Adsorption isotherm equations associated with the gamma micropore-size distribution and their application for characterizing microporous solids

Materials

Chemistry

and Physics,

24

(1989) I - 12

ADSORPTION ISOTHERM EQUATIONS ASSOCIATED WITH THE GAMMAMICROPORE-SIZE DISTRIBUTIClrJ AND THEIR APPLICATION FOR CHARACTERIZING MICROPOROUSSOLIDS

M. JARONIEC,* R. MADEY and X. LU Department Kent, J.

of

Ohio

Physics,

44242

Kent State

University

(U.S.A.)

CHOMA

Institute

of

Received

Chemistry,

February

WAT, 00908

6,

1989;

Warsaw (Poland)

accepted

March

17.

1989

ABSTRACT The gamma micropore-size the

overall

very

fine

adsorption

isotherm

of

the

equation

gives

characterize

that

the

a symmetrical

this the

simplified solids

to

a simpler

representation

that

that

a four-parameter

structurally-heterogeneous

reduces

a good

reveal

on heterogeneous

resembles

generates

For

equation

studies

show also

adsorption

this

that

Model

parameters

studies

isotherm.

micropores,

isotherms.

distribution

equation

solids

experimental

with

micropores

These

for

if

adsorption

permits

structure.

may be used

large

for

three-parameter

equation

microporous

with

distribution

of

simplified

equation

evaluation model

describing

their

size-distribution

function.

INTRODUCTION In 1977 Stoeckli overall

[l]

adsorption

significant

progress

microporous

solids

were

made to

to

represent

[4,8,9] for

that

example,

has occurred

the

the

micropore

the

Gaussian

[l]

in

the

size

theory

distribution

distribution

assumes

of

does

of

[2.3]

Chemistry,

for used

function. not

satisfy value

Since

then,

attempts

various the It

the

on heterogeneous

Several

equations

a non-zero

representing solids.

adsorption

[2-201.

Dubinin

distribution

for

microporous

isotherm

and next

*Permanent address: Institute 20031, Lublin (Poland)

0254-0584/89/%3.50

equation

characterization

adsorption

Stoeckli

this

an integral

on heterogeneous

and their

derive

distributions.

proposed

isotherm

[2-71

micropore-size

Gaussian

distribution

was shown elsewhere

some physical when the

M. Curie-Skaodowska

requirements:

micropore

size

University,

0 Elsevier Sequoia/Printed in The Netherlands

2 is

equal

micropore

value

size

microporous the

to

and in contrast

al

[7,10-141

to

well

the

for

the

overall

equation

for

showed the

Gaussian

The gamma micropore

equation

three-parameter

paper,

these

suitable

isotherm

distribution

of

the

structural

model

equations

size

that

the

heterogeneity

adsorption

isotherm,

gamma of

distribution,

it

distribution

the satisfies

generates

which

structurally-heterogeneous

studies

that

are

The aim of

function.

limitations the

et L

describes

a

reduces

solids

to

a

with

very

compare

both

micropores. In this

of

Jaroniec function

requirements.

four-parameter

fine

[9].

structure,

physical

simple

zero

distribution

simple

are

these

of

in

is

to

isotherm

order

show the

solids.

the

isotherm

size

advantages

for

equation

microporous

to

gamma micropore

by the

studies

three-parameter

heterogeneities

presented

generated

and

characterizing

ADSORPTION ISOTHERM EQUATIONS The integral

representation

heterogeneous

microporous

of

solids

has

overall the

following

for

form

gas

adsorption

on

[S]:

00

e=

s

exp

P(B)

[-B(A/13)2]

dB

(1)

BO

where A = RT In

(p,/p)

Here A is

the

filling T,

of

p.

is

depends

the

structural the

size

isotherm, by the

=

;

on the

vapor

micropores

the

Note

that

smallest in

on+1

(B-B,)”

parameters

dispersion

the

to

relative

equilibrium

pressure,

adsorption

pressure

B is

nature

of

[3],

B, is

micropores. integral

B.

It

the

the

minimal

(1)

the

in uniform

is

the

[Zl],

value the

function

degree

that B is

that

of

exponential

[6]

is

distribution

factor

that

a

characterizes

B that

that

are the

is

the

DR

characterized gamma

F(B):

exp[-q(B-B,)]

q > 0 and n 2 -1

of

temperature

micropore

micropores

was shown elsewhere

represent

the

coefficient

adsorbate (DR) equation

and F(B)

eqn.

6 is

p and absolute

a similarity

Dubinin-Radushkevicb

adsorption

parameter

may be used

P(n+l)

the

represents

structural

the

physicochemical

uniform with

which

Here the the

saturation

at

in the

distribution F(B)

potential,

micropores

parameter of

function.

adsorption

the

only

associated

(2)

(3)

are

associated

with

the

average

value

B and

oB:

= B, + gfl = B, + (n+l)/q

(4)

3

OB

(n+l)‘h/q

=

Solving

the

for

overall

the

(5)

integral

eqn.

(1)

adsorption

with

eqn.

(3),

we have

the

four-parameter

equation

isotherm: n+l

8 = 61.6~

=

exP[-Bo(A/B)21

Here @I denotes factor

the

in this

micropore J(x)

exponential

equation.

size

factor

The F(B)

distribution

= 2 E

(6)

(s]

in eqn.

distribution

function

x(x~-x~)~

(6),

J(x)

and e2 denotes

function

the

power

may be converted

to

the

[18]:

exp[-q*(x2-xg)]

(7)

where B = cx2,

=

BO

Here c is

and

c(xo)2.

an empirical

constant

q* = cq

[3,7],

(8)

and x is

the

half-width

of

the

slit-like

with

eqn.

micropores. The adsorption is

given

X(A)

distribution

= 2A13-~ exp[-Bo(A/B)2]

The X(A)

distribution

microporous its

potential

Because

the

to

l

function

solid,

structural

reduces

X(A)

= -dB/dA

associated

(6)

by:

whereas

[

n+l

-1 q+(A/8)’

l

(9)

q+(A/13)2

represents

the

[B. + &l

the

energetic

heterogeneity

micropore

size

distribution

small,

may be replaced

J(x)

of

a

characterizes

heterogeneity. parameter

a simple

B,

is

three-parameter

it

isotherm

equation

by zero;

then

eqn.

(6)

[7]:

n+l e=e2=[

The functions

F(B),

respectively, section (10) solids

(10)

-1 q+(A/13j2

of

gives

J(x),

and X(A)

by means of this

paper,

a good

eqns.

we will

(3),

instead

with

eqn.

(7)

and

(9)

show that

the

simplified

representation

and may be used

associated

of of

the

the

with

(10) 8,

= 0.

may be calculated, In the

three-parameter

adsorption

isotherms

four-parameter

eqn.

next eqn.

on microporous (6).

4 MODEL STUDIES We generated eqn.

(6)

for

Table I. eqn. (6).

the

values

adsorption

of

of

the

C

d e

*

Bo,

q,

model

0.218 0.278 0.278 1.736 1.736

and n used

in the

according

and n given

adsorption

5.0 5.0 5.0 5.0 5.0

by dividing q,

B,,

BoxlO (mole/kJ)2

maximum amount adsorbed

obtained

a,,

isotherms

(mmole/q)

a0

a b

a0

model

Parameters

Isotherm Code

the

the

in

isotherms

to

Table

the I;

four-parameter

here

calculated

a,

denotes

according

to

n (kJ/ile)2

825 382 124 382 124

0.65 1.90 0.08 1.90 0.08

The relative

micropores.

the

adsorbed

amount at by the

to

generate

the

model

quantity

adsorption

adsorption ao.

isotherms

6 is

The values

of

shown in Fig.

1.0

0.6

-15

-10

Relative Pressure,

-5

0

ln(p/po)

1. Overall adsorption isotherms 6 (solid lines) calculated according to given in Table I_(lines a, b. and c). The eqn . (6) for the model parameters dotted line denotes the function 61 = exp[-B,(A/B)“], .whereas the dashed lines denote the e2 curves calculated according to the simplified eqn. (10).

1

5 are

analogous

to

those

[11,12.14].

Figure

calculated

according

line)

and e2

parameters (8)

to

value

equal

in Table

0.2

nm. it

which

isotherm

= 0.278

the

a small

given

to

the

experimental

isotherms

this

8,

1;

the

in this

simplified

for

the

according

corresponds

(dotted to

line) overall

case

eqn.

lines

61 (dotted

calculated

value

isothel,ms

6 [solid

functions

x 10m3 (mole/kJ)2;

6I function

by the

the

and c were

contribution

b and c in Fig.

is

6 is

b,

x nm)2 13.71,

Because

curves

62 function,

adsorption

I and B,

a,

of

adsorption

in comparison

Curves

gives

an analysis

model

(6)],

(mole/kJ

I especially

6 [cf. the

to

x0 value,

the

eqn.

lines).

and c = 0.00694

a small

from

1 presents

(dashed

given

obtained

the

(lo),

is

to to

near

unity

adsorption difference and the

eqn.

the

xo for

isotherm between

overall

small].

0.6

Relative

0

-5

-10

-15

Pressure,

ln(p/po)

Fig. 2. Overall adsorption isotherm 6 (solid line) calculated according (6) for the model parameters given in Table 1 for system d in comparison 81 function (dotted line) and the e2 function (dashed line).

Figure letter

2 presents

b but

corresponds curves case,

refer which

significantly between

e2

the

calculated to

the

to

value

a solid

differs the (dashed

from

analogous for

curves

B, = 1.736

of with that

x0 equal

presented

adsorption

line)

and 6 (solid

those

marked

x 10m3 (mole/kJ)2; to

micropores

overall

to

0.50 of

nm, which sizes

line)

1, 8;

is

this

the

than

significant.

0.50

61 function

consequently,

1 by the

value

means that

greater

in Fig. isotherm

in Fig.

the

to eqn to the

of

these nm.

B, isotherm In this

influences difference

In contrast

to

Fig.

1, the 81 function

is a rapidly

presented

increasing

function

The model adsorption

isotherms

in Table I were described

2 differs

generated

significantly

by eqn.

by the simplified

was to answer the following evaluating

in Fig.

eqn.

(6) for the parameters given

(10).

The aim of this

is the siaplified

question:

the parameters that characterize

eqn.

the microporous

Table II. Parameters obtained by fitting the simplified adsorption isotherms described in Table I. Isotherm Code

(mmole/q)

(mole/kJf2

4.996 4.978 4.960 4.985 4.896

1051 443 151 561 260

(10) to the model isotherms from TableIIthat

1.37 2.41 0.29 4.15 1.41

(10) reproduces

eqn.

a0 is reproduced

with an accuracy

A comparison

Table T.

summarized in Tables means of eqn. difference obvious

eqn.

decreases

= 0; then el = 1. in spite

(10)

is connected

[c_f..

eqns.

difference

directly

5 values

Table I).

given in

This conclusion eqn.

pairs

is reproduced well

by eqn.

with eqns.

(10) reproduces

of eqn.

(10) and (8).

is (10) is that

of q and n given (10).

meaningful quantities feature

this

by assuming 8,

between the suitable

this

by

the model isotherms:

(IO),

This 5 and o8 we compared

This comparison

shows that the simplified

eqn.

for the isotherm e, which was generated

is observed

the value of

of q and n obtained

greater

error

eqn.

It follows

model isotherms

of the simplified

with the physically

associated

the simplified

decreases.

feature

To illustrate

(4) and (5)).

in Table IIthe

(n+l)fq

2.278 7.870 8.988 9.328 10.446

case of eqn (6) obtained

A most interesting ratio

eqn. (6) ,_._......_._..._..I..I

of the parameters q and n, which are

that the values

is a special

(10)

the parameter a,;

from those used to generate

of the observed

ratio

sets

when the B,-parameter

I and 11,the

by fitting

1 and 2 (cf.,

of about 2% for all

I and II,shows

Table II

(10) to the model

2.255 7.698 8.543 9.180 9.269

very well

of the suitable

(10) differ

because eqn.

in Tables

shown in Figs.

structure?

for

~x103(mole/kJ)2

the parameters ao, q, and n obtained

contains

description

(10) useful

eqn.

n

4

a0

from unity and

of In (p/p,).

well the average value 5.

A

for a higher

value of B, and a low value of n. Additional P(B).

and J(x)

associated

figures

present

generated

a comparison of the distribution

with the model adsorption

q and n listed

functions

by the parameters q and n given inTable

in Table I).

In Fig.

isotherms

f&.

3. the solid

generated lines

represent

IIwith

X(A), those

by the parameters the adsorption

Adsorption

Potential,

A (kJ/mole)

Fig. 3. Adsorption potential distributions X(A) calculated according to eqn. (9) for the systems a, b, and c. The solid lines are obtained for the parameters Bo, q, and n given in Table I. whereas the dotted lines are calculated for the parameters q and n listed in Table II and B, = 0.

potential

distributions

parameters those

calculated

(dotted (10)

q,

B,,

for This

line).

reproduce

the

model

the

systems

well

b.

simplified

eqn.

Fig.

of

distribution

which

B,

are

according

that

to

distributions

q and n listed the

in Table

parameters

(9)

for

conpared

IIand

B, = 0

the

simplified

of

X(A)

comparison

eqn.

are

distributions

an analogous

Table

F(B)

curves

(solid

size

distribution

that F(B) 8,.

which

for

the

with

eqn.

associated

was obtained

curves

the

to

worse

this

with

also

and reproduced system

nm.

between

compared

for

the

an asymmetrical

for

a

that

lines)

the

similar for

in Fig.

to

the 5.

a relatively

distribution model

the parameters

It

distribution

d with

To illustrate size

for the

was obtained

model for

are

for

obtained

(dotted

result

F(B)

(3)

comparison

which

micropore

for

eqn.

isotherms from

distributions

x0 = 0.5

the

to

J(x),

also

The agreement

becomes

model

A similar

observed

functions

according

follows

F(B)

lines).

of

distribution

the

it

the

corresponds

d.

For I),

between is

a comparison system

the

functions

agreement and J(x)

of

calculated

I and II.

(cf.,

generates

6 presents

evaluated

these

potential

and c;

(10)

surprising

value

calculated I;

shows

adsorption b,

in Tables

small

functions

parameters

a comparison

and c,

relatively

model

are

in Table

comparison

a,

4 presents

q and n given

the

which

d and e.

a,

micropore

the

the

isotherms

Figure systems

X(A),

and n given

this

large

observation,

curves

J(x)

and reproduced

nicropore

size

distribution;

is

Structural Parameter, B (mole/kJ)2 Micropore distribution functions F(B) calculated according to eqn. (3) Fig. 4. The solid lines are obtained for the parameters for the systems a, b, and c. q, and n given in Table I, whereas the dotted lines are calculated for B, = B O’kth the parameters q and n given in Table II

0.0

0.5

1.0

1.5

2.0

2.5

Micropore Dimension, x (nm) Micropore size distribution functions J(x) calculated according to eqn. Fig. 5. The solid lines are calculated for the (7) for the systems a. b, and c. parameters B,, q, and n given in Table I, whereas the dotted lines are obtained for go = 0 with the parameters q and n given in Table II.

9

0.0

0.5

Micropore Comparison Fig. 6. system d; the solid Fig. 5.

1.5

1.0

Dimension,

of the micropore and dotted lines

2.0

2.5

x (nm)

size distribution are denoted in

the

curves J(x) for the same way as those in

.. .. . .

0.000

0.005

Structural and dotted

0.010

Parameter,

0.015

0.020

B (mole/kJ)’

Comparison of the P(B) distribution curves for the system lines are denoted in the same way as those in Fig. 4.

e:

the

solid

10

for

example,

the

F(B)

and J(x)

curves

for

system

e are

shown in Figs.

7 and 8,

respectively. The above evaluating but

also

the

micropores

studies

model

not

only

the

show that

parameters

distribution

(small

by a symmetrical

the that

functions

value

of

micropore

Micropore

F(B).

Ro) or size

simplified

eqn.

characterize J(x),

solids

larger (cf.,

Dimension,

is

for

for

structure

solids

micropores Fig.

useful

microporous

and X(A)

with

distribution

(10)

the

with

but

fine

characterized

6).

x (nm)

Comparison of the J(x) distribution curves for the system Fig. 8. and dotted lines are denoted in the same way as those in Fig. 5.

e;

the

solid

EXPERIMENTAL ILLUSTRATION To provide benzene published activated by the

of

also for

III the

verification

form

carbon

a small

shows

by the

this

surfaces,

adsorption

parameters values

isotherm

the a,, of

[22].

structural

mesopore

parameters 8, Bo;

eqn.

His

in our (S)].

we used

(h,

the

of

at

studies,

studies

first

these

the

nine

that

the was

this

and may be characterized this

to

equation

neglect

(p/p0

experimental

by means of

eqns. obtained

is

benzene

low pressure

DR equation.

parameters

isotherm

showed

notation,

the

we analyzed

293 K; this

In order only

a0 and B, of

set

model

heterogeneity

and oB calculated the

the

carbon

(DR) equation; [cf.,

contains

different

by Dubinin

81 function

on the

of

on AC.3 activated

in a tabular

adsorption

Table

isotherm

Dubinin-Radushkevich

expressed

part

an experimental

adsorption

< 0.02)

points). This

(61, for

table (4)‘ B,

contains

and (5) = 0

Table III. Adsorption parameters obtained for the benzene isotherm on ACS activated carbon at 293 K for different values of Bo.

Equation

Box104 (mole/kJ)2

Bx103

a0

x0

(nm)

i&j

(mole/kJ) 2

(mmole/g)

SD

+03

(nm) (mole/kJ.)2

(mmole/g)

. __

. . . .._......

10

0

0

5.14

1.18

0.41

4.39

0.0709

6

2.78 4.34 6.25 8.50 9.50

0.20 0.25 0.30 0.35 0.37

5.14 5.14 5.15 5.15 5.16

1.18 1.18 1.19 1.21 1.24

0.41 0.41 0.41 0.42 0.42

4.51 4.65 4.91 6.07 8.38

0.0709 0.07og 0.0710 0. on1 0.0712 .._.,...........__

DR

10.69

0.39

4.99

-

-

0

The quantity GB is defined: xB = (ii/c)%. SD denotes the standard deviation between the experimental adsorbed amounts. relates

to the simplified

Chona (JC) equation experimental

special

The values

and calculated

and (10) are similar. values

isotherm eqn.

[7].

form of eqn.

microporous

and calculated

which is known as the Jaroniec-

of the standard deviation

(SD) between the

adsorbed anounts for the DR equation

In the region

are almost identical

(lo),

o.0740

of B, from 0 to 0.37,

means that the simplified

eqn.

and eqns.

the fact (lo),

(6) for 8, = 0, may be used to evaluate

(6)

that the SD

which is a

the parameters of

structure.

CONCLUSION Model studies evaluate

the minimum values

adsorbate: as well

and an experimental

example show that it is difficult

of 8, and x0 from the adsorption

however, the other parameters

as the distribution

means of the JC eqn (10). heterogeneous

solids

are characterized

functions

F(B),

This equation

with fine

micropores

by a symmetrical

that characterize J(x),

or solids size

isotherm for one microporous

structure

and X(A) may be evaluated

is applicable

micropore

to

by

to structurally

with large

micropores

that

distribution.

ACKNOWLEDGMENT This work was supported grant CBT-8721495.

in part by the National

Science

Foundation under

12

REFERENCES 1 H.F.

Stoeckli,

J.

Colloid

2 M.M. Dubinin

and H.P.

3 M.M. Dubinin,

Carbon,

4 M. Rozwadowski 5 R. Wojsz

Interface J.

23 (1985)

373.

Carbon -9

22 (1984)

363.

Carbon,

24 (1986)

225.

and J.

Piotrowska,

and J.

Choma, Mater.

Choma, If. Jankowska,

J.

Monatsh. Chen.

Piotrowska

Chem..

117

Phys.,

Sci.,

75 (1980)

(1986)

15 (1986)

and M. Jaroniec,

34.

7. 521.

Monatsh.

Chem.,

-118

315.

9 M. Jaroniec,

X. Lu and R. Madey,

Monatsh.

10 M. Jaroniec,

Langmuir,

3_ (1987)

11 M. Jaroniec,

R. Madey,

X. Lu and J.

12 M. Jaroniec

and J.

Choma,

13 J.

Choma and M. Jaroniec,

14 J.

Choma, M. Jaroniec

15 M. Jaroniec

Carbon,

and J.

s

Carbon,

and R. Madey,

Chem. Sot.

and R. Madey,

Separation

and R. Madey,

J.

20 M. Jaroniec

and R. Madey,

Physical

Amsterdam,

22 M.M. Dubinin,

Chem.,

Surface

Carbon,

25 (1987)

26 (1988) Trans.

II,

1. 84 (1988) -

579. u

(1987)

103.

Technol.,

22 (1987)

92 (1988)

3986.

Adsorotion

Membrane Sci., 593.

911.

179.

on Heterorreneous

1988.

Pronress

889.

4_ (1988)

(1988)

Carbon,

Sci.

Phys.

@

Faraday

Chem. Phys.,

18 M. Jaroniec

Dubinin,

(1988)

747.

25 (1987)

Choma. Mater.

19 M. Jaroniec

Elsevier,

(1988)

Piotrowska,

J.

and J.

119 -

Choma. Langmuir,

Chem. Phys.,

and R. Madey,

16 M. Jaroniec

Chem.,

795.

Mater.

17 M. Jaroniec

21 M.M.

184.

Interface

and R. Wojsz,

7 M. Jaroniec

(1987)

59 (1977)

Colloid

and M. Rozwadowski.

6 M. Jaroniec

8 J.

Sci.,

Stoeckli,

9_ (1975)

1.

2367.

Solids,

1139.