in a fictitious value of the vibration frequency estimated with a TPD technique. However, this study was performed for systems without readsorption, so this conclusion must be applied to conventional TPD systems with caution. To analyze the effect of the energetic heterogeneity in our system, the model was modified considering that the activation energy of desorption decreases linearly with loading, according to the following equation:
(
Ea,des(θ) ) Ea,des - b θ -
)
θ0 2
(26)
where b is a constant and Ea,des is the average value of the activation energy of desorption. The model was fitted to the TPD peak of H-mordenite for different values of the parameter b, setting the parameter Ea,des to the value estimated with b ) 0. It was observed that the effect of the variation of the activation energy of desorption is negligible for bθ0 < 0.8 kJ mol-1. For larger values of b, the peak becomes broader, and the resulting average frequency decreases slightly; for bθ0 ) 5.0 kJ mol-1, ν ) 4.4 × 1011, instead of 5 × 1011 s-1. The calculated curve for this case is shown in Figure 3. It is observed that the homogeneous model performs better than the heterogeneous one. This result may be attributed to the fact that the desorption parameters estimated from TPD measurements are given by the sites in which most of the adsorbate is adsorbed. Thus, the validity of the assumption of constant activation energy of adsorption suggests that the amount of sites with adsorption energies significantly higher and lower than the average value is not very high, which has been previously observed in the TPD of ammonia from acid zeolites.1 It is also deduced that the energetic heterogeneity of the sorbent does not affect the estimated vibration frequency dramatically in our system. Moreover, the estimated values of the average vibration frequency in Na- and H-mordenite are of the same order of magnitude as the ones estimated in the literature for the adsorption of carbon dioxide on other sorbents;5 9.95-1.2 × 1012 for Na-Y and 7.1-8.5 × 1011 for cellulose carbon, which is consistent with the previous conclusion. Conclusions A model is proposed for the estimation of the adsorption parameters from TPD thermograms when the adsorption cell can be modeled as a well-mixed reactor, evaluating the adsorption and desorption rate constants from statistical thermodynamics. The study of the effect of readsorption in this system, using a quantitative criterion proposed in a previous work,9 reveals that this effect must be taken into account in most cases. When high carrier flow rates or high pumping rates are used to reduce the partial pressure of adsorbate, the absence of diffusional resistance must be verified. Only with high activation energies of adsorption may the effect of readsorption be negligible. The model has been used to estimate the adsorption parameters of the systems CO2-Na-mordenite and CO2-H-mordenite. These parameters include the average vibration frequency, the activation energy of desorption, and the degrees of freedom in the adsorbed phase. The estimated values of the average vibration (19) Soler, J. M.; Garcı´a, N. Surf. Sci. 1983, 124, 563.
frequency, ranging between 5 × 1011 and 2 × 1012 s-1 are consistent with physical adsorption in these systems. The adsorption enthalpies estimated by application of the van’t Hoff equation to the adsorption equilibrium constant derived from statistical thermodynamics are in good agreement with the ones estimated from adsorption equilibrium data. The model proposed in this work is based on the assumption of constant activation energy of desorption. The validity of this assumption suggests that the amount of sites with adsorption energies significantly higher and lower than the average value is not very high. The analysis of the state of the adsorbed CO2 in Namordenite and H-mordenite suggests that localized CO2 without rotation is the most probable state in Namordenite, whereas a higher mobility of the adsorbate is possible in H-mordenite. Nomenclature β ) heating rate, K s-1 c ) concentration, mol m-3 D ) diffusivity, m2 s-1 Ea ) activation energy, J mol-1 p ) particle porosity f ) partition function h ) Planck’s constant, 6.626 × 10-34 J s k ) Boltzmann’s constant, 1.38 × 10-23 J K-1 ka ) adsorption rate constant, Pa-1 s-1 kd ) desorption rate constant, s-1 m ) mass of a molecule of adsorbate, kg N ) number of molecules or sites N.S. ) normalized signal ν ) vibration frequency, s-1 p ) partial pressure, Pa q ) adsorbed concentration, mol kg-1 Q ) volumetric flow rate, m3 s-1 θ ) surface coverage Θr ) characteristic rotational temperature, K R ) gas constant, 8.31 J mol-1 K-1 r ) rotational degrees of freedom lost upon adsorption rdf ) rotational degrees of freedom rp ) particle radius, m σ ) symmetry number Fp ) particle density, kg m-3 T ) temperature, K t ) time, s; translational degrees of freedom lost upon adsorption ta ) translational degrees of freedom in the adsorbed phase V ) volume, m3 W ) sample weight, kg x ) dimensionless spatial coordinate -∆H ) adsorption enthalpy, J mol-1 Subscripts sat ) maximum adsorption capacity av ) average 0 ) initial ads ) adsorption des ) desorption diff ) diffusion a ) adsorbed phase ac ) activated complex s ) adsorption site pore ) intracrystalline pore space g ) gas Superscripts rot ) rotational vib ) vibrational trans ) translational LA050966U
