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Monomolecular mixed-gas sorption on polymers
Volume
120, number 4.5
CHEMICAL
MONOMOLECULAR M
MIXED-GAS
PHYSICS
SORPTION
18 October
LETTERS
1985
ON POLYMERS
PYDA
Insrrrure 01 PhJ
SICS
and Chemrsrn, Academy of rlgrrculrure, 60 - 637 Po:non
Poland
and
Recewed 15 Apnl
1985
,n final form 18 July 1985
Although m the last 15 years consderable progress has been made in the theoretrcal studres of mixed-gas sorptron on sohds (see refs [l--8], and references therem), thrs problem IS strJJ msufticrently understood and further studies m this drrectron would be desrrable The maJorrty of papers devoted to mrxed-gas sor-ptron presents a theory ofthrs phenomenon on homogeneous and heterogeneous surfaces, which are assumed to be inert sohds durmg the adsorptron process. In thrs paper a sirnple descnphon of mrxed-gas sorption on flexible lmear chains of a polymer LSconndered. Tlus descnptron IS an cxtensron of the treatment presented in a prevrous paper [9] and rt mvolves changes m the internal structure of the polymer cham, e g , different conformatronal states of lmkages between monomers of a grven cham Let us consider monomolecular sorptron from an &f-component gas mrxture on a flevlble lmear cham consrstmg of Iv ldentrcal monomers, which are sorptron centres m the sorptron process Smularly, as m ref. [9], let us assume that the conformatronal states of lmkages between monomers of the cham are drscrete according to the rotatronal rsomerrc model [IO] and that these lmkages occur m two states only This assumption and neghgence of excluded volume effects enables one ’ Permanent address II-ISIIIUI~ of Chemlswy Sktodowsha Unnerslry 20031 Lublrn Poland
416
ICI
Cune-
to descrrbe a conformation of the whole cham III terms of a onedrrnennonal lsmg model [ 1 l] _ Next, if we assume that each sorptron centre mteracts wrth the confor-matron of the nearest lrnkage and conformatrons of drfferent linkages are mutually dependent, the sorbed molecules interact indrrectly through the conformatronal subsystem of the polymer. This mteraction leads to a cooperatrve effect in the mixed-gas adsorptron, which produces a devratron of the system from Langmulr’s behavlour [I 21 Tim effect is sun&~ to that appearmg m adsorptron of single gases on flexrble lmear chams of a polymer [9J _ As mentioned above, the sorptron process occurs on a flexrble lmear chain consrstmg ofN rdentrcal monomers, wluch are assumed to be sorption centres Each linkage can be found m one of two conformat~onal states, the ground state characterized by zero energy and the exerted state characterized by the energy B 2 0 The value of B is modded by A, dependmg on the states of neighbourmg h&ages Smularly as m ref. [9] the parameter A descrrbes the cooperatrve behaviour of conforrnatronal degrees of freedom The characteristrc sorptron potenttal 1s denoted by E, (E, < 0 and i = 1,2, _ , M) for a molecule of the Ith gas sorbed on a cham being at the ground state and E, + C, for thrs molecule sorbed on the same cham centre
0 009-2614/85/S (North-Holland
03 30 0 Elsener Physics PubMung
Scrence Pubhshers B V. Dwmon)
Volume
CHEMICAL
120, number 4.5
PHYSICS
being at the excited state. The parameters C, descnbe the couphng between conformational and sorptlonal degrees of freedom Conadenng jomtly the hnkage and its nearest sorption centie as one ste we can write the Hanultoruan for sorption ofM gases on a bnear chain
of M sites as follows N
N
H=As~lmsms,l +BC s=l
m, N
N
M
where S,!, 1s a delta-fun&on Each ste of the polymer chain can be m 2 (M t 1) states characterized by a pau of “occupation numbers” (m,,n,) taking the values nz,=O, 1 andn,=O, 1,2,...,M ThevaluesOand 1 of the conformatlonal number nz, refer to the ground and excited states of the sth lmkage, respectively Smularly,rz,=iforl= 1,2, , M denotes that the sth ste is occupied by a molecule of the Ith gas and )zs = 0 denotes that the sth gte s unoccupied by a molecule Smce, only one molecule may be adsorbed on a ste, the 11~values 0 and I = 1,2,._ , M refer to empty and occupied sites, respectwely In the above, y denotes the chemical potential of the lth gas To obtam the thermodynamic hmlt, we assume cychc boundary condltlons [9] mN+s = ms
r?N+s = ns_
and
(2)
The free energy F per site for a one-dunennonal sys tern with the Hamdtonian gven by eq (1) and the boundary con&tions defied by eq (2) can be found exactly in the thermodynamic lust employmg the transfer matnx method [ 131, whuzh was used successfully m ref. [9] This method grves F= -(flN)-1
ln{Tr[exp(-OH)])
=-p-1 ln{O S[Uhf +((I$ -4vg"q),
(3)
where M _._
r/M = 1 +ab
+c k&(1 I=1
+dc,),
(
l+Ck,p, r-l
)(
1 +$,P,C1
a = exp(-DA),
b = exp (-/3B),
k, Pi = 9
-
[--P(E,
1985
c, = exp(-DC,),
&I-
(4c)
Ln the above,fl IS the well-known quantity with the absolute temperature, exp[-_B(E,
connected
- &,)I has the meamng of the sorbate activity, and pr IS the pressure of the rth gas UtAzmg the well-known relationstip between the function F and total relative surface coverage 8 191, we have
(5) where M (6)
e=F14
1s the total relative coverage defined as a sunple of the partial relatwe coverages 0,, and n,,f=0.5+0.5
$lklP,(.bct
sum
-
denotes the population of the excited conformatIona states [9] In the absence of any coupling between subsystems,C,=Oandc,=l fori=1,2,. ,M,the Isotherm eq (5) becomes equation describmg mrxedgas adsorptlon according to LangmuIr’s model [ 121 and the population coefficient even by eq. (7) becomes that predlcted by the Ismg model [ 1 l] _ Let us consider the special cases of eq. (5). For M = 1 (smgle-gas adsorption) eq (5) becomes the adsorption isotherm denved m the previous paper [9] _
It 1s
hf
M V,=(ab-b)
(4a)
18 October
LETTERS
1
)
(4b)
8, = [klpll(l
+klpl)l[l
+q(q
- 1)/(1
+k,plcl)l (8) 417
Volume
k;
where
7),=05+05[ktp~(n6ct-l)+ab-l]/(~~-441r~)’~~ and
v, =W-
b)(l
(10)
+ktp,)(l
(11)
+Qrcr)
Extensrve model studies for the adsorption rsotherm eq (8) have been discussed m the previous paper [9] In the case ofAf = 2 (adsorption from bmary gas rmytures) eq (5) gives
x 11 +~17[kIpl(cl--I)+klP1.(C3 x
[(klP]
+
k*Pa)(l + k,Plcl+
-111 k2PICZr’l
=-PI--P(EI
-&I;
= exp[--P(E?
- ~211
=k,Pz
k; = k?pz (9)
U, = 1 +ob+Fi,p,(l+obcl)
18 October1985
PHYSICS LEX-TEl%S
CHEMICAL
120, number 4.5
(17)
and (18)
PI/P2
P12 =
Then the functron
u(p,z)
is defmed as follows
U(P11) = 0 5
1 kiP12(CI - l)+k;@, - 1) (1% 1 (k;p,z +k;)(l +k;Plzq +k;+) kipll(obc,
x
X
[
-
-
1) +nb
-
1
1+
where (12)
where
CJ, = 1 +ab +kiplz(l I’? =
b)(l
(~b -
+abcl)
+ kip,,
+k;(l
+k;)(l
+abc2)
+k;~,~c,
(20)
+k;+) (21)
+k2~2(obc2
-
1) +
ob-
I]/&
- 4v,)“’
Smce the mlxed-gas
(13)
ally performed
adsorption measurements are usuassummg constancy of one variable,
and U?=
1 +ab +k,p,(l
+abc,)+k+(l
+abc,)
(14) a-05
oo-Ol(15) Eq (12) IS a product of two expressrons The first expresnon IS identical with Langmurr’s rsotherm for mrxed-gas adsorptron, however, the second expression IS equal to uruty plus a function of the partial pressures of components If thus functron IS equal to zero, eq (12) becomes Langmutr’s adsorptron Isotherm Thus, this function describes devratron of the Isotherm grven by eq (13) from the behavrour predrcted by Langmurr’s adsorptIon model The lack of expertmental data of mixed-gas adsorption on polymenc adsorbents makes unpossrble vertfiatron of eq (12) To illustrate devratron of thts rsotherm equation from Langmuir’s behaviour we performed the model studres by usmg eq (12) written rn a slightly different form 0 = Kk;Plz where
418
+kZ)/(1 +kfPIZ
+~;)l[++dl
(16)
-09 t
-*o I
1
2
3
G
5
6
7
PI2 fig 1 The functlon~(Pt~)Qlculatedaccordvlgto foro=80.b=30,cl=0008,kf=100,cr=05and~=03, OSand20
eq
(19)
Volume 120. number 43
0
12
CHEMICAL PHYSCIS LETTERS
3
4 42
5
6
7
6
18 October
1985
adsorbent mteractlons of both components wrth the adsorptron sites of the polymeric adsorbent. However, the function u(P,~) IS a httle senatlve on changes of the parameter Q, wluch reflects a change of the couphng between conformational and sorptlonal degrees of freedom due to molecules of different sorbates The model studies presented m figs 1 and 2 show that a greatest devlatron of the isotherm eq. (16) from Lang~~~uir’s behavlour IS observed for adsorptlon of gas rruxtures contaunng components of ddferent sorption propertles. Thus devlatlon mcreases when the pressure ratio p12 = pI/p2 decreases, rt means that the partial pressure p 1 decreases too because p2 was assumed to be constant The model studres suggest that Langmuir’s isotherm equatron may gve a good representation of rmxed-gas adsorption data on polymeric absorbents at lugher values of the partlal pressure p I , whereas, at low values ofpl an isotherm equation takmg into account the specrfc features of gas adsorptlon on polymers should be applied to describe the experunental data
fig 2 The functmn u(P,~) calculated nccordmfi to eq (19) = lO,~=2 0 and_r=O 3, fora= 0. b=3 0.q =0008.k; 05and20
References
e g , constancy of the total pressure, partlal pressure of one component, mole fractions of components m the bulk phase, adsorbed amount of one component, etc [7], we calculated the finctron u(p12) for p2 = const Moreover, we used the foUoHrlng relatIonAlps to defme the constants k; and cl
k; =rk;,
7=kl/k2;
Cl
= cur22
-
W)
The function u@,2), showrng devlatlon of the ~sotherm eq. (16) from Langrnulr’s behavlour, has been calculated for different values of-y and a (see figs 1 and 2) It follows from these figures that a change III
the value of y changes strongly course of the firnctlon u@~?)_ The parameter y 1s the ratio of the constants k, and k,, which are connected wth the sorptron potent& of 1st and 2nd components, respectively Thus, tlus parameter reflects difference m the adsorbate-
[I] H C Van Ness, Ind Eng Chcm Fundam 8 (1969) 464 121 hf. Billow. A Grossmann and W Schlrmcr, 2 Cbem 12 (1972) 161 [3] S SIrcar and A L h!yers, Chem Tng Sn 28 (1973)489 [4] hf Jaroruec, Thm Sohd Fdms 50 (1978) 163 [S] hl Jaromec. -Run Sohd F~lnu 71 (1980) 273 161 M Jaroruec. A Patryhe~ew and hf Bor6wko. Progxss Surface hiembrane Sa 14 (I 98 1) 1 [7] hi Jaroruec, Advances ColloId interface SCI 18 (1983) 149 [8] D Valcnzuck and A L Myers, Sepamtlon & Punficatlon Methods 13 (1984) 153 191 hf Pyda and hl Kurzynsl.~. Chem Phys 67 (1982) 7 [lo] P J Flory. Stanstlal hlechamcs of Cham Molecules (Whey. New York. 1969) ]I I] K Huanp, Statancal Mechamcs (Mley, New York, 1963) [ 121 D.hI Young and A D CrowelI. Phyncal Adsorption of Gases (Buttenvorths, London, 1962) [ 131 H E. Stanley. Introduction to Phase Transttlon and Crttlcal Phenomena (Clarendon Press. Oxford. 197 1)
419
120, number 4.5
CHEMICAL
MONOMOLECULAR M
MIXED-GAS
PHYSICS
SORPTION
18 October
LETTERS
1985
ON POLYMERS
PYDA
Insrrrure 01 PhJ
SICS
and Chemrsrn, Academy of rlgrrculrure, 60 - 637 Po:non
Poland
and
Recewed 15 Apnl
1985
,n final form 18 July 1985
Although m the last 15 years consderable progress has been made in the theoretrcal studres of mixed-gas sorptron on sohds (see refs [l--8], and references therem), thrs problem IS strJJ msufticrently understood and further studies m this drrectron would be desrrable The maJorrty of papers devoted to mrxed-gas sor-ptron presents a theory ofthrs phenomenon on homogeneous and heterogeneous surfaces, which are assumed to be inert sohds durmg the adsorptron process. In thrs paper a sirnple descnphon of mrxed-gas sorption on flexible lmear chains of a polymer LSconndered. Tlus descnptron IS an cxtensron of the treatment presented in a prevrous paper [9] and rt mvolves changes m the internal structure of the polymer cham, e g , different conformatronal states of lmkages between monomers of a grven cham Let us consider monomolecular sorptron from an &f-component gas mrxture on a flevlble lmear cham consrstmg of Iv ldentrcal monomers, which are sorptron centres m the sorptron process Smularly, as m ref. [9], let us assume that the conformatronal states of lmkages between monomers of the cham are drscrete according to the rotatronal rsomerrc model [IO] and that these lmkages occur m two states only This assumption and neghgence of excluded volume effects enables one ’ Permanent address II-ISIIIUI~ of Chemlswy Sktodowsha Unnerslry 20031 Lublrn Poland
416
ICI
Cune-
to descrrbe a conformation of the whole cham III terms of a onedrrnennonal lsmg model [ 1 l] _ Next, if we assume that each sorptron centre mteracts wrth the confor-matron of the nearest lrnkage and conformatrons of drfferent linkages are mutually dependent, the sorbed molecules interact indrrectly through the conformatronal subsystem of the polymer. This mteraction leads to a cooperatrve effect in the mixed-gas adsorptron, which produces a devratron of the system from Langmulr’s behavlour [I 21 Tim effect is sun&~ to that appearmg m adsorptron of single gases on flexrble lmear chams of a polymer [9J _ As mentioned above, the sorptron process occurs on a flexrble lmear chain consrstmg ofN rdentrcal monomers, wluch are assumed to be sorption centres Each linkage can be found m one of two conformat~onal states, the ground state characterized by zero energy and the exerted state characterized by the energy B 2 0 The value of B is modded by A, dependmg on the states of neighbourmg h&ages Smularly as m ref. [9] the parameter A descrrbes the cooperatrve behaviour of conforrnatronal degrees of freedom The characteristrc sorptron potenttal 1s denoted by E, (E, < 0 and i = 1,2, _ , M) for a molecule of the Ith gas sorbed on a cham being at the ground state and E, + C, for thrs molecule sorbed on the same cham centre
0 009-2614/85/S (North-Holland
03 30 0 Elsener Physics PubMung
Scrence Pubhshers B V. Dwmon)
Volume
CHEMICAL
120, number 4.5
PHYSICS
being at the excited state. The parameters C, descnbe the couphng between conformational and sorptlonal degrees of freedom Conadenng jomtly the hnkage and its nearest sorption centie as one ste we can write the Hanultoruan for sorption ofM gases on a bnear chain
of M sites as follows N
N
H=As~lmsms,l +BC s=l
m, N
N
M
where S,!, 1s a delta-fun&on Each ste of the polymer chain can be m 2 (M t 1) states characterized by a pau of “occupation numbers” (m,,n,) taking the values nz,=O, 1 andn,=O, 1,2,...,M ThevaluesOand 1 of the conformatlonal number nz, refer to the ground and excited states of the sth lmkage, respectively Smularly,rz,=iforl= 1,2, , M denotes that the sth ste is occupied by a molecule of the Ith gas and )zs = 0 denotes that the sth gte s unoccupied by a molecule Smce, only one molecule may be adsorbed on a ste, the 11~values 0 and I = 1,2,._ , M refer to empty and occupied sites, respectwely In the above, y denotes the chemical potential of the lth gas To obtam the thermodynamic hmlt, we assume cychc boundary condltlons [9] mN+s = ms
r?N+s = ns_
and
(2)
The free energy F per site for a one-dunennonal sys tern with the Hamdtonian gven by eq (1) and the boundary con&tions defied by eq (2) can be found exactly in the thermodynamic lust employmg the transfer matnx method [ 131, whuzh was used successfully m ref. [9] This method grves F= -(flN)-1
ln{Tr[exp(-OH)])
=-p-1 ln{O S[Uhf +((I$ -4vg"q),
(3)
where M _._
r/M = 1 +ab
+c k&(1 I=1
+dc,),
(
l+Ck,p, r-l
)(
1 +$,P,C1
a = exp(-DA),
b = exp (-/3B),
k, Pi = 9
-
[--P(E,
1985
c, = exp(-DC,),
&I-
(4c)
Ln the above,fl IS the well-known quantity with the absolute temperature, exp[-_B(E,
connected
- &,)I has the meamng of the sorbate activity, and pr IS the pressure of the rth gas UtAzmg the well-known relationstip between the function F and total relative surface coverage 8 191, we have
(5) where M (6)
e=F14
1s the total relative coverage defined as a sunple of the partial relatwe coverages 0,, and n,,f=0.5+0.5
$lklP,(.bct
sum
-
denotes the population of the excited conformatIona states [9] In the absence of any coupling between subsystems,C,=Oandc,=l fori=1,2,. ,M,the Isotherm eq (5) becomes equation describmg mrxedgas adsorptlon according to LangmuIr’s model [ 121 and the population coefficient even by eq. (7) becomes that predlcted by the Ismg model [ 1 l] _ Let us consider the special cases of eq. (5). For M = 1 (smgle-gas adsorption) eq (5) becomes the adsorption isotherm denved m the previous paper [9] _
It 1s
hf
M V,=(ab-b)
(4a)
18 October
LETTERS
1
)
(4b)
8, = [klpll(l
+klpl)l[l
+q(q
- 1)/(1
+k,plcl)l (8) 417
Volume
k;
where
7),=05+05[ktp~(n6ct-l)+ab-l]/(~~-441r~)’~~ and
v, =W-
b)(l
(10)
+ktp,)(l
(11)
+Qrcr)
Extensrve model studies for the adsorption rsotherm eq (8) have been discussed m the previous paper [9] In the case ofAf = 2 (adsorption from bmary gas rmytures) eq (5) gives
x 11 +~17[kIpl(cl--I)+klP1.(C3 x
[(klP]
+
k*Pa)(l + k,Plcl+
-111 k2PICZr’l
=-PI--P(EI
-&I;
= exp[--P(E?
- ~211
=k,Pz
k; = k?pz (9)
U, = 1 +ob+Fi,p,(l+obcl)
18 October1985
PHYSICS LEX-TEl%S
CHEMICAL
120, number 4.5
(17)
and (18)
PI/P2
P12 =
Then the functron
u(p,z)
is defmed as follows
U(P11) = 0 5
1 kiP12(CI - l)+k;@, - 1) (1% 1 (k;p,z +k;)(l +k;Plzq +k;+) kipll(obc,
x
X
[
-
-
1) +nb
-
1
1+
where (12)
where
CJ, = 1 +ab +kiplz(l I’? =
b)(l
(~b -
+abcl)
+ kip,,
+k;(l
+k;)(l
+abc2)
+k;~,~c,
(20)
+k;+) (21)
+k2~2(obc2
-
1) +
ob-
I]/&
- 4v,)“’
Smce the mlxed-gas
(13)
ally performed
adsorption measurements are usuassummg constancy of one variable,
and U?=
1 +ab +k,p,(l
+abc,)+k+(l
+abc,)
(14) a-05
oo-Ol(15) Eq (12) IS a product of two expressrons The first expresnon IS identical with Langmurr’s rsotherm for mrxed-gas adsorptron, however, the second expression IS equal to uruty plus a function of the partial pressures of components If thus functron IS equal to zero, eq (12) becomes Langmutr’s adsorptron Isotherm Thus, this function describes devratron of the Isotherm grven by eq (13) from the behavrour predrcted by Langmurr’s adsorptIon model The lack of expertmental data of mixed-gas adsorption on polymenc adsorbents makes unpossrble vertfiatron of eq (12) To illustrate devratron of thts rsotherm equation from Langmuir’s behaviour we performed the model studres by usmg eq (12) written rn a slightly different form 0 = Kk;Plz where
418
+kZ)/(1 +kfPIZ
+~;)l[++dl
(16)
-09 t
-*o I
1
2
3
G
5
6
7
PI2 fig 1 The functlon~(Pt~)Qlculatedaccordvlgto foro=80.b=30,cl=0008,kf=100,cr=05and~=03, OSand20
eq
(19)
Volume 120. number 43
0
12
CHEMICAL PHYSCIS LETTERS
3
4 42
5
6
7
6
18 October
1985
adsorbent mteractlons of both components wrth the adsorptron sites of the polymeric adsorbent. However, the function u(P,~) IS a httle senatlve on changes of the parameter Q, wluch reflects a change of the couphng between conformational and sorptlonal degrees of freedom due to molecules of different sorbates The model studies presented m figs 1 and 2 show that a greatest devlatron of the isotherm eq. (16) from Lang~~~uir’s behavlour IS observed for adsorptlon of gas rruxtures contaunng components of ddferent sorption propertles. Thus devlatlon mcreases when the pressure ratio p12 = pI/p2 decreases, rt means that the partial pressure p 1 decreases too because p2 was assumed to be constant The model studres suggest that Langmuir’s isotherm equatron may gve a good representation of rmxed-gas adsorption data on polymeric absorbents at lugher values of the partlal pressure p I , whereas, at low values ofpl an isotherm equation takmg into account the specrfc features of gas adsorptlon on polymers should be applied to describe the experunental data
fig 2 The functmn u(P,~) calculated nccordmfi to eq (19) = lO,~=2 0 and_r=O 3, fora= 0. b=3 0.q =0008.k; 05and20
References
e g , constancy of the total pressure, partlal pressure of one component, mole fractions of components m the bulk phase, adsorbed amount of one component, etc [7], we calculated the finctron u(p12) for p2 = const Moreover, we used the foUoHrlng relatIonAlps to defme the constants k; and cl
k; =rk;,
7=kl/k2;
Cl
= cur22
-
W)
The function u@,2), showrng devlatlon of the ~sotherm eq. (16) from Langrnulr’s behavlour, has been calculated for different values of-y and a (see figs 1 and 2) It follows from these figures that a change III
the value of y changes strongly course of the firnctlon u@~?)_ The parameter y 1s the ratio of the constants k, and k,, which are connected wth the sorptron potent& of 1st and 2nd components, respectively Thus, tlus parameter reflects difference m the adsorbate-
[I] H C Van Ness, Ind Eng Chcm Fundam 8 (1969) 464 121 hf. Billow. A Grossmann and W Schlrmcr, 2 Cbem 12 (1972) 161 [3] S SIrcar and A L h!yers, Chem Tng Sn 28 (1973)489 [4] hf Jaroruec, Thm Sohd Fdms 50 (1978) 163 [S] hl Jaromec. -Run Sohd F~lnu 71 (1980) 273 161 M Jaroruec. A Patryhe~ew and hf Bor6wko. Progxss Surface hiembrane Sa 14 (I 98 1) 1 [7] hi Jaroruec, Advances ColloId interface SCI 18 (1983) 149 [8] D Valcnzuck and A L Myers, Sepamtlon & Punficatlon Methods 13 (1984) 153 191 hf Pyda and hl Kurzynsl.~. Chem Phys 67 (1982) 7 [lo] P J Flory. Stanstlal hlechamcs of Cham Molecules (Whey. New York. 1969) ]I I] K Huanp, Statancal Mechamcs (Mley, New York, 1963) [ 121 D.hI Young and A D CrowelI. Phyncal Adsorption of Gases (Buttenvorths, London, 1962) [ 131 H E. Stanley. Introduction to Phase Transttlon and Crttlcal Phenomena (Clarendon Press. Oxford. 197 1)
419