Kinetics of Gas Adsorption in Activated Carbons, Studied by Applying

6858

J. Phys. Chem. B 2001, 105, 6858-6866

Kinetics of Gas Adsorption in Activated Carbons, Studied by Applying the Statistical Rate Theory of Interfacial Transport Wladyslaw Rudzinski* and Tomasz Panczyk Department of Theoretical Chemistry, Faculty of Chemistry UMCS, pl. Marii Curie-Sklodowskiej 3, Lublin, 20-031, Poland ReceiVed: NoVember 10, 2000; In Final Form: May 1, 2001

The Statistical Rate Theory of Interfacial Transport (SRTIT) is an approach that makes it possible to deduce the basic features of adsorption kinetics from the behavior of the system at equilibrium. In the case of activated carbons, the Dubinin-Astakhov (DA) isotherm has been known as the simplest and very efficient formula to correlate equilibrium adsorption isotherms, and their temperature dependence. Thus, using the SRTIT approach, we have developed the rate equation corresponding to the DA isotherm equation. The new kinetic equation has successfully been used to correlate experimental data for the kinetics of sorption of benzene by an activated carbon. That successful attempt has brought us to postulate another form of sorption kinetics, than those already recognized, i.e., “molecular”, “Knudsen”, and “surface”. Namely, that in the case of some adsorption systems, the Knudsen’s diffusion may be fast, and the rate-determining step is the exchange of mass between the pore walls and the interior of the pores.

Introduction The kinetics of gas adsorption in activated carbons is of a fundamental importance for their application in the environmental protection and in gas separation processes, as well as in other numerous applications. So, it is no surprise that many papers have been published that reported experimental data and proposed their theoretical interpretation. Finally, at least four excellent monographs, (Ruthven,1 Yang,2 Do,3 and Suzuki4) have already been published summarizing the obtained results, and bringing an exhaustive review of literature. Very interesting comments can be found in the recent feature article by Sircar and Hufton.5 These reviews suggest that the kinetics of sorption by activated carbons (porous solids in general) can be governed by three transport mechanisms. When pore diameters are much larger than the mean free path of the adsorbate molecules, the so-called molecular (like in bulk phase) diffusion will likely govern the kinetics of sorption. When, on the contrary, the pore diameters are much smaller than the mean free path, they determine the frequency of the collisions between the adsorbate molecules and the pore walls. Then, the transport in pores acquires a much different mechanism, called the Knudsen diffusion. When the pores are very narrow (comparable with the dimensions of the adsorbing molecules), the transport in the pores proceeds via surface diffusion. There have been published dozens of papers bringing advanced theoretical treatments of both Knudsen’s transport, and surface diffusion transport, in particular. Most recent treatments went toward considering geometric heterogeneity of pores, as well as the surface energetic heterogeneity of the pore walls.3-7 Still more and more complicated equations are being developed to calculate the time dependence of adsorption, but their growing complexity hinders their practical applications.5 * To whom the correspondence should be addressed. Tel: +48 81 5375633. Fax: +48 81 5375685. E-mail: [email protected]

In practice, some simple empirical formulas have been used, as the Linear Driving Force (LDF) model, for instance, and its various modifications.1-5 According to LDF the rate of adsorption dNt/dt is given by8

dNt ) X[N/t - Nt(t)] dt

(1)

where Nt is the average adsorbate concentration (the monitored adsorbed amount at a time t), and N/t is the concentration (amount adsorbed) at equilibrium, Nt(tf∞). This formula, and its further modifications, have commonly been used in practical applications. Then, it is interesting to note that the LDF model is used in both chromatography where mainly nonporous column packings are used, and in the pressure-swing-adsorption (PSA) separation processes that uses molecular sieves, for instance. So, it is surprising that until the recently published paper by Sircar and Hufton,5 no fundamental research was reported on the theoretical basis of LDF. Their paper is oriented toward numerical studies that have the objective of showing under what conditions the LDF, or its modifications will simulate the kinetics of surface transport. It is implicitly assumed in their approach that the transport mechanism determines when the LDF model is applicable. But then, there remains still the unsolved question of why the LDF model also describes the kinetics of adsorption on nonporous solid surfaces. This fact was strikingly demonstrated in the paper by Pajares et al.,9 published in 1978, but that paper was forgotten later on. Herein, we will propose an answer for this question. Our proposal is based on assuming fast diffusion to the solid surface and next a slower kinetics of adsorption on an energetically heterogeneous lattice of adsorption sites. These sites are assumed to be characterized by the Dubinin-Astakhov adsorption energy distribution. Also, we will show that this mechanism may govern the kinetics of adsorption in pores in some systems. This proof is based on applying the new theoretical approach to the kinetics

10.1021/jp004166y CCC: $20.00 © 2001 American Chemical Society Published on Web 06/29/2001

Kinetics of Gas Adsorption in Activated Carbons

J. Phys. Chem. B, Vol. 105, No. 29, 2001 6859

of adsorptionsthe Statistical Rate Theory of Interfacial Transport, SRTIT,10-13 generalized by us to describe the kinetics of adsorption/desorption on/from energetically heterogeneous solid surfaces.14-24 Theory Our present theoretical study, has been based on applying the Statistical Rate Theory of Interfacial Transport (SRTIT), which makes it possible to predict the features of adsorption kinetics from the behavior of adsorption equilibria. Therefore, our analysis of experimental data will involve simultaneous consideration of both the kinetic adsorption isotherms, and of two observables monitored under equilibrium conditions. These will be equilibrium adsorption isotherm, and the corresponding heats of adsorption measured at equilibrium. The SRTIT approach belongs to the modern trend in adsorption theories, linking the rate of adsorption/desorption processes to the chemical potentials of the adsorbate molecules in the free gas phase, µg, and in the adsorbed phase, µs. It is based on quantum mechanics, and uses a first-order perturbation analysis, the Boltzmann definition of entropy and a hypothesis to determine the probability of a spontaneous change in the molecular distribution of a macroscopic system. The hypothesis is that the exchange rate between the possible quantum mechanical states of an isolated system has the same value for all combinations of possible quantum mechanical states. The SRTIT approach leads to the following expression for the rates of simultaneously occurring adsorption, Ra, and desorption, Rd, respectively13,25

Ra ) K′gs exp

(

)

(

)

µg - µ s µs - µ g , Rd ) K′gs exp kT kT

(2)

K′gs is the exchange rate between the gas phase and the adsorbed phase once an isolated system has reached equilibrium. To a good approximation, K′gs can be written as the following product13,25

K′gs ) Kgsp(e)(1 - θ(e))

(3)

where the equilibrium state (e) is defined as the one to which the system would evolve if it were isolated at the surface coverage θ and the bulk pressure p. The objective of the SRT approach was to obtain explicit expressions for the dependence of Ra and Rd on the (relative) surface coverage θ. That dependence was an obscure problem in the classical theories of adsorption/desorption kinetics. From the beginning of the 20th century, Absolute Rate Theory (ART) has, almost exclusively, been used for theoretical interpretation of adsorption/desorption kinetic data.26 That approach was based on applying some ideas from the field of chemical reactions, even for the case of physisorption where the adsorption of a gas molecule onto a solid surface can hardly be viewed as a chemical reaction. So, it is not surprising that dramatic discrepancies were reported between theoretical predictions and the experimentally monitored kinetics of adsorption/ desorption. As a result, many scientists started to use empirical equations (e.g., the Elovich, Power-Law) to correlate experimental data.27 Others attempted various improvements of ART, introducing the concept of a precursor state, for instance.28-30 The attempts to generalize the ART approach for the case of energetically heterogeneous solid surfaces met one big barrier related to the fundamentals of the ART approach. This is the

appearance of the “activation energies” for adsorption and desorption, a and d, which are so fundamental for that approach. According to ART, adsorption of an adsorbate molecule is viewed as a chemical reaction between that molecule and a solid surface (adsorption site). In the simplest case of one-siteoccupancy adsorption, and in absence of interactions between the adsorbed molecules, ART offers the following rate expressions

( )

( )

a d Ra ) Kap(1 - θ) exp - , Rd ) Kdθexp kT kT

(4)

At equilibrium, when Ra ) Rd Equation 4 then yields the Langmuir isotherm

(kT ) exp( ) kT

Kp(e) exp θ ) (e)

1 + Kp(e)

(5)

where K ) Ka/Kd and ) d - a. In the theories of adsorption equilibria, the surface heterogeneity is usually considered as a variation of value from one adsorption site to another. Let Nt(p) denote the adsorbed amount expressed in certain units, and Nm the maximum adsorbed amount that can be adsorbed at θ ) 1. Then, Nt(p) can be considered as a result of the following integration31,32

Nt(p,T) ) Nm

∫0∞θ(, p, T)χ()d

(6)

where θ(,p,T) is the relative coverage of the adsorption sites characterized by a certain adsorption energy , and χ() is the differential distribution of the number of the adsorption sites among their corresponding adsorption energies, normalized to unity. Most frequently, the Langmuir equation was applied to represent θ(,p,T). However, things get much more complicated when the kinetics of adsorption is taken into consideration. The main difficulty lies in the physical meaning of a and d, and their interrelations when the surface is energetically heterogeneous. And such is the vast majority of adsorption systems of practical importance. Not one fundamental paper was published that reported how a and d may change from one adsorption site to another. Do they vary independently, or do some correlations exist between a and d when going from one site to another? The lack of an answer for that fundamental question has overshadowed for decades the studies of both isothermal adsorption kinetics, and of the kinetics of thermodesorption. On the contrary, application of the SRTIT approach eliminates considering such obscure problems, and offers an easy way to describe the kinetics of adsorption/desorption processes. Moreover, that description is consistent with the description of adsorption equilibria. Depending on the conditions at which the kinetic experiment is carried out (dθ/dt) can take various forms. In their recent works, Rudzinski et al. considered the case when the experimental conditions are not far from equilibrium. Then

1 - θ(e) ≈ 1 - θ, and p(e) ≈ p

(7)

At such conditions the rate equation takes the form

(1 - θ) dθ ) K a p2 exp - Kdθ exp dt θ kT kT 2

( )

( )

(8)

6860 J. Phys. Chem. B, Vol. 105, No. 29, 2001

Rudzinski and Panczyk

The rate of adsorption/desorption appears to be a function of , the same physical quantity which is also used to describe adsorption equilibria. This is an important feature of all the theoretical approaches linking the rate of adsorption/desorption kinetics to the chemical potentials of the adsorbate molecules in the adsorbed, and in the free gas phase. The SRTIT appro ach is not the first one of that kind which has ever been proposed so far. The assumption that Rd ≈ exp(µs/kT) was launched first by de Boer33 in 1956. It next received further rationalization in the works of Nagai that was published in 1985. Also Kreuzer’s approach leads to such an expression for Rd.35-37 However, despite the impressive success of that new approach this very different view of the adsorption/desorption kinetics did not receive an easy, fast acceptance. This is especially true in the case of the newest SRTIT approach, where the proper understanding of the fundamentals involves considering the related quantum-mechanical derivation. On the contrary, the concepts of ART are relatively simple. The problem begins with attempts to apply the related ART expressions to real adsorption systems, because the ART approach fails to describe the simulataneously appearing adsorption and desorption.20 Meanwhile, the SRTIT approach has already been successfully applied to describe the interfacial transport in various physical processes like gas absorption at a liquid/gas interface,38-41 dissociative hydrogen adsorption,42 hydrogen absorption by metals,43 electron exchange between ionic isotopes in solution,11 permeation of ionic channels in biological membranes,44 nondissociative gas adsorption on solid surfaces,10,12,13,45,46 beam dosing adsorption kinetics,47 temperature-programmed thermal desorption,25 the solid crystal dissolution rate,48 and rates of liquid evaporation.49-51 More recently, the SRTIT approach has successfully been used for the theoretical interpretation of both isothermal adsorption kinetics and the kinetics of thermodesorption in the case of adsorption systems with energetically heterogeneous surfaces.14-24 In the case of the isothermal adsorption kinetics, for the first time, the full form of the rate expression was developed, taking into account both the rate of adsorption, and the rate of the simultaneously occurring desorption.23 This was done for the systems characterized by a symmetrical Gaussianlike adsorption energy distribution, which inserted into eq 6 yields the Langmuir-Freundlich isotherm at equilibrium. Here, we consider the nonsymmetrical adsorption energy distribution, leading to the Dubinin-Astakhov isotherm, describing the equilibria of adsorption in activated carbons. We begin our study by reviewing briefly some well-known facts about the equilibria of adsorption in activated carbons. Since approximately the 1940s, the Dubinin-Raduskevich (DR) equation has been extensively used to describe the equilibria of gas adsorption in activated carbons.31,32 Later, it was modified by Astakhov with the objective of achieving better flexibility in describing equilibrium adsorption isotherms. The Dubinin-Astakhov (DA) isotherm equation is usually written in the following form

{ [ ( )] }

p Nt(p) ) Nm exp - -βkT ln p0

r

(9)

The values of the constants β, r are usually associated with the features of the adsorbent (activated carbon), whereas p0 was associated with the properties of adsorbate and was commonly related to the saturated vapor pressure of the adsorbate. This association reflected the general belief that the adsorption in

activated carbons is a transition of the adsorbate molecules from the gaseous state, to a liquidlike state in the pores of activated carbons. Computer simulations now carried out so commonly show, that the state of the adsorbate in pores may be far different from that in the free gaseous phase. So, while considering the theoretical origin of eq 1, Rudzinski and Everett32 suggested that p0 should, in general, be considered as another parameter characterizing the gas/solid system under investigation. The DA eq 9 was originally proposed as an empirical formula to correlate experimental adsorption isotherms for systems in equilibrium. The enormous success of that simple analytical formula to represent the behavior of such systems encouraged several scientists to investigate the theoretical basis of the DA equation. It is now well established that this theoretical background is to be related to the geometric and energetic heterogeneity of activated carbons. Below, we define some functions which will be used in our forthcoming consideration. We start by writing the Langmuir isotherm equation in the following form

( ) ( )

- c kT θ(, p, T) ) - c 1 + exp kT exp

(10)

where c ) -kT ln Kp, and K is the Langmuir (Henry) constant. When the surface is strongly heterogeneous, i.e., when the variance of χ() is larger than that of the derivative (∂θ/∂), the CA (Condensation Approximation) approach can be applied to evaluate the integral (6)32

θt(p, T) )

Nt(p,T) ) Nm

∫∞χc()d

(11)

c

where the function χc() calculated from the experimental adsorption isotherm θt(p,T)

χc(c) ) -

( ) ∂θt ∂c

(12)

has the following relation to χ()

χc(c) )

χ()d ∫0∞(∂θ ∂ )

(13)

The CA function χc(c) is an approximation to χ(), and when the surface is strongly energetically heterogeneous χc()fχ(). This condition is usually satisfied in the case of activated carbons which are believed to represent the most strongly energetically heterogeneous solid surfaces with which surface scientists have to deal. According to eqs 9 and 12, the CA function characterizing the activated carbons takes the following form

χc(c) )

{ [ ]}

- 0 r ( - 0)r-1exp r δ δ

r

(14)

where β ) 1/δ, and p0 ) 1/K exp(0/kT) has not longer the meaning of the saturated vapor pressure. Now we rewrite θt(p,T) to the following form

{(

θt(p, T) ) θt(c) ) exp -

)}

c - 0 δ

r

(15)

The corresponding equation for θt(p,T,t) can be obtained by



integrating eq 8, and next calculating the following average

θt(p, t) )

∫θ(,p,t)χ()d

(15a)

where θ(,p,t) is the solution of the differential eq 8. These calculations may be made easily with a computer, but elegance lies in obtaining compact analytical solutions. So, we are going to show that one may arrive at such compact solution for the systems where the adsorption equilibria are described by the Dubinin-Astakhov isotherm equation. We have shown it in our previous publication,23 that having known χc(c) and θt(c), one can calculate the rate of adsorption/ desorption kinetics from the following equation

[

( )]

()

c c dθt 1 1 - Kd exp ) 4kTχc(c) Kap2 exp dt 2 kT 2 kT

(16)

Equation 16 was obtained from Statistical Rate Theory of Interfacial Transport (SRTIT) by generalizing that approach so that it could be applied to energetically heterogeneous solid surfaces. This generalization was accomplished by introducing the Condensation Approximation.23 Our new procedure has been successfully used to develop kinetic adsorption isotherms corresponding to the LangmuirFreundlich and Temkin equilibrium adsorption isotherms.23 It has been shown rigorously, that the well-known Elovich kinetic equation describes the kinetics of adsorption, where the equilibrium adsorption isotherm follows the Temkin isotherm equation. Similarly, it has been shown, that the well-known Power-Law kinetic equation corresponds to the Freundlich isotherm at equilibrium. Provided that the kinetics of adsorption in carbon pores would be governed by the rate of interfacial mass transport, let us see which form would then acquire the related kinetic equation. That kinetic equation is obtained by calculating c(θt) from eq 15

c(θt) ) δ(-ln θt)1/r + 0

(17)

expressing χc(c) as χc(θt) from eqs 14 and 17

r χc(θt) ) (-ln θt)(r-1)/r θt δ

(18)

and by substituting c in eq 16 by eq 17. Then, we obtain the following equation for (dθt/dt)

dθt δ r ) 2kT (-ln θt)(r-1)/r θt Kap2e0/kT exp (-ln θt)1/r dt δ kT δ Kde-0/kTexp - (-ln θt)1/r (19) kT

[

(

) )]

(

After solving the differential eq 19 with the boundary condition θt(t ) 0) ) 0, we arrive at the following integral form of the equation to describe the kinetics of adsorption in the activated carbons that have not very narrow pores

{ ( [x

kT θt(t) ) exp - - ln δ

Ka 0/kT pe tanh{2pxKaKd t} Kd

]) } r

(20)

or, in another form

{(

)}

r kT θt(t) ) exp - - ln[Kpe0/kT tanh{2pKgs t}] δ

(21)

At equilibrium, i.e., when tf0, eq 21 takes the form of the

Dubinin-Astakhov eq 9 in which β ) 1/δ and p0 ) (K exp(0/kT))-1. The physical meaning of the constants Ka, and Kd has been considered in detail in our previous paper23

Ka ) Kgsqs0 exp

()

( )

µg0 µg0 1 , Kd ) Kgs s exp kT kT q 0

(22)

where qs0 is the molecular partition function of the adsorbed molecules, µg0 is the standard chemical potential of adsorbate, and Kgs is a constant proportional to the frequency of the collisions of adsorbate molecules with the solid surface. Provided that the adsorption system would have infinite dimensions, this collision rate could be found from the Maxwell-Boltzmann distribution to be proportional to (2πmkT)-1/2 at a given pressure p.13 Let us mention at this point one important assumption accepted in the Statistical Rate Theory of Interfacial Transport, (SRTIT). This is the assumption that at every θt(t), all the surface correlation functions are the same as they would be at full equilibrium, and the same surface coverage θt.13 Equation 21 is applied herein to study the kinetics of benzene adsorption by an activated carbon F4. The data that we examine was previously reported by Wojsz.52 In addition to the kinetic adsorption isotherms, Wojsz also has reported the corresponding equilibrium adsorption isotherms, and the corresponding isosteric heats of adsorption. Certain experimental isotherms can be fit well with a variety of adsorption models. In other words, based only on an analysis of adsorption isotherms, it is difficult to distinguish between the true (actual) adsorption in a given adsorption system, and different models. Also, predicted calorimetric effects are more sensitive to the model (mechanism) than are the measured adsorption isotherms.32 So, a simultaneous fit of experimental adsorption isotherm, and of the corresponding heats of adsorption measured at various surface coverages creates a much better chance to select the true (actual) model of adsorption. With the same values of the adsorption parameters that appear in both the theoretical expressions for adsorption isotherm and in the expression for the isosteric heat of adsorption one should arrive at a simultaneous fit of the data obtained from the two independent adsorption experiments. This strategy is not common. Despite the large number of published papers that consider adsorption phenomena, only a few reports demonstrate simultaneous fits of both the adsorption isotherms and the heats of adsorption. One can see a somewhat similar situation in the studies of adsorption kinetics. The strategy of verifying an assumed model of adsorption by simultaneously fitting adsorption data that had been obtained from various experiments is still rare. To our knowledge, only three papers have been published reporting a simultaneous quantitative fit,25,46,47 (using the same set of parameters), of the following set of experimental data: Kinetic adsorption isotherms measured at various pressures, Equilibrium adsorption isotherms, Isosteric heats of adsorption measured at various surface coverages. The above-mentioned three papers by Ward and co-workers25,46,47 concerned adsorption/desorption on well-defined surfaces. If the CA approximation is accepted, here, for the case of the heterogeneous carbon surface, the isosteric heat of adsorption Qst(θt) is given by the expression

Qst(θt) ) Q0st + c(θt)

(23)



where Q0st is the usually a small nonconfigurational contribution to Qst

Q0st ) -k

d ln K d1/T

(24)

which is assumed to be equal to zero in our fitting of the experimental data in Qst(θt). Results and Discussion The physicochemical characteristics of the activated carbon F4, studied here by us are given in the monograph by Wojsz.52 We mention only that the total pore volume of carbon F4 was 0.28 dm3/kg. The strategy of our best-fit exercises assumed that using the same set of parameters 0, r, δ, K should lead us to fit simultaneously the experimental data measured in three independent experiments: the equilibrium adsorption isotherm, described by eq 9, in which β ) 1/δ, and p0 ) exp(-0/kT)/K the isosteric heats of adsorption, measured at various surface coverages, and described by eq 23 the set of the kinetic adsorption isotherms described by eq 21 In the case of the kinetic isotherms, we have yet another bestfit parameter Kgsp appearing only in the kinetic adsorption isotherms. Provided that the state of the nonadsorbed molecules in the interior of the pores would be similar to their status in the bulk gaseous phase having infinite (thermodynamically) dimensions, we should arrive in our best-fit exercises at one constant value of Kgs. This happened in our previous work,23 where we fitted the kinetic data for CO2 adsorption on a nonporous surface of scandia. But now, in the case of porous carbons, we expect to face a somewhat different physical situation. In the interior of pores the Kgsp parameter will not longer describe the pressure dependence of the rate of collisions with pore walls. We will treat, therefore, Kgs as another best-fit parameter. Its value is expected to depend on the diameter of the pore. In the bulk phase, the rate of collisions with the surface Kgsp is related to the mean free path λ of molecules

λ)

kT

x2πσ2p

(25)

TABLE 1: Values of the Best-Fit Parameters Found in Our Computer Exercises, to Fit Simultaneously Equilibrium Adsorption Isotherm, Kinetic Isotherm of Adsorption, and the Heat of Adsorption of Benzene in the Activated Carbon F4 0 kJ/mol

r

δ kJ/mol

K hPa-1

42.5

0.856

10.0

1.58 × 10-9

TABLE 2: Values of the Parameters Kgs and X Corresponding to Four Non-Equilibrium Adsorbate Pressures, P, and Found by Fitting the Kinetic Experimental Isotherms Using Eq 21 and 22, Respectively pressure hPa

2.32

4.41

8.61

15.53

Kgs hPa-1 s-1 1.11 × 10-3 5.0 × 10-4 2.50 × 10-4 1.35 × 10-4 X s-1 1.79 × 10-2 9.95 × 10-3 1.01 × 10-2 8.21 × 10-3

Figure 1 shows the agreement between the experimental and our theoretical equilibrium isotherm, whereas Figure 2 shows that this experimental adsorption isotherm could not be correlated by the Langmuir equation developed for an energetically homogeneous lattice of adsorption sites. The commonly accepted way to verify theoretical approaches is to compare theoretical predictions with an experiment by using possible linear relationships. In the case of the Langmuir isotherm, the commonly accepted way is the following linear representation

p p 1 + ) Nt NmK exp(/kT) Nm

(27)

whereas the Dubinin-Astakhov isotherm is written usually in the following form

[

p ln Nt(p) ) ln Nm - -βkT ln p0

]

r

(28)

Figure 2 demonstrates that the linear representation (eq 28) yields a much better correlation of the experimental data for that equilibrium adsorption isotherm. Figure 3 shows the agreement between the theoretical and the experimental heats of adsorption, measured under equilibrium conditions.

where σ is the collision diameter. Thus

Kgsp )

p

x2πmkT

)

kT 1 λ x2πσ2 x2πmkT

(26)

It means, the rate of the gas-surface collisions is proportional to λ-1. Inside the pores the mean free path will roughly be proportional to pore diameter, R, so, the best-fit parameter Kgsp should be expected as roughly proportional to the inverse pore diameter, R-1. In our picture of the adsorption starting in the narrowest (most energetic) pores, and then gradually proceeding to larger and larger pores, Kgsp ≈ R-1 should decrease in the course of adsorption. And this has been found indeed in our fitting exercises. Thus, we assumed that for every nonequilibrium pressure the best-fit parameter 2pKgs will acquire another value. In that way, using the same set of parameters collected in Table 1, and Table 2 we arrived at an excellent simultaneous fit of the kinetic isotherms, equilibrium isotherm, and of the heat of adsorption.

Figure 1. Agreement between the experimental equilibrium isotherm of benzene adsorption in carbon F4 (o o o o), measured at 298.2 K, and our theoretical isotherm (s) calculated by eq 9, using the parameters collected in Table 1, and assuming that the maximum adsorbed amount Nm ) 5.961 mol/kg.



Figure 2. Approximations of the equilibrium isotherm of benzene adsorption in carbon F4, using the linear representation (27) offered by Langmuir equation, (o o o o), and the linear representation (28), (] ] ] ]) offered by Dubinin-Astakhov equation. The solid lines (s) represent the theoretical values. The broken line (- - -) is a polynomial approximation of the experimental data (o o o o), to emphasize the nonlinear trend in these experimental data.

Figure 3. Agreement between the experimental heats of benzene adsorption in carbon F4 measured at equilibrium, (bbbb), and our theoretical heats of adsorption (s) calculated from eqs 23 and 24, using the parameters collected in Table 1, and assuming that Q0st ) 0.

Figure 4(A,B) shows the comparison between the experimental kinetic isotherms of benzene adsorption in the carbon F4 and the theoretical kinetic isotherms calculated from the simple eq 21, developed in this paper. For better clarity, this comparison is shown in Figure 4A and 4B. Also, to better display the behavior of θt(t) at small coverages, this function is also displayed by using the logarithmic time scale. The results shown in Figure 5 confirm our expectations. The estimated value of the parameter 2pKgs is not a linear function of pressure, as it would be in the case of adsorption on nonporous surfaces. The rate of collisions with surface walls is now different, and depends on the dimensions of the pores, in which the adsorption is running at a certain surface coverage θt. Looking next for the possible (expected) correlations between the kinetic parameter Kgsp, and the geometric and energetic

Figure 4. (A,B) Comparison of the experimental kinetic isotherms of benzene adsorption in F4 with the theoretically calculated ones, using eq 21 and the values of the parameters collected in Table 1. The values of the 2pKgs parameter used in these best-fit exercises are shown in Figure 5.

features of carbon pores, we consider the adsorption energy distribution χ() characterizing the energetic heterogeneity of benzene/F4 system. This function is shown in Figure 6. In Figure 7, the function c(θt) is compared with that approximating the estimated values of the parameter Kgsp considered as a function of θt. One can see in Figure 7 obvious correlations between c and Kgsp. In Figure 8 the Kgsp values are plotted against c to see these correlations better.



Figure 5. Dependence on pressure, of the values of the parameter 2pKgs, found to fit the kinetic isotherms in Figure 4(A,B).

Figure 7. Comparison of the functions c(θt), and Kgsp vs θt characterizing the benzene/F4 system.

Figure 6. Adsorption energy distribution χc(), calculated from eq 12, by using the parameters collected in Table 1.

Figure 8 confirms what has been known for a long time, and shown rigorously in a number of theoretical works,53-60 namely, that the (average) free energy of adsorption in a micropore decreases with pore dimension. In our picture of gradually proceeding adsorption, c is directly related to the free energy of adsorption. The c(R) interrelations have been used in many papers to determine the pore size distribution from experimental adsorption isotherms.61 The most essential is that the simultaneous good fit presented in Figures 1-4, provides a solid support for the mechanism of the benzene sorption kinetics assumed by us, while developing the kinetic eq 21. This is fast Knudsen’s diffusion into the interior of pores, next followed by a much slower exchange of mass between the pore walls and the interior of pores. Now, let us consider finally the applicability of the Linear Driving Force (LDF) approximation to represent the combined Knudsen-SRTIT kinetics. While accepting the condition that Nt(t ) 0) ) 0, and integrating the LDF expression for dNt/dt, one arrives at the following linear dependence on time

ln

N/t N/t - Nt

) Xt

(29)

Figure 9 shows our efforts to correlate the kinetic isotherms

for the benzene adsorption in the activated carbon F4 by using the LDF eq 29. For the purpose of clarity, that dependence is shown only for a few randomly selected values of the nonequilibrium pressure p. That effectiveness of the LDF eq 29 to correlate the experimental data for adsorption kinetics is next compared with a linear fit offered by our SRTIT eq 21, and presented in Figure 10. While comparing the linear fits in Figure 9 and Figure 10, one can see, that the simple LDF equation is not able to represent the combined Knudsen-SRTIT kinetics of adsorption in the investigated activated carbons, and that some more complicated modifications of LDF would be necessary. On the contrary, our simple SRTIT eq 21 leads to an excellent fit of the kinetic adsorption isotherms. So, using that simple eq 21, having welldefined physical meaning, might be an attractive option for many practical applications, including gas separation processes. Let us remark that the size of the carbon particles studied by Wojsz and analyzed here by us was 0.6 - 1.2 mm, i.e., is comparable with those used in separation processes. In the case of the Knudsen-SRTIT kinetics investigated here, the shape of the adsorption energy distribution seems to be essential for the applicability of the LDF approximation. This is evidenced by the following example. While publishing their numerous data on the kinetics of CO2 adsorption on scandia, Pajares et al.9 demonstrated a good applicability of the LDF approximation to correlate their experimental data. (Unfortunately, they called this - “A New Rate Equation”). Then, in our recent publication we have shown,23 that these kinetic data can be correlated very well by using the SRTIT equation developed for the fully symmetrical



Figure 8. Dependence of the parameter values Kgsp on c in the adsorption benzene/F4 system.

Figure 10. Linear regression of the experimental data for adsorption kinetics, offered by our eq 21. The points (g,O,[,1) denote the experimental data, whereas the solid lines are the theoretically calculated values collected in Tables 1 and 2.

We do not consider here the applicability of LDF to represent kinetics of sorption governed by the surface diffusion. This has already been done in the feature article by Sircar and Hufton.5 Conclusions

Figure 9. Presentation of the applicability of the LDF approximation to represent the kinetics of the combined Knudsen-SRTIT sorption, by the activated carbon F4. The points (g,O,[,1) denote the experimental data for four randomly selected pressures, whereas the solid lines (s) represent their best linear approximation obtained by using the LDF eq 29. The estimated from this linear regression parameters are collected in Table 2.

Gaussian-like adsorption energy distribution

( ) ( )]

- 0 1 exp c c χc() ) - 0 1 + exp c

[

2

(30)

where 0 is the value of at which the function reaches its maximum, and c is proportional to the variance of that adsorption energy distribution. It means that the LDF approximation might successfully be used to represent the SRTIT kinetics only in the adsorption systems characterized by symmetrical Gaussian-like energy distributions. While applying eq 6 to calculate N/t corresponding to the Gaussian-like function (30), one arrives at the well-known Langmuir-Freundlich isotherm.31,32 This isotherm equation has been found to be probably the best formula to describe gas separation on real nonporous surfaces. This would explain the excellent applicability of the LDF approximation in chromatography, where nonporous column packings are commonly used.

So far only three sorption kinetics, (and their combinations), have been considered to govern the kinetics of sorption by activated carbons, and by porous media, in general. These are the molecular diffusion, the Knudsen’s diffusion, and the surface diffusion. The successful application of the equations developed from the Statistical Rate Theory of Interfacial Transport to analyze kinetics of sorption by an activated carbon, along with the corresponding equilibrium adsorption isotherms and heats of adsorption, has brought us to postulate another yet possible sorption mechanism. This would be a fast Knudsen’s diffusion into the interior of pores, and next a slower rate determining exchange of mass between the pore walls and the interior of pores. Such a mechanism might exist in mesopores or in wider micropores. The kinetics of sorption is then described by a simple equation, attractive for practical applications. Moreover, the essential features of sorption kinetics can be deduced from the behavior of an adsorption system at equilibrium. It is to be expected, that the new postulated mechanism of sorption cannot be assumed in the case of microporous adsorbents, where the ratio of the area of pore walls to the total pore volume is high. In such a case, the already existing theoretical approaches assuming a great role of surface diffusion should be applied. Acknowledgment. This work was supported by the Polish State Committee for Scientific Research (KBN) Grant No. 3 T09A 066 18. References and Notes (1) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Willey and Sons: New York, 1984. (2) Yang, R. T. Gas Separation by Adsorption Processes; Butterworth Publisher: Stoneham, USA: 1987. (3) Do, D. D. Adsorption Anaysis: Equilibria and Kinetics; Imperial College Press: London: 1998. (4) Suzuki, M. Adsorption Engineering; Elsevier, New York, 1990. (5) Sircar, S.; Hufton, I. R. Adsorption 2000, 6, 137.

6866 J. Phys. Chem. B, Vol. 105, No. 29, 2001 (6) Zgrablich, G. Surface Diffusion of Adsorbates on Heterogeneous Substrates. In Equilibria and Dynamics of Gas Adsorption on heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1996. (7) YuDong, C.; Yang, R. T. Multicomponent Diffusion in Zeolites and Multicomponent Surface Diffusion. In Equilibria and Dynamics of Gas Adsorption on heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1996. (8) Gleuckauf, E. Trans. Faraday Soc. 1955, 51, 1540. (9) Pajares, J. A.; Garcia Fierro, J. L.; Weller, S. W. J. Catal. 1978, 52, 521. (10) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599. (11) Ward, C. A. J. Chem. Phys. 1983, 79, 5605. (12) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 5615. (13) Elliott J. A.; Ward, C. A. Statistical Rate Theory and the Material Properties Controlling Adsorption Kinetics on Well-Defined Surfaces. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier, New York: 1996. (14) Rudzinski, W.; Aharoni, C. Polish, J. Chem. 1995, 69, 1066. (15) 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: New York, 1996. (16) Rudzinski, W.; Aharoni, C. Langmuir 1997, 13, 1089. (17) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1997, 13, 3445. (18) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T.; Gryglicki, J. Polish, J. Chem. 1998, 72, 2103. (19) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1999, 15, 6386. (20) Rudzinski, W.; Panczyk, T.; Surface Heterogeneity Effects on Adsorption Equilibria and Kinetics: Rationalizations of the Elovich Equation. In Surfaces of Nanoparticles and Porous Materials, Schwarz, J.; Contescu, C.; Eds.; Marcel Dekker: 1999. (21) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. AdV. Coll. and Interface Sci. 2000, 84, 1. (22) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. J. Phys. Chem. B 2000, 104, 1984. (23) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (24) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Langmuir 2000, 16, 8037. (25) Elliott J. A.; Ward, C. A. J. Chem. Phys. 1997, 106, 5667. (26) Clark, C. A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (27) Low, M. J. Chem. ReV. 1960, 60, 267. (28) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95. (29) King, D. A. Surf. Sci. 1977, 64, 43. (30) Gorte, R.; Schmidt, L. D. Surf. Sci. 1978, 76, 559. (31) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1988.

Rudzinski and Panczyk (32) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (33) de Boer, J. H. AdV. Catal. 1956, 8, 1. (34) Nagai, K. Phys. ReV. Lett. 1985, 54, 2159. (35) Kreuzer, H. J.; Payne, S. H. Surf. Sci. 1988, 198, 235. Kreuzer, H. J.; Payne, S. H. Surf. Sci. 1988, 200, L433. (36) 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. (37) 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. (38) Ward, C. A. J. Chem. Phys. 1977, 67, 229. (39) Ward, C. A.; Rizk, M.; Tucker, A. S. J. Chem. Phys. 1982, 76, 5606. (40) Ward, C. A.; Tikuisis, P.; Tucker, A. S. J. Colloid Interface Sci. 1986, 113, 388. (41) Tikuisis, P.; Ward, C. A. In Transport Processes in Bubbles, Drops and Particles; Chabra, R.; DeKee, D., Eds.; Hemisphere, New York, 1992. (42) Findlay, R. D.; Ward, C. A. J. Chem. Phys. 1982, 76, 5624. (43) Ward, C. A.; Farabakhsk, B.; Venter, R. D. Z. Phys. Chem., Neue Folge 1986, 147, S 89-101, 7271. (44) Skinner, F. K.; Ward, C. A.; Bardakjian, B. L. Biophys. J. 1993, 65, 618. (45) Ward, C. A.; Elmoseli, M. B. Surf. Sci. 1986, 176, 457. (46) Elliot, J. A. W.; Ward, C. A. Langmuir 1997, 13, 951. (47) Elliott J. A.; Ward, C. A. J. Chem. Phys. 1997, 106, 5677. (48) Dejmek, M.; Ward, C. A. J. Chem. Phys. 1998, 108, 8698. (49) Ward, C. A.; Fang, G. Phys. ReV. E 1999, 59, 429. (50) Fang, G.; Ward, C. A. Phys. ReV. E 1999, 59, 441. (51) Fang, G.; Ward, C. A. Phys. ReV. E 1999, 59, 417. (52) Wojsz, R., (in Polish) Charakterystyka Heterogenicznoœci Strukturalnej i Energetycznej Mikroporowatych Adsorbento´w Weˆglowych z Uwzgleˆdnieniem Adsorpcji Substancji Polarnych, (Rozprawa Habilitacyjna) Wydawnictwo UMK, Torun˜: 1989. (53) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (54) Horvath, G.; Kawazoe, K. J. Chem. Eng. Japan 1983, 16, 470. (55) Choma, J.; Jaroniec, M.; Klinik, J. Karbo-Energochemia-Ekologia 1994, 10, 258. (56) Cheng, L. S.; Yang, R. T. Chem. Eng. Sci. 1994, 16, 2599. (57) Saito, A.; Foley, H. C. AIChE 1991, 37, 429. (58) Mariwala, R. K.; Foley, H. C. Ind. Eng. Chem. Res. 1994, 16, 2314. (59) Kaminsky, R. D.; Maglara, E.; Conner, W. C. Langmuir 1994, 10, 1556. (60) Cheng, L. S.; Yang, R. T. Adsorption 1995, 1, 187. (61) Jaroniec, M.; Choma, M. Characterization of Geometrical and Energetic Heterogeneities of Active Carbons by Using Soprtion Measurements. In Equilibria and Dynamics Of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W.; Steele, W. A.; Zgrablich, G., Eds.; Elsevier: Amsterdam: 1997.

Kinetics of Gas Adsorption in Activated Carbons, Studied by Applying

Recommend Documents