Application of the Statistical Rate Theory of interfacial transport to investigate the kinetics of mixed-gas adsorption onto the energetically homogeneous and strongly heterogeneous surfaces

Applied Surface Science 255 (2009) 4627–4635

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Application of the Statistical Rate Theory of interfacial transport to investigate the kinetics of mixed-gas adsorption onto the energetically homogeneous and strongly heterogeneous surfaces Krzysztof Nieszporek Department of Theoretical Chemistry, Maria Curie-Sklodowska University, Pl. M. Curie-Sklodowskiej 3, 20-031 Lublin, Poland

A R T I C L E I N F O

A B S T R A C T

Article history: Received 22 October 2008 Received in revised form 28 November 2008 Accepted 29 November 2008 Available online 7 December 2008

Kinetics of mixed-gas adsorption by using the Statistical Rate Theory is studied. Applying the adsorption lattice model two cases are investigated: when adsorption occurs like on the homogeneous surface and when energetic heterogeneity of adsorption system can be described by the rectangular adsorption energy distribution function. The model of calculations offers possibility of theoretical prediction of the rate of adsorption/desorption of mixture components by using the single-gas equilibrium and kinetic data. Possible changes of adsorbate concentration near the adsorbing surface are also taken into account. The obtained theoretical expressions were verified using real adsorption systems. ß 2008 Elsevier B.V. All rights reserved.

Keywords: Adsorption Kinetics Statistical thermodynamics

1. Introduction The use of adsorption phenomenon can be helpful in designing of modern chemical technologies which are friendlier for the environment. One of the examples are the gas separation and purification processes commonly used in industry. A very popular cryogenic method proposed by von Linde in 1895 can be replaced by modern technologies based on the idea of Skarstrom [9]. By passing gas mixture through the adsorption columns and suitably manipulating the pressures and directions of gas flow the mixture can be separated. This technique named Pressure Swing Adsorption (PSA) has been developed lately. In comparison to the Linde method PSA is less energy consuming, involves applying ‘‘soft’’ technological regimes and the purity of separated gases is similar to that obtained by the cryogenic method. PSA systems are controlled by computers so they are fully automated. Industrial gas separation of gases by adsorption processes is usually based on the equilibrium properties of mixture components. Separation can be also performed by utilizing the kinetics of adsorption/desorption of mixture components. For example, if large differences between adsorption rates of components exist, one of them is quickly removed from a mixture. In the present paper we focus our attention on the theoretical description of the time dependence of partial surface coverage of gas mixture components. To our knowledge only few papers

E-mail address: [email protected]. 0169-4332/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2008.11.087

devoted to this subject can be found [3]. The obtained results can be used in the computer software controlling PSA apparatus. As the conditions of gas separation process change quickly, it is necessary to determine very fast how to change the physical conditions of the process. The accuracy and quickness of this determination have major influence on the purity of separated gases. As the theoretical tool we use the Statistical Rate Theory (SRT) proposed by Ward et al. [1,2]. SRT has been applied successfully to represent the rates of various interfacial transports, like the rate of gas adsorption at the liquid–gas interface [12–14], hydrogen adsorption by metals [15], electron exchange between ionic isotopes in solution [16], permeation of ionic channels in biological membranes [17], rate of liquid evaporation [18–20] and kinetics of adsorption on energetically heterogeneous surfaces [21–23]. Recently Azizian et al. [27] used SRT approach to study the kinetics of competitive adsorption at the solid/ solution interface. It is still one more reason for which we took interest in the problem of theoretical description of the kinetics of mixed-gas adsorption. Similarly to the adsorption equilibrium, the experimental measurement of adsorption rate of gas mixture components is difficult and time consuming in comparison to analogous measurements for pure gases. Theoretical predictions of mixed-gas adsorption equilibrium have been studied by us recently [4–6]. Generally, to predict theoretically predict the mixed-gas adsorption equilibrium, only information about single-gas isotherms is necessary. Such rule we try to adopt for the case of theoretical description of kinetics of mixed-gas adsorption.

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The present paper shows new theoretical expressions describing the rate of adsorption of mixture components for the case of homogeneous and strongly heterogeneous surfaces. While considering the heterogeneity effects we used the rectangular adsorption energy distribution function and the adsorption isotherms calculated by the Integral Equation approach and Condensation Approximation [6]. Finally, the proposed model of calculations will be verified by using the experimental kinetic data. 2. Kinetics of competitive gas adsorption on the energetically homogeneous surface The Statistical Rate Theory approach is based on the assumption that adsorption rate is determined by the difference between the chemical potentials in gas and the adsorbed phases. While considering the case when two-component mixture of gases is adsorbed, the adsorption rate of component ‘1’ can be described by the following equation: du 1 ðeÞ ðeÞ ¼ K gs p1 ð1  u1 dt   g   s  m1  ms1 m1  mg1 ðeÞ  exp  u2 Þ exp kT kT

(3)

Next, the change of the adsorbate concentration in the nearsurface region caused by adsorption nSA can be written: nSA ¼

du N 0 SA dt

(4)

where N0 is the number of adsorption centers per unit of surface area. The mass balance should be fulfilled dn ¼ n0  nSG  nSA dt

(1)

(2)

(5)

By adopting the ideal gas law dn d p SG h ¼ dt dt kT

(6)

the pressure changes in the near surface region can be written as follows [3]: dp SG kT du kT ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð p0  pÞ  N0 SA dt SG h dt SG h 2pmkT

where ui is the surface coverage of component ‘1’, t is the time, ms1 and mg1 are the chemical potentials of given component in adsorbed and gas phases, respectively, Kgs is the so-called equilibrium exchange rate. The equilibrium state (e) is defined as the one in which a system isolated at the non-equilibrium pressure p1 and the coverage u1 would evolve. Theoretical investigations by Panczyk [3] show, that the correct description of the adsorption kinetics should take into account also the change of pressure of gas phase near the surface region. It is due to the fact that the surface removes the particles from near region and, after adsorption, they no longer contribute to the pressure in the slice of gas phase. This effect is growing especially when the adsorption process is fast. Below we show how to determine the expression for pressure change near the surface region. For that purpose we use the model of adsorption system proposed by Panczyk [3] (see Fig. 1). It can be stated that the time-change of the number of molecules in the near surface region dn/dt is controlled by the bulk gas phase n0 (the reservoir of the adsorbate) and the molecules which leave the near-surface region through the interface SG, nSG , or are captured/released by the surface area SA, nSA . Kinetic theory of gases predicts that the number of molecules passing through a given area SG per unit time reads: SG n0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 2pmkT

SG nSG ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2pmkT

(7)

The above equation can be written in the simplified form: dp du ¼ að p0  pÞ  b dt dt

(8)

where a and b are the temperature-dependent parameters and p0 is the bulk phase pressure of a given component. From the practical point of view a and b can be treated as the best-fit parameters improving agreement between theory and experimental data. In the simplest case when adsorption occurs like on the energetically homogeneous surface, the chemical potential of component ‘1’ in the adsorbed phase can be expressed as follows:

ms1 ¼ kTln

u1 ð1  u1  u 2 ÞK 1

(9)

where K 1 ¼ qs01 expfe1 =kTg and qs01 is the molar partition function of the adsorbed molecules. If we assume that the gas phase is ideal, the chemical potential has the well-known form:

mg1 ¼ mgo1 þ kT ln p1

(10)

To build up a set of equations describing the process of twocomponent adsorption it is necessary to concentrate on the experimental procedure which is to be described. Rudzinski and Panczyk [7] have considered the three extreme cases: 1. ‘‘Volume-dominated’’—in the non-equilibrium adsorption process the gas in the gaseous phase above the surface exceeds the amount of adsorbed portion significantly. In that case, after the system is isolated and equilibrated, the gas pressure p does not change much so p(e) = p, 2. ‘‘Surface-dominated’’—in the non-equilibrium process the adsorbed amount strongly prevails the amount of gas in the gas phase. After isolating and equilibrating u remains practically unchanged i.e. u(e) = u, 3. ‘‘Equilibrium-dominated’’—the process is carried out under such conditions that the gas/solid system is close to equilibrium. In this case it may assume u(e) = u and p(e) = p. If we assume that the adsorption process is performed, the volume-dominated model should be used. It implies the following ðeÞ ðeÞ form of the functions u1 and p1 : ðeÞ

Fig. 1. Model of adsorption system showing different pressure regions during the adsorption process [3].

ðeÞ

1  u1  u2 ¼ ð1 þ K 1 p1 þ K 2 p2 Þ1 and

ðeÞ pi

¼ pi .

(11)

K. Nieszporek / Applied Surface Science 255 (2009) 4627–4635

Then, the time-change of partial coverage of the component ‘1’ is following: "  g  m01 K 1 p21 ð1  u1  u2 Þ du 1 1 ¼ K gs ð1 þ K 1 p1 þ K 2 p2 Þ exp dt kT u1 #  g  m u1  exp  01 (12) kT K 1 ð1  u 1  u 2 Þ or,   du 1 ð1  u1  u2 Þ u1 ¼ ð1 þ K 1 p1 þ K 2 p2 Þ1 K a1 p21 K d1 dt u1 ð1  u1  u2 Þ (13) g 01 =kTg

where K a1 ¼ K gs1 K 1 expfm and K d1 ¼ K gs1 =ðK 1 expfmg01 =kTgÞ.In the case of desorption uptake curves, the theoretical description is based on the solid-dominated ðeÞ ðeÞ model. The functions u 1 and p1 are as follows: ðeÞ

p1 ¼

1 u1 K 1 1  u 1  u2

(14)

ðeÞ

solution of IE with (18) leads to the well-known Tiemkin isotherm. Also the SRT approach with function (18) applied to the single-gas adsorption leads to Elovich equation [8]. While considering the mixed-gas adsorption, the solutions of IE with distribution (18) can be performed by considering the correlation effects between the adsorption energies of mixture components. Two extreme cases can be considered: a lack of correlations between adsorption energies of mixture components and the case when the functional relationship exists. The case of the lack of correlations does not impose restrictions for the shape of the energy distribution functions characterizing adsorption species. The model of high correlations, frequently described by e2 = e1 + D12, is synonymous with the assumption that the energy distribution functions of mixture components have similar (identical) shapes (in the case of rectangular AED—widths) and are only shifted on the energy axis. When assuming that there are no correlations between adsorption energies, the solution of IE with distribution (18) leads to the following theoretical partial isotherm [6]:

u1t ðf pg; TÞ ¼

and ui ¼ ui . Then we have: 2

du1 K a1 K u1 ¼ p ð1  u1  u2 Þ  d1 dt K1 1 K 1 p1 ð1  u1  u2 Þ

(15)

Finally, for the case of ‘‘equilibrium-dominated’’, the adsorption rate is expressed as follows: 2

du 1 ð1  u1  u2 Þ ¼ K a1 p21  K d1 u1 dt u1

(16)

It is interesting to note that when the equilibrium is achieved i.e. @u1/@t = 0 Eq. (16) leads to the Benton equation, well known as the multi-component Langmuir adsorption isotherm. 3. Kinetics of mixed-gas adsorption on the energetically heterogeneous surface The Integral Equation (IE) method is one of the most popular approaches used to study adsorption of gases on heterogeneous solid surfaces [24]. According to the IE formalism, the adsorption isotherm for a heterogeneous surface in the one-component system can be expressed by the following fundamental equation Z u1t ¼ ul ð p1 ; T; e1 Þx1 ðe1 Þde1 (17) V

where e1 is the adsorption energy in the interval V, x1(e1) is the distribution function of the adsorption energy and ul is the local model of adsorption isotherm. In principle, Eq. (1) can be solved for arbitrary functions x1 and ul. However, analytical solutions of Eq. (1) are known only for a few pairs x1–ul [25]. The vast majority of applications of the IE relates to the local model of adsorption that is given by the Langmuir equation (LA). This local model, combined with different distribution functions x1, leads to expressions for u1t that are well-known empirical isotherm equations inherent to adsorption on heterogeneous surfaces. One of the functions x1 is the rectangular adsorption energy distribution (AED): 8 1 < ; for e 2 ðel1 ; em 1Þ x1 ðe1 Þ ¼ em (18)  el1 : 1 0; elsewhere where ðel1 ; em 1 Þ is the physical domain of e1. Although Eq. (18) is a very simple model of the energetic heterogeneity of adsorption system, it is one of few distributions for which analytical solutions of the Integral Equation (IE) approach are known [6]. In the case of single-gas adsorption the

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KT=De1 lnðK 1 p1 Þ½KT=De2 ðlnðK 1 p1 Þ  1Þ KT=De1 lnðK 1 p1 Þ KT=De2 ðlnðK 1 p1 Þ  1Þ

(19)

Eq. (19) was calculated by using the Condensation Approximation so it can be used in the case of strongly energetically heterogeneous surfaces. It is worth mentioning that IE with the rectangular energy distribution (18) for the case of mixed-gas adsorption can be calculated also in the exact way [6]. Now the chemical potential of the component ‘1’ in the adsorbed phase is following:

ms1 ¼ De1

u1 1  u2

 kT ln K 1

(20)

If the volume-dominated model of the experimental process is ðeÞ ðeÞ assumed, the functions u1 and p1 have the following form: ðeÞ

ðeÞ

1  u1  u2 ¼

ðkT=De1 lnðK 1 p1 Þ  1ÞðkT De2 lnðK 2 p2 Þ  1Þ 1  kT=De1 lnðK 1 p1 Þ kT=De2 lnðK 2 p2 Þ

(21)

ðeÞ

and, pi ¼ pi . Then, the rate of adsorption of component ‘1’ is following: ðK d1 expfðDe1 =kTÞðu1 =1  u2 Þg  K a1 p21 expfðDe1 =kTÞ du 1  ðu1 =1 u2 ÞgÞðkT=De1 lnðK 1 p1 Þ  1ÞðkT=De2 lnðK 2 p2 Þ  1Þ ¼ dt ðkT=De1 lnðK 1 p1 ÞÞ ðkT=De2 lnðK 2 p2 Þ  1Þ (22) The theoretical description based on the solid-dominated model leads to the following replacements in Eq. (1):   1 De1 u1 exp (23) p1 ¼ K1 kT 1  u2 ðeÞ

and u i

¼ ui . Then,    du 1 K a1 K De1 u1 ¼ ð1  u1  u2 Þ p1  d1 exp 2 dt K1 K 1 p1 kT 1  u2

(24)

When the adsorption/desorption process is carried out in close to equilibrium conditions, the rate of adsorption of a given component is following: du 1 ¼ ð1  u1 dt      De1 u1 De1 u1  u2 Þ K a1 p21 exp   K d1 exp kT 1  u2 kT 1  u2 (25) Eqs. (22), (24) and (25) are the theoretical uptake curves of adsorption/desorption of a two-component system. Because these

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equations were obtained with the assumption of the lack of correlations between adsorption energies, they can be used independently of the shapes of energy distribution functions of adsorption species. The parameters necessary to use Eqs. (22), (24) and (25) can be obtained from the theoretical analysis of the equilibrium single-gas adsorption isotherms by using the Tiemkin equation:

u1t ð p; TÞ ¼

kT

De1

lnK 1 þ

kT

De1

ln p1

(26)

and treating Ka1 and Kd1 as the best-fit parameters improving the agreement of the theoretical curves with the experimentally measured kinetic data. As we mentioned above, the theoretical description of the mixed-gas adsorption equilibria by using the Integral Equation approach is based on the consideration of the energy correlation effects. Unfortunately, while assuming the model of high correlations between adsorption energies, due to the mathematical complexity of the theoretical partial isotherms corresponding to rectangular distribution (18), it is not possible to obtain the analytical expressions for chemical potentials in the adsorbed phase. In such a case only numerical calculations are possible. 4. Calculations The previous sections include a set of equations describing the rate of mixed-gas adsorption system. Now, the readers can address a question; how can they be used? Due to the complicated mathematical form, it seems that the practical calculations will be difficult. Really, the problem is to solve the set of differential equations describing the time-change of coverage and pressure in the adsorption system. We cope with it by using the Mathematica 6.0 software. Determination of the best-fit parameters in the presented equations seems to be easier. Namely, the proposed theoretical description of mixed-gas adsorption rate is consistent with our previous studies concerning the theoretical description of mixedgas adsorption equilibria and enthalpic effects accompanying them. The most valuable in our theoretical studies, is possibility of theoretical prediction of the properties of mixed-system by using single-gas experimental data. For example, to calculate the adsorption rate curves by using the Langmuirian model of adsorption, Eqs. (13), (15) and (16), we need to know the values of Kai, Kdi, Ki and the maximum adsorbed amount Mi for each mixture component. Ki and Mi can be obtained from the theoretical analysis of pure gas adsorption isotherms by using the Langmuir equation. The other parameters Kai and Kdi can be determined by analyzing the measured single-gas adsorption kinetics experimentally. To do it, the experimental data should be adjusted by the single-gas versions of SRT Eqs. (13), (15) and (16). In the case of the Langmuir model of adsorption, they can be obtained by assuming that pj and uj are equal to zero. Summing up, at this stage of calculations, we can determine eight best-fit parameters characterizing the single-gas adsorption systems: K1, K2, M1, M2, Ka1, Ka2, Kd1 and Kd2. During the calculations of time-dependence of partial surface coverages of mixture components, only four best-fit parameters exist: a and b for each of mixture components. 5. Results and discussion In the previous sections we showed the generalizations of the SRT approach to study the kinetics of mixed-gas adsorption. Two extreme cases were considered: when adsorption occurs like on the homogeneous surface and when the energetic heterogeneity of adsorption system can be described by the rectangular distribution

(18). Due to the clarity of presentations, in the present paper we focus our attention on the Langmuir model of adsorption. The studies concerning the theoretical description of the rate of competitive adsorption of gas mixture on an energetically heterogeneous surface will be the matter of further publication. We realize that the Langmuirian model of adsorption presupposes the lack of interactions between adsorbed molecules. From the physical point of view, it is not too realistic because that interaction effects play more or less important role in adsorption phenomena. Taking into account the interaction effects between the adsorbed molecules should lead to better theoretical description of real adsorption systems. However, it results in additional best-fit parameters. For example, it can be done by using the Fowler–Guggenheim adsorption isotherm. In our case using the Langmuir model of adsorption we assume that the interactions are so small that can be neglected. The simplest way to check the accuracy of proposed solutions is the comparison with the real adsorption system. To perform such studies, we use the experimental data published by Qiao and Hu [10]. The authors published the kinetic isotherms of ethane and propane on the Norit activated carbon. They report also the adsorption equilibrium isotherms which are necessary to apply for the model of calculations proposed by us. The adsorption equilibrium data were collected using a volumetric isotherm measurement rig and the kinetic responses were obtained using a differential adsorption bed. The kinetics measurements were performed by using a gas chromatograph to determine the mole fractions of ethane and propane in the samples. The adsorbent was kept under gas flow passing through the adsorber. The pressure of gas phase was constant, equal to one atmosphere, for different concentrations of alkanes with helium as the diluting gas. Such experimental technique suggests the use of volume-dominated model. The results of investigations performed by Qiao and Hu [10] are not surprising and showed that in the case of a highly non-uniform adsorbent which is the Norit activated carbon, the rate of adsorption is, to a greater extent, controlled by surface diffusion. Nevertheless, the theoretical studies showed [3] that the SRT approach can well describe the adsorption systems when the variances of pressure near the surface region were taken into account. Authors [10] applied the HMSD (Heterogeneous Macropore and Surface Diffusion) model of adsorption kinetics. The model takes into account the effects of surface energetic heterogeneity on both sorption kinetics and equilibrium and allows for the diffusion in both bulk and adsorbed phases. In our opinion, the results of calculations of kinetic partial isotherms showed by Qiao and Hu are not satisfactory; some discrepancy between the theoretical curves and the experimental data exists. To assure that we can analyze the kinetic data reported by Qiao and Hu [10], the criteria proposed by Rudzinski and Plazinski [11] can be used. The authors showed the simple way to resolve whether surface reaction or intraparticle diffusion is the ratedetermining step. It can be done by using the linear Lagergren regression for the kinetic data points close to the equilibrium values N(e): lnðNðeÞ  NðtÞÞ ¼ b  k t

(27)

(e)

The N values can be determined from the equilibrium isotherm. To adjust the above equation we plotted the experimental ln(N(e)  N(t)) values versus time. This procedure can be used only for those N(t) data points that are close to the equilibrium value N(e) which we determined by analyzing the equilibrium isotherms of ethane and propane by the Langmuir–Freundlich equation: kT=c1

u1t ðf pg; TÞ ¼

½K 1 p1 expfe01 =kTg

1 þ ½K 1 p1 expfe01 =kTg

kT=c1

(28)

K. Nieszporek / Applied Surface Science 255 (2009) 4627–4635

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Fig. 2. Application of LF equation (28) to the equilibrium isotherms of ethane and propane on the Norit activated carbon reported by Qiao and Hu [10].

Fig. 3. The Lagergren plots (27) drawn for the experimental kinetic data reported by Qiao and Hu [10]. The kinetics of adsorption was studied for three concentrations of adsorbates as indicated in the figure.

We decided to use the LF equation due to very good applicability for energetically non-uniform adsorption systems. The results are shown in Fig. 2 and Table 1. In the case of the kinetic data reported by Qiao and Hu [10], it seems that N(t) values are close to equilibrium for the times larger than 1000 s. The obtained results are shown in Fig. 3 whereas Table 2 collects the values of best-fit b and k parameters determined in this way. The satisfactory linearity in ln(N(e)  N(t)) )) versus t observed in Fig. 3 indicates good applicability of the Lagergren equation. To decide whether the rate of adsorption is determined by surface reaction or diffusion effects we should concentrate on the best-fit values of b. Two theoretical interpretations are possible [11]:

bD ¼ ln NðeÞ þ ln

6

(29)

p2

Table 1 The best-fit values of the parameters determined by applying LF Eq. (28) to the experimental equilibrium isotherms of ethane and propane reported by Qiao and Hu [10].

or

bL ¼ ln N ðeÞ

(30)

Except for the case of 10% concentration of ethane in the bulk phase, the determined values of b are distinctly closer to bL than to bD. It means that the model of the surface reaction describing the rate of adsorption can be applied [11]. It is beyond any doubt that in the case of discussed adsorption systems diffusion effects play a substantial role, but the results of Lagergren equation application justify the use of the SRT approach. As we mentioned above, taking into account the variances of pressure in the near surface region improves the theoretical description of kinetics of adsorption by the SRT approach. Equipped with the reason to accept the data reported by Qiao and Hu to verify the proposed model of calculations, we can start Table 2 Values of the best-fit parameters found by applying the Lagergren Eq. (27) to the kinetic data reported by Qiao and Hu [10]. Adsorbate

Bulk phase concentrations

k [s1]

bL

bD

b

N(e) [mmol/g]

1.09  102 6.09  103 6.78  103

C2H6

2% 5% 10%

0.003879 0.003500 0.003437

0.409 0.045 0.365

0.907 0.452 0.132

0.162 0.239 0.524

0.664 1.046 1.441

3.32  102 2.38  102 1.91  102

C3H8

2% 5% 10%

0.003127 0.003209 0.003142

0.388 0.682 0.881

0.109 0.184 0.383

1.301 1.105 0.866

1.475 1.977 2.413

Adsorbate

T [8C]

Mi [mmol/g]

kT/ci

K i expfe0i =kTg [kPa1]

C2H6

30 60 90

6.50 5.80 4.07

0.57 0.63 0.76

C3H8

30 60 90

6.31 5.40 4.54

0.44 0.47 0.55

K. Nieszporek / Applied Surface Science 255 (2009) 4627–4635

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Table 3 The best-fit values of the parameters determined by applying the Langmuir equation to the experimental equilibrium isotherms of ethane and propane, measured at 30 8C, reported by Qiao and Hu [10]. Adsorbate

Mi [mmol/g]

K i expfe0i =kTg [kPa1]

Error, residual sum of squares

C2H6 C3H8

3.67 3.75

6.452  102 2.957  101

0.232468 0.369383

the analysis of experimental data. As the first step of calculations we performed the theoretical analysis of single-gas adsorption equilibrium isotherms by the Langmuir equation. As the kinetic partial isotherms of mixed adsorption systems were measured at 30 8C, we analyzed only the equilibrium single-gas adsorption isotherms determined at the same temperature. Table 3 collects the best-fit parameters obtained in this way. Though the analyzed adsorption systems have distinctly heterogeneous character of adsorption, it can be seen in Fig. 4 that we obtained satisfactory agreement of the Langmuir equation with the experimental data. In the next step we can calculate the kinetic isotherms of the investigated adsorption systems. The main idea of the presented studies is to calculate the rate of adsorption of mixture components. To do it the following equation system should be

Fig. 4. Application of the Langmuir equation to the equilibrium isotherms of ethane and propane at 30 8C on the Norit activated carbon reported by Qiao and Hu [10].

Fig. 5. The results of calculations of partial kinetic isotherms by (31) for the experimental data reported by Qiao and Hu [10]. The kinetics of adsorption was studied for different gas-phase compositions (denoted in figures) at 30 8C. Theoretical curves were calculated by using the best-fit parameters obtained by adjusting the Langmuir equation to the equilibrium isotherms and collected in Table 3.

K. Nieszporek / Applied Surface Science 255 (2009) 4627–4635

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Table 4 The best-fit parameters obtained by applying equations system (31) to the experimental data reported by Qiao and Hu [10]. Bulk phase concentrations

Ka1

Kd1

Ka2

Kd2

a12

a21

b12

b21

Error (residual sum of squares)

2% C2H6 + 5% C3H8 5% C2H6 + 5% C3H8 10% C2H6 + 5% C3H8 5% C2H6 + 2% C3H8 5% C2H6 + 10% C3H8

4.26  105 4.38  105 1.96  105 1.31  105 1.24  104

9.72  104 4.48  104 2.09  105 9.17  104 2.06  104

3.71  105 6.34  105 1.75  103 3.35  105 5.39  105

1.99  104 1.32  104 1.12  104 6.26  105 2.69  105

1.81  104 1.01  105 1.11  104 97.18 8.82  104

5.58  101 1.05  102 1.31  102 1.03 2.00  101

5.05  101 3.12 1.77 1.15  105 2.13

427.87 18.25 66.81 557.77 137.99

1.93  102 2.40  102 9.90  103 1.53  102 5.61  102

solved (volume-dominated model):   8 du1 u1 1 > 2 ð1 u 1  u 2 Þ > ¼ ð1þ K p þK p Þ K p K > 1 2 a1 d1 1 2 1 > dt u1 ð1  u1  u2 Þ > > >   > > du2 ð1 u  u Þ u2 > 1 2 1 2 < ¼ ð1 þ K 1 p1 þ K 2 p2 Þ K a2 p2 K d2 dt u2 ð1 u1  u2 Þ > > d p1 du1 > > ¼ a ð p  p Þ  b > 12 01 1 12 > dt > dt > > > : d p2 ¼ a ð p  p Þ  b du2 21 02 2 21 dt dt (31) with the following boundary conditions: p1(0) = p01, p2(0) = p02, u1(0) = 0 and u2(0) = 0. In the above equations system, there occur only four parameters for each component of the mixture: Kai, Kdi, aij and

bij, whose values can be determined by the calculation procedure optimizing the agreement of (31) with the experimental data. The remaining quantities, i.e. Mi and Ki are those obtained from the single-gas adsorption equilibrium isotherms (Table 3). Authors [10] determined kinetic isotherms as fractional uptake versus time. The fractional uptake is defined as the uptake at any time divided by its value at final equilibrium (t ! 1). In order to compare the results of calculations with the experimental data the values of @ui/@t. should be multiplied by Mi/Ni1, and Ni1 can be determined by using the Langmuir equation for pi = p0i. The results of calculations are shown in Fig. 5 and the best-fit parameters are collected in Table 4. As can be seen, Fig. 5 shows excellent agreement between the theoretical curves and the experimental data. Although the theoretical model is relatively simple (assuming homogeneous

Fig. 6. The results of calculations of kinetics of single-gas adsorption isotherms by (32) for the experimental data reported by Qiao and Hu [10]. The kinetics of adsorption was studied for different gas-phase compositions (denoted in the figures) at 30 8C. The theoretical curves were calculated by using the best-fit parameters collected in Tables 3 and 4.

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Table 5 The best-fit parameters obtained by applying equation system (32) to the single-gas kinetic adsorption isotherms reported by Qiao and Hu [10]. Bulk phase concentrations

Kai

Kdi

ai

bi

Error (residual sum of squares)

2% C2H6 5% C2H6 10% C2H6 2% C3H8 5% C3H8 10% C3H8

4.26  105 1.31  105 1.96  105 3.35  105 3.71  105 5.39  105

9.72  104 9.17  104 2.09  105 6.26  105 1.99  104 2.69  105

1.90  103 1.50  103 3.20  104 5.28  104 9.41  104 3.05  104

40.00 24.06 28.92 2.35 4.15 11.82

8.34  102 5.44  102 7.38  102 1.57  102 9.10  103 2.04  102

character of the adsorption system) the calculations quantitatively reflect the experimental data. Also the assumption that diffusion effects are not a very important factor in the kinetics of adsorption of C2H6 + C3H8 on the Norit activated carbon, as reported by Qiao and Hu [10], proves to be advisable. Unquestionably, including into calculations the partial pressure change near the surface region during the adsorption process corresponds to such best agreement with the experimental data. The main aim of this paper, as frequently mentioned above, is the theoretical description of kinetics of mixed-gas adsorption. Just for this reason we start the calculations from the case of mixed-gas adsorption kinetics but we do not apply the calculation technique based on the theoretical predictions of the time-dependence of partial isotherms by using the single-gas equilibrium and kinetic data. The model of calculations shown above requires only the equilibrium of single-gas adsorption isotherms and all other values of best-fit parameters in (31) can be obtained by adjusting (31) to the experimental kinetic data. The following question arises: how will the results of calculations be obtained by using the same parameters, collected in Table 4, for the case of single-gas kinetics isotherms? The answer can be found in Fig. 6 and Table 5. In this case, to describe the adsorption kinetics of pure gases the following equation system should be solved: 8   du ð1  u 1 Þ u1 > > < 1 ¼ ð1 þ K 1 p1 Þ1 K a1 p21 K d1 dt u1 ð1  u1 Þ (32) > d p1 du 1 > : ¼ a1 ð p01  p1 Þ  b1 dt dt As can be seen the theoretical description of the rate of adsorption of single gases can be obtained by assuming in Eq. (13) that u2 = 0 and p2 = 0. In the above equation the parameters Ka1 and Kd1 are the same as in the case of gas mixture, i.e. those collected in Table 4. We decided to insert a1 and b1 as the quantities different from a12 and b12. We feel that a1 and b1 represents the diffusion of a given component in the environment of vacuum or another component— it is the reason for which we inserted different parameters a12, a21, b12 and b21 in Eq. (22). So, in the case of single-gas adsorption kinetics during our calculations we have only the two best-fit parameters: a1 and b1. Still another problem should be solved: which values of Ka1 and Kd1 can be used while calculating the rate of single-gas adsorption? For example, to describe theoretically the kinetic isotherm for the case of 5% C2H6, it seems that Ka1 and Kd1 can be taken from the results of calculations corresponding to the following mixture components: 5% C2H6 + 2% C3H8, 5% C2H6 + 5% C3H8 and 5% C2H6 + 10% C3H8. We decided to use those parameters which refer to the mixture with the lowest content of the second component to minimize its influence on adsorption of ethane. In our example, this is the mixture 5% C2H6 + 2% C3H8. Such a rule seems to be important in the quantitative description of kinetic curves of single-gases. Fig. 6 shows very good agreement between the theoretical curves (32) and the experimental single-gas kinetic data. Especially in the case of ethane adsorption, the theoretical curves agree with the experimental data.

Comparing Figs. 5 and 6 we can see that the shape of kinetic curve for ethane is different for mixtures and single gases whereas for propane the shape is the same. By calculating the adsorption energy distribution functions Qiao and Hu showed [26] that propane molecules are preferentially adsorbed. Since the adsorption system consists of fast moving/less strongly adsorbed ethane and slower moving/more strongly adsorbed propane, the overshoot maximum in the ethane uptake curve is observed. The kinetic curves for ethane are different in single and binary systems because preferentially adsorbed propane dislodges already adsorbed ethane molecules. 6. Summary Summing up, we can point to very good applicability of the presented theoretical model of calculations with the experimental data reported by Qiao and Hu [10]. Taking into account simplicity of the adsorption equilibrium model used (energetically homogeneous, lattice model of monolayer adsorption), the results of application of the SRT approach are very satisfactory. Although we studied only the case of volume-dominated model, it seems that in other cases theoretical kinetic equations should work. In our opinion, the reason for such effectiveness is taking into account the variances of partial pressures of mixture components near the surface region. Our results confirm the finding by Panczyk [3]. Besides the homogeneous model of adsorption, the case of strongly energetically heterogeneous one, which can be described by the rectangular AED, was studied. The obtained theoretical kinetic expressions were not verified on the real adsorption system due to very good agreement of the Langmuir-SRT model with the experimental data by Qiao and Hu [10]. The detailed studies concerning this case will be the matter of further publication. The present paper can be treated as opening of the series devoted to the theoretical description of kinetics of mixed-gas adsorption by using the SRT approach. Taking into account different theoretical models of energetic non-ideality of the adsorbed phase, interaction effects between the adsorbed molecules or other important physical factors, the model of calculations can be further developed. Acknowledgement The author expresses his gratitude to Dr. Tomasz Panczyk for valuable discussions. References [1] [2] [3] [4]

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