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Thermodynamics of adsorption from nonideal solutions of nonelectrolytes
Pergamon Press.
Chemical Engineering Science, 197 1, Vol. 26, pp. 1025-1030.
Thermodynamics
Printed in Great Britain.
of adsorption from nonideal solutions of nonelectrolytes
0. G. LARIONOV Institute of Physical Chemistry, U.S.S.R. Academy of Sciences, MOSCOW, U.S.S.R. and A. L. MYERS Department ChemicalEngineering, University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.
of
(Received 17 July 1970) Abstract-The thermodynamics of nonideal adsorbed solutions is extended to the case of mixtures of molecules of unequal size. Experimental data for the adsorption of three liquid pairs (benzene+ isooctane, carbon tettachloride + isooctane, benzene + carbon tetrachloride) on aerosil at 20°C are compared with the theory. Two solutions show ideal behavior for the adsorbed phase; data for the third solution (benzene + isooctane) cannot be explained by adsorbed-phase nonidealities.
SEVERAL theories[1,3,4,7,13,14] have been proposed to explain adsorption from solutions of nonelectrolytes. All of these theories share the assumption that there is an adsorbed phase. Actually the “adsorbed phase” has microscopic dimensions and therefore its properties, such as composition and entropy, are not real thermodynamic variables [6]. The advantage of the adsorbed-phase model of adsorption is that the multilayer structure of the liquid may be ignored. The multilayer character of the interface is explained by the decay, with distance, of the potential energy for interactions of the adsorbate molecules with the adsorbent. The lack of quantitative information on intermolecular potentials, to say nothing of the complexity of a multilayer model of a dense-fluid adsorbate, is the reason for pushing the adsorbedphase concept as far as it will go. The Gibbsian model to be described is for adsorbate molecules of different size and allows for arbitrary nonidealities in the adsorbed and bulk phases. THEORY
The surface excess of component defined by [5] :
number 1 is
4
e= n”(xlo--xl)
(1)
where no is the total number of moles of liquid of composition xIo before the solution is contacted with the adsorbent and x1 is the bulk solution composition measured after the establishment of equilibrium. The critical assumption of the adsorbed-phase model is that the solution may be separated by a dividing surface into an adsorbed phase containing n’ molecules and a bulk phase containing n molecules so that: ?P= n’+n.
(2)
A material balance on component 1 shows[5] that the surface excess may be written in terms of the composition of the adsorbed phase (xi) : nl e = n’(xi
-x1).
(3)
The location of the dividing surface is defined so that the number of moles in the adsorbed layer. For a phase (n’) is a monomolecular binary solution: 1 = xi ; 4 (4) n’
1025
m,
m,’
0.
G. LARIONOV
Equation (4) follows from the assumption that the area change upon mixing is zero. The monomolecular layer assumption, used by Elton[3], is not essential. For example, the adsorbed phase might correspond more closely to 14 molecular layers. But it is advantageous to trade the more restrictive nature of the monolayer assumption for a parameter (ml) which is reasonably welldefined by other measurements [ 151. The application of solution thermodynamics to the adsorbed phase requires the definition of a standard state for the pure adsorbates. The standard state is defined as the pure adsorbate at the same temperature and surface tension as that of the adsorbed solution. For a fixed temperature, let o? be the surface tension for the interface between the adsorbent and pure i liquid. The effect of surface tension upon the chemical potential of pure component i is given by the Gibbs adsorption isotherm [ 121: A do =-n;
dp.t.
(5)
In terms of the fugacity, Eq. (5) becomes: A do = -nlRT
d lnfi.
(6)
Integration of Eq. (6) under the constraint of monolayer surface coverage (n; = mi) yields the fugacity of pure i(E) at the state (o, T): fl =fi” exp
-A(a--c+$‘) miR T
1
and A. L. MYERS
state (ff ) : j-i’ =_fi*y;x;.
The fugacity in the bulk liquid is given by: fi = The
selectivity
ponent number by:
P:yiXi.
(10)
(S) of the adsorbent for com1 of a binary solution is defined
s=- &lx; x1/-%-
(11)
The adsorption selectivity was used by Schiessler and Rowe [I I]. who called it the adsorption separation factor. For preferential adsorption of component number 1, S > 1. Combination of Eqs. (3), (4) and (11) yields: 4
e
_
mlxlx2
0 - 1)
sx,+$x, .
(12)
Equation (12) is for nonideal adsorbed and bulk phases composed of molecules of different size (ml + mz). At equilibrium, fi’ =fi. Equations obtain:
(13)
(7)-( 11) and (13) may be combined to
(7)
fro, the fugacity of the pure adsorbate at (cr:, T) , is equal to the fugacity of the equilibrium bulk liquid: fi” = Pt.
(9)
(8)
Equation (8), which neglects vapor-phase imperfections, is adequate for the usual subatmospheric pressure of interest in adsorption. Equation (7), which accounts for the effect of surface tension upon the fugacity of the pure adsorbate, is analogous to the Poynting correction[9], which accounts for the effect of pressure upon the fugacity of a bulk liquid. The fugacity of component i in the adsorbed phase (4’) is defined in terms of the activity coefficient (-yi> and the fugacity in the standard
(14) Since the activity coefficients and surface tension are functions of composition, it follows from Eq. (14) that the selectivity is a function of composition. The bulk-phase activity coefficients (rl, y2) obey the Gibbs-Duhem equation[2] for isothermal data: xldlny,+x,dIny,=O.
(15) The equation corresponding to Eq. (15) for the adsorbed phase may be derived from the Gibbs adsorption isotherm [ 121:
1026
Thermodynamics
A do = -ni d lnfi -n; RT
Substitution
of adsorption from nonideal solutions of nonelectrolytes
d In&.
(16)
of Eqs. (7)-(9) into (16) gives:
1
=x:dlnyl+x6dln-y~. (17)
Substitution of Eq. (4) into (17) shows that the adsorbed-phase activity coefficients obey a Gibbs-Duhem type of equation, too: xldlny~+x~dlny;=O.
(18)
The desired function is nle(xl). According to Eq. (14) the selectivity depends upon the activity coefficients in the adsorbed phase (ri, 7;) which depend, in turn, upon the composition of the adsorbed phase (xi). The composition of the adsorbed phase is obtained from the selectivity by Eq. (I 1):
SXl
x;=sx,+
(19)
The surface tension of the solid-liquid interface is given in terms of the surface excess by an integration of the Gibbs adsorption isotherm [IO, 121: A((+-cr,O) RT
_ --
x’ I
Consider the differential equation (21) and the two algebraic equations (14) and (19), for which the variables are u, S, x1 and xi_ The activity coefficients are functions of composition (x,, xi). After elimination of S and xi using Eqs. (14) and (19), Eq. (21) may be integrated (numerically) for a(~~). If the adsorbed phase is ideal, then the composition of the adsorbed phase (xi) is not needed for the integration of Eq. (21). Having determined u(xl), the surface excess is given by Eqs. (12) (14) and (19). Calculation of activity coeficients mental data
In&=
A(u-ulO) m,RT
lnB+lnSYl
d(ylx,)
s
II
I x1=0
’
The selectivity is given by Eq. (12):
x1=1
RT
+A(u--2’) m,RT
(22)
or
A((+--crzO) =_
experi-
If the experimental function nle(xl) and the bulk-phase activity coefficients (rl, yz) are known, then the calculation of activity coefficients in the adsorbed phase is straightforward. The monomolecular layer values (m,, m,) are assumed to be measured independently [I 51. The surface tension at the bulk composition (x1) of interest is obtained from the experimental data using Eq. (20). The ratio of the activity coefficients in the adsorbed phase is obtained from Eq. ( 14):
r;
&
from
&d(y,x,).
-m,(S-
d(y&
=75[%+ h/m)(1-xdl
1)
x2 (XI+ x1 (x2-
(20)
The surface excess is a function of the properties m,, m2, A(czo- ulo)/RT and the activity coefficients for the adsorbed phase (r;, 76) and the bulk phase (rl, yz). According to Eqs. (12), (14) and (20), nie is a function of S, which is a function of o, which is a function of rile. u may be found independently by substituting nle from Eq. (12) into Eq. (20); after differentiation of Eq. (20): d(aA/RT)
=
helm) he/ml) .
(23)
Equations (19) and (22), taken together, provide the function In ($Jri) in terms of the adsorbedphase composition (xi). The excess Gibbs free energy of the adsorbed phase is defined by: Age ’ RT F-1
= xl In 7; + xi In 7;.
(24)
It follows from Eqs. (18) and (24) that:
. (21) 1027
[$z]‘=
j! In2 .zf=O
dx;.
(25)
0. G. LARIONOV
Finally, the individual activity coefficients are obtained by differentiation of the excess Gibbs free energy:
and A. L. MYERS
(a) Nonideal bulkphase, ideal adsorbedphase, molecules of equal size. The selectivity is still a function of composition s
lny; = [$$]‘+&&[~]:
(26)
In $ = [$$I’
(27)
-x;&[$$]:.
The adsorbed-phase activity coefficients obey the differential and integral consistency tests devised for bulk-phase activity coefficients [2]. A comparison of Eq. (17) with the unrestricted form of the Gibbs-Duhem equation suggests a generalization of these thermodynamic equations. In general, the quantity A (l/n’ -xi/m1 -xi/m,) in Eq. (17) is the area change upon mixing along the locus of the monolayer. If Eq. (4) is not obeyed, the excess Gibbs free energy of the adsorbed phase varies with surface tension as well as composition. However, the area change upon mixing, like the volume change upon mixing for bulk liquids, is probably small and it seems reasonable to ignore this refinement.
=ylexp
Special
Substitution of Eq. (28) into (12) yields explicit expression for the surface excess:
Soal-x1 (Soal+ a,) (Soal + a2)
4 e=m
where So = exp
A (a20-Ucr,o) m,RT
1*
(30)
Substitution of Eq. (29) into (20) gives, after some algebraic manipulation, a simple equation for the surface tension in terms of bulk phase activities (a,, a,) : e-uAlm~RT
=
a,
e-d’Alm~RT
+
a2
e-uzoAlm~RT
.
(31)
(b) Ideal bulk phase, ideal adsorbed phase, molecules of equal size. Equation (29) reduces to:
-
(So- 1)
wlx2
with earlier work
Equation (12) is a general expression for the surface excess for nonideal bulk and adsorbed phases and molecules of different size. The following special cases, discussed previously [ 1,3,4,13], are added for completeness.
an
(29)
1
&++x2
cases
.
m,RT
e _
For the special case of liquid molecules of equal size (ml = m,), the activity coefficients given by Eq. (22) are the same as those defined by Everett[4]. For molecules of unequal size, the phase-exchange reaction model of Everett [4] yields unequal fugacities of the liquids in the adsorbed and bulk phases. We prefer to preserve the equality of fugacity of the liquids in the adsorbed and’bulk phases at equilibrium, according to Eq. (13), in order to apply the principles of solution thermodynamics. This work is an extension of the thermodynamics of nonideal adsorbed solutions to the case of molecules of unequal size ( m1 # m2).
1 (28)
A(~2°-alo)
Y2
n1 Comparison
but reduces to:
(32) .
COMPARISON OF THEORY WITH EXPERIMENT
Values for bulk-phase activity coefficients (rl, ‘y2) may be derived from vapor-liquid equilibrium data. In the simplest case for nonideal adsorbed solutions, (Age) ’ is symmetric about the equimolar composition and quadratic in form:
1-l Ag” ’
RT
= cx;x;.
(33)
For this case, Eq. (12) contains four constants: and C. Naturally, using ml, m,, AAulRT adjustable values for four constants, there is sufficient freedom to fit experimental data for a wide variety of adsorption isotherms. But if the adsorbed-phase activity coefficients are to have a physical meaning, it is necessary that the set of three constants (m,, m2 and AAuIRT) be evaluated independently.
1028
Thermodynamics
of adsorption
from nonideal solutions of nonelectrolytes
We followed this procedure using experimental data [8] for the adsorption of benzene + carbon tetrachloride, carbon tetrachloride + isooctane (2,2,4_trimethylpentane) and benzene + isooctane on aerosil at 20°C. The monolayer capacity for benzene (060 mmol/g) was determined from the adsorption isotherm for benzene vapor at 20°C by means of the BET equation [15]. The monolayer capacities for carbon tetrachloride and isooctane were calculated by the relation [ 121: --m-
ez.
ml
[ Pl
1
Eq. (12), irrespective adsorbed phase. That nor negative deviations the data for benzene +
CONCLUSIONS It
I.
z/3
(34)
Monolayer capacities aerosil at 20°C
on
mmol/g m
Substance, Benzene Carbon tetrachloride lsooctane
has been shown [7] that (35)
p is the molar density of the bulk liquid at 20°C. The monolayer values for the three liquids are listed in Table 1. Table
of the nonideality in the is to say, neither positive from Raoult’s law explain isooctane.
0600 0.568 0.399
The differences in the surface tensions for the pure liquids were obtained from the experimental data by means of Eq. (20) and are reported in Table 2. Any one of these differences can be calculated from the other two; a comparison of the calculated and experimental value is a test of the thermodynamic consistency of the data [12]. These data are thermodynamically consistent. For the systems benzene+ carbon tetrachloride and carbon tetrachloride + isooctane, it was found that the adsorbed phases are ideal solutions. For the system benzene + isooctane, the experimental data cannot be described by
According to the experimental data for the benzene-isooctane system on Fig. 1, the minimum value of ml predicted by Eq. (35) is O-8 mmol/g. Thus there is a discrepancy between the monolayer capacity obtained from the mixedliquid isotherm (0.8 mmol/g) and the monolayer capacity obtained from the pure vapor isotherm (0.6 mmol/g). This discrepancy between independent determinations of monolayer capacity obtained from the pure vapor isotherm (0.6 mmol/g). This discrepancy between independent determinations of monolayer capacity stresses the elusive character of the size of a monolayer; the monolayer model is an approximation to the actual multilayer structure of the interface. If the adsorbed-phase model does not provide a quantitative description of the experimental observations, it is nevertheless a reasonable and
04U
Table 2. Differences in surface tension of solid-liquid interface for pure liquids on aerosil at 20°C. For each pair, component number I is listed first
0
02
04
Mole fraction Solution
Benzene + isooctane Benzene + carbon tetrachloride Carbon tetrachloride + isooctane
A(
06
bulk solution,
06
0
xl
uZO- a,O), mmol/g
1.325 0.754 0.571
IFig. I. Adsorption from solution on aerosil at 20°C. 2 -benzene + carbon benzene + isooctane. tetrachloride. 3 -carbon tetrachloride + isooctane. For each liquid pair, component number I is listed first. Circles-experimental, lines - Eq. ( 12).
1029
0. G. LARIONOV
useful model. This conclusion is verified by Fig. 1, where the experimental data are compared with Eq. (12) for an ideal adsorbed solution (?I = $ = 1). The constants for Eq. (12) are given in Tables 1 and 2. The bulk-phase activity coefficients were obtained from vapor-liquid equilibrium measurements. Therefore the solid lines drawn on Fig. 1 were calculated solely on the basis of the properties of the pure adsorbates.
ana ’
’
A.
L. MYERS
SO a constant, Eq. (30) T
X
absolute temperature mole fraction
Greek symbols y activity coefficient p chemical potential p density of bulk liquid (T surface tension of solid-liquid interface Subscripts
NOTATION
refers to component 1 2 refers to component i refers to component
activity
A”surface
area per unit mass of adsorbent c a constant, Eq. (3 3) f fugacity 42” excess molar Gibbs free energy layer m number of moles in monomolecular per unit mass of adsorbent n number of moles per unit mass of adsorbent P” vapor pressure of bulk liquid R gas constant S selectivity, Eq. (1 1)
number 1 number 2 number i
Superscripts O refers
to pure adsorbate; refers to adsorbate solution before contacting adsorbent ’ refers to adsorbed phase e refers to surface excess quantity * refers to pure adsorbate at (v, T)
REFERENCES [l] BLACKBURN A., KIPLING J. J. andTESTER D. A., J. them. Sot. (Lend) 2373 (1957). [2] DENBIGH K., The Principles ofChemical Equilibrium, p. 284. Cambridge University Press, London 1966. [3] ELTON G. A. H.,J. them. Sot. (Lo&) 3813 (1954). 141 EVERETT D. H.. Trans. Faraday Sot. 1965 612478. [5] KIPLING J. J.,Adsorptionfrom Solutions of Non-Electrolytes. Academic Press, N.Y. 1965. 161 LANDAU L. and LIFSHITZ E.. Statistical Physics, p. 48. State Publishing House, Moscow, U.S.S.R. 1951. i7] LARIONOV 0. G., CHMUTOV K. V. and YUDILEVICH M. D.,Zh.&. Khim. 1967 41 1011. [8] LARIONOV 0. G., CHMUTOV K. V. and YUDILEVICH M. D.,Zh.fiz. Khim. 1967 412616. [9] PRAUSNITZ J. M., Molecular Thermodynamics of Fluid-Phase Equilibrium, p. 37. Prentice-Hall, Englewood Cliffs, N.J. 1969. [lo] SCHAY G., NAGY L. G. and SZEKRENYESY T.. Periodica Polytechnica, Budapest 1962 6 9 1. [1 1] SCHIESSLER R. W. and ROWE C. N.,J.Am. them. Sot. 1953 754611. 1121 SIRCARS.andMYERSA. L.,A.I.Ch.E.J. 1971 17 186. c13] SIRCAR S. and MYERS A. L., J.phys. Chem. 197074 2828. 1141 YELOVICH S. Yu. and LARIONOV. 0. G., Izu. Akad. Nauk SSR, Ord. Khim. Nauk 1962 209. i15j YOUNG D. M. and CROWELL A. D., PhysicalAdsorption of Gases, p. 150, p. 184. Butterworths, London 1962. Resume- La thermodynamique de solutions adsorb&es non id&ales est &endue au cas de melanges de molecules de tailles differentes. Les don&es experimentales pour l’adsorption de trois paires de liquides (benzene + iso-octane, tetrachlorure de carbone + iso-octane, benzene + tetrachlorure de carbone) sur un aerosol a 20°C sont comparees a la theorie. Deux solutions montrent un comportement ideal pour la phase adsorbee; les don&es pour la troisieme solution (benzene + iso-octane) ne peuvent s’expliquer par les conditions non ideales en phase adsorb&e. Zusammenfassung
- Die Thermodynamik nicht-idealer adsorbierter Losungen wird auf den Fall von ischungen von Moleklilen ungleicher Grosse ausgedehnt. Die Versuchsergebnisse fur die Adsorption van drei Fllssigkeitspaaren (Benzol+ Iso-oktan, Tetrachlorkohlenstoff + Iso-oktan, Benz01 + Tetrachlorkohlenstof an Aerosil bei 20°C werden mit der Theorie verglichen. Zwei der Losungen zeigen ideales Verhalten fur die adsorbierte Phase; die Ergebnisse fur die dritte Losung (Benz:]+ Iso-oktan) kiinnen nicht durch nichtideales Verhalten der adsorbierten Phase erklart werden.
1030
Chemical Engineering Science, 197 1, Vol. 26, pp. 1025-1030.
Thermodynamics
Printed in Great Britain.
of adsorption from nonideal solutions of nonelectrolytes
0. G. LARIONOV Institute of Physical Chemistry, U.S.S.R. Academy of Sciences, MOSCOW, U.S.S.R. and A. L. MYERS Department ChemicalEngineering, University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.
of
(Received 17 July 1970) Abstract-The thermodynamics of nonideal adsorbed solutions is extended to the case of mixtures of molecules of unequal size. Experimental data for the adsorption of three liquid pairs (benzene+ isooctane, carbon tettachloride + isooctane, benzene + carbon tetrachloride) on aerosil at 20°C are compared with the theory. Two solutions show ideal behavior for the adsorbed phase; data for the third solution (benzene + isooctane) cannot be explained by adsorbed-phase nonidealities.
SEVERAL theories[1,3,4,7,13,14] have been proposed to explain adsorption from solutions of nonelectrolytes. All of these theories share the assumption that there is an adsorbed phase. Actually the “adsorbed phase” has microscopic dimensions and therefore its properties, such as composition and entropy, are not real thermodynamic variables [6]. The advantage of the adsorbed-phase model of adsorption is that the multilayer structure of the liquid may be ignored. The multilayer character of the interface is explained by the decay, with distance, of the potential energy for interactions of the adsorbate molecules with the adsorbent. The lack of quantitative information on intermolecular potentials, to say nothing of the complexity of a multilayer model of a dense-fluid adsorbate, is the reason for pushing the adsorbedphase concept as far as it will go. The Gibbsian model to be described is for adsorbate molecules of different size and allows for arbitrary nonidealities in the adsorbed and bulk phases. THEORY
The surface excess of component defined by [5] :
number 1 is
4
e= n”(xlo--xl)
(1)
where no is the total number of moles of liquid of composition xIo before the solution is contacted with the adsorbent and x1 is the bulk solution composition measured after the establishment of equilibrium. The critical assumption of the adsorbed-phase model is that the solution may be separated by a dividing surface into an adsorbed phase containing n’ molecules and a bulk phase containing n molecules so that: ?P= n’+n.
(2)
A material balance on component 1 shows[5] that the surface excess may be written in terms of the composition of the adsorbed phase (xi) : nl e = n’(xi
-x1).
(3)
The location of the dividing surface is defined so that the number of moles in the adsorbed layer. For a phase (n’) is a monomolecular binary solution: 1 = xi ; 4 (4) n’
1025
m,
m,’
0.
G. LARIONOV
Equation (4) follows from the assumption that the area change upon mixing is zero. The monomolecular layer assumption, used by Elton[3], is not essential. For example, the adsorbed phase might correspond more closely to 14 molecular layers. But it is advantageous to trade the more restrictive nature of the monolayer assumption for a parameter (ml) which is reasonably welldefined by other measurements [ 151. The application of solution thermodynamics to the adsorbed phase requires the definition of a standard state for the pure adsorbates. The standard state is defined as the pure adsorbate at the same temperature and surface tension as that of the adsorbed solution. For a fixed temperature, let o? be the surface tension for the interface between the adsorbent and pure i liquid. The effect of surface tension upon the chemical potential of pure component i is given by the Gibbs adsorption isotherm [ 121: A do =-n;
dp.t.
(5)
In terms of the fugacity, Eq. (5) becomes: A do = -nlRT
d lnfi.
(6)
Integration of Eq. (6) under the constraint of monolayer surface coverage (n; = mi) yields the fugacity of pure i(E) at the state (o, T): fl =fi” exp
-A(a--c+$‘) miR T
1
and A. L. MYERS
state (ff ) : j-i’ =_fi*y;x;.
The fugacity in the bulk liquid is given by: fi = The
selectivity
ponent number by:
P:yiXi.
(10)
(S) of the adsorbent for com1 of a binary solution is defined
s=- &lx; x1/-%-
(11)
The adsorption selectivity was used by Schiessler and Rowe [I I]. who called it the adsorption separation factor. For preferential adsorption of component number 1, S > 1. Combination of Eqs. (3), (4) and (11) yields: 4
e
_
mlxlx2
0 - 1)
sx,+$x, .
(12)
Equation (12) is for nonideal adsorbed and bulk phases composed of molecules of different size (ml + mz). At equilibrium, fi’ =fi. Equations obtain:
(13)
(7)-( 11) and (13) may be combined to
(7)
fro, the fugacity of the pure adsorbate at (cr:, T) , is equal to the fugacity of the equilibrium bulk liquid: fi” = Pt.
(9)
(8)
Equation (8), which neglects vapor-phase imperfections, is adequate for the usual subatmospheric pressure of interest in adsorption. Equation (7), which accounts for the effect of surface tension upon the fugacity of the pure adsorbate, is analogous to the Poynting correction[9], which accounts for the effect of pressure upon the fugacity of a bulk liquid. The fugacity of component i in the adsorbed phase (4’) is defined in terms of the activity coefficient (-yi> and the fugacity in the standard
(14) Since the activity coefficients and surface tension are functions of composition, it follows from Eq. (14) that the selectivity is a function of composition. The bulk-phase activity coefficients (rl, y2) obey the Gibbs-Duhem equation[2] for isothermal data: xldlny,+x,dIny,=O.
(15) The equation corresponding to Eq. (15) for the adsorbed phase may be derived from the Gibbs adsorption isotherm [ 121:
1026
Thermodynamics
A do = -ni d lnfi -n; RT
Substitution
of adsorption from nonideal solutions of nonelectrolytes
d In&.
(16)
of Eqs. (7)-(9) into (16) gives:
1
=x:dlnyl+x6dln-y~. (17)
Substitution of Eq. (4) into (17) shows that the adsorbed-phase activity coefficients obey a Gibbs-Duhem type of equation, too: xldlny~+x~dlny;=O.
(18)
The desired function is nle(xl). According to Eq. (14) the selectivity depends upon the activity coefficients in the adsorbed phase (ri, 7;) which depend, in turn, upon the composition of the adsorbed phase (xi). The composition of the adsorbed phase is obtained from the selectivity by Eq. (I 1):
SXl
x;=sx,+
(19)
The surface tension of the solid-liquid interface is given in terms of the surface excess by an integration of the Gibbs adsorption isotherm [IO, 121: A((+-cr,O) RT
_ --
x’ I
Consider the differential equation (21) and the two algebraic equations (14) and (19), for which the variables are u, S, x1 and xi_ The activity coefficients are functions of composition (x,, xi). After elimination of S and xi using Eqs. (14) and (19), Eq. (21) may be integrated (numerically) for a(~~). If the adsorbed phase is ideal, then the composition of the adsorbed phase (xi) is not needed for the integration of Eq. (21). Having determined u(xl), the surface excess is given by Eqs. (12) (14) and (19). Calculation of activity coeficients mental data
In&=
A(u-ulO) m,RT
lnB+lnSYl
d(ylx,)
s
II
I x1=0
’
The selectivity is given by Eq. (12):
x1=1
RT
+A(u--2’) m,RT
(22)
or
A((+--crzO) =_
experi-
If the experimental function nle(xl) and the bulk-phase activity coefficients (rl, yz) are known, then the calculation of activity coefficients in the adsorbed phase is straightforward. The monomolecular layer values (m,, m,) are assumed to be measured independently [I 51. The surface tension at the bulk composition (x1) of interest is obtained from the experimental data using Eq. (20). The ratio of the activity coefficients in the adsorbed phase is obtained from Eq. ( 14):
r;
&
from
&d(y,x,).
-m,(S-
d(y&
=75[%+ h/m)(1-xdl
1)
x2 (XI+ x1 (x2-
(20)
The surface excess is a function of the properties m,, m2, A(czo- ulo)/RT and the activity coefficients for the adsorbed phase (r;, 76) and the bulk phase (rl, yz). According to Eqs. (12), (14) and (20), nie is a function of S, which is a function of o, which is a function of rile. u may be found independently by substituting nle from Eq. (12) into Eq. (20); after differentiation of Eq. (20): d(aA/RT)
=
helm) he/ml) .
(23)
Equations (19) and (22), taken together, provide the function In ($Jri) in terms of the adsorbedphase composition (xi). The excess Gibbs free energy of the adsorbed phase is defined by: Age ’ RT F-1
= xl In 7; + xi In 7;.
(24)
It follows from Eqs. (18) and (24) that:
. (21) 1027
[$z]‘=
j! In2 .zf=O
dx;.
(25)
0. G. LARIONOV
Finally, the individual activity coefficients are obtained by differentiation of the excess Gibbs free energy:
and A. L. MYERS
(a) Nonideal bulkphase, ideal adsorbedphase, molecules of equal size. The selectivity is still a function of composition s
lny; = [$$]‘+&&[~]:
(26)
In $ = [$$I’
(27)
-x;&[$$]:.
The adsorbed-phase activity coefficients obey the differential and integral consistency tests devised for bulk-phase activity coefficients [2]. A comparison of Eq. (17) with the unrestricted form of the Gibbs-Duhem equation suggests a generalization of these thermodynamic equations. In general, the quantity A (l/n’ -xi/m1 -xi/m,) in Eq. (17) is the area change upon mixing along the locus of the monolayer. If Eq. (4) is not obeyed, the excess Gibbs free energy of the adsorbed phase varies with surface tension as well as composition. However, the area change upon mixing, like the volume change upon mixing for bulk liquids, is probably small and it seems reasonable to ignore this refinement.
=ylexp
Special
Substitution of Eq. (28) into (12) yields explicit expression for the surface excess:
Soal-x1 (Soal+ a,) (Soal + a2)
4 e=m
where So = exp
A (a20-Ucr,o) m,RT
1*
(30)
Substitution of Eq. (29) into (20) gives, after some algebraic manipulation, a simple equation for the surface tension in terms of bulk phase activities (a,, a,) : e-uAlm~RT
=
a,
e-d’Alm~RT
+
a2
e-uzoAlm~RT
.
(31)
(b) Ideal bulk phase, ideal adsorbed phase, molecules of equal size. Equation (29) reduces to:
-
(So- 1)
wlx2
with earlier work
Equation (12) is a general expression for the surface excess for nonideal bulk and adsorbed phases and molecules of different size. The following special cases, discussed previously [ 1,3,4,13], are added for completeness.
an
(29)
1
&++x2
cases
.
m,RT
e _
For the special case of liquid molecules of equal size (ml = m,), the activity coefficients given by Eq. (22) are the same as those defined by Everett[4]. For molecules of unequal size, the phase-exchange reaction model of Everett [4] yields unequal fugacities of the liquids in the adsorbed and bulk phases. We prefer to preserve the equality of fugacity of the liquids in the adsorbed and’bulk phases at equilibrium, according to Eq. (13), in order to apply the principles of solution thermodynamics. This work is an extension of the thermodynamics of nonideal adsorbed solutions to the case of molecules of unequal size ( m1 # m2).
1 (28)
A(~2°-alo)
Y2
n1 Comparison
but reduces to:
(32) .
COMPARISON OF THEORY WITH EXPERIMENT
Values for bulk-phase activity coefficients (rl, ‘y2) may be derived from vapor-liquid equilibrium data. In the simplest case for nonideal adsorbed solutions, (Age) ’ is symmetric about the equimolar composition and quadratic in form:
1-l Ag” ’
RT
= cx;x;.
(33)
For this case, Eq. (12) contains four constants: and C. Naturally, using ml, m,, AAulRT adjustable values for four constants, there is sufficient freedom to fit experimental data for a wide variety of adsorption isotherms. But if the adsorbed-phase activity coefficients are to have a physical meaning, it is necessary that the set of three constants (m,, m2 and AAuIRT) be evaluated independently.
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Thermodynamics
of adsorption
from nonideal solutions of nonelectrolytes
We followed this procedure using experimental data [8] for the adsorption of benzene + carbon tetrachloride, carbon tetrachloride + isooctane (2,2,4_trimethylpentane) and benzene + isooctane on aerosil at 20°C. The monolayer capacity for benzene (060 mmol/g) was determined from the adsorption isotherm for benzene vapor at 20°C by means of the BET equation [15]. The monolayer capacities for carbon tetrachloride and isooctane were calculated by the relation [ 121: --m-
ez.
ml
[ Pl
1
Eq. (12), irrespective adsorbed phase. That nor negative deviations the data for benzene +
CONCLUSIONS It
I.
z/3
(34)
Monolayer capacities aerosil at 20°C
on
mmol/g m
Substance, Benzene Carbon tetrachloride lsooctane
has been shown [7] that (35)
p is the molar density of the bulk liquid at 20°C. The monolayer values for the three liquids are listed in Table 1. Table
of the nonideality in the is to say, neither positive from Raoult’s law explain isooctane.
0600 0.568 0.399
The differences in the surface tensions for the pure liquids were obtained from the experimental data by means of Eq. (20) and are reported in Table 2. Any one of these differences can be calculated from the other two; a comparison of the calculated and experimental value is a test of the thermodynamic consistency of the data [12]. These data are thermodynamically consistent. For the systems benzene+ carbon tetrachloride and carbon tetrachloride + isooctane, it was found that the adsorbed phases are ideal solutions. For the system benzene + isooctane, the experimental data cannot be described by
According to the experimental data for the benzene-isooctane system on Fig. 1, the minimum value of ml predicted by Eq. (35) is O-8 mmol/g. Thus there is a discrepancy between the monolayer capacity obtained from the mixedliquid isotherm (0.8 mmol/g) and the monolayer capacity obtained from the pure vapor isotherm (0.6 mmol/g). This discrepancy between independent determinations of monolayer capacity obtained from the pure vapor isotherm (0.6 mmol/g). This discrepancy between independent determinations of monolayer capacity stresses the elusive character of the size of a monolayer; the monolayer model is an approximation to the actual multilayer structure of the interface. If the adsorbed-phase model does not provide a quantitative description of the experimental observations, it is nevertheless a reasonable and
04U
Table 2. Differences in surface tension of solid-liquid interface for pure liquids on aerosil at 20°C. For each pair, component number I is listed first
0
02
04
Mole fraction Solution
Benzene + isooctane Benzene + carbon tetrachloride Carbon tetrachloride + isooctane
A(
06
bulk solution,
06
0
xl
uZO- a,O), mmol/g
1.325 0.754 0.571
IFig. I. Adsorption from solution on aerosil at 20°C. 2 -benzene + carbon benzene + isooctane. tetrachloride. 3 -carbon tetrachloride + isooctane. For each liquid pair, component number I is listed first. Circles-experimental, lines - Eq. ( 12).
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0. G. LARIONOV
useful model. This conclusion is verified by Fig. 1, where the experimental data are compared with Eq. (12) for an ideal adsorbed solution (?I = $ = 1). The constants for Eq. (12) are given in Tables 1 and 2. The bulk-phase activity coefficients were obtained from vapor-liquid equilibrium measurements. Therefore the solid lines drawn on Fig. 1 were calculated solely on the basis of the properties of the pure adsorbates.
ana ’
’
A.
L. MYERS
SO a constant, Eq. (30) T
X
absolute temperature mole fraction
Greek symbols y activity coefficient p chemical potential p density of bulk liquid (T surface tension of solid-liquid interface Subscripts
NOTATION
refers to component 1 2 refers to component i refers to component
activity
A”surface
area per unit mass of adsorbent c a constant, Eq. (3 3) f fugacity 42” excess molar Gibbs free energy layer m number of moles in monomolecular per unit mass of adsorbent n number of moles per unit mass of adsorbent P” vapor pressure of bulk liquid R gas constant S selectivity, Eq. (1 1)
number 1 number 2 number i
Superscripts O refers
to pure adsorbate; refers to adsorbate solution before contacting adsorbent ’ refers to adsorbed phase e refers to surface excess quantity * refers to pure adsorbate at (v, T)
REFERENCES [l] BLACKBURN A., KIPLING J. J. andTESTER D. A., J. them. Sot. (Lend) 2373 (1957). [2] DENBIGH K., The Principles ofChemical Equilibrium, p. 284. Cambridge University Press, London 1966. [3] ELTON G. A. H.,J. them. Sot. (Lo&) 3813 (1954). 141 EVERETT D. H.. Trans. Faraday Sot. 1965 612478. [5] KIPLING J. J.,Adsorptionfrom Solutions of Non-Electrolytes. Academic Press, N.Y. 1965. 161 LANDAU L. and LIFSHITZ E.. Statistical Physics, p. 48. State Publishing House, Moscow, U.S.S.R. 1951. i7] LARIONOV 0. G., CHMUTOV K. V. and YUDILEVICH M. D.,Zh.&. Khim. 1967 41 1011. [8] LARIONOV 0. G., CHMUTOV K. V. and YUDILEVICH M. D.,Zh.fiz. Khim. 1967 412616. [9] PRAUSNITZ J. M., Molecular Thermodynamics of Fluid-Phase Equilibrium, p. 37. Prentice-Hall, Englewood Cliffs, N.J. 1969. [lo] SCHAY G., NAGY L. G. and SZEKRENYESY T.. Periodica Polytechnica, Budapest 1962 6 9 1. [1 1] SCHIESSLER R. W. and ROWE C. N.,J.Am. them. Sot. 1953 754611. 1121 SIRCARS.andMYERSA. L.,A.I.Ch.E.J. 1971 17 186. c13] SIRCAR S. and MYERS A. L., J.phys. Chem. 197074 2828. 1141 YELOVICH S. Yu. and LARIONOV. 0. G., Izu. Akad. Nauk SSR, Ord. Khim. Nauk 1962 209. i15j YOUNG D. M. and CROWELL A. D., PhysicalAdsorption of Gases, p. 150, p. 184. Butterworths, London 1962. Resume- La thermodynamique de solutions adsorb&es non id&ales est &endue au cas de melanges de molecules de tailles differentes. Les don&es experimentales pour l’adsorption de trois paires de liquides (benzene + iso-octane, tetrachlorure de carbone + iso-octane, benzene + tetrachlorure de carbone) sur un aerosol a 20°C sont comparees a la theorie. Deux solutions montrent un comportement ideal pour la phase adsorbee; les don&es pour la troisieme solution (benzene + iso-octane) ne peuvent s’expliquer par les conditions non ideales en phase adsorb&e. Zusammenfassung
- Die Thermodynamik nicht-idealer adsorbierter Losungen wird auf den Fall von ischungen von Moleklilen ungleicher Grosse ausgedehnt. Die Versuchsergebnisse fur die Adsorption van drei Fllssigkeitspaaren (Benzol+ Iso-oktan, Tetrachlorkohlenstoff + Iso-oktan, Benz01 + Tetrachlorkohlenstof an Aerosil bei 20°C werden mit der Theorie verglichen. Zwei der Losungen zeigen ideales Verhalten fur die adsorbierte Phase; die Ergebnisse fur die dritte Losung (Benz:]+ Iso-oktan) kiinnen nicht durch nichtideales Verhalten der adsorbierten Phase erklart werden.
1030