METHOD FOR THE CALCULATION
OF EFFECTIVE
DISTRIBUTION
IN MOLECULAR SIEVE CARBON
PORE SIZE
Geza HORVATHand Kunitaro KAWAZOE
Department of Chemical Engineering, University of Tokyo, Tokyo 113
Amethod for the calculation of effective pore size distribution from adsorption isotherms in molecular-sieve
carbon is described. This method is more exact theoretically as well as practically than previously described
methods. An average potential function has been determined inside the slit-like pores. With the help of this function
the doubtful use of the Kelvin equation can be avoided at the scale of molecular dimensions. The method gives poor
values for the larger pores but can be combined with the well-known Dollimore-Heal method at a pore size of
1.34 nm. Calculation is possible over a wide range of pore sizes. The calculation is shown through two examples
from N2 isotherms at 77.4K. The model can be extended to other pore shapes as well as to other
adsorbent-adsorbate pairs.
Introduction
Molecular-sieve activated carbon usually contains
slit-like pores of the order of0.1 nm and the distances
between the walls of slits are of the interest from the
standpoint
of separation processes. Kawazoe et
^ i,i2-i4) repOrtec[
studies carbons.
of severalHowever,
properties theseof
activated
molecular-sieve
authors found no reported theoretical works on pore
size distribution at the scale of molecular dimensions.
Dollimore
and Heal2) have given a review and
criticism of previous methods used for the calculation
of pore size distribution.
They pointed out that their
improved method also had some problems
when
The calorimetric
^diff
face" of carbon atoms may be regarded as the
"effective
radius of the carbon atom."4) Therefore,
the pore size itself must be called the "effective pore
size." This definition by Everett is in use in the case of
micropores. For larger pores, the problem of the
uncertain size of the "outer surface" is negligible.
1.
the molar integral
change of the free
energy (Gibbs function for T=constant):
AGads=AH*ds- TASads
(1)
The molar integral change of enthalpy on adsorption
AHads=
-qdm-RT+K(Tp/9)(dn/dT)d
(2)
is16)
Received February 6, 1983. Correspondence concerning this article should be addressed
to K. Kawazoe, Dept. ofInd. Admin., Sci. Univ. ofTokyo, Noda 278. G. Horvath is now
at Dept. of Unit Operations, Univ. of Chem. Eng., Veszprem, Hungary.
470
= AHv*p_ RT_ (dhf/dna)T
(3)
ASads=AStr+ASrot+ASyih
(4)
Except for ASl\ the terms of this equation are nearly
constant, and we can write
ASads= AStr(w/wJ +AS0
(5)
The free energy change can be calculated from the
gas-phase pressure. Combining this with the previous
equation:
AGad* = RTln {p/p0)
=AHads-
T(ASu(w/wJ
+AS0)
(6)
If we consider the limit of zlGads as p approaches p0,
lim AStT(w/wJ=0
lim p->Po
RT\n(p/po)=AHoads-
TASo=0
(7)
AHoads= TASo= -AHwap= - TASyaip
(8)
then
Wecan also write
Theory
Consider
of qdlff is
The molar integral change of entropy is
pores approach molecular dimensions, because the
Kelvin equation had been used. This problem has
been reported elsewhere also.9)
In the micropore domain the so-called "outer sur-
definition
RTln (p/p0)
= - qdá"- RT+K(Tp/e)(8n/dT)d
-
T(AS\w/wJ+AS0)
Supposing that K(T/3/6)(dn/dT)d
can be simplified to
RT\n(p/po)-AHvaP=
(9)
^ RT, the expression
-qdm-TASir{wlwJ
(10)
In the range of p/p0^pjpo>0, the above general
equation is approximated into Eq. (12), since the
adsorbed phase is considered similar to the liquid
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phase and then
Having
molecular
I TAS^w/wJKlq^l
(ll)
-qdiff=RT\n(p/p0)-AHá"»
(12)
considered
level16):
the definition
surfaces. The potential, originated from the interaction of the adsorbate molecules in a pore, increases
the interaction
energy. The potential
NnAn+NAA,
(13)
l-r
we used the samelogic to obtain:
RTln (p/po) = Uo +Pa
(14)
'<m «
where NAis the numberof molecules per unit area of
the adsorbate
where
interactions,
- (fHfr
2<r4
Uo is the expression
and
of the adsorbent-adsorbate
Aa
and
6mc2ocn(x A
Pa is an implicit
function of the adsorbateadsorbate-adsorbent
interactions.
Some authors4'18'19*
Xa
the so-called (10 : 4) potential is likely more realistic
than the (9:3) potential,
but it is not possible to
choose between these models on the basis of the
precision with which they represent the adsorption
data. Our aim was to develop a suitable method for
the calculation and not to decide which is the most
adequate model among those mentioned above.
The micropores of molecular-sieve materials are
usually considered to be slits between two graphitized
carbon layer planes.1* Supposing that the potential
fields of these layers can be approximated by a
potential of the graphite layer, simple mathematical
expressions can be used.
The potential
function20* over a graphite
(interaction
energy between a gas molecule
infinite graphite layer) is
(<j/r)*
+ ((j/r)10]
<P* = (NaAa)/(1.06)(2a*)
surface at zero interaction
energy, r is the distance
light, oca is the polarizability and %a is the magnetic
susceptibility of an adsorbent atom, <xAand %Aare the
polarizability and magnetic susceptibility of an adsorbate
molecule.
Equation (18) consists of two parts. One of them,
NaAa\2o^
multiplied
by the expression inthe parentheses,
corresponds
to K~X'UO (Eq. (14));
other corresponds to K'1 Pa (Eq. (14)).
Having
taken into account Eq. (14),
RT\n (p/Po)=K
2a\l-d)
G\10
HtJ
-l^J
d= da
Kirkwood-Muller, London, etc., equations.
RTln (p/Po)=
K
by the
for the present case of adsorption
VOL.
16
NO.
6
1983
on two parallel
J*
(21)
+dA
(22)
a\l-d)
9(1-d/2)
3(rf/2)
3
(23)
(17)
layers. The Frenkel-Halsey-Hill theory5 "8'15* uses
the Kirkwood-Muller expressions for the calculation
of adsorption potential. A similar expression is used
\1(T
,.10
3(1-d/2)3
Where / is the distance between the nuclei of the two
+^J
^
r10
in the case of one adsorbate molecule is:
0 = [(NaAaW^)I ~ {°lrf + Mr)10
W(/-r)}10]
/
NaAa + NAAA
The potential function between two parallel layers
- W(/-r)}4+
\4
da is the diameter of an adsorbent atom, and dA is the
diameter of the adsorbate molecule. Integration gives
result:
given
[-V
/
where ^ is Avogadro's number, and
the following
Aa is a constant
an average
potential value that depends on the absolute values of
distances between the two layers can be calculated:
NnA»+NAA,
r}d/2 d/2
from the surface, Na is the number of atoms per unit
area of surface,
(20)
where m is the mass of an electron, c is the velocity of
( 1 6)
where a is the distance between a gas atom and the
Xa
3á"c2*aXa
surface
and an
(15)
(19)
AA
have determined potential func-
tions over a carbon layer and inside a pore having
either slit-like or cylindrical shape. They suggest that
<P= 3.06<P*[-
be-
tween the two carbon layers filled with adsorbates is
of qdl{{ on the
-qdi({=U0+Pa-AEyib-AEtr-AErot
function
where / should be larger than d. When /=d, Eq. (21)
gives a finite value of p/p0, pc/p0. So the above
equation is considered
to be valid for p/po^Pc/Po>
which satisfies
(12).
the condition assumed for deriving Eq.
Wenow have a function ofp with respect to /.
471
Having used the data of a nitrogen isotherm at liquid
nitrogen
temperature,
expression
where
W^=#(/?),
the
for adsorption.
A schematic diagram of the experimental apparatus
w/Woo =f(l-
da)
(24)
can be obtained. This function gives us the effective
pore size distribution since w is considered as the mass
of the nitrogen adsorbed into pores smaller than
(l-da), and w^ is the maximumamount of nitrogen
adsorbed into the pores.
2. Model Parameters
The necessary physical data can be seen in Table 1.
Everett and Pawl4) give an equation
lation of a values:
for the calcu-
<7=0.858d/2
In
(99.999%, Takachiho Kagakukogyo Ltd.) was used
(25)
our case,
g=0.275nm
is shown in Fig. 2.
For measurement of the N2 isotherms at liquid N2
temperature
a sample (~0.3g)
was put into a sample
holder
and degassed
at 200°C ,and 10"5 Torr
(1.33x10~3Pa)
pressure
Cahn electrobalance
measurement. For
ULVAC ionization
Baratron sensors
1.33x l(T6-6.65x
for at least
48 hours.
A
provided highly accurate mass
the measurement of pressure
vacuum gauges and MKS
were used (pressure
ranges
KT1Pa; 1.33x 10"1-105Pa).
The data obtained are shown in Table 4 and Fig. 3.
4. Comparison of qdmValues from Different Sources
Using the N2 isotherm of HGS 638 at liquid N2
temperature the expression
qdm = RTln (po/p) + AHvap
(27)
According to Walker20) the value ofNa=3.845 x 1015
molec-cm"2 while NA=6.7 x 1014 molec-cm"2 calculated from the liquid N2 density. From the numerical
data substituted into Eq. (23) the following expression
can be calculated
can be obtained,
temperatures. The empirical results confirm our supposition that the TASXr{wlwao) term is really negligible in the circumstances mentioned above.
5. Calculation of Pore Size Distribution Function
Having determined w^, the maximumamount of
N2 adsorbed into the pores at p/p0=0.9, the calcu-
w
.
.
ln (p//?o)
where / is in nm.
62.38
= /3o64
1.895xlO"3
2.7087x10
(/-0.32)3
-7
(/-0.32)9
- 0.05014
(26)
as a function
of w/w^- If we
compare the data of this calculation with the data of
Kawai,n) good agreement can be seen (Fig. 4). Kawai
determined the qst values from isotherms at different
lation
of pore size distribution
can be carried
out.
From the theoretical part it follows that if we know
thep-l functions, we know the;?-(/- da) functions also
Suitable (l-da) values between 0.35 and 1.4nm are
chosen. Substituting these values of/ into Eq. (26), the
according to Dollimore can be found. The two curves
measured isotherms, the w values corresponding to
relative pressure of//=5 x lO~2. At this pressure the
slopes of the two curves are nearly equal. At the point
of intersection the two methods could be combined.
Beyond p' the Dollimore method can be used while
belowpf the new method may be used.
w/w^ against the (l-da), the pore size distribution
(Table
(Fig.
3.
2). In the same table
1) intersect
pore size calculated
at a pore size of 1.34nm
and a
Experimental
Table 3 gives the physical properties of the carbon
samples used for this investigation. HGM366 and
HGS 638 respectively
are molecular
sieve carbons
provided by Takeda Ind. Co., Ltd. Research grade N2
values of p are obtained.
the chosen (l-da)
Using the data
of the
pore sizes can be found. Plotting
obtained.
The procedure itself is very simple. Table 5 shows
the results of the calculation
and HGM 366.
in the case of HGS638
In Fig. 5 the molecular probe data3) ofHGS638 are
compared by the calculated pore size distribution. In
Fig. 6 the effective pore size distribution
curves of
HGM 366 and HGS 638 can be seen. The main
effective
pore size of HGM366 is 0.42nm, obtained
from molecular probe data by Takeda Co.
Table 1. Physical properties for calculation of model parameters
C arbon
Diameter
Liquid density
Polarizability
Susceptibility
472
[nm]
[g/cm3]
[cm3]
[cm3]
is
Reference
Nitr ogen
Reference
0.3
0. 34
0.808
1.46x
2xl0~29
1.02 x 10 "24
13.5 x 10 "29
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Table 2. The values of corresponding (plp0),
according to the new model and Dollimore
Relative
Effective
p/Po l-]
New model (/-Ja)
pressure
l(T7
0.4
6.47
2.39x
1.05x
1.54x
8.86x
1.46x
x
0.43
0.46
0.5
0.6
0.7
2.95x
2.22x
4.61
lO""3
lO~2
x lO-2
l(T7
lO"6
lO~5
l0~4
l0~4
(l-da)
pairs
HGS 638 w0O=173.2mgN2/gC (p/Po=0.9)
pore size
[nm]
Dollimore model
-
w
PlPo
[mg N2/gC]
1.16
1.32
l0~2
1.5
1.46
3.15x
lO"1
3.0
2.23
10.0
3.34
6.7
8.1xl0~4
1.08X10'1
13.7
1.07xl0"6
0.079
1.1xlO~3
1.47X10"1
18.7
1.45xlO~6
0.108
2.1xlO~3
2.80X10"1
38.4
2.76xlO~6
0.222
3.5xlO"3
4.67xlO"1
52.1
4.61x10~6
0.301
1.01xlO~2
1.35
95.3
1.33xlO~5
5.09
4.3xlO~2
5.73x10
1.6
114.0
2.13xlO2
10.0
47.0
1.09xlO4
Table 3. Physical properties
investigation
of carbon samples used in this
HGS 638
HGM 366
Particle
density
[g/cm3]
True
density
[g/cm3]
Micropore
volume* [cm3/g]
0.90
1.8
0.21
1.09
1.80.084
Main pore size**
0.5
0.42
[nm]
0.550
0.658
2.11xlO~3
0.814
1.32xlO~2
6.18x10~2
163.1
0.880
0.907
1.08xlO"1
2.20xlO4
390.0
5.20xlO4
170.8
5.13xlO"1
0.986
535.0
7.13xlO4
171.8
7.04xlO"1
0.992
640.0
8.53xlO4
173.1
8.42xlO"1
9.87x
166.8
0.942
165.0
[Torr]
size vs. pressure.
0.019
0.039
5.66xlO'4
152.4
157.1
lO4
665.0
530.0
420.0
228.0
150.0
110.0
94.0
48.0
13.8
7.0
2.2
0.56
2.17xlO"1
173.5
Desorption
P
Pore
3.03xlO~7
5.00xlO~7
141.0
1.33xlO3
6.27xlO3
740.0
1.
w/W oo
[-]
3.07xlO~2
5.07xlO"2
82.0
Fig.
[-]
2.3xl(T4
3.8xlO~4
-
0.8
1.1
1.3
lO"1
Adsorption
P
[Torr]
[Pa]
-
7.59x
7.24x
Table 4. Data of isotherm
9.74x
w
[Pa]
8.87xlO4
7.07xlO4
5.60xlO4
3.04xl04
2.12xlO4
1.47xlO4
1.25x
lO4
6.40xlO3
1.84xlO3
9.33xlO2
2.93xlO2
7.47x10
HGM 366
woo=67.5mgN2/gC
0.999
lO"1
p/po
[mg N2 /gC]
173.1
173.1
172.8
170.1
168.4
167.0
166.1
163.1
157.1
152.4
146.4
137.7
0.963
-
wjw^
[-]
[-]
8.75xlO"1
6.97xlO"1
5.52xlO"1
3.00X10"1
2.09xlO"1
1.45xlO"1
1.24xlO"1
6.32xlO"2
1.82xlO~2
9.21x10~3
2.89x1Q~3
7.37x10~4
0.999
0.999
0.998
0.982
0.972
0.964
0.959
0.942
0.907
0.880
0.845
0.795
(p/po=0.9)
* At liq. N2 temperature.
** From molecular probe data of Takeda Co.
Desorption
P
[Torr]
700.0
VOL
16
NO.
6
1983
has been shown to have
5.26xlO"1
1.32xlO'1
1.32x10~2
1.33xlO2
65.3
1.32xlO"3
1.33xlO1
64.5
1.32xlO"4
l.OxlO"2
described
67.4
66.7
66.1
9.21x10"1
1.0xlO"1
3.0xl0-3
Discussion
The method
5.33xlO4
1.33xlO4
1.33xlO3
67.5
[-]
1.0
5.0xlO"2
6.
w
PlPo
[mg N2 /gC]
9.33xlO4
400.0
100.0
10.0
Fig. 2. Schematic picture of experimental apparatus.
1, microbalance; 2, glass tube; 3, weight; 4, sample; 5, N2bath; 6, gas bomb; 7, vacuum pump; 8, N2 inlet; 9, control
unit; 10, recorder; 1 1, ionization vacuum gauge (ULVAC);
12, measuring unit (ULVAC); 13, 14, Baratron sensors; 15,
measuring unit (Baratron); 16, diffusion vacuum pump.
[Pa]
1.25xlO~3
5.0xlO"4
2.3xlO-4
6.67
1.33
4.00X10-1
1.67X10"1
6.67xlO~2
3.07xlO-2
64.4
6.58xlO"5
62.6
61.5
48.1
37.0
26.1
1.32xlO"5
3.95xlO"6
1.64xlO~6
6.58xlO~7
3.03x10~7
W^ oo
[-]
1.00
0.998
0.988
0.970
0.967
0.956
0.954
0.927
0.911
0.713
0.548
0.387
1.2xlO-4
1.60X10"2
12.3
1.58xlO~7
0.182
l.OxlO"4
1.33xlO~2
10.6
1.32xlO~7
0.157
advantages over previous methods, both in theoretical exactness and ease of use. Previous methods did
not consider the material properties of the adsorbents.
473
Fig. 3. N2 isotherms
temperature.
ofHGS 638 and HGM366 at liq.
N2
Fig. 5. Calculated effective pore size distribution of HGS
638 compared to molecular probe data (#). , calculated
curve (1-w/vO %.
N2
CH2C12
CH3(CH2)2CH3
CH3(CH2)3CH3
CH3(CH2)5CH3
CH3(CH2)8CH3
CH3(CH2)CH(CH3)2
CHC13
CH2-CH2x
O
CH2-Cu{
CH,
Fig.
4. qAm vs. w/w^ diagram
CH,
ofHGS 638.
CH,
CH,-CH-CH,
14
<
H-s
potelze
[n
0.4
0.43
0.46
0.5
0.6
0.7
0.8
1.1
1.3
1.5
m ]
W/W"HGM366
[-]
0. 163
0.51
1
0.726
0.926
0.956
0.958
0.970
0.977
0.978
0.985
/
>H, CH2-CH2
.CH
17
19
18
CH/
21
20
^H2 ^ch;
NCHXCH2
/CH2
hi
XCH2
^rt2^CH2-CH2/
/CH2CH2./H
CH3-C(CH3)2-CH2-CH(CH3)2
CH3C(CH3)2-OH
^NH2
CC14
W/W-HGS638
[-]
0.0087
0.0473
0.0895
0.531
0.739
0.808
0.866
0.906
0.924
0.935
Agreement with the data obtained from molecular
probes is surprisingly good.
Dollimore and Heal state2) that about 1.5nm is the
lower limit of their method in the case of cylindrical
pores, and in the lower interval it gives undoubtedly
poor values. Our model gives poor values for larger
pores. We can state that the two models together
cover the full range of pore sizes.
The near-equality of the slopes of the two pore size
vs. pressure curves at their intersection ensures the
possibility
of a relatively continuous calculation
474
CH3
15
CH2^CH2CH9
16
CH
-CH
Table 5. Calculated values of effective pore size distribution
Fig. 6. Calculated effective
pore size distributions.
across a wide range of pressures.
Acknowledgment
The authors are indebted to the Takeda Co. and personally to
Mr. Kyoshi Itoga for providing the adsorbents and the molecular
probe data.
N omenclature
Aa = constant in Lennard-Jones potential [J/molec]
AA = constant in Lennard-Jones potential [J/molec]
da
= diameter of an adsorbent atom
[nm]
dA = diameter of an adsorbate molecule
[nm]
Eyih
= vibrational
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ENGINEERING
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a
[J/mol]
[J/mol]
[J/mol]
[J/mol]
[J/mol]
enthalpy of adsorption
[J/mol]
limited enthalpy of adsorption
[J/mol]
enthalpy of vaporization
[J/g]
residual heat of immersion
Avogadro's number
[molec./mol]
[nm]
distance between nuclei of two layers
[J]
kinetic energy of an electron
translational
energy
rotational energy
free energy
enthalpy
H
G
Erot
Hads
tt ads
#0
Hvap
hf
K
I
me2
Na
= distance between a gas atom and the nuclei of
0*
la
Xa
Literature Cited
1) Chihara,
numberof atoms per unit area of
[atom/cm2]
adsorbent
number of molecules per unit area of
NA
adsorbate
(1978).
Pa
[molec. /cm2]
1,
q«
R
r
S
£ads
entropy
entropy of adsorption
entropy of rotation
entropy of translation
entropy of vaporization
entropy of vibration
limited entropy value of adsorption
srot
svap
5vib
^0
T
[J/(mol - K)]
[J/(mol - K)]
[J/(mol - K)]
P/(mol - K)]
P/(mol - K)]
[J/(mol - K)]
P/(mol - K)]
Oxford
7)
Halsey,
8)
Soc,
Hill,
compound
polarizability
polarizability
*a
aA
G. D.:
74, 1082
T. L.: /.
79,
737
/. Chem.
Phys.,
(1952).
Chem. Phys.,
17,
16, 931
590,
(1948);
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/. Am. Chem.
(1949).
H. and H. Seewald: Ber. Bunsenges. Physik. Chem.,
(1975).
10) "Kagaku Binran," Chem. Soc. of Japan, Maruzen (1966).
1 1) Kawai, T.: "Kitai no Kyuchakubunri ni Kansuru Kenkyu,"
Doctoral Thesis, Univ. of Tokyo (1976).
12) Kawazoe, K., V. A. Astakhov, T. Kawai and Y. Eguchi:
Kagaku Kdgaku, 35, 1006 (1971).
13) Kawazoe, K., V. A. Astakhav and T. Kawai: Seisan-Kenkyu,
(Journal
Tokyo),
adsorptive potential adsorbed amount of N2
maximumamount of N2 adsorbed into
the
pores
maximumadsorbed amount of /-th
(1946).
9) Jiintgen,
temperature
Uo
w
619 (1976).
6) Frenkel, J.: "Kinetic Theory of Liquids," Clarendon Press,
pressure corresponding to l=d (=da+dA)
saturation pressure of N2
differential heat of adsorption
isosteric heat of adsorption
gas constant
[J/(mol à" K)]
distance from the nuclei of adsorbent layer [nm]
Pc
Po
qdm
72,
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potential energy of adsorbate-adsorbate-
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[nm]
the surface at zero interaction energy
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interaction energy
characteristic interaction energy
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magnetic susceptibility of an adsorbent
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density of z-th compound
VOL.
16
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1983
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