Separation and Purification Technology 34 (2004) 165–176

Study of isosteric heat of adsorption and activation energyfor surface diffusion of gases on activated carbon

using equilibrium and kinetics information

Kean Wang1, Shizhang Qiao2, Xijun Hu∗

Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Abstract

Pure component adsorption equilibria and kinetics of ethane and propane were measured on two activated carbon samplesat various temperatures. The isosteric heat of adsorption was derived from the equilibrium information using the Clapeyronequation, while the activation energy for surface diffusion was derived from the surface diffusivity, which was extracted fromthe kinetics data using the macropore and surface diffusion (MSD) model. Two isotherm models, Unilan and Toth equations areused in the analysis of equilibrium and kinetics data. It is found that the choice of adsorption isotherm has a significant influenceon the calculated values of isosteric heat of adsorption, but has less influence on the activation energy for surface diffusion.The ratio of activation energy to isosteric heat of adsorption is found between 0.25 and 0.6 for ethane and propane on the twocarbons, depending on the choice of isotherm equation. This ratio is a weak function of surface loading.© 2003 Elsevier B.V. All rights reserved.

Keywords: Adsorption; Heat of adsorption; Activation energy; Surface diffusion; Equilibrium, Kinetics

1. Introduction

Activated carbon is a highly microporous materialwith large surface area (>1000 m2/g). In the ad-sorption process of gases/vapors on such adsorbents,surface diffusion can make significant contribution tothe overall mass transfer of strongly adsorbed speciesinside the carbon particle. Surface diffusion is very

∗ Corresponding author. Tel.: +85-2-2358-7134;fax: +85-2-2358-0054.

E-mail address: kexhu@ust.hk (X. Hu).1 Present address: School of Mechanical Production Engineer-

ing, Nanyang Technological University, Singapore.2 Present address: Department of Chemical Engineering,

University of Queensland, Brisbane 4072, Australia.

complicated in nature, but has been known to be anactivated process with surface diffusivity presentinga strong dependence on surface loading [1,2] andfollowing an Arrhenius-type temperature dependence[3]. Studies also revealed that the activation energyfor surface diffusion is related to and is a fraction ofthe energy of adsorption [4,5]. Literature on surfacediffusion showed that the ratio of the activation energyfor surface diffusion to the energy of adsorption couldrange from 0.33 to 1 [1]. Sladek et al. [6] pointed outthat, for the diffusion of physically adsorbed species,this ratio can be 1 or 1/2. Although one would expectthat this ratio should be dependent on local adsorptionenergy, it has been treated as a constant in many stud-ies, e.g. Kapoor and Yang [7–9] investigated the effect

1383-5866/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S1383-5866(03)00190-4

166 K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176

Nomenclature

a ratio of activation energy for surface diffusion to adsorption energya0 ratio of activation energy for surface diffusion to adsorption energy at zero surface loadingb adsorption isotherm parameter (per kPa in eq.4, kPat in eq.5)Cb bulk phase concentration in adsorber (kmol/m3)Cp gas phase concentration in macropore (kmol/m3)Cp0 initial gas phase concentration in macropore (kmol/m3)Cµ adsorbed phase concentration (kmol/m3)Cµ0 initial adsorbed phase concentration (kmol/m3)Cµs saturation adsorption capacity (kmol/m3)Dp pore diffusivity (m2/s)Dµ apparent surface diffusivity or Fickian diffusivity (m2/s)D0

µ surface diffusivity at zero loading (m2/s)D∗

µ surface diffusivity at zero energy level (m2/s)Ea activation energy for surface diffusion (kJ/mol)E0

a activation energy for surface diffusion at zero surface loading (kJ/mol)f fugacity (kPa)H Henry’s law constant (kmol/m3 per kPa)JP diffusion flux in macropore (kmol/m2 per s)Jµ diffusion flux in adsorbed phase (kmol/m2 per s)km mass transfer coefficient of film (1/s)MSD macropore and surface diffusion (model)qst isosteric heat at a given specific loading (kJ/mol)q0st isosteric heat of adsorption at zero loading (kJ/mol)

r particle coordinate (mm)s isotherm parameter (eq. 4) geometric factor of particle, 0, 1, 2 for slab,

cylinder and sphere, respectivelyt isotherm parameter (eq. 5), time (s)θ surface coverageµ chemical potential (kJ/mol)εM porosity of adsorbent particle

of surface heterogeneity on adsorption kinetics withthis ratio being taken as 0.5 or 1. Do and co-workers[10–13] used the value of 0.5 for this ratio in most oftheir studies of adsorption kinetics on activated carbon.

The isosteric heat of adsorption for pure gas andgas mixture is a critical parameter for adsorptive sep-aration processes. Dunne et al. [14–16] pointed outthat, for binary mixture, the isosteric heats are directlyrelated to the temperature coefficient of selectivityat fixed loading. They measured the heat of adsorp-tion with a calorimeter (calorimetric heat) and theisotherm simultaneously for a series of gas adsorbateson a series of adsorbents, which present different

heterogeneity. Their results showed good agreementbetween the heat of adsorption measured and the isos-teric heat of adsorption calculated using the Clapeyronequation. Sircar [17,18] and Dunne et al. [16] inves-tigated the isosteric heat of adsorption of binary gasmixtures on heterogeneous adsorbent and found thatthe isosteric heat for a component in gas mixture canbe very much different from that of a pure species.

The isosteric heat of adsorption can be conve-niently calculated from the Clapeyron equation withthe assumption of an isotherm, Valenzuela and Myers[19] derived isosteric heat of adsorption for manyadsorbate–adsorbent systems using two isotherm

K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176 167

equations: Unilan and Toth. Their results revealed thatthe isosteric heat versus surface loading is very sen-sitive to the choice of the isotherm equation as wellas the pressure range of the equilibrium data. Dunneet al. [14] pointed out that the difference between theisosteric heats of adsorption at the limit of zero load-ing reported by different investigators might vary bya large margin (up to 40%). Pan et al. [20] employedthe non local density function (NLDF) to generateisotherms at different temperatures and examined thetwo assumptions used in the Clapeyron equation forthe calculation of the heat of adsorption, i.e. (1) idealbulk gas phase and (2) a negligible adsorbed phasemolar volume. They concluded that these two as-sumptions are reasonable at the low-pressure regionbut may have a significant effect on the result at highrelative pressure region or for heavy adsorbates.

This article will study the activation energy for sur-face diffusion and the isosteric heat of adsorption fortwo adsorbates, ethane and propane, on two activatedcarbon samples (Ajax, and Norit). These two impor-tant parameters will be derived from pure componentadsorption equilibrium and kinetic data, respectively.The effect of the adsorption isotherm on the calcula-tion of these two parameters will also be investigated.

2. Theory

2.1. Isosteric heat of adsorption

The isosteric heat of adsorption of a pure gas, qst , ata given specific surface loading, Cµ, can be calculatedfrom the adsorption isotherms of that adsorbate at dif-ferent temperatures using the Clapeyron equation:

qst = RT2[∂ ln f

∂T

]Cµ

(1)

where R is the gas constant, T is temperature, and f isthe fugacity, which is equal to the pressure, Cp, for anideal gas. At the low pressure (Henry’s Law) region,the isotherm should be linear, i.e.:

Cµ = H × Cp (2)

where H is Henry’s law constant. Since H is a func-tion of temperature only, the isosteric heat of adsorp-tion of a species will represent a characteristic of the

adsorbate–adsorbent system. By substituting eq. (2)into eq. (1), the isosteric heat of adsorption at zeroloading limit, q0

st , can be expressed in terms of Henry’slaw constant as:

q0st = −RT2

[∂ ln H

∂T

](3)

The isosteric heat of adsorption calculated fromClapeyron equation is sensitive to the choice of theisotherm equation [17–19]. Two frequently usedisotherm equations, Unilan and Toth, are employed inthis article to calculate qst . The Unilan isotherm takesthe form of:

Cµ = Cµs

2sln

(1 + besCp

1 + be−sCp

)(4a)

where Cµs is the adsorption saturation capacity, b ands are the affinity and the heterogeneity parameters ofthe adsorbent. The Henry’s law constant for Unilanequation is:

H = Cµsbes − e−s

2s(4b)

The Toth isotherm equation is in the form of:

Cµ = Cµs

Cp

(b + Ctp)1/t

(5a)

where t is the degree of the heterogeneity of the adsor-bent. The Henry’s law constant for Toth equation is:

H = Cµsb−1/t (5b)

2.2. Adsorption kinetics and activation energy forsurface diffusion

Surface diffusion is an activated process and thetemperature dependence of surface diffusivity can becorrelated by an Eyring equation:

Dµ = D∗µ exp

(−Ea

RT

)(6)

where Ea is the activation energy for surface diffusion.D∗

µ is the surface diffusivity at zero energy level.With the driving force for surface diffusion being

taken as the gradient of chemical potential, µ, thesurface flux can be written as:

Jµ = −D0µCµ

∂µ

∂r= −D0

µ

d ln Cp

d ln Cµ

∂Cµ

∂r

= −Dµ

∂Cµ

∂r(7)

168 K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176

where D0µ is the surface diffusivity at zero loading,

r is the coordinate of the diffusion path, and Dµ isthe ‘apparent’ surface diffusivity, which is stronglyconcentration dependent, represented by the Darkenrelationship:

Dµ = D0µ

d ln Cp

d ln Cµ

(8)

When a Langmuir isotherm is used, Eq. (8) reducesto the HIO model [21], which has been widely usedin the past to correlate Dµ and D0

µ:

Dµ = D0µ

1 − θ(9)

where θ is the fractional surface loading. Modifica-tions of this relation are also seen in the literature, e.g.Yang et al. [22] modified the HIO model with the con-sideration of the second layer adsorption. Hu et al. [2]found that the concentration dependence of the sur-face diffusivity is much stronger than that predictedby the Darken relationship and later Do [23] proposeda theory to explain that observation. In this study, theDarken relationship will be used as the approximationfor the concentration dependence of surface diffusiv-ity.

Since the structure of activated carbon particle isgenerally bi-dispersed, the macropore and surface dif-fusion or MSD model [2] is a reasonable choice forthe simulation of adsorption kinetics on such an ad-sorbent. With the MSD model, however, the adsorbentis treated as being effectively homogeneous and thederived kinetics parameters will represent some kindof average kinetic behavior of the system. The overallmass balance over a particle with geometric factor of‘s’ (0, 1, 2 for slab, cylinder, and sphere, respectively)can be expressed in term of MSD model as:

εM

∂Cp

∂t+ (1 − εM)

∂Cµ

∂t

= − 1

rs

∂

∂r[εMrsJP + (1 − εM)rsJµ] (10a)

where εM is the macroporosity of the adsorbent, andCp is the bulk phase concentration. Jp is the diffusionflux in the bulk phase, which is described by the Fick’slaw:

JP = −DP

∂CP

∂r(10b)

where the pore diffusivity, Dp, is the combina-tion of the Knudsen and molecular diffusivity [24].The boundary and initial conditions for Eq. (10a)are described in the reference [2] and are brieflyre-introduced here as:

t = 0, Cp = Cp0, Cµ = Cµ0, Cb = Cb0 (10c)

r = 0, εMJP + (1 − εM)Jµ = 0 (10d)

r = R, εMJP + (1 − εM)Jµ = km(CP − Cb) (10e)

Using the MSD model, where the concentrationdependence of surface diffusivity is described by theDarken relationship (Eq. (8)), we can analyze thekinetic data of an adsorbate–adsorbent system at dif-ferent bulk concentrations simultaneously and deriveD0

µ. With D0µ at different temperatures, the activation

energy for surface diffusion at zero surface loading,E0

a, is then derived from the slope of the plot of ln(D0µ)

against 1/T. Taking q0st as the heat of adsorption at

zero surface loading, the ratio of activation energy forsurface diffusion to heat of adsorption at zero surfaceloading, a0, can be calculated as:

a0 = E0a/q

0st (10)

Meanwhile, the effect of isotherm equation on thisratio can also be examined.

3. Experimental

The pure component adsorption equilibrium ofethane and propane were measured on Ajax activatedcarbon and Norit activated carbon, respectively, usingvolumetric method. For Ajax carbon, the isotherm ofeach species were measured at three temperatures:283, 303, and 333 K [11] while for Norit carbon, theisotherm of each species were measured at 303, 333,and 363 K [25,26]. The pure component adsorptionkinetics of ethane and propane on each carbon sam-ple were measured using a differential adsorbed bed(DAB) rig. The kinetics data of each adsorbate weremeasured on Ajax carbon in 4.4-mm full-length slabparticles at the temperatures of 283, 303, and 333 K[11], respectively. At each temperature, sorption ki-netics at three bulk phase molar fractions: 5, 10, and20% were measured, at a total pressure of 1 atm bal-anced by nitrogen. The adsorption kinetics on Norit

K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176 169

Table 1Experimental equilibrium and kinetics conditions of ethane and propane on two activated carbons

Equilibrium Kineticsa

Ethane Propane Ethane Propane

Ajax 283, 303, and 333 K 283, 303, and 333 K 5, 10, and 20% 5, 10, and 20%Norit 303, 333, and 363 K 303, 333, and 363 K 2, 5, and 10% 2, 5, and 10%

a At each molar fraction the data were measured for the corresponding three temperature used in equilibrium.

carbon was recently measured on 8.82-mm full-lengthslab particles at 303, 333, and 363 K, respectively[25]. The bulk phase molar fractions of each speciesare 2, 5, and 10% for each temperature, also at a totalpressure of 1 atm but balanced by high purity helium(99.999%). Thus, we have twelve isotherm curves and36 kinetic uptake curves in total for this study. The ex-perimental conditions for the equilibrium and kineticdata are listed in Table 1. The detailed experimentalprocedures can be found in the related references.

4. Results and discussion

4.1. Isosteric heat of adsorption

The pure component experimental equilibrium dataof ethane and propane collected on two carbon sam-ples at different temperatures are first employed, re-spectively, to optimize the parameters of each isothermequation. The relative average error (ARE) betweenthe model fitting and the experimental data is used asthe objective function for optimization. Fig. 1a andb show the experimental data of ethane and propaneon Norit carbon (symbols), respectively. Also shownin these two figures are the model fittings of Unilanequation (dashed lines) and Toth equation (solid lines).It is seen that either isotherm can adequately fit theexperimental data, but the Toth equation fits the dataslightly better. Fig. 2a and b present the experimentaldata (symbols) of ethane and propane on Ajax carbon.The model fittings from the Unilan and Toth equationsare presented in these two figures as dashed and solidlines, respectively. It is seen again that both models fitthe data well.

Table 2 lists the isotherm parameters derived fromUnilan equation for ethane and propane on the twocarbons. With these parameters, the Henry’s law con-stant, H, of each adsorbate is calculated for each

temperature using Eq. (4b) and is tabulated in Table 2.By plotting ln(H) against 1/T, the isosteric heat of ad-sorption at zero loading, q0

st is derived from the slope.Fig. 3a shows the ln(H) ∼ 1/T plot for each species onNorit carbon as an example. The same good linearityof the plot is also observed on Ajax carbon althoughthe plot is not shown here, to save space. The derivedq0st is listed in Table 2. The order of q0

st for the twospecies is in the range of 24–40 kJ/mol, which is com-parable to the value reported by Valenzuela and Myers[19] for the two adsorbates on Nuxit carbon, that is, q0

st

is approximately 25 kJ/mol for ethane and 38 kJ/molfor propane (derived with Unilan equation). Some

Fig. 1. Adsorption equilibrium data and model fittings on Noritcarbon. (--- Unilan isotherm, — Toth isotherm).

170 K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176

Table 2Adsorption isotherm parameters for ethane and propane on two carbons derived from the Unilan equation

Sample T (K) Cµs (mmol/g) b (per kPa) s H (mmol/g per kPa) q0st (kJ/mol)

C2 C3 C2×102 C3×102 C2 C3 C2 C3 C2 C3

Norit 303 6.41 6.45 1.07 2.85 4.03 4.83 0.478 2.38 27.9 24.1333 4.71 6.38 1.00 1.17 3.20 4.60 0.180 0.810363 3.50 5.36 0.867 0.953 2.57 4.46 0.0768 0.496

Ajax 283 10.3 8.34 0.701 5.62 4.36 5.70 0.646 12.3 24.7 40.6303 9.01 7.74 0.544 3.66 3.85 4.73 0.300 3.40333 7.37 7.01 0.420 2.08 3.36 3.83 0.133 0.876

abnormality is also observed in the value of q0st on

Norit carbon, that is, ethane has slightly higher isos-teric heat than propane. A possible reason could be thatNorit carbon has some smaller pores, which presentstrong adsorption affinity for ethane but are inacces-sible to propane due to its larger molecule size [26].

To study the effect of isotherm equation on thecalculation of q0

st , the Henry’s law constants of thetwo adsorbates are then derived from Toth isotherm(Eq. (5b)). The plot of ln(H) ∼ 1/T on Norit carbon is

Fig. 2. Adsorption equilibrium data and model fittings on Ajaxcarbon. (--- Unilan isotherm, — Toth isotherm).

shown in Fig. 3b as an example. The plot on Ajax car-bon presents similar linearity but is not shown here.The derived q0

st for each adsorbate–adsorbent systemare listed in Table 3. Comparing with the related val-ues in Table 2, it can be seen that Toth equation givesslightly higher values of q0

st than Unilan equation. Thevalues of q0

st for ethane and propane on Ajax carbonare also very close to those obtained on Nuxit carbon

Fig. 3. Plot of Clapeyron equation, ln(H) ∼ 1/T; (a) with Unilanisotherm, (b) with Toth isotherm.

K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176 171

Table 3Adsorption isotherm parameters for ethane and propane on two carbons derived from the Toth equation

Sample T (K) Cµs (mmol/g) b (kPat) t H (mmol/g per kPa) q0st (kJ/mol)

C2 C3 C2 C3 C2 C3 C2 C3 C2 C3

Norit 303 9.27 6.47 1.66 0.840 0.314 0.331 1.87 10.9 38.2 33.9333 10.5 5.29 2.75 1.51 0.328 0.415 0.480 1.96363 10.6 5.28 4.41 1.75 0.350 0.383 0.153 1.25

Ajax 283 13.1 9.03 1.91 0.490 0.340 0.267 1.97 11.6 31.7 41.1303 13.2 9.32 2.35 0.697 0.335 0.278 1.02 4.02333 14.4 11.8 4.36 1.06 0.368 0.264 0.262 1.07

with Toth equation (i.e. 31 kJ/mol for ethane and 42kJ/mol for propane, by Vanlenzuela and Myers [19]).The q0

st derived on Norit carbon for ethane is againslightly higher than that of propane.

The isosteric heats of adsorption at other surfaceloadings, qst , are calculated using Eq. (1) with Uni-lan or Toth isotherm, which are plotted in Fig. 4. Withthe Toth isotherm, the maximum adsorption capacity(Cµs) ranges from 9.27 to 10.6 mmol/g for ethane,

Fig. 4. Isosteric heat of adsorption as a function of surface loading.

and 5.28 to 6.47 mmol/g for propane, in Norit car-bon at different temperatures between 303 and 363K. That is, the difference in Cµs is not significant fordifferent temperatures. This is also true for Ajax car-bon. Therefore, when Toth isotherm is used, as ex-pected, the value of qst decreases as surface loadingincreases, for both ethane and propane on both Noritand Ajax carbons. However, when Unilan isotherm isused, because Cµs decreases significantly as tempera-ture increases, e.g. from 3.50 at 303 K to 6.41 mmol/gat 363 K (83% increase) for ethane in Norit carbon,qst increases as surface loading increases. The changeof Cµs with temperature is less severe for Ajax car-bon, so the isosteric heat of adsorption changes onlymarginally as surface loading increases for Ajax car-bon, when Unilan isotherm is applied.

4.2. Adsorption kinetics

The adsorption kinetics is simulated by the MSDmodel, which employs Unilan or Toth equation todescribe the equilibrium isotherm. The pore diffu-sivity of the two adsorbates are listed in Tables 4and 5, respectively, which are temperature and adsor-bent dependent but concentration independent [24].With those parameters defined, we are ready to studythe surface diffusivity at zero loading, D0

µ, for eachadsorbate–adsorbent system at different temperatures.

First, the adsorption kinetics of ethane on Noritcarbon is studied. Fig. 5a shows the experimentaldata (symbols) of ethane on Norit carbon in 8.82mm full-length slab particles at the temperature of303 K. The experimental bulk phase molar fractionsof ethane are 2, 5, and 10%, respectively, at a totalpressure of 1 atm. All data in Fig. 5a were employedsimultaneously to optimize the surface diffusivity at

172 K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176

Table 4Diffusivities and activation energy of ethane and propane on Norit activated carbon

T (K) Dp×10−6 (m2/s) Dµ0×108 (m2/s) E0a (kJ/mol) a0

303 333 363 303 333 363

Ethane 2.12 2.26 2.38 1.55 2.33 3.24 11.3 Unilan 0.401.32 1.92 2.80 11.4 Toth 0.30

Propane 1.75 1.86 1.97 0.641 0.943 1.12 8.70 Unilan 0.360.597 0.846 1.02 8.36 Toth 0.25

zero coverage of the MSD model using the Unilanisotherm. The optimal model fittings are shown inFig. 5a as dashed lines. It is seen that the Darkentype concentration dependence for surface diffusivityis adequate for the MSD model to describe the up-take of ethane on Norit carbon. The derived surfacediffusivity at zero loading, D0

µ, is listed in Table 4.Toth isotherm is then used in the MSD model to

study the above kinetic data of ethane again. Themodel fittings are shown in Fig. 5a as solid lines whilethe derived D0

µ is listed in Table 4. It can be seen that,contrary to what we saw in the calculation of q0

st , thetwo isotherm equations have similar fitting goodnessfor the kinetic data and give similar values of D0

µ, in-dicating that the Unilan and Toth isotherms have thesimilar capability in simulating the adsorption kinet-ics in the MSD model, because both are able to fit theisotherm data well.

Next, the above procedures are applied in the studyof the adsorption kinetics of ethane on Norit carbonat two other temperatures, 333 and 363 K. The bulkphase molar fractions are again 2, 5 and 10% for eachtemperature. Fig. 5b and c show graphically the exper-imental data (symbols) and the model fittings (solidlines for MSD model with Toth isotherm and dashedlines for MSD model with Unilan isotherm) at thosetwo temperatures, respectively. It can be seen that the

Table 5Diffusivities and activation energy of ethane and propane on Ajax activated carbon

T (K) Dp×106 (m2/s) Dµ0×109 (m2/s) E0a (kJ/mol) a0

283 303 333 283 303 333

Ethane 1.51 1.68 1.96 2.37 3.21 5.88 14.4 Unilan 0.582.16 2.71 5.07 13.6 Toth 0.43

Propane 1.20 1.30 1.56 0.761 0.950 1.78 13.4 Unilan 0.330.778 0.868 1.48 10.3 Toth 0.25

fittings are reasonably good. The derived D0µ with the

two isotherms are listed in Table 4.Having obtained D0

µ for ethane on Norit carbon atthree different temperatures, we are ready to calcu-late its activation energy for surface diffusion at zerosurface loading, E0

a. Fig. 6 shows the plot of ln(D0µ)

against 1/T. Good linearity of the plot is observed, sup-porting the consistency of D0

µ derived with the MSDmodel. The value of E0

a for ethane is found to be 11.3kJ/mol with Unilan isotherm and 11.4 kJ/mol withToth isotherm. We note that the choice of isothermequation does not have a significant influence on thevalue of E0

a. However, since q0st is highly isotherm de-

pendent, the value of ‘a0’ (E0a/q0

st) is also isotherm de-pendent. The value of ‘a0’ for ethane on Norit carbonis found to be 0.4 for Unilan isotherm and 0.3 for Tothisotherm.

The adsorption kinetics of propane was also mea-sured on Norit carbon under the same experimentalconditions as those of ethane. Thus the proceduresdiscussed above are repeated to derive D0

µ and E0a of

propane on Norit carbon. Fig. 7 shows the experimen-tal data (symbols) and the model fittings (solid lines forMSD model with Toth isotherm and dashed lines forMSD model with Unilan isotherm) of the adsorptionkinetics of propane on Norit carbon at 303, 333, and363 K, respectively. The agreement between model

K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176 173

Fig. 5. Adsorption kinetics data and model fittings for ethane onNorit carbon at various temperatures. (--- with Unilan isotherm,— with Toth isotherm).

fittings and experimental data is good. Fig. 8 presentsthe plot of ln(D0

µ) against 1/T for propane. The derivedkinetics parameters, D0

µ and E0a are listed in Table 4.

Since propane is more bulky than ethane, the surfacediffusivity is smaller for propane. The value of ‘a0’for propane in Norit carbon is found to be 0.36 forUnilan isotherm and 0.25 for Toth isotherm.

Finally, the adsorption kinetics of the two adsor-bates on Ajax carbon was studied. The pure compo-nent adsorption kinetics of ethane and propane weremeasured on Ajax carbon in 4.4 mm full-length slab

Fig. 6. Plot of ln(D0µ) ∼ 1/T of ethane on Norit carbon.

Fig. 7. Adsorption kinetics data and model fittings for propane onNorit carbon at various temperatures. (--- with Unilan isotherm,— with Toth isotherm).

174 K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176

Fig. 8. Plot of ln(D0µ) ∼ 1/T of propane on Norit carbon.

Fig. 9. Adsorption kinetics data and model fittings for ethane onAjax carbon at various temperatures. (--- with Unilan isotherm,— with Toth isotherm).

particles at three temperatures: 283, 303, and 333 K,respectively. The bulk phase molar fractions of eachadsorbate are 5, 10, and 20% for each temperature, ata total pressure of 1 atm. These data were then em-ployed to optimize D0

µ with respect to the MSD modelusing either Unilan or Toth isotherm. The proceduresof driving D0

µ and E0a for each adsorbate are the same

as those described for Norit carbon.Fig. 9 shows the kinetic data (symbols) and the fit-

tings of MSD model (solid lines for MSD with Tothand dashed lines for MSD with Unilan isotherm) for

Fig. 10. Adsorption kinetics data and model fittings for propaneon Ajax carbon at various temperatures. (--- with Unilan isotherm,— with Toth isotherm).

K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176 175

ethane on Ajax carbon at 283, 303, and 333 K, andFig. 10 shows the data (symbols) and model fittings(lines) of propane on Ajax carbon. The agreement be-tween experimental data and the model fittings areagain good for both adsorbates under various experi-mental conditions, which seems to support the use ofDarken relation in the experimental bulk molar frac-tion range (5–20%). The derived D0

µ of each speciesare listed in Table 5. The plots of ln(D0

µ) against 1/Tfor ethane and propane on Ajax carbon present samegood linearity as those on Norit carbon and hence arenot shown here. The derived Ea of each adsorbate onAjax carbon are tabulated in Table 5. The value of ‘a0’on Ajax carbon ranges from 0.43 to 0.58 for ethaneand 0.25 to 0.33 for propane.

With D0µ obtained, the surface diffusivities at other

surface loadings, Dµ, are calculated according to theDarken relationship, Eq. (8), for both Unilan andToth isotherms at three temperatures. The results areshow in Fig. 11 for Norit carbon and Fig. 12 for Ajax

Fig. 11. Surface diffusivity of ethane and propane in Norit activatedcarbon as a function of surface loading.

carbon. As expected, the surface diffusivity increasesas surface loading increases, for both adsorbates inboth carbons, and for both Unilan and Toth iso-therms.

The derived surface diffusivity at any surface load-ing at various temperatures is then used to computethe activation energy at that surface loading, usingEq. (6). The obtained activation energy, together withthe isosteric heat of adsorption shown in Fig. 4, arethen used to calculate the ratio of activation energy ofsurface diffusion to the heat of adsorption, a = Ea/qst ,at any surface loading, which is plotted in Fig. 13.Although both qst and Dµ are functions of surfaceloading (Figs. 4, 11 and 12), the ratio of activationenergy of surface diffusion to the heat of adsorption,a, does not change much with surface loading. Thissupports the constant value of ‘a’ used in variousprevious studies [10,11,13,26–28]. In general, for thesame species the value of ‘a’ from Unilan isotherm islarger than that from Toth isotherm, while for differ-

Fig. 12. Surface diffusivity of ethane and propane in Ajax activatedcarbon as a function of surface loading.

176 K. Wang et al. / Separation and Purification Technology 34 (2004) 165–176

Fig. 13. Ratio of activation energy of surface diffusion to the heatof adsorption as a function of surface loading.

ent species, the ‘a’ for ethane is larger than that forpropane.

5. Conclusions

The isosteric heat of adsorption, qst , and the acti-vation energy for surface diffusion, Ea, of ethane andpropane are derived from their equilibrium and kinet-ics information measured on two activated carbons. Itis found that the choice of adsorption isotherm has asignificant influence in the calculation of qst , but hasless influence on Ea. The value of ‘a’ (Ea/qst) is foundbetween 0.25 and 0.60 for the two adsorbates on thetwo carbons and this ratio depends on the choice ofthe kinetics as well as isotherm models. It is shownthat Toth isotherm resulting in a smaller value of ‘a’than Unilan isotherm. The pore and surface diffusionmodel with a Darken type concentration dependencefor surface diffusivity is found adequate to describe the

sorption kinetics of the two adsorbates on the two car-bons under the experimental conditions investigated.

Acknowledgements

Financial support from the Croucher Foundation,and the Research Grants Council of Hong Kong(Project No. HKUST6114/97P) is greatly acknowl-edged.

References

[1] A. Kapoor, R.T. Yang, C. Wong, Catal. Rev. Sci. Eng. 31(1989) 129.

[2] X. Hu, D.D. Do, G.N. Rao, Chem. Eng. Sci. 49 (1994) 2145.[3] J.W. Ross, R.J. Good, J. Phys. Chem. 60 (1956) 1167.[4] A.J. Robell, E.V. Ballou, M. Boudart, J. Phys. Chem. 68

(1964) 2749.[5] E.R. Gilliland, R.F. Baddour, G.P. Perkinson, K.J. Sladek,

Ind. Eng. Chem. Fundam. 13 (1974) 95.[6] K.J. Sladek, E.R. Gilliland, R.F. Baddour, Ind. Eng. Chem.

Fundam. 13 (1974) 100.[7] A. Kapoor, R.T. Yang, AIChE J. 35 (1989) 1735.[8] A. Kapoor, R.T. Yang, Chem. Eng. Sci. 45 (1990) 3261.[9] A. Kapoor, R.T. Yang, Chem. Eng. Sci. 46 (1991) 1995.

[10] X. Hu, D.D. Do, AIChE J. 39 (1993) 1628.[11] X. Hu, G.N. Rao, D.D. Do, AIChE J. 39 (1993) 249.[12] D.D. Do, in: W. Rudzinski, W.A. Steele, G. Zgrablich

(Eds.), Equilibria and Dynamics of Gas Adsorption onHeterogeneous Solid Surfaces, Elsevier, Amsterdam, 1997,pp. 777–835.

[13] D.D. Do, K. Wang, Carbon 36 (1998) 1539.[14] J.A. Dunne, R. Mariwala, M. Rao, S. Sircar, R.J. Gorte, A.L.

Myers, Langmuir 12 (1996) 5888.[15] J.A. Dunne, R. Mariwala, M. Rao, S. Sircar, R.J. Gorte, A.L.

Myers, Langmuir 12 (1996) 5896.[16] J.A. Dunne, R. Mariwala, M. Rao, S. Sircar, R.J. Gorte, A.L.

Myers, Langmuir 13 (1997) 4333.[17] S. Sircar, Langmuir 7 (1991) 3065.[18] S. Sircar, Ind. Eng. Chem. Res. 31 (1992) 1813.[19] D.P. Valenzuela, A.L. Myers, Adsorption Equilibria Data

Handbook, Prentice Hall, New Jersey, 1989.[20] H. Pan, J.A. Ritter, P.B. Balbuena, Langmuir 14 (1998)

6323.[21] K. Higashi, H. Ito, J. Oishi, J. At. En. Soc. Jpn. 5 (1963) 846.[22] R.T. Yang, J.B. Fenn, G.L. Haller, AIChE J. 19 (1973) 1052.[23] D.D. Do, Chem. Eng. Sci. 51 (1996) 4145.[24] D.M. Ruthven, Principles of Adsorption and Adsorption

Processes, Wiley, New York, 1984.[25] S. Qiao, X. Hu, Separation Purif. Technol. 16 (1999) 261.[26] S.Z. Qiao, X. Hu, Chem. Eng. Sci. 55 (2000) 1533.[27] X. Hu, S.Z. Qiao, D.D. Do, Langmuir 15 (1999) 6428.[28] S.Z. Qiao, X. Hu, AIChE J. 46 (2000) 1743.