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Adsorption from binary liquid mixtures of unequal adsorbate sizes on heterogeneous asorbents
489
Surface Science 148 (1984) 489-498 North-Holland, Amsterdam
ADSORPTION ADSO~A~
FROM BINARY LIQUID SIZES ON ~~R~ENEOUS
MIXTURES OF UNEQUAL ASORBENTS
Shivaji SIRCAR Air Products and Chemicals, Inc., Allentown, Penns_ylvonia iSiO5,
Received
8 February
1984; accepted
for publication
14 August
USA
1984
A new analytic surface excess isotherm is derived for adsorption of a binary liquid mixture of dissimilar adsorbate sizes on a heterogeneous adsorbent. The model assumes that (a) the monolayer adsorption model with an ideal adsorbed phase describes the local adsorption on the homogeneous sites of the adsorbent and (b) the adsorbent heterogeneity can be described by a uniform distribution of the site adsorbate selectivity. The theory is tested using data for adsorption of benzene + cyclohexane mixtures on silica gel. A parametric study of the effects of the adsorbate sizes and the degree of surface heterogeneity on the shapes of the surface excess isotherm is carried out to demonstrate the complexity of interactions between these variables.
1. Introduction An analytic equation to describe the surface excess isotherm for adsorption of a binary liquid mixture on a heterogeneous adsorbent was recently derived by assuming that (a) the surface heterogeneity can be described by a uniform distribution of the site adsorbate selectivity, (b) the adsorbed phase is ideal, and (c) the adsorbates have the same molecular sizes [l]. The model was tested using experimental data for adsorption of various binary liquid hydrocarbon mixtures on silica gel and graphon. It was shown that the model was very flexible to describe various types of “U” and “S” shaped surface excess isotherms despite the rather simplified assumption regarding the adsorbent heterogeneity. The objective of the present work is to extend the above model to account for unequal adsorbate sizes. This can be very useful to design engineers since most systems of practical interest have dissimilar adsorbate sizes.
2. Theory of adsorption on homogeneous
adsorbent
We first need an analytic surface excess isotherm for adsorption mixture with unequal adsorbate sizes on a homogeneous adsorbent 0039-602S/g4/$03.~ Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
of a liquid which can
490
be used to describe the local adsorption on the sites of a heterogeneous adsorbent. The monolayer adsorption model of Larionov and Myers [2] was chosen for this purpose. According to this model, the adsorbed phase mole fraction (xi) of component i of a binary liquid mixture may be written as: x;=o,exp($y),
i=l,2,
a, = xi-r;,
(2)
xx: = 1,
(3)
where Q;, xi, and y, are, respectively, the activity, the mole fraction and the activity coefficient of component i in the bulk liquid phase. (Au) and (Au,‘) are, respectively, the Gibbs free energy of adsorption for a liquid mixture of composition xi and that for the pure liquid i. m, is the monolayer or saturation adsorption (pore filling) capacity of pure liquid i. A is the surface area of the adsorbent per unit mass. R is the gas constant. T is the system temperature. Eq. (1) assumes that the adsorbed phase is ideal. The individual amounts adsorbed (n:) of component i are related by: C(O?)
= I,
(4)
xi = $/C
nj,
(5)
i is defined
and the surface excess (N,“), of component
by [3].
(6) Larionov and Myers defined the selectivity of adsorption (S) for component 1 as S = x;xZ/x;x, and obtained an expression for (N;)/ by combining eqs. U-16). We define the selectivity
of adsorption
for component
S = x;a&X;ai. Eqs. (l)-(7) (Y=),=
1 as (7)
can then be combined
M,(S%%
- %x1)/(%
to obtain + Pa*).
(8)
and ’
(9)
where B = ml/m2 is the ratio of the monolayer or saturation capacities of the two components of the liquid mixture. Eq. (8) represents the surface excess isotherm on a homogeneous adsorbent for adsorbates of nonequal sizes (m, + m2). It reduces to the expression for (Nf), used earlier [l] for adsorbates of equal size (m, = m, = m, /? = 1) which can also be derived by the statistical
S. Sircar / Adsorption from binary liquid mixtures
thermodynamics
[4] of adsorption.
The selectivity
491
in that case is given by
Eq. (9) shows that S is a function of the liquid composition when /3 # 1. However, the advantage of defining S by eq. (7) is that an explicit expression for S as a function of a, can be obtained by the application of the adsorption thermodynamics. The isothermal Gibbs adsorption equation may be written in the differential form as [3].
(11)
1, and the Gibbs-Duhem xx,
equation
for the bulk liquid phase may be written
d In a, = 0.
Eqs. (9) (ll),
and (12) may be combined
s/s, = (Sa, + “2)(p-‘)‘8,
as [5] (12)
to obtain (13)
where
(14) Eq. (13) gives S as a function of a,. S, is the selectivity of adsorption at the limit of a, + 0 (a + (~20). An expression similar to eq. (13) was previously derived for an ideal bulk liquid mixture [6]. Eqs. (8) and (13) show that the parameters of the surface excess isotherm are m,, @, and S, [or (A/RT)(uf - a:)]. The parameter (A/RT)(u,O - up) can be estimated from the surface excess data by integration of the Gibbs adsorption equation [3]: (15) Thus, for a homogeneous adsorbent, there are only two adjustable parameters in eq. (8), m, and /3, if complete surface excess data are available to carry out the integration of eq. (15). Furthermore, the Gurvitch rule [7] approximately relates p with the liquid molar volumes (up) of the pure adsorbates i as p = v;/u;.
(16)
The rule is valid for the pore filling mechanism of adsorption and it is generally applicable for adsorption on porous adsorbents. This reduces the number of adjustable parameters in eq. (8) to one.
492
S. Sircar / Adsorption from binary liquid mixtures
of adsorption on heterogeneous
3. Theory
adsorbent
The heterogeneous adsorbent can be considered to be a distribution of homogeneous sites and each site can be characterized by a value of the parameter S,. Then the overall surface excess isotherm (Np) on the heterogeneous adsorbent can be obtained by [l] (17) where A(&) is the probability density function for the parameter S,,. The integration in eq. (17) is carried out between the lower (SOL) and the higher (S,,) limits of the parameter S,,. We assume that A(&) is a uniform distribution function [8]. h(S*)=C,
S($l.SS,S&.
where C is a constant.
%‘i(So) d&
/ %L
The normalization
118) requirement
= 1,
(19)
gives the value of C as C = 1/(%,
(20)
- SJ,).
Eqs. (8), (18). and (20) can be substituted (13) to obtain
into eq. (17) and integrated
using eq.
where S,/S,,
= (&a,
f .JB-Y
(22)
S,/S,,
= (&_a, + LzJp-l?
(23)
Eq. (21) is the analytic surface excess isotherm for adsorption of a binary liquid mixture having unequal adsorbate sizes on a heterogeneous adsorbent with uniform energy dist~bution. The parameters of eq. (21) are m,, p, S,,, constraint described below. and SoLI which are related by the thermodynamic The integral form of the Gibbs adsorption equation can also be applied to the overall surface excess isotherm to obtain: (24)
S. Sirear / Adsorptionfrombinaryliquidmixtures
493
where (Au,“)* is the Gibbs free energy of adsorption of pure adsorbate i on the heterogeneous adsorbent. Eqs. (12) and (21)-(24) can be solved simultaneously by using L’Hospital’s rule to get:
Eq. (25) shows that the parameters m,. S,,, , and S,,, are related to ( A/RT)( CT; - a:)* which can be estimated from the experimental Nf data by using eq. (24). This constraint and the Gurvitch rule reduce the number of adjustable parameters in eq. (21) to two.
4. Properties of the new surface excess isotherm It can be shown from eqs. (21)-(23) N;-0,
when
a,-+0
that:
(i=1,2),
(26)
which is the correct limit for the surface excess isotherm. Deduction of eq. (26) from eq. (21) is trivial when a, + 0. Application of L’Hospital’s rule is required to obtain the limiting value of Nf when al 4 0. The slope of the surface excess isotherm at the limit of u1 --+ 1 (x, -+ 1) can be shown to be:
($!j_,=-J--l+ soHy!rso, &(W - w7]?
(27)
where yp” is the activity coefficient of component i at the infinite dilution of that component. Eq. (27) provides an analytic relationship between the limiting slope of the isotherm and the parameters of the model (m,, SoH, and S,,). Thus, if the limiting slope of the isotherm can be measured accurately, which is the case for most “U” shaped isotherms, eqs. (25) and (27) can be used to obtain S,,, and S,, for a given ml. The limiting form of eq. (21) as /3 + 1 (S, + So”, S, + S,,) can be shown to be:
(28) which is identical to the expression for NF derived earlier for equal sized adsorbates [l]. Furthermore, eq. (21) reduces to eq. (8) at the limit of S,, = S,, which represents the limiting case of the homogeneous adsorbent. The distribution function X(S,) in this case reduces to the Dirac delta function.
5. Analysis of experimental data The surface excess data for adsorption of benzene (1) + cyclohexane (2) mixture on silica gel at 30°C [3] was analyzed by the proposed modet. Fig. 1 shows the experimental data (circles). The isotherm was “U” shaped and benzene was selectively adsorbed at all compositions. The saturation adsorption capacities for both adsorbates were available from independent vapor adsorption isotherms {61. They were respectively 3.91 and 3.06 mmol/g for benzene and cyclohexane. This gave a p value of 1.278 for the system which compares well with the p value of 1.217 obtained by the Gurvitch rule. The liquid mixture was nonideal [3] and the bulk phase activity coefficients at 30°C could be described fairly well by the regular solution model using y, = exp[0.472(1 -xi)‘]. The value of (A/RT)(a,O - ep)* for this system calculated by using eq. (24) and the above functionality for y, was 6.8 mmolj’g. Eqs. (25) and (27) were then used to calculate S,, (= 0.352) and S,,, (= 14.09) using the above-mentioned values of WZ, and /3. These values of S,, indicate that the silica gel surface consisted of sites which were selective to both benzene (S, > 1) and cyclohexane (So < 1). Eqs. (22) and (23) were then used to obtain S, and S, as functions of cti and the complete surface excess isotherm was generated using eq. (21). Fig. 1 compares the calculated isotherm with the experimental data. tt can be seen that the fit is very good at the low and the high concentration levels of
TEROGENEGUS / m. = 3.91
HOMOGENEOUS
0
0.2
0.6
0.4 Xl
0.8
1.0
-
Fig. 1. Adsorption of benzene-eyclohexane
mixtures on silica gel at 3WC.
495
S. Sircar / Adsorpiion from binaty liquid mixtures
benzene but the model slightly overpredicts the surface excess in the 0.1 I xi I 0.3 region. The figure also shows the calculated isotherm using the homogeneous equations (8) and (13) with S, (= 5.69) obtained by eq. (14). The homogeneous model underpredicts the experimental isotherm in the low xi region and overpredicts the isotherm in the high x, region. This demonstrates that (a) the monolayer adsorption model with an ideal adsorbed phase provides a satisfactory description of the adsorption behavior for this system when the surface heterogeneity is taken into account and (b) the proposed simplified model for describing surface heterogeneity is adequate. Earlier, House had analyzed the role of the surface heterogeneity for this system by numerical inversion of the Fredholm integral equation of the first kind like eq. (17) written in terms of an adsorption energy parameter [9]. The resulting distribution energy function was Gaussian-like with one or two peaks depending on the method of calculation used.
6. Parametric study The mean (cl) and the variance P = (%I a* = (S,,
(a*) for the uniform
distribution
are [8]: (29)
+ &)/2, - S&*/12.
(30)
It follows from eqs. (29) and (30) that &I = I*(1 + *k),
&I_ = CL0 - *I,
(31)
where 9 = &r/p.
(32)
The parameter q can be considered to be a measure of the surface heterogeneity. V’ + 0 (u + 0) corresponds to the homogeneous surface, where the uniform distribution function reduces to the Dirac delta function. \k + 1 represents the maximum surface heterogeneity by this model because S,,, + 0 at this limit, which is the minimum possible value for Se,_. The adsorbent heterogeneity increases as !P increases. We evaluated the effects of p and 9 on the surface excess isotherm for given values of p by using eqs. (21)-(23). The bulk liquid is assumed to be ideal. Figs. 2-4 show the results respectively for /3 = 0.5, 1.0, 2.0, and 1_1=1.5. Fig. 5 shows the isotherms for p = 0.5 and I_L = 5.0. The quantity Nf/m, is plotted in the ordinates of these figures. It may be seen from these figures that: (a) Nf decreases for any given x, and the maximum of N; occurs at a lower value of xi as the surface heterogeneity increases. The effect can be striking when p is low, irrespective of the value of j3. The isotherm can even change
496
S. Sirear / Adsorption from binary liquid mixtures
from “U” shape to “S” shape when the adsorbent is substantially heterogeneous. The effect of the surface heterogeneity on the isotherm shape is, however, much less pronounced when p is large. This was found to be true for a11values of /3. (b) The value of x, at which Nf changes sign in case of an “s” shaped isotherm increases as j3 increases for a given p and 9.
1.0
Fig. 2. Effect of surface heterogeneity on the surface excess isotherm. B = 0.5; p = 1.5.
Fig. 3. Effect of surface heterogeneity on the surface excess isotherm, /3 = 1.0; p = 1.5.
S. Sircar / Aakorption from binary liquid mixtures
497
The isotherm shape at the limit of x, + 0 is a strong function of j3 but it is practically independent of the degree of surface heterogeneity. The isotherm shape at the limit of x, + 1, on the other hand, is strongly dependent on both /3 and ‘Pk. (c)
0.10
.O
Fig. 4. Effect of surface
0
0.2
heterogeneity
0.4
on the surface
0.6
0.6
excess isotherm,
/3 = 2.0; p = 1.5.
1.0
Xl + Fig. 5. Effect of surface
heterogeneity
on the surface
excess isotherm,
B = 0.5; p = 5.0.
598
The above parametric study shows that both the size of the adsorbates and the adsorbent heterogeneity play major roles in determining the shape of the overall surface excess isotherm and the interactions between these variables are rather complex. Consequently, all models on adsorption of liquid mixtures must include the effects of these variables for meaningful interpretation of the data.
References [I] [2] [3] [4] {S] [6] [7] [8] [9]
S. Sircar, Surface Sci. 148 (1985) 478. O.G. Larionov and A.L. Myers, Chem. Eng. Sci. 26 (1971) 102.5. S. Sircar and A.L. Myers, AIChE J. 17 (1971) 186. S. Sircar and A.L. Myers, J. Phys. Chem. 74 (1970) 2828. K. Denbigh, The Principtes of Chemical ~quilib~um (Cambridge University Press. London, 1966). S. Sircar and A.L. Myers, AIChE J. 19 (1973) 159. L. Gurvitch, J. Phys. Chem. Sot. Russ. 47 (1915) 805. M. Abramowitz and LA. Stegan, Handbook of Mathematical Functions (Appl. Math. Ser. 55) (Nat]. Bur. Std., US Government Printing Office, Washington, DC, 1972). W.A. House, Chem. Phys. Letters 60 (1978) 169.
Surface Science 148 (1984) 489-498 North-Holland, Amsterdam
ADSORPTION ADSO~A~
FROM BINARY LIQUID SIZES ON ~~R~ENEOUS
MIXTURES OF UNEQUAL ASORBENTS
Shivaji SIRCAR Air Products and Chemicals, Inc., Allentown, Penns_ylvonia iSiO5,
Received
8 February
1984; accepted
for publication
14 August
USA
1984
A new analytic surface excess isotherm is derived for adsorption of a binary liquid mixture of dissimilar adsorbate sizes on a heterogeneous adsorbent. The model assumes that (a) the monolayer adsorption model with an ideal adsorbed phase describes the local adsorption on the homogeneous sites of the adsorbent and (b) the adsorbent heterogeneity can be described by a uniform distribution of the site adsorbate selectivity. The theory is tested using data for adsorption of benzene + cyclohexane mixtures on silica gel. A parametric study of the effects of the adsorbate sizes and the degree of surface heterogeneity on the shapes of the surface excess isotherm is carried out to demonstrate the complexity of interactions between these variables.
1. Introduction An analytic equation to describe the surface excess isotherm for adsorption of a binary liquid mixture on a heterogeneous adsorbent was recently derived by assuming that (a) the surface heterogeneity can be described by a uniform distribution of the site adsorbate selectivity, (b) the adsorbed phase is ideal, and (c) the adsorbates have the same molecular sizes [l]. The model was tested using experimental data for adsorption of various binary liquid hydrocarbon mixtures on silica gel and graphon. It was shown that the model was very flexible to describe various types of “U” and “S” shaped surface excess isotherms despite the rather simplified assumption regarding the adsorbent heterogeneity. The objective of the present work is to extend the above model to account for unequal adsorbate sizes. This can be very useful to design engineers since most systems of practical interest have dissimilar adsorbate sizes.
2. Theory of adsorption on homogeneous
adsorbent
We first need an analytic surface excess isotherm for adsorption mixture with unequal adsorbate sizes on a homogeneous adsorbent 0039-602S/g4/$03.~ Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
of a liquid which can
490
be used to describe the local adsorption on the sites of a heterogeneous adsorbent. The monolayer adsorption model of Larionov and Myers [2] was chosen for this purpose. According to this model, the adsorbed phase mole fraction (xi) of component i of a binary liquid mixture may be written as: x;=o,exp($y),
i=l,2,
a, = xi-r;,
(2)
xx: = 1,
(3)
where Q;, xi, and y, are, respectively, the activity, the mole fraction and the activity coefficient of component i in the bulk liquid phase. (Au) and (Au,‘) are, respectively, the Gibbs free energy of adsorption for a liquid mixture of composition xi and that for the pure liquid i. m, is the monolayer or saturation adsorption (pore filling) capacity of pure liquid i. A is the surface area of the adsorbent per unit mass. R is the gas constant. T is the system temperature. Eq. (1) assumes that the adsorbed phase is ideal. The individual amounts adsorbed (n:) of component i are related by: C(O?)
= I,
(4)
xi = $/C
nj,
(5)
i is defined
and the surface excess (N,“), of component
by [3].
(6) Larionov and Myers defined the selectivity of adsorption (S) for component 1 as S = x;xZ/x;x, and obtained an expression for (N;)/ by combining eqs. U-16). We define the selectivity
of adsorption
for component
S = x;a&X;ai. Eqs. (l)-(7) (Y=),=
1 as (7)
can then be combined
M,(S%%
- %x1)/(%
to obtain + Pa*).
(8)
and ’
(9)
where B = ml/m2 is the ratio of the monolayer or saturation capacities of the two components of the liquid mixture. Eq. (8) represents the surface excess isotherm on a homogeneous adsorbent for adsorbates of nonequal sizes (m, + m2). It reduces to the expression for (Nf), used earlier [l] for adsorbates of equal size (m, = m, = m, /? = 1) which can also be derived by the statistical
S. Sircar / Adsorption from binary liquid mixtures
thermodynamics
[4] of adsorption.
The selectivity
491
in that case is given by
Eq. (9) shows that S is a function of the liquid composition when /3 # 1. However, the advantage of defining S by eq. (7) is that an explicit expression for S as a function of a, can be obtained by the application of the adsorption thermodynamics. The isothermal Gibbs adsorption equation may be written in the differential form as [3].
(11)
1, and the Gibbs-Duhem xx,
equation
for the bulk liquid phase may be written
d In a, = 0.
Eqs. (9) (ll),
and (12) may be combined
s/s, = (Sa, + “2)(p-‘)‘8,
as [5] (12)
to obtain (13)
where
(14) Eq. (13) gives S as a function of a,. S, is the selectivity of adsorption at the limit of a, + 0 (a + (~20). An expression similar to eq. (13) was previously derived for an ideal bulk liquid mixture [6]. Eqs. (8) and (13) show that the parameters of the surface excess isotherm are m,, @, and S, [or (A/RT)(uf - a:)]. The parameter (A/RT)(u,O - up) can be estimated from the surface excess data by integration of the Gibbs adsorption equation [3]: (15) Thus, for a homogeneous adsorbent, there are only two adjustable parameters in eq. (8), m, and /3, if complete surface excess data are available to carry out the integration of eq. (15). Furthermore, the Gurvitch rule [7] approximately relates p with the liquid molar volumes (up) of the pure adsorbates i as p = v;/u;.
(16)
The rule is valid for the pore filling mechanism of adsorption and it is generally applicable for adsorption on porous adsorbents. This reduces the number of adjustable parameters in eq. (8) to one.
492
S. Sircar / Adsorption from binary liquid mixtures
of adsorption on heterogeneous
3. Theory
adsorbent
The heterogeneous adsorbent can be considered to be a distribution of homogeneous sites and each site can be characterized by a value of the parameter S,. Then the overall surface excess isotherm (Np) on the heterogeneous adsorbent can be obtained by [l] (17) where A(&) is the probability density function for the parameter S,,. The integration in eq. (17) is carried out between the lower (SOL) and the higher (S,,) limits of the parameter S,,. We assume that A(&) is a uniform distribution function [8]. h(S*)=C,
S($l.SS,S&.
where C is a constant.
%‘i(So) d&
/ %L
The normalization
118) requirement
= 1,
(19)
gives the value of C as C = 1/(%,
(20)
- SJ,).
Eqs. (8), (18). and (20) can be substituted (13) to obtain
into eq. (17) and integrated
using eq.
where S,/S,,
= (&a,
f .JB-Y
(22)
S,/S,,
= (&_a, + LzJp-l?
(23)
Eq. (21) is the analytic surface excess isotherm for adsorption of a binary liquid mixture having unequal adsorbate sizes on a heterogeneous adsorbent with uniform energy dist~bution. The parameters of eq. (21) are m,, p, S,,, constraint described below. and SoLI which are related by the thermodynamic The integral form of the Gibbs adsorption equation can also be applied to the overall surface excess isotherm to obtain: (24)
S. Sirear / Adsorptionfrombinaryliquidmixtures
493
where (Au,“)* is the Gibbs free energy of adsorption of pure adsorbate i on the heterogeneous adsorbent. Eqs. (12) and (21)-(24) can be solved simultaneously by using L’Hospital’s rule to get:
Eq. (25) shows that the parameters m,. S,,, , and S,,, are related to ( A/RT)( CT; - a:)* which can be estimated from the experimental Nf data by using eq. (24). This constraint and the Gurvitch rule reduce the number of adjustable parameters in eq. (21) to two.
4. Properties of the new surface excess isotherm It can be shown from eqs. (21)-(23) N;-0,
when
a,-+0
that:
(i=1,2),
(26)
which is the correct limit for the surface excess isotherm. Deduction of eq. (26) from eq. (21) is trivial when a, + 0. Application of L’Hospital’s rule is required to obtain the limiting value of Nf when al 4 0. The slope of the surface excess isotherm at the limit of u1 --+ 1 (x, -+ 1) can be shown to be:
($!j_,=-J--l+ soHy!rso, &(W - w7]?
(27)
where yp” is the activity coefficient of component i at the infinite dilution of that component. Eq. (27) provides an analytic relationship between the limiting slope of the isotherm and the parameters of the model (m,, SoH, and S,,). Thus, if the limiting slope of the isotherm can be measured accurately, which is the case for most “U” shaped isotherms, eqs. (25) and (27) can be used to obtain S,,, and S,, for a given ml. The limiting form of eq. (21) as /3 + 1 (S, + So”, S, + S,,) can be shown to be:
(28) which is identical to the expression for NF derived earlier for equal sized adsorbates [l]. Furthermore, eq. (21) reduces to eq. (8) at the limit of S,, = S,, which represents the limiting case of the homogeneous adsorbent. The distribution function X(S,) in this case reduces to the Dirac delta function.
5. Analysis of experimental data The surface excess data for adsorption of benzene (1) + cyclohexane (2) mixture on silica gel at 30°C [3] was analyzed by the proposed modet. Fig. 1 shows the experimental data (circles). The isotherm was “U” shaped and benzene was selectively adsorbed at all compositions. The saturation adsorption capacities for both adsorbates were available from independent vapor adsorption isotherms {61. They were respectively 3.91 and 3.06 mmol/g for benzene and cyclohexane. This gave a p value of 1.278 for the system which compares well with the p value of 1.217 obtained by the Gurvitch rule. The liquid mixture was nonideal [3] and the bulk phase activity coefficients at 30°C could be described fairly well by the regular solution model using y, = exp[0.472(1 -xi)‘]. The value of (A/RT)(a,O - ep)* for this system calculated by using eq. (24) and the above functionality for y, was 6.8 mmolj’g. Eqs. (25) and (27) were then used to calculate S,, (= 0.352) and S,,, (= 14.09) using the above-mentioned values of WZ, and /3. These values of S,, indicate that the silica gel surface consisted of sites which were selective to both benzene (S, > 1) and cyclohexane (So < 1). Eqs. (22) and (23) were then used to obtain S, and S, as functions of cti and the complete surface excess isotherm was generated using eq. (21). Fig. 1 compares the calculated isotherm with the experimental data. tt can be seen that the fit is very good at the low and the high concentration levels of
TEROGENEGUS / m. = 3.91
HOMOGENEOUS
0
0.2
0.6
0.4 Xl
0.8
1.0
-
Fig. 1. Adsorption of benzene-eyclohexane
mixtures on silica gel at 3WC.
495
S. Sircar / Adsorpiion from binaty liquid mixtures
benzene but the model slightly overpredicts the surface excess in the 0.1 I xi I 0.3 region. The figure also shows the calculated isotherm using the homogeneous equations (8) and (13) with S, (= 5.69) obtained by eq. (14). The homogeneous model underpredicts the experimental isotherm in the low xi region and overpredicts the isotherm in the high x, region. This demonstrates that (a) the monolayer adsorption model with an ideal adsorbed phase provides a satisfactory description of the adsorption behavior for this system when the surface heterogeneity is taken into account and (b) the proposed simplified model for describing surface heterogeneity is adequate. Earlier, House had analyzed the role of the surface heterogeneity for this system by numerical inversion of the Fredholm integral equation of the first kind like eq. (17) written in terms of an adsorption energy parameter [9]. The resulting distribution energy function was Gaussian-like with one or two peaks depending on the method of calculation used.
6. Parametric study The mean (cl) and the variance P = (%I a* = (S,,
(a*) for the uniform
distribution
are [8]: (29)
+ &)/2, - S&*/12.
(30)
It follows from eqs. (29) and (30) that &I = I*(1 + *k),
&I_ = CL0 - *I,
(31)
where 9 = &r/p.
(32)
The parameter q can be considered to be a measure of the surface heterogeneity. V’ + 0 (u + 0) corresponds to the homogeneous surface, where the uniform distribution function reduces to the Dirac delta function. \k + 1 represents the maximum surface heterogeneity by this model because S,,, + 0 at this limit, which is the minimum possible value for Se,_. The adsorbent heterogeneity increases as !P increases. We evaluated the effects of p and 9 on the surface excess isotherm for given values of p by using eqs. (21)-(23). The bulk liquid is assumed to be ideal. Figs. 2-4 show the results respectively for /3 = 0.5, 1.0, 2.0, and 1_1=1.5. Fig. 5 shows the isotherms for p = 0.5 and I_L = 5.0. The quantity Nf/m, is plotted in the ordinates of these figures. It may be seen from these figures that: (a) Nf decreases for any given x, and the maximum of N; occurs at a lower value of xi as the surface heterogeneity increases. The effect can be striking when p is low, irrespective of the value of j3. The isotherm can even change
496
S. Sirear / Adsorption from binary liquid mixtures
from “U” shape to “S” shape when the adsorbent is substantially heterogeneous. The effect of the surface heterogeneity on the isotherm shape is, however, much less pronounced when p is large. This was found to be true for a11values of /3. (b) The value of x, at which Nf changes sign in case of an “s” shaped isotherm increases as j3 increases for a given p and 9.
1.0
Fig. 2. Effect of surface heterogeneity on the surface excess isotherm. B = 0.5; p = 1.5.
Fig. 3. Effect of surface heterogeneity on the surface excess isotherm, /3 = 1.0; p = 1.5.
S. Sircar / Aakorption from binary liquid mixtures
497
The isotherm shape at the limit of x, + 0 is a strong function of j3 but it is practically independent of the degree of surface heterogeneity. The isotherm shape at the limit of x, + 1, on the other hand, is strongly dependent on both /3 and ‘Pk. (c)
0.10
.O
Fig. 4. Effect of surface
0
0.2
heterogeneity
0.4
on the surface
0.6
0.6
excess isotherm,
/3 = 2.0; p = 1.5.
1.0
Xl + Fig. 5. Effect of surface
heterogeneity
on the surface
excess isotherm,
B = 0.5; p = 5.0.
598
The above parametric study shows that both the size of the adsorbates and the adsorbent heterogeneity play major roles in determining the shape of the overall surface excess isotherm and the interactions between these variables are rather complex. Consequently, all models on adsorption of liquid mixtures must include the effects of these variables for meaningful interpretation of the data.
References [I] [2] [3] [4] {S] [6] [7] [8] [9]
S. Sircar, Surface Sci. 148 (1985) 478. O.G. Larionov and A.L. Myers, Chem. Eng. Sci. 26 (1971) 102.5. S. Sircar and A.L. Myers, AIChE J. 17 (1971) 186. S. Sircar and A.L. Myers, J. Phys. Chem. 74 (1970) 2828. K. Denbigh, The Principtes of Chemical ~quilib~um (Cambridge University Press. London, 1966). S. Sircar and A.L. Myers, AIChE J. 19 (1973) 159. L. Gurvitch, J. Phys. Chem. Sot. Russ. 47 (1915) 805. M. Abramowitz and LA. Stegan, Handbook of Mathematical Functions (Appl. Math. Ser. 55) (Nat]. Bur. Std., US Government Printing Office, Washington, DC, 1972). W.A. House, Chem. Phys. Letters 60 (1978) 169.