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Modified GAB model for correlating multilayer adsorption equilibrium data
Longhui zou, Linghui Gong, Peng Xu, Guochao Feng, Huiming Liu
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S1383-5866(16)30026-0
http://dx.doi.org/10.1016/j.seppur.2016.01.026
SEPPUR 12808
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Separation and Purification Technology
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16 November 2015
14 January 2016
18 January 2016
Please cite this article as: L. zou, L. Gong, P. Xu, G. Feng, H. Liu, Modified GAB model for correlating multilayer
adsorption equilibrium data, Separation and Purification Technology (2016), doi: http://dx.doi.org/10.1016/
j.seppur.2016.01.026
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Modified GAB model for correlating multilayer adsorption
equilibrium data
Longhui zoua,b, Linghui Gonga,*, Peng Xua , Guochao Fenga,b ,Huiming Liua
a).Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry, Chinese Academy
of Sciences, Beijing 100190,PR China
b).University of Chinese Academy of Sciences, Beijing 100049, PR China
*Corresponding author. Tel.: +86-010-82543527
E-mail address: [email protected] (L. H.Gong)
Abstract
A new multilayer gas adsorption model is built based on the GAB (Guggenheim-Anderson-de
Boer) model and L-F isotherm (Langmuir-Freundlich isotherm). Accounting for the heterogeneity of
the adsorption system, the adsorption rate is assumed to beαth power of the surface area as
demonstrated in L-F isotherm; the modified GAB model has the same form as that of the GAB
equation with the difference lying on the relative pressure form; the nominal relative pressure is the
α
th power of the relative pressure. Different adsorption isotherm models are applied to correlate the
adsorption data of microporous materials. The modified GAB model has the best conformity with the
experimental data and can get almost the same so called “BET monolayer capacity” with that of the
original BET equation using consistence criteria. The parameters in the modified GAB model were
calculated as well; the non-unity of
α indicated that the heterogeneity of the adsorption system is
un-negligible even on the highly homogeneous surface.
Keywords:
Multilayer adsorption; Heterogeneity; BET; Microporous material
1. Introduction
Adsorption is a common technology to study the characteristic of porous material and adsorption
isotherm is mostly used to describe the adsorption process. There are many famous models available to
describe the pure component adsorption, such as the Langmuir model[1], Freundlich, BET[2],
Langmuir-Freundlich isotherm[3], Toth model[4] and DR (DA) equation[5], etc. The BET model is
usually used to evaluate the specific area of the non-porous and mesoporous material and it’s
applicable when the relative pressure is 0.05-0.35. Because the existence of micropores can lead to
non-linear BET plot[6] and capillary condensation may occur in high pressure. A modified BET
equation is built by Anderson [7] and Brunauer et al[8] respectively by employing a third parameter k,
which is a measurement of the attractive force field of the adsorbent. And it greatly extends the
applicability of the BET model with
P P 0 or c c 0 up to at least 0.9[9] while the BET areas are
almost the same for the two models. Timmermann[10] did more research about the parameters in GAB
(Guggenheim-Anderson-de Boer) isotherms(that is the modified BET model derived by Anderson and
Brunauer) and found that although the GAB isotherm can reproduce the results of the BET equation,
the GAB monolayer value is about 15% higher than the BET value and the GAB energy constant C
reduces 35%-40% or more than the BET constant. That is huge difference. And these two models are
unable to explain and represent the sorption at very low activities (relative pressure P P0  0. 1).
Bashiri and Orouji[11] derived a new multilayer adsorption model based on the BET model and
accounting for the heterogeneity by assuming the first layer being heterogeneous and the lateral layers
being homogeneous. And it shows it has a good conformity with experimental data in some system.
BET equation is traditionally used to evaluate the specific surface area of non-microporous
materials. But still, it’s routinely used in microporous system. There’re several problems applying the
BET model to microporous materials[12];(1)The BET monolayer capacity is doubtful,(2) monolayer
structure is different for different materials,(3) Localized monolayer adsorption or micropore filling
could occur in low relative pressure. Rouquerol et al[13]did further study about the applicability of
BET equation in microporous adsorbent. They found that two other criteria (consistency criteria)
besides the relative pressure range being 0.05-0.35 should be included if the BET equation was used to
microporous adsorbent and the monolayer capacity was more like “BET strong retention capacity”.
Bae et al[12] and Walton, Snurr[14]did molecular simulation to predict nitrogen adsorption in MOFs and
zeolite with micropores and even ultra-micropores .The BET surface area calculated with the consistency
criteria agreed well with the accessible surfaces areas obtained from crystal structures, indicating that BET
theory can be applied to microporous materials on heterogeneous surface.
In this work, a new multilayer adsorption isotherm is built based on the GAB model and the BET
model; the adsorption surface is heterogeneous, not just for the first layer. And it shows that the new
modified GAB model can be applied to describe the adsorption process in microporous material and
can get the almost the same so called “BET monolayer capacity” with original BET model in the full
relative pressure range.
2. Theory
Assume that one molecule occupies  sites when being adsorbed, then the rates of adsorption
and desorption rate are proportional to
1   

and   respectively (  is the surface coverage);
Similar to the derivation of Langmuir model, from adsorption and desorption kinetics:
d
 kaP(1   )  kd  
dt
When adsorption equilibrium is arrived，
(1)
d
 0
dt
Thus：
1
n  nma x
bP 

1  bP 


This is the L-F isotherm[3], b  ka kd

1

,
V V
n

,

Vm V m n ma x
(2)
. ka is the adsorption rate while kd is the desorption
rate. n is the adsorption amount when adsorption equilibrium is reached. nmax is the saturation
capacity.
When  is unity, the Langmuir-Freundlich isotherm reduces to Langmuir model. Thus  can
be a symbol of heterogeneity. Based on the concept of multisite occupation of one molecule when
being adsorbed, the adsorption rate is proportional to surface coverage. And the new multilayer
adsorption model is built. There are several assumptions need to make before the derivation.
⑴ One molecule occupies  sites when being adsorbed.
⑵ The adsorption occurs at second and further layers before the completion of the first one[15].
⑶ The heat of adsorption differs from that of liquefaction by a constant amount d from the
second layer and above[7].
⑷ The number of adsorbed layer is infinite when adsorption equilibrium is reached.
Accounting for the heterogeneity of the system whether it is the heterogeneous surface or the
force between adsorbed molecules, all is displayed in the parameter  . When adsorption
equilibrium is reached, the adsorption rate on the free surface (not occupied by molecules) is equal
to the desorption rate of the first layer, that is[11]:
1
1
 E 
a1PS0  b1S1 exp   1 
 RT 
(3)
Here a1 and b1 are constants. P is the equilibrium pressure, E1 is the heat of adsorption. Si is the
number of adsorption sites on each layer of the multilayer adsorption.
For the second layer：
1
1
 E d
a2PS1  b2S2 exp   2

RT 

(4)
The quantity d will be added to the adsorption heat of the first layer as well for convenience [7].
For the i th layer：
a a
Si   1  2
b b
2
 1
 E +d E  d
ai
exp  1  2

bi
RT
 RT
E d
 i
RT
We know from the derivation of BET equation, E2 , E3 ,

 i 
  P S0

(5)
Ei is assumed to be equal to the
heat of liquefaction. From the second layers and above, the adsorption is mainly due to the
attracting force among adsorbed molecules, the interaction among adsorbed molecules is similar
(not equal) to the interactions in a pure liquid, so it is expected that d
Ei and ai bi is
constant.
a2
a
 3  
b2
b3
 
ai
 
bi

a
b
(6)
Define：
x 
 E +d 
a2
exp  2   P
b2
 RT 
(7)
y 
 E +d 
a1
exp  1   P
b1
 RT 
(8)
C1 
y
x
(9)
The total amount of adsorption is：
V

i S
(10)
i
i 0
Si is the monolayer adsorption capacity：
Vm 

S
(11)
i
i 0
Surface coverage is defined:

V
n
θ


Vm
nmax


i Si
i 0

Si
i 0
(12)
x  1 , we get:
θ
C1 x 

1  x

1  x   C1x 
(13)

When adsorption saturation is reached, the adsorption layers is infinite and the adsorption
amount is infinite at the liquefaction pressure of the adsorbate[7], seen from Eq.(7), x must be
infinitely close to unity and d=0 [7]. That is:
E 
a
exp     P0 =1
b
 RT 
(14)

P
d 
Combine Eq.(7) and Eq.(14), we get: x    exp 
.
P
R
T


 0
 d 

 C=C1
 RT  ,
Define: k  exp 

P
Vm  C  k   
 P0 
V




P 
P
P 
1  k     1  k     C  k    

 P0   
 P0 
 P0  

(15)
There’s a variant form of Eq.(16), that is:
Vm  C  K   P 

V

1  K  P


1  K  P  C  K  P



(16)
There’re four adjustable parameters in the modified GAB model, that is nmax ,C, k,  . The
monolayer capacity nmax and energy constants C vary with the temperature, adsorbate and
adsorbent. Anderson[7] and
[8] found that by multiplying a constant less than
unity with the relative pressure can greatly improve the applicability of BET model, situation is
the same here. The parameter k accounts for the adsorption heat deviation from that of
liquefaction. The parameter  suggests the heterogeneity of the adsorption system. The only
parameter which can be compared to that of other adsorption isotherms is the monolayer
capacity. It will differ from different adsorption isotherms, but the difference is small enough to
be neglected. That’s also a standard self-checking principle when we use different adsorption
isotherm in the fitting process. In the next section, adsorption data from already published
literature [16, 17]is applied to the modified GAB isotherm model. It shows that the new modified
GAB model has good conformity with experimental data.
3. Application of Eq.(15) and Eq.(16) to Microporous material
3.1 Physisorption of ethane on Template carbon
The first system is the ethane adsorption on the Template carbon, as the literature
described[16]; the Templated carbon has a good microporous volume and a relatively
homogeneous microporous structure with the pore sizes distributed around 1.0 nm[16]. To fit the
experimental data of ethane adsorption with the modified GAB model, nonlinear curve fitting
method is used. The adsorption data of ethane on Template carbon at 263K, 273K and 303K are
also correlated by Eq.(16), displayed in Fig. 1 and it shows a good match.
Fig. 1 Adsorption of ethane on a Template carbon respectively at 263K, 273K and 303K. Solid black
Squares, solid red circles and solid blue triangles are respectively experimental data at 263K,273K and
303K from[16]. Red solid line, magenta solid line and olive solid line are correspondingly predictive
data using modified GAB model derived in this work. Black dash line, blue dot line and pink dash dot
line are correspondingly predictive data using GAB model derived by Anderson[7].
From Tab. 1, we can see that parameter  varies from 0.4-0.6 at different temperature and
the saturation capacity decreases as temperature increases since the adsorption process is
exothermal. The terms  and nmax calculated by Eq.(15) and Eq.(16) stay put but the parameter
k(or K) differ. The parameter K in Eq.(16) is always reduced an order of magnitude compared to
that of the parameter k in Eq.(15).We can see from Eq.(15) and Eq.(16), the parameter K in Eq.(16)

is k P0 .
Tab. 1 Correlated Parameters of Eq.(15) and Eq.(16) with data from [16]

C
k(or K)
nmax/(mol·kg-1)
Adj.R-Square
263K
Eq.(15)
263K
Eq.(16)
273K
Eq.(15)
273K
Eq.(16)
303K
Eq.(15)
303K
Eq.(16)
0.43169
215.1713
0.00156
55.01412
0.998287
0.43067
216
0.0004358
55.7216
0.998286
0.5674
176.6126
0.00941
18.1814
0.99947
0.5674
176.5703
0.00156
18.1797
0.99947
0.59841
105.5281
0.024629
13.5659
0.999965
0.5984
105.52798
0.002478
13.5659
0.999965
In order to investigate the accuracy of the modified GAB model, Langmuir-Freundlich
isotherm, the GAB model are also used to correlate the experiment data and it’s showed in Fig. 1.
The Langmuir-Freundlich isotherm has the same effect as the modified GAB model did. And the
two lines overlap with each other into one line, and you couldn’t find the difference. So it’s not
displayed in Fig. 1.The monolayer capacities obtained from the two models are almost identical.
It’s showed in Tab. 2. Although the GAB model can fit the experimental data well, the “BET
monolayer capacity” calculated from the GAB model is much different from the modified GAB
model and the L-F isotherm. From the Fig. 1 and Tab. 2, we can see the modified GAB model
built in this work has the best conformity with the experimental data.
Tab. 2 Comparison of L-F isotherm, GAB and Modified GAB model in correlating adsorption data
Langmuir-Freundlich
Modified GAB model
GAB
isotherm
(This work)
(Anderson)
Temp
nmax/(mol·kg-1)
Adj.R-square
nmax/(mol·kg-1)
Adj.R-square
nmax/(mol·kg-1)
Adj.R-square
263K
56.39186
0.99786
55.01412
0.99755
10.01543
0.97493
273K
18.66808
0.99934
18.1814
0.99923
9.0874
0.98391
303K
14.23082
0.99995
13.5659
0.99994
6.76205
0.99124
The Clausius–Clapeyron equation[18] is usually used to calculate the isosteric enthalpy. In
order to look into more details about the modified GAB model, isosteric enthalpy of ethane
adsorption is calculated with the modified GAB model and the Langmiur-Freundlich model.
 d l n P  
Qst  RT 2 

 dT 

N
(17)
As show in Fig. 2, the isosteric enthalpy of ethane adsorption on Templated Carbon
calculated with the Clausius–Clapeyron equation is very high at low coverage and decrease
gradually until the loading reaches 6mol/kg, it then increases. It shows that the Templated
carbon has energetically heterogeneous surface. This phenomenon is due to the two
assumptions that an ideal bulk gas phase and a negligible adsorbed phase molar volume when
using the Clausius-Clapeyron equation. As Pan, Ritter et al[19] indicated in their work, the largest
effect comes from the ideal gas assumption at high loadings. Choi, Lee et al[20]did calculation of
isosteric heat of ethane in the temperature range of 293.15K-313.15K and found the same
phenomenon. The heterogeneity of the surface is the dominant effect at low surface loading,
and then the interaction between adsorbate and adsorbate becomes the main effect, resulting in
the decrease of isosteric enthalpy at low surface loading and then increase. The isosteric
enthalpy of ethane is about 60kJ/mol at low surface loading, the same as what is indicated in Fig.
2 with the calculation using adsorption data at 263K and 273K. Other calculations using the
adsorption data at temperature 263K and 303K or using the combination of adsorption data at
273K and 303K is misleading.
(a)
(b)
Fig. 2 The isosteric enthalpy of ethane adsorption on Templated Carbon (a) isosteric enthalpy
plotted as lnP vs 1/T. (b) isosteric enthalpy plotted as isosteric enthalpy vs loading. The solid
lines with black square, red circle and blue triangle are the isosteric enthalpy obtained by
Langmuir-Freundlich model using adsorption data respectively at 263K and 273K, 263K and
303K, 273K and 303K. While the solid lines with magenta up-down triangle, olive diamond and
navy blue left triangle are obtained by the Modified GAB model.
3.2 Physisorption of Carbon Dioxide adsorption on Graphitized Carbon Black
The second system comes from literature[17]. It measured the adsorption data of dioxide and
ethane on STH-2 graphitized carbon black using volumetric method and the relationship between
the relative pressure P P0 and the adsorption equilibrium amount at different temperature displays
a characteristic curve as the potential theory predicts. We correlated the adsorption equilibrium
data with several multilayer adsorption models. Through Fig. 3, we can see that the model built by
Anderson[7] and the modified GAB all have good conformity with the experimental data and
almost the same effect. But seen from the enlarged drawing, the modified GAB model built in this
article fits experimental data the best. The model derived by Anderson is a special case of the
model derived in this paper in the case that alpha(  ) reaches unity. The Graphitized Carbon
Black surface is regarded to be homogeneous[17], the term  is approaching unity ranging from 1
to 1.3, displaying some heterogeneity of the surface.
(a)
(b)
(c)
Fig. 3 Comparison of model prediction with experimental data for CO2 adsorption on graphitized
carbon black. Fig. (a), (b) and (c) represents adsorption at 263.2K, 273.2K 303.2K respectively. Black
squares represent experimental data. Blue dash line represents the data predicted by BET model[2];
black dot line represents the data predicted by Bashiri[11]; Red solid line represents the data predicted
by the modified GAB model built in this work; Oliva dash dot line represents the data predicted by the
GAB model[7].
Tab. 3 Correlation Parameters of Eq.(15) and Eq.(16)with adsorption data at different temperature
from [17]

C
k
nmax/ (mol·kg-1)
Adj.R-Square
263.2K
273.2K
303.2K
1.31775
16.06288
0.7871
0.59695
0.99985
1.25905
14.13714
0.84528
0.544432
0.99935
1.04088
5.14199
0.85644
0.69365
0.9999
One of the usages of BET isotherm is to calculate the specific surface area of an adsorbent,
but when it comes to microporous material, Rouquerol. et.al[13] did a further study about how we
can get the so called “monolayer content” in use of BET isotherm in microporous material.
Including the condition that the relative pressure range should be 0.05-0.35, two other conditions

should be considered. The value of C should be positive and the term of n  P0  P

should
increase with the relative pressure. We took the adsorption data of carbon dioxide on graphitized
carbon at 263.2K to demonstrate how the modified GAB model can applied to calculate the so
called BET monolayer capacity in the full relative pressure range. The saturation capacity was
calculated by using the BET model with the two added conditions and by using the modified GAB
model (Eq.(15)).


Fig.4 Plot of the term n P0  P vs. P P0


From Fig.4 we can see, in the relative pressure range of 0.05-0.35, the term n P0  P increases
with the relative pressure, so we choose the data from relative pressure range 0.05-0.35, if the
linear criterion of the BET plot is not satisfied, the relative pressure range should be narrowed
down until the determination coefficient of the non-linear fitting process is larger than 0.99. The
saturation capacity was calculated using the original BET equation and the modified GAB model
in this work. Adsorption data of other temperature was also analyzed, the comparison is as
follows:
Tab. 4 the saturation capacity of carbon dioxide adsorption on graphitized carbon at 263K calculated
by BET model and Eq.(15)
adsorption
model
nmax /(mol·kg-1)
Adj.R-square
263.2K
BET
Eq.(15)
0.54359
0.59695
0.9926
0.99985
273.2K
BET
Eq.(15)
0.56213
0.54443
0.99376
0.99935
303.2K
BET
Eq.(15)
0.65409
0.69365
0.99979
0.9999
We can see from Tab. 2 that the saturation retention capacity (“so called BET monolayer
capacity”) in microporous material, calculated from the modified GAB model and the original
BET equation is almost the same, the error was less than 9%.Once the saturation capacity nmax
was known, the specific surface area can be calculated from Eq.(18)
A = nmax  NA  am
(18)
NA is the Avogadro number and am is the molecular projected area[11, 21] which in this
 2

circumstances is the projection area of carbon dioxide of 20  A / mol ecul e  .




4. Conclusions
The paper built a new multilayer adsorption model based on the original BET model and the
GAB model; the modified GAB model has the same form with the original BET model, but the so
called relative pressure is different, the nominal relative pressure is αth power of the relative
pressure and it can improve the applicability of the BET model with the relative pressure being up
to at least 0.9. BET model and the models supposed by Anderson[7] and Bashiri[11] are special
cases of the model in this work. Microporous material system adsorption data are applied to the
modified GAB model; it shows the model has good conformity with the experimental data.
Comparison between several adsorption models is provided, and the model supposed by
Anderson[7] and the model derived here has almost the same effect correlating experimental data,
but the modified BET model derived in this work is the most accurate regarding the coefficient of
determination. The term alpha (  ) in the modified GAB model is larger or smaller than unity
accounting for the heterogeneity of the adsorption system. The so called “BET monolayer capacity
“or more like the “BET strong retention capacity “of microporous materials calculated by the
modified GAB model in the full relative pressure range and the original BET model with linearity
criterion and two other criteria proposed by Rouquerol et al[13] is almost the same. It’s a new way
to characterize the microporous materials.
Nomenclature
ai
BET constant ( i  1, 2,
, )
am
projected area occupied by one molecule( nm molecule )
A
the specific surface area( m kg )
b
equilibrium constant(1/Pa)
bi
evaporation coefficients in BET equation ( i  1, 2,
c
solute concentration in the liquid phase (mole/kg)
2
2
, )
c0
solute saturation concentration in the liquid phase(mole/kg)
d
C
adsorption enthalpy deviation from that of liquefaction(J/mole)
energy constant
Ei
adsorption enthalpy in the i th layer(J/mole, i  1, 2,
GAC
k
Guggenheim-Anderson-de Boer
GAB model constant
ka
the adsorption rate(1/(s  Pa))
kd
the desorption rate(1/s)
K
GAB model constant ( k P0 )
L- F
Langmuir-Freundlich
NA
Avogadro number
n
adsorption equilibrium amount(mole/kg)
nmax
monolayer saturation capacity(mole/kg)
V
adsorption equilibrium amount(m3/kg)
Vm
monolayer saturation capacity(m3/kg)
P
gas pressure (Pa)
P0
saturated vapor pressure (Pa)
Qst
adsorption enthalpy(kJ/mole)
R
universal gas constant( J  mole  K )
Si
number of adsorption sites in the i th layer( i  1, 2,
T
temperature (K)
, )

1
1
, )
Greek symbol


display of the heterogeneity of the adsorption system
surface coverage
Subscripts
N
equivalent of surface loading when calculating the isosteric heat
Notes
The authors declare no competing financial interest.
ACKNOWLEDGEMENTS
Project supported by the National Natural Science Foundation of China (Grant No. 51406217) and
by the National Key foundation (from National Ministry of Finance of the People’s Republic of
China) for developing Major Scientific Instruments under Grant No.ZDYZ2014-1.
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Highlights
 A new multilayer adsorption isotherm is built
 The new model is found to correlate adsorption data in microporous material well
 The model extends the applicability of the BET equation