Langmuir 2003, 19, 1173-1181
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A Simultaneous Description of Kinetics and Equilibria of Adsorption on Heterogeneous Solid Surfaces Based on the Statistical Rate Theory of Interfacial Transport Tomasz Panczyk and Wladyslaw Rudzinski* Group for Theoretical Problems of Adsorption, Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Krakow, Poland Received February 21, 2002. In Final Form: November 14, 2002 The statistical rate theory predicts that the observed kinetics of gas adsorption on solid surfaces may be affected by the construction of an experimental setup, and by the conditions at which the kinetic experiment is carried out. The current paper is a further extension of our recent theoretical studies of the kinetics of adsorption on/from energetically heterogeneous surfaces for the conditions when, at any instant of time, the investigated system is far from equilibrium (Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149). It is shown that under certain experimental conditions, concave kinetic isotherms might be observed in some adsorption systems. As such data are rarely reported in the literature, the conclusion is that the majority of the reported kinetic experiments were carried out under the conditions at which the adsorption systems could be considered as close to equilibrium. This is because for such experimental conditions the theory predicts existence of only convex kinetic isotherms for both energetically homogeneous and heterogeneous solid surfaces. For all these experiments an easy way is shown how to deduce the form of the kinetic isotherms from the equilibrium adsorption isotherms.
Introduction In many adsorption systems crucial for life, technology, and science, the time dependence of an adsorption process is at least as much important as the features of the adsorption system at equilibrium. However, the number of papers treating adsorption kinetics is incomparably smaller than those on adsorption equilibria. This is, particularly, true in the case of theoretical papers. Looking for a possible explanation, we consider the following fundamental reason. Namely, the methods of classical statistical thermodynamics provided a convenient tool for description of adsorption equilibria. On the contrary, in the case of the adsorption/desorption kinetics, such a convenient tool based on employing the statistical thermodynamics has been offered only recently. Beginning from the 1920s up to the end of the 20th century, the description of the kinetics of adsorption/ desorption on/from solid surfaces was commonly based on the absolute rate theory and its further conceptual developments.1
dθ ) Kap(1 - θ)se-a/kT - Kdθse-d/kT dt
(1)
where θ is the fractional surface coverage, t is the time, p is the nonequilibrium gas pressure, a and d are the so-called activation energies for adsorption and desorption, and Ka and Kd are some related constants. Furthermore, s is the number of the surface sites involved into an elementary adsorption/desorption process. The models of localized adsorption were usually used to describe adsorption/desorption kinetics, and the process was viewed * Corresponding author. Department of Theoretical Chemistry, Faculty of Chemistry, UMCS, pl. Marii Curie-Sklodowskiej 3, Lublin, 20-031, Poland. Phone: +48 81 5375633. Fax: +48 81 5375685. E-mail:
[email protected]. (1) Clark, C. A The Theory of Adsorption and Catalysis; Academic Press: New York, 1970.
as a kind of chemical reaction between the adsorbate molecules and these surface sites. The ART equation was extensively used to describe the adsorption/desorption kinetics also in the physisorption systems, where the gassolid interactions can hardly be considered as chemical reactions. So, no surprise that the inapplicability of the ART expressions to correlate experimental data were reported from the very beginning. The attempts to improve the applicability of the ART approach went into three directions: •taking into account the possible interactions between the adsorbed molecules (MFA, quasi-chemical approach, and computer simulations), •considering the energetic surface heterogeneity of the actual solid surfaces, (analytical descriptions, and computer modeling of the surface energetic heterogeneity), and •modifying the fundamentals of the ART approach (sticking coefficient, precursor states, etc.) Sometimes we used a combination of two from the above-listed strategies. However, all these improvements could not solve some questions which appeared in a theoretical analysis of the experimental kinetic data. In particular, very difficult problems are faced when the surface energetic heterogeneity of the real solid surfaces is taken into consideration. These problems have been discussed thoroughly in our very recent publications.2,3 So, at the beginning of the 1980s a certain feeling started to grow that the fundamentals of adsorption/desorption kinetics have to be deeply reconsidered. Although apparently different from the first viewpoint, the efforts went toward considering the chemical potentials of the adsorbed molecules µs and of the bulk molecules µg as the most fundamental quantities determining the features of adsorption/desorption kinetics.4-10 (2) Rudzinski, W.; Panczyk, T. Langmuir 2002, 18, 439. (3) Rudzinski, W.; Panczyk, T. J. Non-Equilib. Thermodyn. 2002, 27, 149. (4) de Boer, J. H. Adv. Catal. 1956, 8, 1. (5) Nagai, K.; Hirashima, A. Chem. Phys. Lett. 1985, 118, 401.
10.1021/la020193z CCC: $25.00 © 2003 American Chemical Society Published on Web 01/18/2003
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In a refined form this new orientation appeared finally in 1982 as the statistical rate theory of interfacial transport (SRT). On the bases of quantum mechanics and thermodynamics, Ward and Findlay have developed the following rate expression:11-13
[ (
)
(
)]
dθ µg - µs µs - µg ) K′gs exp - exp dt kT kT
(2)
where K′gs describes the exchange rate between the gas and adsorbed state at equilibrium.14-16 Its more detailed discussion will be given soon. Meanwhile, it is very important to emphasize that the SRT is the first ever theoretical approach that makes it possible to determine the features of adsorption/desorption kinetics from the knowledge of the behavior of an adsorption system at equilibrium.14,16 So far, only the reverse operation was possible. The equilibrium features of an adsorption system were found from the condition dθ/dt ) 0, and the expression for dθ/dt had to be assumed first. Now, assuming a certain adsorption model, and developing, by using the methods of statistical thermodynamics, the related expression for µs(θt,T), one arrives simultaneously at the equilibrium isotherm equation from the condition µs ) µg, and to the corresponding kinetic isotherm by inserting that expression for µs into eq 2. This has been demonstrated by Ward and co-workers studying the kinetics and equilibria of adsorption in the gas/solid systems with well-defined surfaces.14,16 One important feature of adsorption kinetics, revealed by SRT is that the experimentally monitored kinetics will depend on the construction of a particular experimental setup and on the conditions under which the kinetic experiment is carried out. In one of our recent papers17 we have published a theoretical analysis showing how the experimental conditions may affect the isothermal kinetics of adsorption/desorption on/from well-defined surfaces. Here we are going to extend this analysis by taking into account the energetic heterogeneity of the really existing solid surfaces.
After inserting in eq 2, the expression for µs calculated by using the methods of statistical thermodynamics, one has to calculate yet the pair p(e), θ(e), corresponding to p and θ, by considering the technical parameters of the applied experimental setup. To see how the construction of an experimental setup may influence the observed features of adsorption/desorption kinetics, Rudzinski et al. have considered the following three extreme cases:17 (1) The adsorption process is essentially a nonequilibrium one, and the features of the system are “volume dominated”, i.e., the amount of the gas in the gas phase above the surface dominates strongly over the adsorbed portion. In that case, after the system is isolated and equilibrated, the gas pressure p does not change much, so that p(e) ≈ p. (2) The process is essentially a nonequilibrium one, but the features of a gas/solid system are “surface dominated”. In this case the adsorbed amount prevails so strongly over the amount in the bulk gas phase that, after isolating the system and equilibrating, θ remains practically unchanged, so that θ ≈ θ(e). (3) The process is carried out under such conditions that the gas/solid system is close to equilibrium. We call such systems, “equilibrium dominated”, i.e., the process is carried out under such conditions, that one may assume θ ≈ θ(e) and p ≈ p(e). Now, let us consider which rate equations will be found in the three above-discussed cases, when at equilibrium the Langmuir isotherm equation is observed. For the Langmuir model of adsorption,
θ µs ) kT ln - kT ln qs 1-θ
The molecular partition function qs is written usually as the product q0s exp(/kT), where is the so-called “energy of adsorption”. (The local minimum in the gas/surface potential, taken with the reverse sign). Assuming that µg is the chemical potential of an ideal gas phase,
µg ) µ0g + kT ln p
(5)
we arrive at the Langmuir isotherm equation,
Theory 14-16
The K′gs in eq 2 has the following meaning:
K′gs ) Kgsp(e)(1 - θ(e))
(4)
θ) (3)
where the equilibrium state “e” is defined as the one to which a system isolated at the nonequilibrium pressure p and coverage θ would evolve. (6) Nagai, K.; Hirashima, A. Surf. Sci. 1986, 171, L464. (7) Nagai, K. Surf. Sci. 1988, 203, L659. (8) Nagai, K. Surf. Sci. 1991, 244, L147. (9) Kreuzer, H. J.; Payne, S. H. Thermal Desorption Kinetics. In Dynamics of Gas-Surface Interactions; Rettner, C. T., Ashfold, M. N. R., Eds.; The Royal Society of Chemistry, Thomas Graham House, Science Park: Cambridge, 1991; p 221. (10) Kreuzer, H. J.; Payne, S. H. Theories of Adsorption-Desorption Kinetics on Homogeneous Surfaces. In Equilibria and Dynamics Of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (11) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599. (12) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 5615. (13) Findlay, R. D.; Ward, C. A. J. Chem. Phys. 1982, 76, 5624. (14) Ward, C. A.; Elmoseli, M. B. Surf. Sci. 1986, 176, 457. (15) Elliott, J. A. W.; Ward, C. A. J. Chem. Phys. 1997, 106, 5667. (16) Elliott, J. A. W.; Ward, C. A. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G.; Elsevier: New York, 1997. (17) Rudzinski, W.; Panczyk, T. Adsorption 2002, 8, 23.
Kp exp(/kT) 1 + Kp exp(/kT)
(6)
in which K ) q0s exp(µ0g/kT). The Langmuir isotherm equation is also obtained from the ART eq 1, when s ) 1, and K ) (Ka/Kd) exp((d-a)/kT). For the Langmuir model of adsorption, the SRT expression for (dθ/dt) takes the following form:
1-θ dθ ) Kap exp dt θ kT 1 θ exp [1 - θ(e)]p(e) (7) Kd p1-θ kT
[
( )
)]
(
where
( )
Ka ) Kgsq0s exp
( )
Kgs µ0g µ0g , Kd ) s exp kT kT q 0
(8)
At equilibrium, when (dθ/dt) ) 0, eq 7 leads to the Langmuir isotherm eq 9, in which
K ) xKa/Kd ) q0s exp(µ0g/kT)
(9)
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Langmuir, Vol. 19, No. 4, 2003 1175
Thus,
1
1 - θ(e) ) 1 + Kp
(e)
(10)
kT
( )
exp
θt(p,T) )
Thus, for the “volume dominated” (V) systems we have
1-θ dθ ) Kap2 exp dt θ kT θ exp 1 + Kp exp Kd 1-θ kT kT
[
( ) ( )][
-1
( )]
(11)
whereas for the “surface dominated” (S) systems, we obtain
K d 1 θ2 2 dθ Ka ) p(1 - θ) exp dt K K p (1 - θ) kT
(
)
(12)
Finally, for the “equilibrium dominated” (E) systems we arrive at the following expression:
(1 - θ)2 dθ - Kdθ exp (13) ) Kap2 exp dt θ kT kT
( )
chemical potential of adsorbed molecules as a function of the experimentally monitored average surface coverage θt, defined as follows:26-28
(
)
Equations 11-13 describe the kinetics of adsorption/ desorption on a hypothetical energetically homogeneous solid surface, where all the adsorption sites are characterized by the same adsorption energy . None of the equations 11-13 is identical with the kinetic ART eq 1, leading to the Langmuir isotherm at equilibrium. The features of the kinetic equations 11-13 have been studied in detail by Rudzinski and Panczyk17 showing that eqs 11-13 can be analytically integrated to yield θ(t) functions exhibiting fully physical behavior when compared to the experimental data reported in the literature. Also, in some of their recent papers, Rudzinski et al.18-21 have shown, how eq 13 can be further generalized to describe both the kinetics of isothermal adsorption/ desorption processes on/from energetically heterogeneous surfaces, and the kinetics of thermodesorption.22-25 That method can be also used to generalize eqs 11 and 12, but here we will show another still simpler procedure. We start by invoking one fundamental assumption made to develop the SRT rate eq 2.11-13 Namely, it is assumed that in the course of the kinetic process leading to equilibrium, the adsorbed phase is still at “quasiequilibrium”. That means, all the surface correlation functions are close to their values which they would have at the same coverage of solid surface being at equilibrium with the gas phase. In the case of energetically heterogeneous surfaces instead of eq 4 we will have another expression for the (18) Rudzinski, W. A New Theoretical Approach to AdsorptionDesorption Kinetics on Energetically Heterogeneous Flat Solid Surfaces Based on Statistical Rate Theory of Interfacial Transport. In Equilibria and Dynamics Of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (19) Rudzinski W.; Panczyk, T. Surface Heterogeneity Effects on Adsorption Equilibria and Kinetics: Rationalisation of Elovich Equation. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J., Contescu, C., Eds.; Marcel Dekker: New York, 1999. (20) Rudzinski W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (21) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2001, 105, 6858. (22) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1999, 15, 6386. (23) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Adv. Colloid Interface Sci. 2000, 84, 1. (24) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Langmuir 2000, 16, 8037. (25) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Appl. Catal. A, 2002, 224, 299.
∫0∞θ(,p,T)χ() d
(14)
where χ() is the differential, normalized to unity distribution of the number of adsorption sites among their corresponding adsorption energies , which is commonly called the “adsorption energy distribution”. Further, θ(,p,T) is the so-called “local” isotherm describing the coverage of the sites characterized by the adsorption energy . This will be Langmuir isotherm in our considerations. Like in our previous publications, we will use here the CA (condensation approximation) method to calculate θt.29-31 Namely, we replace in eq 14 the Langmuir isotherm by the step-function θc, traditionally called the “condensation isotherm”, θc(p,T),
{
0 θc(p,T) ) 1
for for
< c > c
(15)
in which the step is located at c such that
( ) ∂2θ ∂2
)0
(16)
)c
Then, c ) -kT ln Kp and θt(p,T) in eq 14 takes the form
θt(p,T) )
∫∞χ() d c
(17)
It was shown in a number of theoretical papers,27,32-34 that eq 17 is a good approximation if the solid surface is strongly heterogeneous, i.e., when the variance of χ() is larger than that of the derivative (∂θ/∂). Then θ() resembles much the step function θc defined in eq 15. That means, at a certain pressure p and temperature T a sharp “adsorption front” is observed. The sites characterized by the adsorption energy > c are covered whereas the sites with < c are empty. That sharp adsorption front is found on the adsorption sites having the surface coverage θ ) θc(c) ) 1/2. The function χc(c), calculated from eq 17, called the “condensation function”,
χc(c) ) -
∂θt ∂c
(18)
is then a good approximation of the actual adsorption energy distribution χ(). It has the following relation to χ():
χc(c) )
χ() d ∫0∞(∂θ ∂ )
(19)
(26) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1989. (27) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Solid Surfaces; Academic Press: New York, 1992. (28) Do, D. D. Adsorption Anaysis: Equilibria and Kinetics; Imperial College Press: London, 1998. (29) Harris, L. B. Surf. Sci. 1968, 10, 129; 1969, 13, 377. (30) Cerofolini, G. F. Surf. Sci. 1971, 24, 391. (31) Cerofolini. G. F. J. Low Temp. Phys. 1972, 6, 473. (32) Rudzinski, W.; Jagiello, J. J. Low Temp. Phys. 1981, 45, 1. (33) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478. (34) Jagiello, J.; Schwarz, J. A. J. Colloid Interface Sci. 1991, 146, 415.
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When the surface is strongly heterogeneous, χc() f χ(). Under kinetic conditions, but when the adsorbed phase is still at quasi-equilibrium, we will observe the sharp “adsorption front” moving with time. The observed adsorption/desorption kinetics is now described by the equation
dc dθt ∂θ dc ) ) -χc(c) dt ∂c dt dt
[
]
(e)
θt (p,T) )
( ) ( )][
c dc ) -2kT Kap2 exp dt kT c c Kd exp 1 + Kp exp kT kT
( )]
( )]
[
( )
( )]
(
)]
kT/c
1 + (K0p)
, K0 ) K exp
( ) 0 kT
(27)
1 - θt θt
(28)
For the χ() function in eq 26, and the CA level of accuracy,
(23)
(24)
Of all the expressions 22-24, only the last expression dc/dt for the equilibrium dominated systems has already been applied in our previous publications18-21 to develop the θt(t) functions corresponding to various (assumed) distributions of adsorption energy χ(). There, we accepted the following strategies to arrive at the final expressions for the experimentally monitored function θt(t). First, we applied a fully accurate procedure, not involving the use of CA approximate method. Namely, having assumed a certain adsorption energy distribution χ(), we calculated next χc(c) by using eq 19. Next, we calculated θt(c), using eq 14, and found the inverse function c(θt). Finally, we expressed both χc(c), and (dc/dt) in eq 20 as the functions of θt, arriving, thus, at the first-order differential equation,
dc(θt) dθt ) -χc(c(θt)) ) F(θt) dt dt
(K0p)kT/c
(22)
Finally, for the “equilibrium dominated” (E) systems, we arrive at the following expression:
c c dc ) -2kT Kap2 exp - Kd exp dt kT kT
[
(26)
2
c ) 0 + c ln
whereas for the “surface dominated” (S) systems, we obtain
[
)
- 0 c - 0 1 + exp c
so commonly used to correlate the data for the equilibria of adsorption on strongly heterogeneous solid surfaces.26-28 In the limit of small adsorbate pressures (surface coverages), eq 27 reduces to Freundlich’s isotherm equation. From eqs 17 and 27, one arrives at the following expression for c(θt), to be inserted in eqs 22-24
-1
Kd 1 Ka 2c dc ) -2kT pexp dt K K p kT
(
exp
centered at ) 0, whose variance σ2 ) c2π2/3. Then, at the same level of CA accuracy, eq 17 yields the well-known Langmuir-Freundlich (LF) isotherm
(21)
)c,θ)1/2
To express (∂θ/∂t) we use one of the expressions 11-13, whereas the derivative (∂θ/∂) is calculated from the Langmuir isotherm equation because the adsorbed phase is assumed to be at the quasi-equilibrium. Then, from eqs 11-13 we arrive at the following three expressions for dc/dt. For the “volume dominated” (V) systems, we have
[
1 χ() ) c
(20)
The derivative (dc/dt) is to be calculated for the adsorption sites with θ ) 1/2. Thus,
dc (∂θ/∂t) )dt (∂θ/∂)
were obtained by using the CA method, i.e., by replacing χc() by χ() in eq 20. In particular, very interesting results were obtained by considering χ() to be the Gaussian-like adsorption energy distribution.
(25)
The differential eq 25 was next solved with the boundary condition θt(t)0) ) 0, to arrive at θt(t). Sometimes another boundary condition θt(t)0) ) constant was used, to account for an already preadsorbed amount. That fully accurate procedure appeared to be necessary to describe the kinetics of adsorption at high surface coverages in the systems characterized by the rectangular (constant) adsorption energy distribution.20 However, most interesting results
1 χ(c) ) χc(c) ) θt(1 - θt) c
(29)
In that way, we arrive at the following three explicit equations for dθt/dt, corresponding to the LangmuirFreundlich equilibrium adsorption isotherm 27. For the volume dominated (V) systems we have
[
2kTKa 2 0/kT(1 - θt)c/kT + 1 dθt ) pe dt c θ c/kT - 1 t
][
2kTKd -0/kT θtc/kT + 1 e 1+ c (1 - θ )c/kT - 1 t
( )
Kpe0/kT
1 - θt θt
]
c/kT -1
(30)
whereas for the surface dominated systems (S), we obtain
2kTKd -20/kT θt2c/kT + 1 dθt 2kTKap ) θt(1 - θt) e dt cK cKp (1 - θt)2c/kT - 1 (31) Then, for the equilibrium dominated systems (E), we arrive at the following equation:
dθt 2kTKap2 0/kT(1 - θt)c/kT + 1 ) e dt c θ c/kT - 1 t
2kTKd -0/kT θtc/kT + 1 e (32) c (1 - θ )c/kT - 1 t
Finally, for the equilibrium dominated systems, the solution of the differential equation (dθt/dt) has already been discussed in our previous paper.20 The integration
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Langmuir, Vol. 19, No. 4, 2003 1177
of that differential equation with the assumed boundary condition θt(t)0) ) 0 yields
θt(t) )
(Kpe0/kT tanh [2pKgst])kT/c 1 + (Kpe0/kT tanh [2pKgst])kT/c
(33)
[ (
) ]
1 Kpe0/kT
kT/c -1
c(T,p,t) )
[
(34)
[ (
1 + C exp(- 4pKgst) +
) ]
kT/2c -1
1 (Kp) e
2 20/kT
(35)
where the integration constant C is to be found from the appropriate boundary condition. However, it appears that the simple boundary condition θt(t)0) ) 0 cannot be applied for these two kinds of systems. This is because that boundary condition leads to the unacceptable value C ) +∝. Such intriguing behavior must suggest that some nonphysical assumptions become essential made in the theoretical development of eqs 34 and 35. It appears that this is nonphysical behavior of the χ() function in eq 26 defined between the nonphysical limits (-∝,+∝). As the adsorption starts at t ) 0 on the sites having the highest adsorption energies, the boundary condition θt(t)0) ) 0 leading to C ) +∝ must be related to the nonphysical upper limit of . To eliminate that problem we introduce the modified Gaussian-like adsorption energy distribution in eq 36, defined between the limits (-∝,m),
{
χ() )
(
)
- 0 exp 1 1 c FN c - 0 1 + exp c
[
(
)]
∈ (-∞,m)
for
2
c(T,p,t) ) kT ln
c(T,p,t) )
)
(
)
[Kp1 + (e
m/kT
-
1 -2pKgst e Kp
)
]
(39)
[
(
)
kT 1 1 + e2m/kT e-4pKgst ln 2 2 2 (Kp) (Kp)
]
(40)
The corresponding kinetics isotherms θt(t) are then calculated from the CA equation
θt(t) )
∫ χ() d m
(41)
c
where χ() is now defined in eq 36. After having done appropriate integrations we arrive at the following functions θt(t), fulfilling the condition θt(t)0) ) 0. For the E-systems we have
(
1 + exp θt(t) )
[
{
)
m - 0 c
1 + Kpe0/kT tanh 2pKgst + arc tanh
{
-
}}] ( ) (42) ( ) -kT/c
1 Kpem/kT
m - 0 c m - 0 1 + exp c
1 + exp -
For the V-systems we arrive at the following θt(t) function:
(
) )
m - 0 c θt(t) ) 1 1 (m - 0)/kT 1+ + e exp(- 2pKgst) Kpe0/kT Kpe0/kT 1 + exp -
[
(
(
-
] ) (43) ) kT/c
m - 0 c m - 0 1 + exp c
1 + exp -
(
1 + exp -
θt(t) )
m - 0 c FN ) m - 0 1 + exp c
(38)
whereas for the S-systems we obtain
(36)
where
(
1 Kpem/kT
(
whereas for the S-systems we obtain
0 for > m
exp
}}]
{
For the V-systems we arrive at the following function:
whereas for the S-systems, after integration in eq 32, we obtain
θt(t) )
{
-kT ln Kp tanh 2pKgst + arc tanh
Equation 33 has been next used successfully to correlate the kinetic data for CO2 adsorption on scandia, measured at various (constant) pressures and temperatures.35 Now let us consider the expressions for θt(t) corresponding to the Langmuir-Freundlich isotherm at equilibrium, provided that the adsorption system would not be close to equilibrium and would be “surface” or “volume” dominated. Then, for the V-systems, the integration in eq 31 yields
θt(t) ) 1 + C exp(- 2pKgst) +
So, first we solve the differential equations 22-24 with the boundary condition c(t)0) ) m. Then, for the equilibrium dominated (E) systems we have
)
m - 0
c kT/2c 1 1 2(m - 0)kT 1+ + e exp(- 4pKgst) 0/kT 2 0kT 2 (Kpe ) (Kpe ) m - 0 1 + exp c (44) m - 0 1 + exp c
[
(
)
]
(
(
(37)
And now we present an alternative and more simpler procedure of arriving at the functions θt(t). (35) Pajares, J. A.; Garcia Fierro, J. L.; Weller, S. W. J. Catal. 1978, 52, 521.
)
)
At equilibrium, i.e., when t f ∝, all these three kinetic isotherms yield the same equilibrium isotherm:
(
)
(
)
m - 0 m - 0 1 + exp c c θt(p,T) ) 0/kT -kT/c 0 1 + [Kpe ] m 1 + exp c (45) 1 + exp -
(
)
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Figure 1. The comparison of the behavior of the kinetic isotherms θt(t) developed for the E-, V- and S-systems, respectively. The parameters used in these calculations were following: 0 ) 13.7kJ/mol, c ) 5.88 kJ/mol, K ) 1.13 × 10-2 Torr, T ) 250 °C, p ) 5 Torr, and Kgs value 0.037 Torr-1 s-1.
which can be considered as the generalized LangmuirFreundlich isotherm 27 taking correctly into account the existence of a maximum adsorption energy m. Thus, having determined the parameters: K exp(0/kT), c, and (m - 0) from fitting equilibrium adsorption isotherms, θt(p,T), one may generate at least a family of the kinetic isotherms θt(t) for a chosen set of values of the parameter Kgs. In the case of E-systems, the related expression 38 for c suggests an easy way to deduce fundamental features of the kinetics of adsorption from the behavior of the adsorption isotherms θt(e)(p(e)) at equilibirum. The kinetic isotherm θt(t) is obtained from the equilibrium isotherm θt(e)(p(e)) by replacing p(e) by p and next multiplying p by the universal function of time: tanh{2pKgst}. And this should be true no matter which is the exact form of the adsorption energy distribution χ() in a particular system under investigation. So, one may arrive at the θt(t) function even when the θt(e)(p(e)) function is an experimentally measured equilibrium adsorption isotherm. Strictly speaking, one may obtain only a family of possible θt(t,Kgs) curves, corresponding to a certain choice of Kgs values. This is because in the case of energetically heterogeneous solid surfaces, an a priori theoretical calculation of Kgs constant represents a very difficult problem. Therefore we have treated Kgs as a best-fit parameter in our hitherto analysis of experimental data based on eq 33.20,21 However, knowing the corresponding family of the θt(t) curves should be interesting for certain studies of adsorption kinetics and equilibria.
Discussion As follows from the above considerations the existence of m does not affect much the behavior of E-systems, i.e., when the kinetic process is carried out in such a way that the system is not far from equilibrium. On the contrary a much different behavior is predicted for V- and Ssystems. We start our analysis by comparing the behavior of the functions θt(t) defined in eqs. (42-44) for the E-, V-, and S-systems, respectively. These three equations contain the same set of parameters; 0, m, c, K, and Kgs. While carrying out our illustrative calculations we used here values of the parameters similar to those found by us in our recent studies of the kinetics of CO2 adsorption on Scandia.20 For this particular system, the experimental kinetic data measured at various pressures and temperatures could be fitted very well by the simplest eq 33, so, there was no need to estimate the value of m for this system. While accepting the other parameters 0, c, K, and Kgs estimated there, we intend to see, how the choice of m affects the behavior of these three functions θt(t), defined in eqs 42-44. The results of our illustrative calculations are shown in Figure 1. In the C and E parts of Figure 1 we can see an intriguing behavior of V- and S-systems, predicted by our eqs 43 and 44 for high values of m ) 35 kJ/mol and m ) 50 kJ/mol. This is the concave shape of the kinetic isotherms θt(t), plotted versus time t. Meanwhile, one can hardly find such a concave shape among the reported experimental data. The reported kinetic isotherms θt(t) are convex as a rule.
Statistical Rate Theory of Interfacial Transport
Figure 2. The kinetic isotherms reported by Garcia and Pajares,36 measured at 648 K, and two pressures: 1.08 × 103 Pa (the lowest investigated pressure), and 8.72 × 103 Pa (the highest investigated pressure). It is to be noted that the kinetic isotherm corresponding to the lowest pressure 1.08 × 103 Pa is clearly concave.
Langmuir, Vol. 19, No. 4, 2003 1179
Figure 4. The results of fitting by eq 43 the experimental kinetic isotherm of hydrogen adsorption on Scandia, measured at 1.08 × 10-3 Pa and 648 K (b), by using the values of the parameters collected in Table 1. The window shows the results of fitting the first seven experimental points, by using the number in parentheses, collected in Table 1. Table 1. The Values of the Best-Fit Parameters Obtained from Fitting by Eq 43 the Kinetic Isotherm of Hydrogen Adsorption on Scandia, Measured at the Lowest Pressure 1.08 × 10-3 Pa and 648 K p [Pa]
K exp(0/kT) [Pa1-]
2Kgs [Pa1-min-1]
1.08 × 10-3 8.73 × 10-5 2.89 × 10-4 (5.31 × 10-5)a (6.29 × 10-4)
(m - 0)/kT 4.10 (5.94)
kT/c
Nm µg/g]
0.88 930 (0.88) (903)
a The values in parentheses are the corresponding values found by fitting the first seven points measured for the lowest surface coverages.
Figure 3. The properties of the volume dominated kinetic isotherm (eq 43) for certain values of parameters. On each graph all the parameters are the same except for these indicated in the corresponding legend. The values of the m are always the same and equal to 50 kJ/mol (this value of m generates a significant concave shape of the kinetic curve). The values of the other parameters were following: T ) 523 K, K ) 1.13 × 10-2 Torr-1, 0 ) 13.7 kJ/mol, c ) 5.88 kJ/mol, Kgs ) 0.037 Torr-1s-1.
A rare example of such a concave shape are the kinetic isotherms of hydrogen chemisorption on Scandia reported by Fierro and Pajares.36 Figure 2 shows two of them corresponding to the lowest and to the highest investigated nonequilibrium pressures p. These authors emphasize that “At pressures below 102 N m-2 the curve is slightly concave”. While looking for an explanation of that the authors suggest that this unusual behavior may be “due to (i) thermomolecular effect and (ii) control of the adsorption by the diffusion in the gas phase”. We do not exclude that their hypothesis may be correct, but we feel that the way in which their experiment was carried out should also be considered. Namely, their experiment might represent typical volume dominated kinetics. An RG Cahn electrobalance connected to a conventional HV system was used. Garcia and Pajares36 emphasize that “Its high dead volume (∼4 dm3) allows operation at constant pressure”. Thus, no doubt that it was a V-system used in their experiment. However, even in the case of the V- and S-systems concave kinetic isotherms will not be observed when the process (36) Garcia Fierro, J. L.; Pajares, J. A. J. Catal. 1980, 66, 222.
is carried out in such a way that the system is not far from equilibrium at any instant of time. Fierro and Pajares reported that their “samples were always degassed to 1.33 × 10-4 N m-2, whereas the lowest investigated pressure was 1.08 × 103 N m-2”. The authors also describe their efforts to minimize the thermomolecular effect. All that indicates that we might have to consider their experiment as a volume dominated kinetics. Thus, we have decided to study the features of such adsorption system to a more detail. Figure 3 shows the results of our illustrative model calculations for such systems. The results of our model investigations shown in Figure 3 suggest that the concave kinetic isotherms recorded at the lowest investigated pressure p ) 1.08 × 10-3 Pa should be reproduced by our eq 43, developed for V-systems. However this experimental kinetic isotherm is not particularly suitable for a quantitative best-fit analysis. This is because these kinetic data were recorded only at small initial surface coverages. This makes difficult to estimate the surface capacity (total number of sites M) necessary to calculate θt) Nt/M values, and to estimate the m parameter. There are also other uncertainties, some of them emphasized by Fierro and Pajares, which make a quantitative interpretation of their experiment difficult. While carrying out such quantitative interpretation of E-systems in our previous publications (CO2 adsorption on Scandia) we had at our disposal kinetic isotherms measured at various pressures, but always reaching plateau.20,21 Nevertheless we went ahead and fitted this experimental kinetic isotherm by eq 43. The results of our bestfit exercises are shown in Figure 4, and the determined best-fit parameters are collected in Table 1. Table 1 and Figure 4 show that the behavior of V-systems is not only sensitive to the existence of m, but also to the exact form of the adsorption energy distribution χ(). A much better fit (concave isotherm) is obtained when only the first seven experimental points are taken into consideration. The explanation for that seems to be simple.
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Panczyk and Rudzinski
Namely, it is to be expected that the actual χ() function in the system H2/Scandia will be much more complicated than the quasi-Gaussian function (36) accepted to develop eq 43. Such quasi-Gaussian function may only approximate to some extent the actual function χ(), within a certain domain of adsorption energies . The smaller is that domain the better approximation. In a CA picture of adsorption a certain domain of corresponds to a certain domain of surface coverages. So, the smaller is the considered region of surface coverages, the better fit of the kinetic data by eq 43. And because the behavior of the V-systems is so extremaly sensitive to the exact form of χ(), and the data for the highest pressure p ) 8.72 × 10-3Pa correspond to entirely different region of surface coverages, we did not attempt to fit these data by eq 43 using the parameters collected in Table 1. A successful fit of θt(t) for V-systems and large regions of surface coverages could only be obtained in the following manner. First the function χc(c) would have to be determined from equilibrium adsorption isotherm measured in such a large region of surface coverages. Next θt(t) should be calculated from eq 41 and eq 39 with a suitably chosen parameter Kgs. Unfortunately, such an equilibrium adsorption isotherm for this system was not measured by Fierro and Pajares.39 They measured only a small initial part of that equilibrium isotherm, corresponding to surface coverages much smaller than those investigated in their kinetic experiments. The E-systems are much less sensitive to the exact form of the adsorption energy distribution. This may explain why eq 33 could be successfully used by us to fit the experimental data for CO2 adsorption kinetics in a fairly large region of surface coverages. Of course it may also be so, that the CO2 function χ() for Scandia better resembles the quasi-Gaussian function 26, but the sensitivity of V-systems may also contribute significantly to the observed behavior of H2 adsorption kinetics. Namely, the observed kinetics is determined by both the thermodynamic features of an adsorption system and the technical conditions at which the experiment is carried out. This seems to explain why the kinetics of CO2 adsorption on Scandia behaves like being an equilibrium dominated one, while H2 adsorption does not. Although both these experiments were carried out in the same experimental setup the thermodynamic features of CO2/Scandia and H2/Scandia systems are much different. The adsorption of CO2 is roughly 100 times faster (the time when platteau on the kinetic isotherm is reached). Next the pressures at which H2 adsorption kinetics was studied were roughly 10 times higher than those applied in the experimental studies of CO2 adsorption kinetics. All that suggests that equilibrium is reached more easily in the case of CO2 adsorption. So, this may explain why the assumption made by us, that this was an equilibrium dominated system, and using the corresponding kinetic eq 33 led us to a good quantitative fit of experimental data. As the appearance of concave kinetic isotherms is a very intriguing prediction of the SRT approach, we decided to study it in more detail. The initially concave isotherm must become convex at the end, so the appearance of the concave shape can be followed by studying the conditions at which inflection point will appear in a kinetic isotherm,
( ) ∂2θt(t) ∂t2
)0 t)tf
(46)
Figure 5. The dependence of the reduced inflection time tr ) 2pKgstf on the dimensionless parameter (m - 0)/kT, calculated by considering various combinations of the dimensionless parameters: kT/c and K0.
where tf is the time when the inflection point is observed. While assuming that θt(t) in eq 46 is the kinetic isotherm 43, from eq 46 we obtain
1-
c tr e - K0e(m - 0)/kT kT K0e(m - 0)/kT + etr - 1
(
K0e
)(
tr
kT/c
c tr e +1kT
)
K0e(m - 0)/kT ) 0 (47) where K0 and tr are the dimensionless quantities
tr ) 2pKgstf
and
K0 ) Kpe0/kT
(48)
The concave shape (also inflection point) will be observed only for such values of the physical (conditions) parameters: K0, kT/c, and exp((m - 0)/kT), when the solution of eq 47 exists for positive values tr. Figure 5 shows the results of our numerical studies based on eq 47. One can see in Figure 5 that the value of the heterogeneity parameter kT/c does not affect much the time when the inflection point will appear. On the contrary, its appearance is strongly influenced by the maximum value of the adsorption energy m. The larger the difference (m - 0)/kT, the longer times at which the inflection point will be observed. Figure 5 also shows that the higher is the nonequilibrium pressure p, the shorter will be the time tf and a higher probability that the concave shape will be found in some adsorption kinetics. When the pressure p falls down below a certain value, concave shape may never be observed. Finally Figure 6 shows how m and p affect the shape of the kinetic isotherm θt(t). Figure 6 shows that, no matter which are the values of all the parameters, the inflection point will always be found within the initial first half of the kinetic isotherm.
Statistical Rate Theory of Interfacial Transport
Figure 6. The dependence of the relative surface coverage θt(tr)/θt(e) on the dimensionless parameter (m - l)/kT, calculated for the same combinations of the dimensionless parameters: kT/c and K0, as those accepted to calculate the results shown in Figure 5.
Next, the lower is the nonequilibrium pressure p, the closer to zero is the inflection point. These conclussions seem to
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be supported by the experimental data shown in Figure 2. One can see there, that in the case of the kinetic isotherm measured at the lower pressure 1.08 × 103 Pa the inflection point is clearly seen, and located at small coverages below 20 µg/g. In the case of the other isotherm measured at the higher pressure 8.72 × 103 Pa, the inflection point lies somewhere below 50 µg/g because the first three experimental points lie on a straight line, and their accuracy is to small to precisely estimate the position of that inflection point. We finish our discussion by certain remarks concerning the classical ART approach. Its applicability to represent the kinetics of adsorption on heterogeneous solid surfaces has been discussed in our previous paper.2 There we demonstrated the failure of that classical approach to describe in a consistent way the kinetics of CO2 adsorption on Scandia. Here, we have made an extensive search based on ART for the parameter values that could lead to reproducing the kinetics of hydrogen adsorption on Scandia. This time, we could not even reproduce the concave shape of this kinetics at the low adsorbate pressure 1.08 × 10-3 Pa. Acknowledgment. One of the authors (T. Panczyk) expresses his thanks and gratitude to the Polish Foundation for Science (FNP) for the Grant for Young Scientistis. LA020193Z
