A continuous site energy distribution function from Redlich–Peterson isotherm for adsorption on heterogeneous surfaces

Chemical Physics Letters 492 (2010) 187–192

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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

A continuous site energy distribution function from Redlich–Peterson isotherm for adsorption on heterogeneous surfaces K. Vasanth Kumar *, M. Monteiro de Castro, M. Martinez-Escandell, M. Molina-Sabio, J. Silvestre-Albero, F. Rodriguez-Reinoso Laboratorio de Materiales Avanzados, Departamento de Química Inorgánica, Universidad de Alicante, Apartado 99, 030080 Alicante, Spain

a r t i c l e

i n f o

Article history: Received 8 March 2010 In final form 18 April 2010 Available online 20 April 2010

a b s t r a c t A site energy distribution function for carbon adsorbents is proposed for the adsorption equilibrium data following a Redlich–Peterson isotherm. The proposed model is successfully applied to determine the site energy distribution of two pitch-based activated carbons (PA and PBA) developed in our laboratory and also for other common adsorbent materials for different gas molecules. According to the proposed model the site energy distribution curve for carbons PA and PBA are found to be unimodal for N2 gas molecules at 77 K. The proposed model successfully predicts the site energy parameters of both homogeneous and heterogeneous adsorbents. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction In recent years, several continuous site energy distribution functions for the adsorption of solutes from liquids on heterogeneous surfaces have been proposed based on different theoretical isotherms [1–5]. A review on these models was recently reported by Umleby et al. [6]. The most widely used site energy distribution functions are the expressions of Umpleby et al. [2] and Szabelski et al. [5], which are based on Freundlich or Langmuir–Freundlich isotherms. A site energy distribution function is proposed in the present study using a Redlich–Peterson isotherm to explain the heterogeneity of the adsorbent materials. The Redlich–Peterson isotherm is described by [7]:

q¼

Ap 1 þ Bpg

ð1Þ

where, A and B are the Redlich–Peterson isotherm constants and g is the exponent reflecting the heterogeneity of the adsorbent, which lies between 0 and 1. The linearized expression of Eq. (1) is given by

ln

  Ap  1 ¼ ln B þ g ln p q

ð2Þ

The site energy distribution function based on Redlich–Peterson isotherm will have several advantages due to the limiting conditions used; it converges to Henry and Langmuir isotherm * Corresponding author. E-mail addresses: [email protected], [email protected] (K. Vasanth Kumar). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.04.044

when g values are equal to zero and one, respectively. Also Redlich–Peterson isotherm will represent a Freundlich isotherm when the isotherm parameters A and B  1 and g < 1. The binding affinity distribution according to a Redlich–Peterson isotherm was obtained using the basic integral equation that explains the theory of heterogeneous surfaces [5]

qðpÞ ¼

Z

Emax

Emin

qh ðE; pÞf ðEÞdE

ð3Þ

where q(p) represents the amount of gas molecules adsorbed on a heterogeneous surface constituted by an addition of different energetically homogeneous isotherms (qh). In the case of Redlich–Peterson isotherm, q(p) corresponds to a heterogeneous non-ideal isotherm, when g < 1 and a homogeneous isotherm, when g = 1. The term f(E) represents the site energy frequency distribution over a range of energy. Emin and Emax are the limits of energy space that are directly related to the maximum and minimum pressure in the adsorption isotherm. The energy of adsorption can be determined using the Polanyi potential theory given by

    E E  Es p ¼ ps exp  ¼ ps exp  RT RT

ð4Þ

where E is the lowest physically realizable energy and Es is the adsorption energy corresponding to p = ps [8]. Incorporation of Eqs. (1) and (4) into Eq. (3) will lead to an approximate site energy distribution, f(E*) [4], which can be obtained by differentiating the Redlich–Peterson isotherm with respect to E*

f ðE Þ ¼ 

 d qe ¼





Aps ½exp ðERT Þ 1þBpgs

dE

½exp ð 

 ERT

Þ

g

ð5Þ

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K. Vasanth Kumar et al. / Chemical Physics Letters 492 (2010) 187–192

The analytical solution to Eq. (5) can be written as

 i  h g  E 1 ps Bps ðg  1Þ exp  gE A exp  RT RT f ðE Þ ¼   2

 E g þ1 RT Bpgs exp  RT

7



PA 6

ð6Þ

Eq. (6) calculates the site energy distributions for the adsorption of gas molecules from the Redlich–Peterson isotherm parameters, A, B and g which can be determined by a simple linear or non-linear regression analysis. In order to demonstrate the applicability of Eq. (6), the experimental isotherms of N2 at 77 K onto two types of activated carbons (i) a pitch-based activated carbon (PA) and (ii) a pitch-based carbon containing boron (PBA) developed in our laboratory are selected as model adsorption systems. The proposed model was also applied to determine the site energy distribution of other adsorbent materials commonly encountered in the literature that include CuBTC, DAY zeolite, 13X zeolite, commercial activated carbon, activated carbon fiber, single wall carbon nanotubes, silica gel and some polymeric adsorbents for different adsorptive molecules.

2. Experimental An aromatic petroleum residue (ethylene tar-R1) [9,10] was mixed with pyridine borane complex (PyB) in an ultrasonic bath for an hour, to give a mixture containing 2 wt.% of boron. Pyridine borane complex is apparently soluble in the petroleum residue. Pyrolysis of the mixture was performed at 440 °C, soak time of 4 h and 1 MPa pressure, thus leading to a pitch containing the heteroatom (labeled PB). A reference pitch P without boron was also prepared. The activated carbons PA and PBA have been prepared from the respective petroleum pitches as follows: KOH and the pitch were mixed in a ball mill during 30 min with a impregnation ratio of KOH/carbon of 3/1 and then thermally treated in a horizontal furnace at 800 °C under nitrogen flow of 100 ml/min, soak time of 2 h. Finally, the activated carbons were washed in a Soxhlet apparatus for 24 h with water and dried at 110 °C for 24 h in a vacuum oven. The chemical compound (boron) was intentionally mixed with the precursor in an attempt to improve the hydrogen storage properties of the pitch-based activated carbon. The detailed physicochemical characteristics of the precursor and chemical characteristics of synthesized carbons and their hydrogen storage capacities are discussed elsewhere [11]. Adsorption isotherms of N2 at 77 K were performed using a homemade automatic volumetric equipment featuring two pressure sensors (0–1.3 kPa and 0–0.1 MPa). The samples were previously degassed at 1  104 Pa and 250 °C.

3. Site energy distributions of pitch-based activated carbons for N2 at 77 K according to the Redlich–Peterson isotherm The sorption of N2 onto PA and PBA following the linearized expression of Redlich–Peterson isotherm can be predicted from a plot of ln(Ap/q  1) versus ln(p). However, this is not directly possible as the linearized Redlich–Peterson isotherm equation contains three unknown parameters, A, B and g. Therefore, an error minimization procedure is adopted that simultaneously optimizes the A value by maximizing the error function, coefficient of determination, r2, between experimental data and Eq. (2). Fig. 1 shows the Redlich–Peterson isotherm plot for N2 at 77 K onto PA and PBA obtained using this optimization procedure. The calculated isotherm parameters and the corresponding r2 values are given in Table 1. The higher r2 values (0.999) suggest that Redlich–Peterson isotherm is the best fit isotherm for the sorption of N2 on stud-

PBA

5

Ln(Ap /q -1)



4

3

2

1

0 0

1

2

3

4

5

6

7

Ln(p ), torr Fig. 1. Redlich–Peterson isotherm plot for N2 onto PA and PBA at 77 K by linear regression analysis.

ied activated carbons. Table 1 indicates that the surfaces of PA and PBA are heterogeneous with the same heterogeneity index (g 0:9Þ. The calculated isotherm parameters are fitted to Eq. (6) to obtain the site energy distributions for N2 molecules on these adsorbents. Though Eq. (6) has no mathematical restrictions, this expression will predict negative energy when E < Es, which has no physical meaning. Fig. 2 shows this concept based on Eq. (4) as a plot of p versus E* for N2 at 77 K. It can be observed from Fig. 2 that the energy values within the range of pmin to ps could be considered valid. Since the isotherm parameters are sensitive to the pressure range covered, only the energy distribution obtained within this pressure range can be considered valid. Fig. 3 shows the site energy distribution plots for the adsorption of nitrogen molecules on carbons PA and PBA at 77 K, respectively. From Fig. 3, it can be observed globally that for both PA and PBA, the energy distribution function based on a Redlich–Peterson isotherm is unimodal, distributed with two characteristic regions: (i) an unimodal peak corresponding to low-energy binding sites and (ii) an asymptotically decaying region corresponding to high energy binding sites. Specifically, the site energies are distributed exponentially and unimodally over the ranges 10–1750 J/mol and 1750–10149 J/mol, respectively. This suggests that PA and PBA have more than one energetic state for the uptake of nitrogen molecules within the studied pressure range. Based on the site energy spectrum, it can be concluded that the binding sites with lower binding site energies are relatively more heterogeneous than the binding sites with higher binding site energies. The lower and higher binding energy sites herein refers to binding sites with E* < 1750 J/mol and E* > 1750 J/mol, respectively. The intensity of energy corresponding to peak maximum of PBA was relatively higher than for PA, confirming the structural differences in these adsorbents. The structural differences in the studied adsorbents was expected as the chemical compound that was dissolved into the petroleum residue to obtain mesophase pitch containing boron upon pyrolysis can alter the reactions taking place during the pyrolysis and, consequently, it could modify the mesophase content and the resulting texture, leading to difference in site energy distributions. Generally, it could be concluded that the site energy distribution can be obtained using a theoretical adsorption isotherm. Sev-

189

K. Vasanth Kumar et al. / Chemical Physics Letters 492 (2010) 187–192 Table 1 Redlich–Peterson isotherm parameters of the studied adsorption systems determined by linear regression analysis.

a

Adsorption system (gas/solid)

References

Redlich–Peterson isotherm parameters

N2/PA @77 K

This study

N2/PBA @77 K

This study

N2/CuBTC @77 K

Synthesized in our lab using the recipe in Ref. [25]

Toluene/dealuminated Y zeolite (DAY) @298 K

[26]

Toluene/dealuminated Y zeolite (DAY) @308 K

[26]

m-Xylene/dealuminated Y zeolite (DAY) @298 K

[26]

m-Xylene/dealuminated Y zeolite (DAY) @308 K

[26]

CO2/SWCNT @273 K

[27]

Toluene/activated carbon fiber @298.15 K

[28]

Dichloromethane/activated carbon fiber @298.15 K

[28]

Trichloroethylene/activated carbon fiber @298.15 K

[28]

Toluene/Ambersorb 600 @298.15 K

[29]

Toluene/Sp 850 @298.15 K

[29]

Toluene/Dowex Optipore V493 @298.15 K

[29]

Methyl acetate/activated carbona @298.15 K

[30]

Methyl acetate/silica gel @298.15 K

[30]

Methyl acetate/13X zeolite @298.15 K

[30]

A (mmol/g)(1/torr) B (1/torr)g g r2 A (mmol/g)(1/torr) B (1/torr)g g r2 A (mol/g)(1/torr) B (1/torr)g g r2 A (mol/kg)(1/Pa) B (1/Pa)g g r2 A (mol/kg)(1/Pa) B (1/Pa)g g r2 A (mol/kg)(1/Pa) B (1/Pa)g g r2 A (mol/kg)(1/Pa) B (1/Pa)g g r2 A (cm3/g)(1/bar) B (1/bar)g g r2 A (mmol/g)(1/kPa) B (1/kPa)g g r2 A (mmol/g)(1/kPa) B (1/kPa)g g r2 A (mmol/g)(1/kPa) B (1/ kPa)g g r2 A (mmol/kg)(1/kPa) B (1/ kPa)g g r2 A (mmol/kg)(1/kPa) B (1/ kPa)g g r2 A (mmol/kg)(1/kPa) B (1/kPa)g g r2 A (mg/g)(1/torr) B (1/torr)g g r2 A (mg/g)(1/torr) B (1/torr)g g r2 A (mg/g)(1/torr) B (1/torr)g g r2

29.632 2.062 0.887 0.999 52.743 2.647 0.877 0.973 53 687 091 7.02E + 09 0.853 0.979 1.45E02 9.12E03 1 0.983 8.55E03 6.12E03 0.997 1.000 1.92E01 1.10E01 0.999 1.000 3.42E01 2.57E01 1.000 1.000 184.082 7.314 0.293 0.980 2.68E+09 7.32E+08 0.607 0.867 48.974 11.646 0.761 0.999 165.138 39.782 0.966 1.000 800.760 413.075 0.776 0.968 37.708 8.736 0.552 0.996 64.124 9.734 0.714 0.915 70.76246 0.569436 0.926413 0.999 53 687 091 7.02E+09 0.853189 0.995 11.384 1.367 0.847 0.999

China Activated Carbon Industries Co.

eral researchers determined the site heterogeneity or binding site energies of carbon based materials and other porous adsorbents using the isosteric heat of adsorption for different surface coverage

values or other thermodynamic parameters like entropy and enthalpy of adsorption [18–24] using the adsorption isotherm obtained at different temperatures. Table 2 shows that some of the

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K. Vasanth Kumar et al. / Chemical Physics Letters 492 (2010) 187–192

4. Applicability of the Redlich–Peterson isotherm for estimating the site energy distribution of some common adsorbents reported in the literature

12000

10000

E *, J/mol

8000 E max

6000

4000

2000 E min

0 p max

p min

-2000 0.00001 0.0001

0.001

0.01

0.1

1

10

100

1000

p , torr Fig. 2. Effect of pressure on net adsorption energy for N2 gas molecules at 77 K (the solid and dotted lines are only as a guide).

0.01

f (E* ), (mmol/g)(g/J)

q¼

Ap q Bp ¼ m 1 þ Bp 1 þ Bp

ð7Þ

Eq. (7) can be used to calculate directly the number of binding sites, qm that has a binding affinity coefficient of B. The binding affinity coefficient, B can be related with the differential heat of adsorption corresponding to the coverage q/qm as [31,32]:

0.012

B ¼ expðQ diff =RTÞ: PA PBA

0.008

0.006

0.004

0.002

0 0

2000

4000

6000

8000

10000

12000

E* , J/mol Fig. 3. Site energy distribution of PA and PBA for N2 gas molecules at 77 K.

Table 2 Site energy distribution of studied activated carbons and isosteric heat of adsorption of some carbon based adsorbents.

a

To study the general applicability of the proposed site energy distribution function (Eq. (6)), the experimental equilibrium data of different gas species onto several adsorbents at different adsorption conditions are fitted to the Redlich–Peterson isotherms to obtain their corresponding site energy parameters. The studied adsorption systems and the experimental conditions and their corresponding Redlich–Peterson isotherm parameters are given in Table 1. The isotherm parameters reported in Table 1 are determined in this study by a trial and error linear regression analysis explained above. It can be observed from Table 1 that the Redlich– Peterson isotherm constant g values for toluene and m-xylene on the DAY zeolite was found to be unity suggesting that this material has a homogeneous surface. In the case of a homogeneous surface the site energy parameters can be determined directly from the isotherm parameters, as the Redlich–Peterson isotherm reduces to Langmuir for g = 1:

Adsorbent/gas

Isosteric heat of adsorption (kJ/mol)

E* (kJ/mol)

References

PA and PB Graphite Activated carbon honeycomb monolith Activated carbon Grade 6 Spheron Graphon Activated charcoal

– 8–10 18–19

1E04–10 – –

This study [12] [13]

0.1–5 7–13 6–13a 15.92

– – – –

[14] [15] [15] [16] as cited by [17]

Calorimetrically determined values.

isosteric heat of adsorption values for carbon based materials are in good agreement with the site energy distribution spectrum reported in this study.

The calculated qm and Qdiff for m-xylene and toluene at 298.15 K for the zeolite was found to be around 22 and 28 kJ/g, respectively. The site energy distribution of other studied adsorption systems for which g < 1, within the limits of analytical window (Emax and Emin) was obtained by substituting the isotherm parameters in Eq. (6) as shown in Fig. 4. The Emin and Emax of the studied adsorption systems are determined from the pmin and pmax of experimental isotherms reported in the literature, as explained in Fig. 2. The value of Emin will be equal to zero for the experimental isotherms covering the pressure range from pmin to ps. The units of the site energy distribution function, f(E*), in Fig. 4 depend on the units of the isotherm parameters. As an example, if the amount of gas adsorbed is expressed in mmol/g, then f(E*) will have the units of (mmol/ g)(mol/J). However, attention should be paid to the units of pressure and they should be maintained constant for both p and ps. The units of f(E*) of the studied systems are given in Table 3. Fig. 4 shows that the site energy distribution of the studied adsorbents that include CuBTC, DAY zeolite, SWCNT, activated carbon fiber, Ambersorb 600, Sp 850 and Dowex Optipore V493, activated carbon, silica gel and 13X zeolite are exponentially distributed (irrespective of the adsorptive molecules and experimental conditions) over the considered range of experimental pressure. The exponential distribution of site energies is expected to be due to one of the limiting cases of the Redlich–Peterson isotherm for the constant B  1. Table 1 shows that for the adsorbents reported in Fig. 4, the Redlich–Peterson isotherm constant, B was found to be much greater than unity (i.e., B  1). When the constant B  1, the denominator of the Redlich–Peterson isotherm can be approximated to Bpg as (Bpg  1) and thus this isotherm reduces to a Freundlich expression as

q¼

A 1g p B

Thus, it can be concluded that the site energy of the adsorbents within the analytical limits of gas molecules is found to be exponentially distributed or, in other words, the surfaces of studied adsorbents are highly heterogeneous. The heterogeneous nature

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K. Vasanth Kumar et al. / Chemical Physics Letters 492 (2010) 187–192

0.0009

0.0008

Ambersorb600/[email protected] K Sp 850/[email protected] K Dowex Optipore V493/[email protected] K

0.0007

Activated carbon fiber/[email protected] K (x 1/100) Activated carbon fiber/[email protected] K

0.0006

Activated carbon fiber/[email protected] K SWCNT/CO2@273 K (x 100) 0.0005

f (E *)

Activated carbon/methyle [email protected] K(x 100) CuBTC/N2@77K(x 1/100)

0.0004

Silica gel/methyl [email protected] K 13X zeolite/methyl [email protected] K

0.0003

0.0002

0.0001

0 0

5000

10000

15000

20000

25000

30000

E *, J/mol Fig. 4. Site energy distribution of some common adsorbents for different gas molecules according to Redlich–Peterson isotherm.

Table 3 The units of f(E*) of the studied adsorption systems. Gas

Adsorbent

f(E*)

N2 N2 N2 Toluene m-Xylene CO2 Toluene Dichloromethane Trichloroethylene Toluene Toluene Toluene Methyl acetate Methyl acetate Methyl acetate

PA PBA CuBTc Dealuminated Y zeolite Dealuminated Y zeolite SWCNT Activated carbon fiber Activated carbon fiber Activated carbon fiber Ambersorb 600 Sp850 Dowex Optipore V493 Activated carbon Silica gel 13X zeolite

(mmol/g)(mol/J) (mmol/g)(mol/J) (mol/g)(mol/J) (mol/kg)(mol/J) (mol/kg)(mol/J) (cm3/g)(mol/J) (mmol/g)(mol/J) (mmol/g)(mol/J) (mmol/g)(mol/J) (mmol/kg)(mol/J) (mmol/kg)(mol/J) (mmol/kg)(mol/J) (mg/g)(mol/J) (mg/g)(mol/J) (mg/g)(mol/J)

of these adsorbents can be due to their complex porous structure which contains meso or micropores of different dimensions and forms. In the case of crystalline adsorbents (CuBTC), the surface heterogeneities could be expected as the CuBTC crystals synthesized in our laboratory seems to have different faces with irregularities and agglomeration and crystal breakages which must be present in such a sample (the SEM pictures are not shown in this study). Site energy heterogeneities are previously reported by Rhodin [33] for polycrystalline copper surfaces. The present study showed that the site energy distribution function obtained from a Redlich–Peterson isotherm can be used to determine the surface heterogeneity of different solid adsorbents differing by physicochemical characteristics for molecules from the gas phase. Due to the limiting cases of the Redlich–Peterson isotherm, the proposed model was found to be useful in estimating site energies of both homogeneous and heterogeneous surfaces. The proposed model is simple and uses only the Redlich–Peterson isotherm parameters to rapidly generate the site en-

ergy distribution of any adsorbent for a particular gas molecule of interest. Though the present study is only limited to gas phase adsorption systems, the site energy distribution equation can be applied to liquid phase adsorption systems provided the experimental equilibrium data are represented by a Redlich–Peterson isotherm. Acknowledgements Support from the Ministerio de Ciencia e Innovacion (Project MAT2007-61734, Fondos FEDER) is acknowledged. KVK would like to thank Ministerio de Ciencia e Innovacion (Spain) for the Juan de la Cierva contract. References [1] R.J. Umpleby, M. Bode, K.D. Shimizu, Analyst 125 (2000) 1261. [2] R.J. Umpleby, S.C. Baxter, Y. Chen, R.N. Shah, K.D. Shimizu, Anal. Chem. 73 (2001) 4584. [3] R.J. Umpleby, S.C. Baxter, M. Bode, J.K. Berc Jr., R.N. Shah, K.D. Shimizu, Anal. Chem. 73 (2001) 4584. [4] M.C. Carter, J.E. Kilduff, Environ. Sci. Technol. 29 (7) (1995) 1773. [5] P. Szabelski, K. Kaczmarski, A. Cavazzini, Y.-B. Chen, B. Sellergren, G. Guiochon, J. Chromatogr. A 964 (2002) 99. [6] R.J. Umpleby, S.C. Baxter, A.M. Rampey, G.T. Rushton, Y. Chen, K.D. Shimizu, J. Chromatogr. B 804 (2004) 141. [7] O. Redlich, D.L. Peterson, J. Phys. Chem. 63 (1959) 1024. [8] G.F. Cerofolini, Thin Solid Films 23 (1974) 129. [9] M. Martinez-Escandell, P. Torregrosa, H. Marsh, F. Rodriguez-Reinoso, R. Santamaría-Ramírez, C. Gomez de Salazar, E. Romero-Palazon, Carbon 37 (1999) 1567. [10] P. Torregrosa-Rodriguez, M. Martinez-Escandell, F. Rodriguez-Reinoso, H. Marsh, C. Gomez de Salazar, E. Romero-Palazon, Carbon 38 (2000) 535. [11] M.M. Castro, M. Martinez-Escandell, M. Miguel-Sabio, F. Rodriguez-Reinoso, Carbon 48 (2010) 636. [12] W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press Ltd., London, UK, 1992. [13] R.P. Ribeiro, T.P. Sauer, F.V. Lopes, R.F. Moreira, C.A. Grande, A.E. Rodrigues, J. Chem. Eng. Data 53 (2008) 2311. [14] B.-U. Choi, D.-K. Choi, Y.-W. Lee, B.-K. Lee, J. Chem. Eng. Data 48 (2003) 603. [15] L.G. Joyner, P.H. Emmett, J. Am. Chem. Soc. 70 (1948) 2353. [16] G.C. Ray, E.O. Box, Ind. Eng. Chem. 42 (1950) 1315.

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