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Surface tension and adsorption of regular solutions on the solid-liquid and vapour-liquid interface
CHEMICALPHYSICS LETTERS
Volume 5. number 1
SURFACE ON
TENSION THE
AND
ADSORPTION
SOLID-LIQUID
AND
of PIqsicnZ
Clzemistq
19’iO
SOLUTIONS INTERFACE
and J. STECKl
of the PolisA
Warszazca 42,
Received
REGULAR
VAPOUR-LIQUID
A. R. ALTENBERGER Institute
OF
L5 February
Academy
of Scirnces,
Poland
24 December
1969
A new method of solving the equilibrium difference equation for a multilayer strictly regular two-component misture mixtures is presented.
adsorption from the
The thermodynamic properties of the surface layer between a strictly regular liquid mixture and a phase such as a gas or solid wall have been studied by many authors [l-9]_ Usually the theory of the interface was formulated in the random mixing approximation of the lattice model of liquids [LO]. Some particular models of the surface phase have been proposed in which the number of monomolecular Iayers with concentrations different from the concentration of the bulk solution was arbitrarily assumed [4.5, it was proved that such an assumption leads to an inconsistency with the thermodynam8,9]. However, ic Gibbs’ adsorption formula [l-3,5] and that we. can deal properly only with an unrestricted number of layers in the surface phase. The determination of the concentration in all the monomolecular layers parallel to the dividiag surface depends on the solution of a nonlinear, second order difference equation. An analytical sotution of numerical solutions of the probtem many the problem has not yet been obtained. Ono [l-3] considered years ago and more recently Lane [13] has published solutions obtained by a successive approximation method. Ash et al. [14] have applied a similar method to mixtures of molecules of different size. In this paper we present a new method for the determination of the concentration which can be applied to the solid-liquid and vapour-liquid interfaces. This method enables us to find the solution of the differwith any required accuracy withir. the range of ence equation in the form of a power series expansion, the convergence of the series. We consider a solid in contact with a mixture of A and B distributed on a rigid lattice. By constructing and maximizing a grand canonical the following difference equation
distribution
.Yco
qyJ,-2BYm=lnl_yi
--
Yi
we obtain
in agreement
2Bui + 27@yi-
yi+l-
with Ono [2] and DeLmas
yf_11,
[S].
(1)
with boundary conditions
In--Y-3
1 -yco
2By,
= In&-
2By1 - mB(1
- 2Y1) -
2mB6’2 -Yl)
- @,
and lim yi =ym, f-00
(3)
B = zzv/kT ,
(4)
where B and Q are given by the relations
(5) 29
Volume 5, number
CHEMICALPHYSICSLETTERS
1
15 February 1970
Here Yi is the mole fraction of component A in the ith monomolecular layer; & is the Boltzmann constant; T is the absolute temperature; Y4) is the bulk concentration of A; z is the coordination number of a molecule; nzz is the number of nearest neighbours in one of the adjoining layers [lo]; zw is a characteristic constant which measures the deviation of the mixture from ideality, defined by (6); yi and J$$ are the surface tensions of pure liquids A and B at the vapour-liquid or solid-liquid interface, and CLis the area per molecule A or B. We have disregarded throughout, interactions between pairs of molecules which are not nearest neighbours. The quantities 211,7% andrg CCUI be expressed by intermolecular pair interaction energies cAA, cBB and EAB (33 and interaction ener gies XA and XB of molecules A and B with the solid phase. l(EAA+EBB) 7 ” = ‘AB - ’
(6) (7)
= (x, - +a11 EAA )/a, yog = (x B - qzn2 EBB)/n *
(8)
For a vapour-liquid interface we usuallv put XA and XB equal to zero. When the recurrence re!ation (1) with (2) and (3) has been solved i.e. whenyi Yi =.f (i, B. Q,Y,)
are all known as (9)
9
the surface tension y is given by
f$
ayz
=YlF+
(1 m
and adsorptions
B g {Yi(l-h)
+YcdY,-Y~)
+y#-Yj)
+“‘(yi+l-yi)2) (10)
I’ are also known as rA=-rB=a
-l g
(11)
~Y~--Y,b
Therefore a practical solution of the problem depends on our ability to generate numerical solutions of the second order difference equation (1). We note that this equation can be solved immediately for the case of an ideal mixture when B = 0 and one obtains [8,10,12] id Y1 = y, We
YW + (1 -Y,) exP(-Q)
have found that it is possible to represent the
id =>:-fori>I. Yi
’
solution
for
B t 0 in the form
(12) of a McLaurin
series
nz,ym, T, Q, = const. , where the successive
derivatives
1
yi(‘I) (0) at the point B = 0 are obtainable from the differential
&ii_1) + 1 2Bn2(6yi+l+ ~Yi -
Y#-Yl)
(13) forms
[B(Y,-Yi) + 2~?~(2Yi-Yi+l-Yi_l)] 6B = 0 ;
- 2BiJ 6yL - 2Bnz6y2 + [2(y,-41~) - nz(l-2Yl)+
2m(yl-y2)]6B
which we obtain by differentiatmg eqs. (1) and (2). We have performed calculations retaining the first seven terms * of the series tion, some of the computed results are shown in tables 1 and 2. In a similar manner we can expand the function exp(-ya/kT) in a power series sion
= 0; l = 1-2222,
(14)
(13). As an illustraand obtain the expres-
* The lengthy algebraic expressions for y_?)(O) are not given here but are available on request.
Volume
5, number
1
CHEMICAL
Contributions Xumbor
0
1
Table 0=-l.
in the surface
.vm = 0.5 Concentration J’i
4.506 592 X 10-2
2.12i
iOi
X 10-l
(13)
4
5
6
2
9.731034
4.576295
3
9.9SY 501 X 10-2
4.9dG636
4
9.999 429 X 10-2
5
._
lnvcr
Q= -1. _YscJ = 0.1 Concentration Yi
B=-1 Ta” 0.5 Concentration 3i
3.445 525 x 10-2
3.068 835 x NL
x 10-l
1.w7444
X 10-2
5.157022
x 10-L
x 10-l
9.987 132 x 10-2
4.973015
x 10-t
1.991221
X 10-l
1.000 053 x 10-l
5.002 537 x 10-l
9.999 972 x 10-2
4.998 $05
X 10-l
9.999 978 x 10-l
6.999 632 x 10-l
G
9.999 998 X 10-2
4.999 554 x 10-l
1.000 000 X 10-l
4.999 995 x 10-l
7
1.000 000 X 10-l
4.999 991 x 10-l
1.000 000 x 10-J
4.999 991 X 10-l
ya,/kT
A 10-2
=
as a function
19iO
2
distribution
1
exp(-ya/kT) We treat
of the expansion
B=l
Ja, = 0.1 Concentration -Yi
I5 February
0= -1. B= 1. )‘Do= 0.1 Power of B pnramctcr 3
2
Concentration hiumber of the Lnyer
LETTERS
Table 1 of the psrticulnr terms
of
the Iayer
PHYSICS
(15) of the type
in the second bracket of (15) give corrections to the formula proper for idctal solutions, and enable us to compute y for non-ideal mixtures. We have found that both series (13) and (15) are represented with excellent accuracy by the first few terms for /BI< 1 and for all Q’s_ Very often only tie Terms
first and second order corrections are important; then yi - y = $
ln[l - y&-eQ)]
+ $B2
In 1 + B 1
(l-Y,)e-QJ2
+ 4nz2y,(l-y,)
(l+e-Q)2
Y,(i-Y,)
[(Zt.
m)y,(l-y_)
(1 +e-T2+2?ze-Q]
[Y, f t~-Y,)e-Ql2
e-29
Y2,(1-yd2
[Y,+
and
+ F
[YW+ (l-Y,)e-Q12
(
BZY,(l-Yd
-
l+e-Q
y-+(1-yde-Q
+ iil(l-2Yd)2
(17) 31
Volume 5. number 1
arA
= Y&-Y,) 1 - 2Yid + . y;d(l-Y\d)
CHEMICAL
l+e-Q Yco+ (I-Yco)e-Q
+ B[yI
(1)
+ 4~7ry,(l-y,)_vv2
(1)
15 February 1970
(1) B2 id id (1) (1) id +Y2 1+ 2 c 4mY1 (l-YI )Y, +4zy1 Y1 (l-Y1 ‘d,
(v(l)~2+~Y,(l_Y,lY()*Y~
- l
PHYSICS LETTERS
1+0(B3).
m
;lyvI,Yp-
4nty,(l-Y,)
(2YF)-Yr))
(18)
where
(1) =
J’1
p Generally. least
favorable
= 2~11_v~(l-_v~)2
2
for larger case.
We have compared
_v,(l-_v,)e-Q [ 2ZYco(l-Y-1 [Ym + (1-_v,)e-Q]2
values
more
1
fVZ(l-2yco)
1 .
+e-Q
(19)
~,+!l-y&-Q
of Q the convergence
than seven
lte-Q yco.t (I-y,)e-Q
terms
of the series
are needed
for
is better.
accurate
results
For Q = 0. that is in the if B = 2 or B = 3.
our
calculations with experimental data for ether+acetone, carbon tetrachloride +benzene and trichloromethane+ benzene mixtures in contact with vacuum. Every set of yi’s has been checked and found to verify the stability conditions. All calculations have been made for a close-packed lattice. The agreenent of the theoretical curves and experimental points is satisfactory.
,
0.5 Fig. 1.
32
.
Ye
Volume 5. number 1
CHEMICAL PHYSICS LETTERS
15 February 1970
Using our method we have also discussed [ll] some trends of the change of surface tension and adsorption curves with the change of the I3 and Q parameters. Some of these results are shown in fig. 1. The authors are indebted to the late Professor helpful discussions.
W. Male&i&i
for his kind interest in this work and
REFERENCES SOno. Mem.Fac. Eng.Kyushu Univ. 10 (1947) 195. [ZJ S. 0110. Mem. Fat. Eng.Kyushu Univ. 12 (1950) 1. [3] S. Ono and S. Kondo, in: Handbuch dcr Physik, Vol. X. ed. S. FlUgge (Springer Verlag. Heidelberrr. 1960). [4l E. A.Gu&nheim~ Trans. Faraday Sot. 41 (1945) 150. [5] R. Defny and I. Prigogine, Trans. Faraday SOc. 4E (1950) 199. IS] G. Dchns and D. Patterson. J. Ph,*s. Chin,. &! (1960) 1927. (71 K. Nishikawa, D. Patterson and G. Delmas. J. Phys. Chem. 65 (1961) 1226. [l]
[S] D. H. Everett,
Tr.ms. Faraday
Sot. GO (1961) 1803.
[9] 0. H. Everett. Trans. Faraday Sot. 61 (1965) 2478. [lo] E. A. Guggenheim, Mixtures (Oxford at the Clnrendon Press, Ill] A. R. Altenberger and J. Stecki, to be published. [12] [13]
A.Schuchowitzky,
Acta Physicochem.
URSS
8 (1935)
531.
1952).
J. E. Lane and C.H. J. Johnson. Autralian J. Chem. 20 (1967) 611; J.E. Lane. Astraiian J. Chem. 21 (1968) 827. [la] S.G. Ash. D. H. Everett and G. H. Findcnegg, Trans. Faraday Sot. 64 (1968) 2&5.
Berlin.
GBttinsen.
Volume 5. number 1
SURFACE ON
TENSION THE
AND
ADSORPTION
SOLID-LIQUID
AND
of PIqsicnZ
Clzemistq
19’iO
SOLUTIONS INTERFACE
and J. STECKl
of the PolisA
Warszazca 42,
Received
REGULAR
VAPOUR-LIQUID
A. R. ALTENBERGER Institute
OF
L5 February
Academy
of Scirnces,
Poland
24 December
1969
A new method of solving the equilibrium difference equation for a multilayer strictly regular two-component misture mixtures is presented.
adsorption from the
The thermodynamic properties of the surface layer between a strictly regular liquid mixture and a phase such as a gas or solid wall have been studied by many authors [l-9]_ Usually the theory of the interface was formulated in the random mixing approximation of the lattice model of liquids [LO]. Some particular models of the surface phase have been proposed in which the number of monomolecular Iayers with concentrations different from the concentration of the bulk solution was arbitrarily assumed [4.5, it was proved that such an assumption leads to an inconsistency with the thermodynam8,9]. However, ic Gibbs’ adsorption formula [l-3,5] and that we. can deal properly only with an unrestricted number of layers in the surface phase. The determination of the concentration in all the monomolecular layers parallel to the dividiag surface depends on the solution of a nonlinear, second order difference equation. An analytical sotution of numerical solutions of the probtem many the problem has not yet been obtained. Ono [l-3] considered years ago and more recently Lane [13] has published solutions obtained by a successive approximation method. Ash et al. [14] have applied a similar method to mixtures of molecules of different size. In this paper we present a new method for the determination of the concentration which can be applied to the solid-liquid and vapour-liquid interfaces. This method enables us to find the solution of the differwith any required accuracy withir. the range of ence equation in the form of a power series expansion, the convergence of the series. We consider a solid in contact with a mixture of A and B distributed on a rigid lattice. By constructing and maximizing a grand canonical the following difference equation
distribution
.Yco
qyJ,-2BYm=lnl_yi
--
Yi
we obtain
in agreement
2Bui + 27@yi-
yi+l-
with Ono [2] and DeLmas
yf_11,
[S].
(1)
with boundary conditions
In--Y-3
1 -yco
2By,
= In&-
2By1 - mB(1
- 2Y1) -
2mB6’2 -Yl)
- @,
and lim yi =ym, f-00
(3)
B = zzv/kT ,
(4)
where B and Q are given by the relations
(5) 29
Volume 5, number
CHEMICALPHYSICSLETTERS
1
15 February 1970
Here Yi is the mole fraction of component A in the ith monomolecular layer; & is the Boltzmann constant; T is the absolute temperature; Y4) is the bulk concentration of A; z is the coordination number of a molecule; nzz is the number of nearest neighbours in one of the adjoining layers [lo]; zw is a characteristic constant which measures the deviation of the mixture from ideality, defined by (6); yi and J$$ are the surface tensions of pure liquids A and B at the vapour-liquid or solid-liquid interface, and CLis the area per molecule A or B. We have disregarded throughout, interactions between pairs of molecules which are not nearest neighbours. The quantities 211,7% andrg CCUI be expressed by intermolecular pair interaction energies cAA, cBB and EAB (33 and interaction ener gies XA and XB of molecules A and B with the solid phase. l(EAA+EBB) 7 ” = ‘AB - ’
(6) (7)
= (x, - +a11 EAA )/a, yog = (x B - qzn2 EBB)/n *
(8)
For a vapour-liquid interface we usuallv put XA and XB equal to zero. When the recurrence re!ation (1) with (2) and (3) has been solved i.e. whenyi Yi =.f (i, B. Q,Y,)
are all known as (9)
9
the surface tension y is given by
f$
ayz
=YlF+
(1 m
and adsorptions
B g {Yi(l-h)
+YcdY,-Y~)
+y#-Yj)
+“‘(yi+l-yi)2) (10)
I’ are also known as rA=-rB=a
-l g
(11)
~Y~--Y,b
Therefore a practical solution of the problem depends on our ability to generate numerical solutions of the second order difference equation (1). We note that this equation can be solved immediately for the case of an ideal mixture when B = 0 and one obtains [8,10,12] id Y1 = y, We
YW + (1 -Y,) exP(-Q)
have found that it is possible to represent the
id =>:-fori>I. Yi
’
solution
for
B t 0 in the form
(12) of a McLaurin
series
nz,ym, T, Q, = const. , where the successive
derivatives
1
yi(‘I) (0) at the point B = 0 are obtainable from the differential
&ii_1) + 1 2Bn2(6yi+l+ ~Yi -
Y#-Yl)
(13) forms
[B(Y,-Yi) + 2~?~(2Yi-Yi+l-Yi_l)] 6B = 0 ;
- 2BiJ 6yL - 2Bnz6y2 + [2(y,-41~) - nz(l-2Yl)+
2m(yl-y2)]6B
which we obtain by differentiatmg eqs. (1) and (2). We have performed calculations retaining the first seven terms * of the series tion, some of the computed results are shown in tables 1 and 2. In a similar manner we can expand the function exp(-ya/kT) in a power series sion
= 0; l = 1-2222,
(14)
(13). As an illustraand obtain the expres-
* The lengthy algebraic expressions for y_?)(O) are not given here but are available on request.
Volume
5, number
1
CHEMICAL
Contributions Xumbor
0
1
Table 0=-l.
in the surface
.vm = 0.5 Concentration J’i
4.506 592 X 10-2
2.12i
iOi
X 10-l
(13)
4
5
6
2
9.731034
4.576295
3
9.9SY 501 X 10-2
4.9dG636
4
9.999 429 X 10-2
5
._
lnvcr
Q= -1. _YscJ = 0.1 Concentration Yi
B=-1 Ta” 0.5 Concentration 3i
3.445 525 x 10-2
3.068 835 x NL
x 10-l
1.w7444
X 10-2
5.157022
x 10-L
x 10-l
9.987 132 x 10-2
4.973015
x 10-t
1.991221
X 10-l
1.000 053 x 10-l
5.002 537 x 10-l
9.999 972 x 10-2
4.998 $05
X 10-l
9.999 978 x 10-l
6.999 632 x 10-l
G
9.999 998 X 10-2
4.999 554 x 10-l
1.000 000 X 10-l
4.999 995 x 10-l
7
1.000 000 X 10-l
4.999 991 x 10-l
1.000 000 x 10-J
4.999 991 X 10-l
ya,/kT
A 10-2
=
as a function
19iO
2
distribution
1
exp(-ya/kT) We treat
of the expansion
B=l
Ja, = 0.1 Concentration -Yi
I5 February
0= -1. B= 1. )‘Do= 0.1 Power of B pnramctcr 3
2
Concentration hiumber of the Lnyer
LETTERS
Table 1 of the psrticulnr terms
of
the Iayer
PHYSICS
(15) of the type
in the second bracket of (15) give corrections to the formula proper for idctal solutions, and enable us to compute y for non-ideal mixtures. We have found that both series (13) and (15) are represented with excellent accuracy by the first few terms for /BI< 1 and for all Q’s_ Very often only tie Terms
first and second order corrections are important; then yi - y = $
ln[l - y&-eQ)]
+ $B2
In 1 + B 1
(l-Y,)e-QJ2
+ 4nz2y,(l-y,)
(l+e-Q)2
Y,(i-Y,)
[(Zt.
m)y,(l-y_)
(1 +e-T2+2?ze-Q]
[Y, f t~-Y,)e-Ql2
e-29
Y2,(1-yd2
[Y,+
and
+ F
[YW+ (l-Y,)e-Q12
(
BZY,(l-Yd
-
l+e-Q
y-+(1-yde-Q
+ iil(l-2Yd)2
(17) 31
Volume 5. number 1
arA
= Y&-Y,) 1 - 2Yid + . y;d(l-Y\d)
CHEMICAL
l+e-Q Yco+ (I-Yco)e-Q
+ B[yI
(1)
+ 4~7ry,(l-y,)_vv2
(1)
15 February 1970
(1) B2 id id (1) (1) id +Y2 1+ 2 c 4mY1 (l-YI )Y, +4zy1 Y1 (l-Y1 ‘d,
(v(l)~2+~Y,(l_Y,lY()*Y~
- l
PHYSICS LETTERS
1+0(B3).
m
;lyvI,Yp-
4nty,(l-Y,)
(2YF)-Yr))
(18)
where
(1) =
J’1
p Generally. least
favorable
= 2~11_v~(l-_v~)2
2
for larger case.
We have compared
_v,(l-_v,)e-Q [ 2ZYco(l-Y-1 [Ym + (1-_v,)e-Q]2
values
more
1
fVZ(l-2yco)
1 .
+e-Q
(19)
~,+!l-y&-Q
of Q the convergence
than seven
lte-Q yco.t (I-y,)e-Q
terms
of the series
are needed
for
is better.
accurate
results
For Q = 0. that is in the if B = 2 or B = 3.
our
calculations with experimental data for ether+acetone, carbon tetrachloride +benzene and trichloromethane+ benzene mixtures in contact with vacuum. Every set of yi’s has been checked and found to verify the stability conditions. All calculations have been made for a close-packed lattice. The agreenent of the theoretical curves and experimental points is satisfactory.
,
0.5 Fig. 1.
32
.
Ye
Volume 5. number 1
CHEMICAL PHYSICS LETTERS
15 February 1970
Using our method we have also discussed [ll] some trends of the change of surface tension and adsorption curves with the change of the I3 and Q parameters. Some of these results are shown in fig. 1. The authors are indebted to the late Professor helpful discussions.
W. Male&i&i
for his kind interest in this work and
REFERENCES SOno. Mem.Fac. Eng.Kyushu Univ. 10 (1947) 195. [ZJ S. 0110. Mem. Fat. Eng.Kyushu Univ. 12 (1950) 1. [3] S. Ono and S. Kondo, in: Handbuch dcr Physik, Vol. X. ed. S. FlUgge (Springer Verlag. Heidelberrr. 1960). [4l E. A.Gu&nheim~ Trans. Faraday Sot. 41 (1945) 150. [5] R. Defny and I. Prigogine, Trans. Faraday SOc. 4E (1950) 199. IS] G. Dchns and D. Patterson. J. Ph,*s. Chin,. &! (1960) 1927. (71 K. Nishikawa, D. Patterson and G. Delmas. J. Phys. Chem. 65 (1961) 1226. [l]
[S] D. H. Everett,
Tr.ms. Faraday
Sot. GO (1961) 1803.
[9] 0. H. Everett. Trans. Faraday Sot. 61 (1965) 2478. [lo] E. A. Guggenheim, Mixtures (Oxford at the Clnrendon Press, Ill] A. R. Altenberger and J. Stecki, to be published. [12] [13]
A.Schuchowitzky,
Acta Physicochem.
URSS
8 (1935)
531.
1952).
J. E. Lane and C.H. J. Johnson. Autralian J. Chem. 20 (1967) 611; J.E. Lane. Astraiian J. Chem. 21 (1968) 827. [la] S.G. Ash. D. H. Everett and G. H. Findcnegg, Trans. Faraday Sot. 64 (1968) 2&5.
Berlin.
GBttinsen.