J. Phys. Chem. B 2001, 105, 6853-6857
6853
Diffusion of Benzene in NaX Zeolite, Studied by Thermal Frequency Response A. Malka-Edery* and Ph. Grenier LIMSI-CNRS, BP 133, F91403 Orsay, France ReceiVed: October 31, 2000; In Final Form: April 2, 2001
The thermal frequency response technique (TFR) has been used to study the diffusion of benzene in NaX zeolite. The corrected diffusion coefficient is found to be independent of loading between 1.5 and 3.6 mol/ cage at 406 K. It is in reasonable agreement with pulsed-field gradient NMR or quasi-elastic neutron scattering methods, but a large discrepancy is observed with the tracer zero length column (TZLC) technique. The activation energy for diffusion determined by TFR, 27 kJ/mol, is in accordance with the TZLC value, but substantially higher than the values reported from the other techniques.
Introduction
is called the characteristic function. It can be written
The diffusivity of benzene in NaX zeolite has been extensively studied, either by macroscopic methods (gravimetry,1 piezometry,2,3 frequency response,4 zero length column, ZLC,5 tracer zero length column, TZLC6) or by microscopic methods (pulsed field gradient NMR, PFG-NMR,7 quasi elastic neutron scattering, QENS8,9). The self-diffusivities directly measured by the microscopic methods are always larger than the corrected diffusivities measured by the macroscopic methods, by up to fifty to hundred times.6,10 For the moment, no satisfactory explanation has been proposed to account for such discrepancies. New macroscopic measurements by the thermal frequency response method (TFR) are reported her. TFR Method The TFR method has been described elswhere.11 Let us briefly recall its main features. The adsorbent sample is situated in a chamber closed by a bellows permitting the volume of the chamber to vary periodically. This periodic variation can be expressed as a sum of sinusoidal variations. For an elementary sinusoidal volume variation, the volume can be written:
V ) Ve(1 - Veiωt)
(1)
where ω is the angular frequency and V is the relative amplitude of the volume variation.12-17 After a transient state, the pressure, P, and the temperature, T, are periodical functions of time at the same frequency:
P ) Pe[1 + pei(ωt+ψ)]
(2)
T ) Te + At ei(ωt+φ)
(3)
where p is the relative pressure amplitude, AT is the temperature amplitude and ψ and φ the pressure and temperature phases, respectively. The complex ratio between the temperature response and the pressure response is independent of time but depends on the frequency. This ratio (multiplied by the volume amplitude V), * To whom correspondence should be addressed. E-mail:
[email protected].
AT θp(ω) ) θpin(ω) + iθpout(ω) ) V ei(φ-ψ) p
(4)
Its amplitude and its phase depend on the thermodynamic and kinetic parameters of the system.11 The characteristic function can be calculated by a model allowing the determination of the kinetic parameters by curve fitting. In fact, at a given frequency the phase lag has the dimension of a time and then depends directly on the kinetics of the system. Thus, the phase is, in many cases, more adapted than the amplitude for kinetic parameters determination. The sample temperature is determined by the heat released by adsorption under varying pressure conditions (see below eq 6). At very low frequencies, the sample temperature is governed by the heat transfer with the wall, and the phase lag is positive, tending toward π/2 at frequency zero. When the frequency increases, the heat transfer has not had time enough to be completed during a period: the system becomes adiabatic and the phase lag tends toward zero, assuming that the mass transfer is much faster than the heat transfer. When the frequency still increases, the mass transfer, and thus the heat released by adsorption, has not had time enough to be completed during a period and the temperature phase lag becomes negative. The frequency at which the phase becomes negative is closely related to the mass transfer characteristic time.11 Model The NaX crystals used in the present study may be considered as spherical with a mean diameter of 50-80 µm. In such small particles, the heat diffusion is very fast (heat diffusion characteristic time less than 1 ms) and the temperature may be considered uniform. The relevant equations of the adsorbentadsorbate system are the following: Mass balance:
(
)
d PV + Vsqj ) 0 dt ReT
(5)
where V and Vs are the volumes of the chamber and of the sample, respectively. qj is the adsorbate mean concentration, and Re is the specific gas constant.
10.1021/jp004025w CCC: $20.00 © 2001 American Chemical Society Published on Web 06/26/2001
6854 J. Phys. Chem. B, Vol. 105, No. 29, 2001
Malka-Edery and Grenier
Heat balance:
Cs
dT 3h dqj + ∆T ) |∆H| dt Rc dt
(6)
where Cs is the overall volumic heat capacity of the sample (adsorbent + adsorbate), Rc is the crystal radius, ∆H is the sorption enthalpy, and h is the overall heat transfer coefficient from the sample to the surroundings. Diffusion equation:
( )
∂q Dc 2 ∂q r ) ∂t r 2 c ∂rc c
(7)
where Dc is the transport diffusivity. Boundary conditions take into account a finite mass transfer rate at the adsorbent surface:
∂q ∂rc
|
rc)0
∂q -Dc ∂rc
)0
|
rc)Rc
) ks(q|rc)Rc - q*)
(8,9)
where rc is the current radius and ks is the surface barrier transfer coefficient. The following mass transfer characteristic times τD and τs are associated with the diffusion coefficient and the surface barrier coefficient: 2
τD )
1 Rc 15 Dc
1 Rc 3 ks
τs )
(10,11)
State equation: The mass adsorbed at equilibrium, q*(P,T), is given by the linearized state equation:
q* - qe ) KP∆P - KT∆T
(12)
where ∆P ) P - Pe and ∆T ) T - Te, the subscript “e” denoting the mean value of the corresponding parameter. KP and KT are the slopes of the isotherm and of the isobar, respectively, at the equilibrium point. These equations allow the explicit calculation of the pressure and of the temperature responses. The diffusivity determined by TFR is the transport diffusivity Dc, corresponding to a mass transfer under a concentration gradient. On the contrary, the microscopic methods, PFG-NMR and QENS, give the diffusivity at equilibrium, or self-diffusivity, D0.18-20 Applying Darken’s law to the transport diffusivity gives a corrected diffusivity which, in principle, is identical to the self-diffusivity D0:
D0 ) Dc
[∂∂ lnln Pq] ) D T
PKP q
c
(13)
Experimental Section The apparatus built at LIMSI consists of a 600 cm3 chamber closed by a bellows.21 This bellows is moved by a cam and a stepper motor at constant frequency, comprised between 1 mHz and 30 Hz. The pressure is recorded by a fast Baratron gauge, and the temperature is measured by infrared detection. The minimum measurable mass transfer characteristic time is 1 ms. The measurements can be performed only in a definite domain of pressure: at very low pressure, the response is too small because the mass of gas displaced by the bellows, and thus the concentration changes in the sample, are very small. Thus, no reliable measurement has been possible under 4 Pa. On the other
Figure 1. Isotherms of NaX (sample I)-benzene. Full symbols and full lines: this work; empty symbols and dotted lines: Bu¨low et al.2 Key: (*) and (O) 373 K; (+) and (0) 403 K.
hand, at high pressure, a spurious effect of heating of the gas appears, limiting the measurement domain to a few kilopascals. Last, the temperature domain is restricted by technological constraints to -20 °C < Te < 150 °C. The NaX zeolite crystals used for the experiments have two origins: sSample I was provided by J. Ka¨rger, Leipzig, and synthesized in Zhdanov’s laboratory, Leningrad. The mean diameter of the crystals is 50 µm with very sharp distribution. The Si/Al ratio is 1.2. sSample II was synthesized by H. Kessler, “Laboratoire des Mate´riaux Mine´raux”, Mulhouse. The diameter of the crystals is between 70 and 80 µm. The Si/Al ratio is also 1.2. The porosity of both samples is close to 0.5. The weight of the sample is approximately 0.1 g. At first the sample is calcinated at 400 °C under oxygen flow for 48 h and then placed in the experimental chamber. Then, experiments are performed without removing the sample from the chamber. Before each experiment, the sample is reactivated overnight: 8 h heating to 400 °C and 3 h of plateau at 400 °C followed by 3 h cooling up to Te. During the reactivation, the chamber is pumped down, and the pressure is approximately 10-4 Pa at the end of the process. Thereafter, a known quantity of benzene is introduced in the chamber, and after a while, the system reaches the equilibrium pressure, Pe. Results Isotherms. As seen previously, the kinetics is obtained by fitting the experimental and calculated phases. To run the model, it is necessary to know the thermodynamic parameters. Thus, isotherms have been measured before experiments, allowing the determination of the differential adsorption capacities, ∂q/∂P and ∂q/∂T, and the differential heat of adsorption, ∆H, has been obtained by the Clausius-Clapeyron equation:
∆H )
ReTe2 [∂q/∂T]P Pe [∂q/∂P]T
(14)
The isotherms have been measured by the volumetric method with a Baratron gauge, 104 Pa Full Scale whose accuracy is approximately 0.02 Pa, using a time scale of about 30 min to establish the end points. As shown in Figure 1, isotherms of sample I are in good agreement with the results obtained elsewhere.2 Isotherms of sample II (Figure 2) are quite different: at 353 K and larger temperatures, a shoulder appears at medium
Diffusion of Benzene in NaX Zeolite
J. Phys. Chem. B, Vol. 105, No. 29, 2001 6855
Figure 2. Isotherms of NaX (sample II)-benzene: (O) 323 K; (*) 337 K; (4) 353 K; (+) 373 K; (0) 393 K; (×) 403 K; (]) 413 K.
Figure 4. Temperature vs pressure phase lag as a function of the frequency. In this example (8th run, sample II, 4 mol/cage, and 353 K), the phase lag tends toward -π/4 at high frequency, indicating a pure diffusion mass transfer control. Key: (*) experimental data; (s) best fit with Dc ) 1.6 × 10-10 m2 s-1; (- - -) Dc ) 2.2 × 10-10 m2 s-1; (- -) Dc ) 1.2 × 10-10 m2 s-1.
Figure 3. Detail of isotherms of NaX (sample II)-benzene showing the crossing of 337 and 353 K isotherms. Same legend as in Figure 2.
TABLE 1: Main Experimental Conditions run sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14
I I I I II II II II II II II II II II
P, Pa
T, K
5.8 4.9 13.2 48.2 55.2 5.78 15.17 67.45 135.1 238.5 334.0 201.0 118.3 389.8
373 406 406 406 353 337.3 337.2 353.0 373.5 392.2 413.3 403.7 392.8 412.7
q, KP, darken kg/m3 mol/cage (kg/m3)/Pa factor 198 111 156 205 270 259 270 269 264 256 246 246 246 246
2.9 1.6 2.3 3.0 4.0 3.8 4.0 4.0 3.9 3.8 3.6 3.6 3.6 3.6
5.3 9.0 3.5 0.55 0.17 1.9 0.61 0.21 0.084 0.050 0.052 0.082 0.128 0.045
6.4 2.5 3.4 7.7 29 24 29 19 23 21.5 14.2 14.9 16.2 14.0
pressure that does not appear at low temperatures (323 and 338 K). Moreover, a (reproducible) singularity is observed at low pressures, as shown in Figure 3: the 338 and 353 K isotherms intersect, indicating that a state transition occurs between 338 and 353 K. It is likely that this singularity has the same origin as the singularity observed by Tezel and Ruthven.22 Diffusivity. The kinetic experiments have been performed under experimental conditions summarized in Table 1. Figures 4 and 5 show typical results of the temperature/pressure phase lag as a function of the frequency. The calculated curves are obtained by adjusting the transport diffusion coefficient, Dc, and the surface barrier coefficient, ks. The error has been evaluated as indicated on the figures. In most cases the estimated upper
Figure 5. Temperature vs pressure phase lag as a function of the frequency. In this example (1st run, sample I, 3 mol/cage, and 373 K), the phase lag becomes smaller than -π/4 at high, frequency indicating the presence of a surface barrier. Key: (*) experimental data; (s) best fit with Dc ) 3.5 × 10-10 m2 s-1 (τD ) 0.12 s); (- - -) Dc ) 2.0 × 10-10 m2 s-1; (- -) Dc ) 6.0 × 10-10 m2 s-1. All fits are obtained with ks ) 1.6 × 10-4 m s-1 (τs ) 0.06 s).
limit of the diffusivity is approximately twice the lower limit. The phase of the run presented in Figure 4 tends approximately toward -π/4 at high frequency. This is characteristic of a pure diffusion mass transfer control.11 On the contrary, the phase of the run presented in Figure 5 becomes smaller than -π/4 at high frequency, denoting the presence of a surface barrier. The whole of the results is presented in Table 2. One can see that the characteristic time corresponding to the surface barrier never exceeds half the diffusion characteristic time. As with the isotherms, there appears a difference between sample I and sample II. In all experiments with sample I, the surface barrier resistance is small, but not negligible. On the contrary, with sample II, the surface barrier when it exists, is very small, especially at high temperature. This difference, in addition to the difference appearing on the isotherms, suggests that a slight structural difference exists between the two samples. Further experiments must be performed to clarify this point. The corrected diffusivity D0 is given also in Table 2. That corrected diffusivity is obtained by applying the Darken correction (eq 12) to the transport diffusivity Dc identified from the experimental data.
6856 J. Phys. Chem. B, Vol. 105, No. 29, 2001
Malka-Edery and Grenier
TABLE 2: Kinetic Results run
sample
1 2 3 4 5 6 7 8 9 10 11 12 13 14
I I I I II II II II II II II II II II
1010Dc, m2/s 2-6 1-4 2-3.5 6-11 2.0-3.2 2.0-5.0 1.4-2.8 1.2-2.2 3.5-7.0 6.0-10 7.0-15 9.0-18 4.0-9.0 7.0-14
darken factor 6.4 2.5 3.4 7.7 29 23.5 29 19 23 21.5 14.2 14.9 16.2 14.0
Figure 6. Corrected diffusivity of benzene in NaX zeolite at 403406 K: (×) sample I; (O) sample II, 12th run; (4) sample II, interpolation between 11th and 13th runs.
Figure 7. Corrected and self-diffusivity at ∼403 K of benzene in NaX zeolite: (*) this work sample I; (+) this work sample II; (×) this work sample II interpolated; (0) PFG-NMR;7 (O) PFG-NMR;23 (]) QENS;9 (4) TZLC.6 The TZLC data have been obtained from measurements at 468 K extrapolated at 403 K assuming 27.2 kJ/mol activation energy.5
The corrected diffusivity as a function of the concentration at 406 K for both samples is shown in Figure 6. One can see that there appears no significant variation of the corrected diffusivity between 1.5 and 3.6 mol/cage. The mean corrected diffusivity at 406 K is D0 ) 8.0 × 10-11 m2 s-1. Figure 7 shows the comparison of the corrected diffusivity obtained by macroscopic methods (TZLC6 and this work) and the self-diffusivity obtained by microscopic methods (PFGNMR7,23 and QENS9). One can see that the corrected diffusivi-
1011D0, m2/s
104ks, m/s
τD, s
τs, s
3-9 4-16 6-10 8-14 0.7-1.1 0.9-2.1 0.5-1.0 0.6-1.2 1.5-3.0 2.8-4.7 4.9-11 6.0-12 2.5-5.5 5.0-10
1-2 0.35-0.5 1-2.5 5-9 ∞ 3 4 3.2 ∞ ∞ ∞ ∞ 5.0 ∞
0.07-0.2 0.1-0.4 0.12-0.2 0.04-0.07 0.29-0.46 0.18-0.46 0.33-0.66 0.41-0.76 0.13-0.26 0.09-0.15 0.065-0.13 0.052-0.10 0.10-0.23 0.064-0.13
0.04-0.08 0.17-0.24 0.03-0.08 0.01-0.02 0 0.04 0.03 0.04 0 0 0 0 0.025 0
Figure 8. Arrhenius plot of the corrected diffusivity of benzene in NaX obtained by TFR: (0) sample I at 3 mol/cage; (*) sample II at 4 mol/cage. The corresponding activation energy is ∼27 kJ/mol for both samples.
ties obtained by the TFR method are in good agreement with the self-diffusivities obtained by microscopic methods. On the contrary, there appears a large discrepancy with TZLC data. Activation Energy. Figure 8 shows the Arrhenius plot for sample I at 3 mol/cage and for sample II at 4 mol/cage. For sample II, the experiments lie between 3.6 and 4 mol/cage. Thus, some extrapolations have been applied to compare all results at 4 mol/cage. The extrapolations have been obtained following the results of Germanus et al.,7 indicating a decrease of the diffusivity from 3.6 to 4 mol/cage by approximately a factor 1.6 at 393 K. The same activation energy is obtained for both samples: 27-28 kJ/mol. This activation energy is close to the value, 27.2 kJ/mol, obtained from the slope of the Arrhenius plot of ZLC5 and TZLC6 data. Nevertheless, it is substantially higher than the values reported from the other techniques at comparable concentration: 16 kJ/mol by QENS,9 20 kJ/mol by PFG-NMR7, and 24 kJ/mol by the FR technique.4 Conclusion The thermal frequency response technique has been used to study the diffusion of benzene in NaX zeolite at various loadings and temperatures. Taken as a whole, it leads to results consistent with the results obtained by microscopic techniques, PFG-NMR and QENS. Nevertheless, the activation energy determined by TFR is 1.4-1.7 times larger than the activation energies determined by these microscopic methods. This shows that the discrepancies between the results obtained by various techniques do not depend on the macroscopic or microscopic character of the technique used.
Diffusion of Benzene in NaX Zeolite Nomenclature AT Cs Dc D0 h KP KT ks P, Pe p q, qe q* Re Rc T, Te V, Ve Vs V ∆H φ ψ ω
temp amplitude, K sample heat capacity, J m-3 K-1 transport diffusivity, m2 s-1 self-diffusivity (or corrected diffusivity), m2 s-1 heat transfer coeff, W m-2 K-1 slope of the isotherm, kg m-3 Pa1slope of the isobar, kg m-3 K-1 surface barrier coeff, m s-1 pressure, its mean value, Pa relative pressure amplitude adsorbed amount, its mean value, kg m-3 adsorbed amount at thermodynamic equilibrium, kg m-3 specific gas constant, J kg-1 K-1 crystal radius, m temp, its mean value, K chamber vol, its mean value, m3 sample vol, m3 relative vol amplitude differential heat of sorption, J kg-1 pressure vs vol phase lag temp vs vol phase lag angular frequency, s-1
References and Notes (1) Ruthven, D. M.; Doetsch, I. H. AIChE J. 1976, 22, 882. (2) Bu¨low, M.; Meitk, W.; Struve, P.; Lorenz,P. J. Chem. Soc., Faraday Trans. 1 1983, 79, 2457. (3) Bu¨low, M.; Mietk, W.; Struve, P.; Schirmer, W.; Kocirik, M.; Ka¨rger, J. In Proceedings of the 6th International Conference on Zeolites,
J. Phys. Chem. B, Vol. 105, No. 29, 2001 6857 Reno, 1983; Olson, D. Bisio, A., Eds.; Butterworth: Guildford, UK, 1984; p 242. (4) Shen, D.; Rees, L. V. C. Zeolites 1991, 11, 666. (5) Eic, M.; Goddard, M.; Ruthven, D. M. Zeolites 1988, 8, 327. (6) Brandani, S.; Xu, Z.; Ruthven, D. M. Microporous Mater. 1996, 7, 323. (7) Germanus, A.; Ka¨rger, J.; Pfeifer, H.; Samulevic, N. N.; Zdanov, S. P. Zeolites 1985, 5, 91. (8) Jobic, H.; Be´e, M.; Ka¨rger, J.; Pfeifer, H.; Caro, J.J. Chem. Soc., Chem. Commun. 1990, 341. (9) Jobic, H.; Fitch, A. N.; ombet, J. J. Phys. Chem. B 2000, 104, 8491. (10) Ka¨rger, J.; Ruthven, D. M. In Diffusion in Zeolites and Other Microporous Materials; Wiley & Sons: New York, 1992; p 449. (11) Bourdin, V.; Grenier, Ph.; Meunier, F.; Sun, L. M. AIChE J. 1996, 42, 700. (12) Evnochides, S. K.; Henley, E. J. J. Polym. Sci. 1970, 8, 1987. (13) Yasuda, Y. J. Phys. Chem. 1976, 80, 1867. (14) Be´temps, M.; Mange, M.; Scavarda, S.; Jutard, A. J. Phys. D: Appl. Phys. 1977, 10, 697. (15) Van Den Begin, N. G.; Rees, L. V. C. in Zeolites: Facts, Figures, Future; Jacobs, P. A., van Santen, R. A., Eds.; Elsevier: Amsterdam, 1989; p 915. (16) Jordi, R. G.; Do, D. D. J. Chem. Soc., Faraday Trans. 1992, 88, 2411. (17) Sun, L. M.; Meunier, F.; Grenier Ph.; Ruthven, D. M. Chem. Eng. Sci, 1994, 49, 373. (18) Pfeifer, H. Phys. Rep., Phys. Lett. Sec. C 1976, 26, 294. (19) Cohen de Lara, E.; Kahn, R. In Spectroscopic and Computational Studies of Supramolecular Systems; Davies, J. E. D., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; p 83. (20) Jobic, H.; Renouprez, A.; Be´e, M.; Poinsignon, C. J. Phys. Chem. 1986, 90, 1059. (21) Bourdin, V.; Gray, P. G.; Grenier, Ph.; Terrier, M. F. ReV. Sci. Instrum. 1998, 5, 2130. (22) Tezel, O. H.; Ruthven, D. M. J. Colloid Interface Sci. 1990, 139, 581. (23) Ka¨rger, J.; Ruthven, D. M. J. Chem. Soc., Faraday Trans. 1 1981, 77, 1485.
