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Determination of the isosteric heat of adsorption by gas adsorption chromatography
JournoI of Chromotogr$h_v, 131 (1977) l-5 0 EbfXier Scientific Publishing Company, Amsterdam
CHROM.
-
Printed ‘in The Netherlands
9398
DETERMINATION OF THE ISOSTERIC HEAT OF ADSORPTION BY GAS ADSORPTION CHROMATOGRAPHY
J. GAWDZTK Department of Physical Chemistry, Institute of Chemistry U&KS,
Nowotki 12,20031 Lublin (Poland)
and M. JARONIEC Department of Theoretical Chemistry, Institute of Chemistry UIUCS, Nowotkil2,20031 (First received March 3lst,
Lublin (Poland)
1976; revised manuscript received June lst, 1976)
SUMMARY
The determinationof the isostericheat of adsorptionby means of polynomial approximation of an experimentallyobtained-function -retention volume W~SLIS adsorbatepressure- is discussed.The equation is’proposed by means of which the isosteric heat of adsorption can easily be calculated by using the approximation coefficients.In order to illustratetheoreticalconsiderations,experimentalgraphs of the isostericheats of adsorption versusadsorbedamounts for benzeneand n-hexane on porous glass beads are presented. . INTRODUCTION
In an earlierpaper’, the following exponentialexpression
VN.~(PI = exp
(*i.Bi P')
-.
(1)
was proposed for approximatingthe retentionvolume VN,r as a function of the adsorbatepressurep.In thisequation, Bi is an opportune coefficient,the value of which may be calculatedby using the polynomial approximation: ln F&t (P) =
i Bt P’ i=o
The best-fit polynomial for approximatingthe experimentalfunction In V,.,(p) be found according to procedurerecommendedby Cole et aZe2 A simplifiedform of eqn. 1, viz., G&9
= exp (B. +
4~)
(2)
may_
(3)
may be obtained from the Jovanovic adsorption isothern?. Eqn. 3 describes the retention mechanism of gas adsorption chromatographyfor homogeneous adsor-
J. GAWDZIK,
2.
M. JARONIEC
bents; heterogeneous adsorbents can bc characterized by eqn. 1. The subscript ‘I” in eqn. 3 relates to homogeneous surfaces. Suprynowicz ct aZ.*discussed the application of eqn. 1 for determining energydistribution functions, virial coefficients, adsorption isotherms and monolayer capacities, and use of this eqn. has been extended to liquid adsorption chromatography3. Recently, eqn. 1 was also used for evaluating the pre-exponential factor of Henry’s constant and the average adsorption energy4. In this paper we shall discuss the determination of the isosteric heat of adsorption (Q=) by means of eqn. 1. For this purpose, we shall propose an equation by means of which Q”’ can easily be calculated from the coefficients &. THEORETICAL The
isosteric heat of adsorption is definedsa by the expression (4) I I
where Nt is the amount adsorbed. By differentiating N&,7’) we obtain dN =~[~]~dlnp+[-/W~
(9
I
By using in eqn. 5 the well-known relationship (6) and assuming the constancy of Nt we obtain the expression
Q-@J-2=
FRZT
P
1
aN, w-3 1
vN.~@,T)
awn
(7)
p
Eqn. 7 may be written in the slightly different form
3
Q’ o, T) = FR’TN, (P,T)
P
vN.t
(PgT)
[
3 ln Nt W? win
I
p
(8)
where F is the James-Martin compressibility factor. In the above equations, V,,,(p,T) may be calculated analytically according to eqn. 1; however, N&T) can be determined by simple numerical integration:
Nr (PST)= WRT)-*
$exp( 2 4
pi)
dp
(9)
f=O
In the case of Iinear chromatography vN.l
=
ew
(Bo)
aIn ( VNJ23
a(i/z-)I
a(Bo- III zr) I NC w/T)
(11): I i
i
ISOSJIERIC HEAT OF ADSORPTION
3
Let us now considerthe correlationbetween @’ and the energy-distribution function X(E). According to Van Dongen and Broekh&, Qsr on a heterogeneous surfacemay be expressedby: _fX(E)-4fl (~ze) - mw In PIT de !Pwf-,l=
(12)
*
or (13) where q”’ is the local isostericheat of adsorption and is definedby:
q”= -R
ahp [a(ljz-)
I
(14)
N1
and (aN,/a lnp), = (FRO-1
- v,,,(p,z4
(15)
-P
In the above calculations,N,(p,T,&) is the local adsorptionisotherm’. The local retentionvolume, V,,, (see eqn. 3) is derivedfrom the JovanoviCadsorptionisotherm6: Nl = N,,, [l -
exp (--a~)]
(16)
where a = (l/K) exp (&/RT)
(17)
N, is the monolayer capacity, K is the pre-exponentialfactor of Henry’s constant and E is the adsorptionenergy. The adsorption isotherm(eqn. 16) may be used for evaiuatingthe local isostericheat of adsorption(eqn. 14): qst=R
dln(llK)
(18)
+E
d (l/T) Thus by replacingthe local retentionvolume V,,, in eqn. 13 by: V,.r = (FRTNJK)
exp (EJRT) exp (-ap)
and substitutingeqn. 18 for qR we obtain: s x(E)- E * eXp dln(IlK) + d
(-a&
(19)
* eXp
(E/m
de
p=R
d (W-l
.Jx(4 - exp (-apI
(20) - exp WW
d&
For the purpose of ihustration,we have appliedeqn 8 to the adsorptiondata for benzeneand n-hexane on porous glass beads at four temperatures(328,343,358 and 373 K); the appropriateexperimentaldetails of these adsorption systems have been repor&&. ’
J. GAWDZIK,
M. JAROVIEC
-8 c
1
1
I
2.7
3.1
2.9 +.103
Fig. 1. Relationship between In iVt (p,T) and l/T for adsorption of benzene (0) and n-hexane (e) on porous glass beads; the numbers denote the adsorbate pressure.
Eqn. 8 was programmed in Algol60 on an Odra 1204computer. Starting from the coefficients B, determining the best-fit polynomial (eqn. 2), we calculated the value of Q= for 2 riquired pressure range. In Fig. 1, graphs of In N&T) against l/T are presented for yarious adsorbate pressures. These graphs are rectilinear, and, if the dependence of In Nt on 1jT is rectilinear, then Qm can be calculated according to eqn. 8.
I
5
I 10
N,*105 ( mole/g 1
I
.
Fig. 2. Values of Q’” for benzene on porous g&s and 373 K (a).
15
beads at temperatures
of 328 (O), 343 (e), 358 ((>)
ISOSTERIC HEAT OF ADSORPTION
I Fig. 3. Values of p
I
5
I
&-lo=
10 ( mole/g)
I
15
I
for n-hexane on porous glass beads (symbols
as in
Fig. 2).
Curves showing the relationship between p calculated from eqn. 8 and Nt for the adsorption of benzene and n-hexane on porous glass beads at four temperatures are shown in Figs. 2 and 3; it can be seen that 0” depends on the temperature. This problem was recently discussed by Koubek et aZ.*,who showed that the isosteric heat of adsorption on heterogeneous adsorbents may be a very variable function of the temperature and of the extent of coverage of the surface. REFERENCES 1 Z. Suprynowicz, M. Jaroniec and J. Gawdzik, Chromufographta, 9 (1976) 161. 2 J. H. Cole, D. H. Everett, C. T. Marshall, A. Ruiz P&ego, J. C. Pow1and F. Rodriguez-Reinoso. J. Chent. Sot_, Farad. Tram_ I, 70 (1974) 2154. 3 Z. Suprynowicz and M_ Jaroniec, J. Chromatogr., 117 (1976) 11. 4 J. Gawdzik, 2. Suprynowicz and M. Jaroniec, J. Chromatogr., 12i (1976) 185. 5 R. H. van Dongen and J. C. P. Broekhoff, Surface Sci., 18 (1969) 462. 6 M. J&roEiec,J. Colloid Interface Sci., 53 (1975) 422. 7 M. Jarcniec, Colloid Polytn. Sci., 254 (1976) 601. 8 J. Koubek, J. P&ek and J. Volf, J_ Colloid Interface Sci., 51 (1975) 491.
CHROM.
-
Printed ‘in The Netherlands
9398
DETERMINATION OF THE ISOSTERIC HEAT OF ADSORPTION BY GAS ADSORPTION CHROMATOGRAPHY
J. GAWDZTK Department of Physical Chemistry, Institute of Chemistry U&KS,
Nowotki 12,20031 Lublin (Poland)
and M. JARONIEC Department of Theoretical Chemistry, Institute of Chemistry UIUCS, Nowotkil2,20031 (First received March 3lst,
Lublin (Poland)
1976; revised manuscript received June lst, 1976)
SUMMARY
The determinationof the isostericheat of adsorptionby means of polynomial approximation of an experimentallyobtained-function -retention volume W~SLIS adsorbatepressure- is discussed.The equation is’proposed by means of which the isosteric heat of adsorption can easily be calculated by using the approximation coefficients.In order to illustratetheoreticalconsiderations,experimentalgraphs of the isostericheats of adsorption versusadsorbedamounts for benzeneand n-hexane on porous glass beads are presented. . INTRODUCTION
In an earlierpaper’, the following exponentialexpression
VN.~(PI = exp
(*i.Bi P')
-.
(1)
was proposed for approximatingthe retentionvolume VN,r as a function of the adsorbatepressurep.In thisequation, Bi is an opportune coefficient,the value of which may be calculatedby using the polynomial approximation: ln F&t (P) =
i Bt P’ i=o
The best-fit polynomial for approximatingthe experimentalfunction In V,.,(p) be found according to procedurerecommendedby Cole et aZe2 A simplifiedform of eqn. 1, viz., G&9
= exp (B. +
4~)
(2)
may_
(3)
may be obtained from the Jovanovic adsorption isothern?. Eqn. 3 describes the retention mechanism of gas adsorption chromatographyfor homogeneous adsor-
J. GAWDZIK,
2.
M. JARONIEC
bents; heterogeneous adsorbents can bc characterized by eqn. 1. The subscript ‘I” in eqn. 3 relates to homogeneous surfaces. Suprynowicz ct aZ.*discussed the application of eqn. 1 for determining energydistribution functions, virial coefficients, adsorption isotherms and monolayer capacities, and use of this eqn. has been extended to liquid adsorption chromatography3. Recently, eqn. 1 was also used for evaluating the pre-exponential factor of Henry’s constant and the average adsorption energy4. In this paper we shall discuss the determination of the isosteric heat of adsorption (Q=) by means of eqn. 1. For this purpose, we shall propose an equation by means of which Q”’ can easily be calculated from the coefficients &. THEORETICAL The
isosteric heat of adsorption is definedsa by the expression (4) I I
where Nt is the amount adsorbed. By differentiating N&,7’) we obtain dN =~[~]~dlnp+[-/W~
(9
I
By using in eqn. 5 the well-known relationship (6) and assuming the constancy of Nt we obtain the expression
Q-@J-2=
FRZT
P
1
aN, w-3 1
vN.~@,T)
awn
(7)
p
Eqn. 7 may be written in the slightly different form
3
Q’ o, T) = FR’TN, (P,T)
P
vN.t
(PgT)
[
3 ln Nt W? win
I
p
(8)
where F is the James-Martin compressibility factor. In the above equations, V,,,(p,T) may be calculated analytically according to eqn. 1; however, N&T) can be determined by simple numerical integration:
Nr (PST)= WRT)-*
$exp( 2 4
pi)
dp
(9)
f=O
In the case of Iinear chromatography vN.l
=
ew
(Bo)
aIn ( VNJ23
a(i/z-)I
a(Bo- III zr) I NC w/T)
(11): I i
i
ISOSJIERIC HEAT OF ADSORPTION
3
Let us now considerthe correlationbetween @’ and the energy-distribution function X(E). According to Van Dongen and Broekh&, Qsr on a heterogeneous surfacemay be expressedby: _fX(E)-4fl (~ze) - mw In PIT de !Pwf-,l=
(12)
*
or (13) where q”’ is the local isostericheat of adsorption and is definedby:
q”= -R
ahp [a(ljz-)
I
(14)
N1
and (aN,/a lnp), = (FRO-1
- v,,,(p,z4
(15)
-P
In the above calculations,N,(p,T,&) is the local adsorptionisotherm’. The local retentionvolume, V,,, (see eqn. 3) is derivedfrom the JovanoviCadsorptionisotherm6: Nl = N,,, [l -
exp (--a~)]
(16)
where a = (l/K) exp (&/RT)
(17)
N, is the monolayer capacity, K is the pre-exponentialfactor of Henry’s constant and E is the adsorptionenergy. The adsorption isotherm(eqn. 16) may be used for evaiuatingthe local isostericheat of adsorption(eqn. 14): qst=R
dln(llK)
(18)
+E
d (l/T) Thus by replacingthe local retentionvolume V,,, in eqn. 13 by: V,.r = (FRTNJK)
exp (EJRT) exp (-ap)
and substitutingeqn. 18 for qR we obtain: s x(E)- E * eXp dln(IlK) + d
(-a&
(19)
* eXp
(E/m
de
p=R
d (W-l
.Jx(4 - exp (-apI
(20) - exp WW
d&
For the purpose of ihustration,we have appliedeqn 8 to the adsorptiondata for benzeneand n-hexane on porous glass beads at four temperatures(328,343,358 and 373 K); the appropriateexperimentaldetails of these adsorption systems have been repor&&. ’
J. GAWDZIK,
M. JAROVIEC
-8 c
1
1
I
2.7
3.1
2.9 +.103
Fig. 1. Relationship between In iVt (p,T) and l/T for adsorption of benzene (0) and n-hexane (e) on porous glass beads; the numbers denote the adsorbate pressure.
Eqn. 8 was programmed in Algol60 on an Odra 1204computer. Starting from the coefficients B, determining the best-fit polynomial (eqn. 2), we calculated the value of Q= for 2 riquired pressure range. In Fig. 1, graphs of In N&T) against l/T are presented for yarious adsorbate pressures. These graphs are rectilinear, and, if the dependence of In Nt on 1jT is rectilinear, then Qm can be calculated according to eqn. 8.
I
5
I 10
N,*105 ( mole/g 1
I
.
Fig. 2. Values of Q’” for benzene on porous g&s and 373 K (a).
15
beads at temperatures
of 328 (O), 343 (e), 358 ((>)
ISOSTERIC HEAT OF ADSORPTION
I Fig. 3. Values of p
I
5
I
&-lo=
10 ( mole/g)
I
15
I
for n-hexane on porous glass beads (symbols
as in
Fig. 2).
Curves showing the relationship between p calculated from eqn. 8 and Nt for the adsorption of benzene and n-hexane on porous glass beads at four temperatures are shown in Figs. 2 and 3; it can be seen that 0” depends on the temperature. This problem was recently discussed by Koubek et aZ.*,who showed that the isosteric heat of adsorption on heterogeneous adsorbents may be a very variable function of the temperature and of the extent of coverage of the surface. REFERENCES 1 Z. Suprynowicz, M. Jaroniec and J. Gawdzik, Chromufographta, 9 (1976) 161. 2 J. H. Cole, D. H. Everett, C. T. Marshall, A. Ruiz P&ego, J. C. Pow1and F. Rodriguez-Reinoso. J. Chent. Sot_, Farad. Tram_ I, 70 (1974) 2154. 3 Z. Suprynowicz and M_ Jaroniec, J. Chromatogr., 117 (1976) 11. 4 J. Gawdzik, 2. Suprynowicz and M. Jaroniec, J. Chromatogr., 12i (1976) 185. 5 R. H. van Dongen and J. C. P. Broekhoff, Surface Sci., 18 (1969) 462. 6 M. J&roEiec,J. Colloid Interface Sci., 53 (1975) 422. 7 M. Jarcniec, Colloid Polytn. Sci., 254 (1976) 601. 8 J. Koubek, J. P&ek and J. Volf, J_ Colloid Interface Sci., 51 (1975) 491.