Colloids and Surfaces, 22 (1987) 317-336 Elsevier Science Publishers B.V., Amsterdam
317 -
Printed
in The Netherlands
Heats of Immersion in Multilayer Adsorption from Binary Liquid Mixtures on Strongly Heterogeneous Solid Surfaces* W. RUDZINSKI**
and J. ZAJAC
Department of Theoretical Chemistry, Institute of Chemistry U.M.C.S., 20-031 Lublin, Nowotki 12 (Poland)
E. WOLFRAM
and I. PASZLI
Department of Colloid Science, Lorand Eiitviis University, H1088, Budapest, Puskin u.ll-13 (Hungary) (Received
28 January
1986; accepted in final form 8 September
1986)
ABSTRACT Equations for heats of immersion are developed, corresponding to the lattice model of multilayer adsorption on heterogeneous solid surfaces proposed recently by Rudziriski and co-workers. The effects of surface topography are taken into account and also the effects of the possible anisotropy of mixing near a solid surface. Two sets of equations are developed; one by means of the crude Condensation Approximation approach, and the other by applying the improved Rudziriski-Jagiello approach. The usefulness of the developed equations in representing the calorimetric effects of adsorption at strongly heterogeneous solid/solution interfaces is shown by analyzing an experimental excess isotherm and the related heat of immersion of (benzene + methanol) mixture adsorbed on a silica gel.
INTRODUCTION
It has been realized for a long time that a model of a monolayer adsorption from solutions onto solid surfaces may not represent the true nature of solid/ solution interfaces in many adsorption systems [l-13]. Recently, Rudzinski et al. [ 141 have proposed a simple theory of multilayer adsorption which takes into consideration all the basic physical factors affecting the adsorption at solid/solution interfaces. These factors are as follows: *Dedicated to the memory of Professor E. Wolfram. **To whom correspondence should be addressed.
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0 1987 Elsevier Science Publishers
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318
(1) The intermolecular interactions in both the adsorbed and the equilibrium bulk phase. (2) The differences in the molecular sizes, and consequently the different surface areas which may be occupied by different admolecules. (3) The perturbations in the structure of the surface solution, and in the nature of the intermolecular interactions in the surface phase, compared to the situation in the bulk phase. (4) The energetic heterogeneity of the actual solid/solution interfaces. (5) The topography of the heterogeneous solid surfaces. To develop such a theory, many simplifications are necessary, and basically the expressions developed can only be used in cases where the surface areas occupied by different admolecules are not much different. Rudzinski et al. [ 141 also carried out a model investigation showing how the various physical factors listed above affect the behaviour of the excess adsorption isotherm. However, they have postponed attempts to fit their theoretical expressions to available experimental data since the present standard of accuracy with which the excess adsorption isotherms are measured, still makes such best-fit exercises risky. Nonetheless, in their recent papers, Rudziliski et al. have shown [ 15,161 that a simultaneous analysis of experimental excess isotherms and the related heats of immersion create a real hope for a reliable estimation of adsorption parameters appearing in more complicated adsorption theories. The importance of measuring the calorimetric effects of adsorption at solid solution interfaces was emphasized long ago by Zettlemoyer and co-workers [.17,18], Schay and co-workers [ 191, Everett [ 20-221, Groszek [ 231, Sircar and Myers [ 241, and Findenegg and co-workers [ 26-281, and Woodbury and No11 [ 251. In their previous works [ l&16,29-33], Rudzinski et al. limited their interest to the relatively simple case of adsorption systems in which a monolayer adsorption mode could be safely assumed. The purpose of the present work is to apply the multilayer adsorption model proposed recently by Rudziriski et al. [ 141, to the development of theoretical expressions for heats of immersion in multilayer adsorption from binary liquid mixtures on strongly heterogeneous solid surfaces. THEORY
Adsorption model Let us consider a two-component liquid mixture (A +B) which is in contact with an insoluble solid adsorbent. As in our previous works, we adopt here a quasi-crystalline model of surface solution. The semi-infinite solution extends from a planar solid surface and becomes identical with the bulk phase at a sufficiently large distance from the solid surface. That surface phase is sliced into L lattice planes parallel to the solid surface, and the numbering starts from
319
the lattice plane closest to the solid surface. Further, N,, is the number of lattice sites in each lattice plane. The lattice structure is assumed to be rigid and influenced by the presence of the solid phase, or changing concentration in the surface phase. As in our previous work [ 141, we consider here the case of binary mixtures of liquids whose bulk volumes are not much different. This means that the molecules A and B occupy one lattice site in the equilibrium bulk phase. However, as in our previous work, we assume that the predominant solid-adsorbate forces may cause some orientation of admolecules near the solid surface, so the number of lattice sites occupied by molecules A and B in the first lattice plane may be different. The solid-adsorbate forces are usually short-ranged, so we will neglect their presence in the second and higher adsorbed layers. Thus, we assume that the concentration change in the second and higher layers is induced via intermolecular forces between adsorbed molecules. To calculate the effect of these interactions, we denote by z the number of nearest neighbours-lattice sites, of which c+,zlies in the same lattice plane, and w in each of the adjacent planes, i.e., g+2a,=l
(1)
Assuming a pair additivity of the neglecting the perturbations due to of adsorbate-adsorbate interactions, tions for the concentration profile adsorbed layer,
interactions in the adsorbed phase, and the presence of the solid in the character one arrives at the following set of equanear the solid surface [ 141: for the first
whereas for the second and higher layers we obtain, ln
&4@B y-lnKi+
(ZCY/RT)[~(l-2#f4)+&(1-
#A@3
(2b)
2&-l) +a,(1-2&+9
- (1-2@A)]
Of course, when i=L, $i+ ’ = #A. are the area1 fractions of A and B in the i-th lattice plane, InEm (2),h,& @A,&are their concentrations in the equilibrium bulk phase, CIis the “interchange energy” parameter, and r is the ratio of the surface areas occupied by one molecule A and B in the first lattice plane, whilst ICIand Ki are appropriate equilibrium constants. R and T stand for the gas constant and absolute temperature, as usual.
320
The equilibrium
constant
for the first adsorbed plane is defined as follows, (3)
where eA and eB are the adsorption the parameter A is defined as, AZRTln
energies
of single molecules
A and B, and (4)
(~it/qA)(~iJ/%?)-’
where q1 and q mean the molecular partition functions in the first lattice plane and bulk phase, respectively. Neglecting the possible difference in the entropy of the molecules adsorbed in the second and higher adsorbed layers, compared to their entropy in the bulk phase, we have, lnKi=0,fori=2,3,.
. .,L
(5)
Everett and Pod01 [ 341, and Kern and Findenegg [ 351 reported that the presence of the solid phase may cause serious perturbations in the character of interactions between molecules close to the solid surface, compared to the equilibrium bulk phase. Similar observations were reported later on by Rudzinski et al. [ 15,311. According to the theoretical derivation presented in our previous paper, these perturbations can be taken into account by distinguishing between the products A,= (aza$T), and A,= (az~,,R2’) associated with every adsorbed layer. For computational reasons, however, we will accept a certain approximation here. While analyzing numerically, in the last past of our work, the experimental excess isotherms and heats of immersion for the (benzene + methanol) mixture adsorbed on silica gel, we accepted in practice a double-layer model. Then, we assumed that A, and A, for the first and the second adsorbed layer were the same, but different from their bulk counterparts. Since A, and A, were determined by fitting appropriate theoretical expressions for excess isotherms and heats of immersion to experimental data, distinguishing between A, and A, related to the first and the second layer would have introduced too many bestfit parameters. In the actual adsorption systems, with the local variations in the stoichiometry and crystallography of solid surfaces, the value of the equilibrium constant will vary from one to another adsorption sites. This effect is known as and is usually described by considering the “energetic surface heterogeneity”, variation of the parameter E, defined as follows, c=eA-reg-A
(6)
To account for the problem quantitatiely, one usually introduces the function X(E) which is the differential distribution of the number of lattice sites in the first lattice plane, among various values of E, normalized to unity. X(e)dt=l R
(7)
321
In Eqn ( 7)) Sz means the physical domain of the variable e. For obvious physical reasons, t must vary between finite integration limits eminand E,,. However, Rudzinski et al. have shown [ 29,361 that one can safely accept the infinite integration limits ( - co, + co), until adsorption at very low (or high) concentrations of one of the components A or B is investigated. Thus, in the case of the actual solid/solution interfaces, the true concentration profile {#A,} is determined by performing the following averaging, (9) of the concentration profile ($4 ( e) >, obtained by solving the equation system ( 2) for a particular value of E.The averaged concentration profile is next used to calculate the theoretical surface excess of A, n_&according to the equation,
(9) where XAis the mole concentration of A in the equilibrium bulk phase. Similarly, instead of the simple equation for the heat of immersion of a homogeneous solid surface Q,,
(10)
Q,=QB+N_A~--N~W+Q~,
its averaged value QWtmust be used to calculate the heat of immersion of the actual solids in binary liquid mixtures.
J -co
(11)
J -co
In Eqns (10) and (11)) QBis the heat of immersion in the pure component B, and Qi, is the interaction term describing the change in the pair interactions A-A, B-B, and A-B, due to the formation of a solid/solution interface, relative to the situation in the bulk phase. In the case of adsorption model considered here, Qintakes the following explicit form [ 141,
+~tih
+q(ti@~ +@A&-$~b)l
(12)
Concerning the problem of the energy distribution function x( E ) , its actual form is expected to be complicated. Thus, to make the problem tractable, this function is usually replaced by its “smoothed” version, reflecting only some
322
basic features of the true distribution function. As in our previous works, we will represent x( e) by the following Gaussian-like function,
(l/c)
ew{tE-~o)/c}
X(t)=[1+exp{(e-eo)/c}]2
(13)
centred about e= co, the spread of which is described by the heterogeneity parameter c. When c tends to zero, the function (13) degenerates into Dirac delta distribution, representing a homogeneous solid surface characterized by E=Eg. The result of the integration in Eqns ( 8) and (11) will depend in addition on the topography of a heterogeneous solid surface. Surfaces with random topography
The “random” model of surface topography was introduced by Hill [ 371 as early as 1949, and later on elaborated by Rudzinski et al. [ 38-43 ] . It is assumed in this model that adsorption sites with the same value of adsorption energy (parameter E ) are distributed on a heterogeneous lattice of sites completely at random. This means that there are no spatial correlations between adsorption sites with similar adsorption properties. Thus, in adsorption systems with random surface topography the local concentration in every adsorbed layer will be the same throughout the whole layer, and equal to the average concentration in that adsorbed layer. Accordingly, the potential of average force acting on an adsorbed molecule from all the other molecules in the system will be the same as in a system with a homogeneous surface, in which the concentration profile is the same as in our actual system with patchwise surface topography. This means that both the expression for Qin, and the form of the interaction terms in Eqn (2 ) will remain unchanged, but the concentrations $i ,& have to be replaced there by their averaged values #it,@&. As to the other integrals appearing in Eqns (8) and (ll), we will evaluate them using the RJ ( Rudzinski-Jagiello) approach [ 291. Equation (8) then takes the following form, (14) where i( E) is the antiderivative of x( c) , which for the particular form of the energy distribution function in Eqn (13 ) reads, X(e)=[l+exp{(e-eo)/c}]-’
(15)
Since in the case of the random surface topography, &Xi z=O,
for i> 1
WI
323
only the function $A( e) has to be averaged. The derivative is a Gaussian-like function of E, centred at the point E= E,, which is found from the obvious condition, (17) After performing appropriate differentiations in the equation system (2)) one arrives at the following explicit expression for E:, in the case of random surface topography, E:’=RT In
(l+r’/2)‘-’
(r)r’2
+RTlny
+~T[A,(1--2~~~) A
+A,tl-2&i,)
-A(($B)2-r(@B)2)1
(18)
The subscript “r” in E:’ refers to random surface topography. In the case of the strongly heterogeneous surfaces considered in our present work, the width of the derivative (@A/&) is much smaller than that of the energy distribution function X(E) . So that derivative behaves like a sampling function with respect to j( E), while performing the integration in Eqn (14). A convenient way to evaluate @it therefore is to expand X(E) into its Taylor series around E= E,, at which the derivative reaches its sharp maximum. By doing so, we obtain,
(19) where Bzm is Bernouli’s number. The expansion (19 ) is quickly convergent, and can practically be truncated after the first or the second term. When it is terminated after the first term, the expression obtained for @it will be identical to that given by the CA (Condensation Approximation) approach. $~=[l+exp(Et-cO)/c]-’
(20)
The additional subscript “ca” at $2 denotes the solution corresponding to the CA approach. With ear defined in Eqn (18)) Eqn ( 20) can be rewritten to the following explicit form,
lnP&_RTln -- ‘A @Et
c
(h3Jr
5+1n c
(l+r1/2)r-1 r r/z +&(1--2&t)
RT -c[4J1-%%)
-A(($~)2-r(~A)2)l
(21)
After solving Eqn (21) together with the equation system (2b) in which all & ,&‘s are replaced by their averaged values &,.,t,#~‘s, one arrives at the concentration profile {&F} correspondingto theCA approach. A still better con-
324
centration profile can be evaluated by means of the RJ approach, which depends on retaining the first correction term &!$ in the expansion (19). &$ = @“t” + q$:
(22)
In Eqn (22)) the additional superscript “rj” denotes the solution corresponding to the RJ approach. For the particular form of the energy distribution function (13)) the first correction term takes the following form, $&Q
K2(RT/C)2~Cta(l-~ficta)(2~Cta-l)
(23)
Using the RJ approach, one can easily evaluate the heat of immersion QWt for surfaces with random topography, denoted hereafter by Q,,. Let us consider to that purpose the first integral on the right-hand side of Eqn (11) . Using the RJ approach, we have, (24) where V(E) is the antiderivative of the product of functions E-,X(E) , which for the particular form of the energy distribution function in Eqn (13) takes the form, q(e)=-[l+exp{(e-e,)/c}]-‘-cln[l+exp{(eo-6)/c}]
(25)
The first term on the right-hand side of Eqn (24) is the result corresponding to the CA approach, and the other term represents the first correction term. Thus, when the CA approach is accepted to evaluate Q,,, the final expression for Qwttakes the following explicit form, Q”,“,=QB+No[~Cta(~o-~)-~(~cta
ln$2+$&
ln#E)]
L-l
+N,,RTi&
[A,~~~~;;a+A,(~Cta~~~l,ca+~~~l,ca~~)]
+NoRT[A,tiAt”“&F +A,M%VB
+h&?)l
The expression for QWrcorresponding easily from the expression, ~~~=QB+No[~ficta(~o-~)-~(~~
(26)
to the M approach can be calculated
ln$E+@g
ln$g)]
(27)
which, however, is valid only for the particular (13).
form of the energy distribution
Surfaces with patchwise topography The case of heterogeneous surfaces with patchwise surface topography appears to be much more complicated. First of all, the explicit analytical form of the derivatives (d&J&) and (d2&/ac2) depends on the number L of adsorbed layers taken account. Secondly, the complexity of the analytical expressions for the first and second derivatives grows quickly as L increases. However, even for the simplest case of the double-layer model, the equation
(28) cannot be solved analytically with respect to @. Here the additional superscript “p” stands for patchwise surface topography. All these complications arise from the fact that all $4’~ are now functions of E, so, the surface activity coefficients are such functions as well. Further, the RJ approach cannot be used now to evaluate the average concentrations in the second and higher layers. This is because for i> 1, #i’s are slowly-varying functions of t, thus the derivatives (d#~/&) do not behave more like sampling functions with respect to j( e) for i> 1. As in our previous publication, we will use the RP approach to evaluate the average values &‘s, for i> 1. According to that approach, the integral (8) is evaluated by expanding the kernel @i ( E) into its Taylor series around the point E= co, where the function X(E) reache s 1‘t s maximum. (Now x( E) behaves like a sampling function with respect to $i ( c ) ) . Thus, for i > 1, we have,
(29) where,
where {@io} is the concentration profile for a homogeneous solid surface, characterized by E= co. The terms under the sum in Eqn (29) ‘represent the perturbation in the concentration of i-th adsorbed layer due to the energetic surface heterogeneity. While retaining only the first perturbation term, we have, (31)
326
As in our previous publication [ 141, we consider here the explicit form of appropriate expressions only for the double-layer model. Our model investigation presented in that previous publication showed that the double-layer model should be accurate enough for many of the experimental adsorption systems reported so far in the literature, and also for the (benzene+methanol) mixture adsorbed on silica gel, investigated in the present work. For the double-layer model, the function D~‘2’ defined in Eqn (30) takes the following explicit form,
11(324 where,
(32b) Thus, the evaluation of the average concentration in the second layer @it is relatively easy. Namely, the starting system of the two Eqns (2a) and (2b) must be solved for the particular value e = eo, and the concentration profile obtained [ @~o,&o] is used to evaluate && in the simple way outlined in Eqns (31) and (32). In order to calculate the average concentration in the first adsorbed layer @it, one has to solve the equation system consisting of Eqns ( 20) and ( 28), together with the starting equation system (2a,2b). After a certain analytical rearrangement, one arrives finally at the following system of three equations to be solved simultaneously,
--8(Ad3 ic B $++A,(l-2&d Bc
A
-[
1
($,,)2
1 ($p,)”
1 =o
(33b)
=o
(33~)
+A#-2&) +Av(l-2$,)
-A(l-2$A)
327
Of course, the solution of the equation system (33) yields the #it value corresponding to the CA approach. To obtain the $it value corresponding to the FLIapproach, one has to replace Eqn (33a) by the equation system consisting of Eqn ( 22)) and Eqn (2a) in which E= E,. We will not write the explicit expression here so as to omit simple repetitions. Having calculated @it, one can evaluate easily the first integral in Eqn (11) . However, in the case of the adsorption systems with patchwise topography, Qin is also a function of E,so we have to evaluate in addition the second integral in Eqn (11) . In particular, the following integrals must be evaluated, +CO s -co
&(e)&(e)X(e)de
(34)
As in our previous work [ 311, we accept here the following approximation, &,& =RT(&Glac)
(35)
which to be precise is only true when r = 1. Applying the FLIapproach, we arrive at the following result,
(36) where the first term on the right-hand side of Eqn (36) represents the CA approach, and the second term is the first correction term. While averaging Qi”, we have another integral still to evaluate.
The function @I( t ) depends on Eonly through the interaction term, which is proportional to $A. So it is expected to be a slowly-varying function of E,even compared to @i ( E) . Therefore, we evaluate the integral on the right-hand side of Eqn (37) by replacing & ( E) by its value taken at the point e = eEP.Doing so, we obtain,
Now we have arrived at the point where we can write the final explicit form of the expressions for the heats of immersion, in the case of the double-layer model considered here. * While accepting the crude CA approach, we have,
p(2) P
-_
-D
1
1 K
l3
1
_p
~P-p-J& >[
MO)"
G&
1 1 m-m
1
11(39b)
The expression for Q&, related to the RJ approach can be obtained easily by including additional terms which arise from the last terms on the right-hand side of Eqns (24)) (36)) and (38). Doing so, we obtain, ~=&“~+N&&(eo--jZ) +
+c(@,f:/2$,“,)
(A,/~c)(RT)~[(~)~+~~,“,-~I(~~~/~)
(40)
-2RTA,&,&~}
RESULTS AND DISCUSSION
In spite of the complexity of the problem, we have arrived at relatively simple, compact expressions for the surface excess n>, , and the heat of immersion QwtiOnce the concentration profile {@it} near a solid surface is evaluated, the related heats of immersion can be evaluated easily according to Eqns (26) and ( 27) or ( 39) and (40). The concentration profile itself it evaluated by solving a system of a few non-linear equations. This can be done easily using many subroutines designed originally by Lane [ 21, by choosing the most useful one for that particular physical problem. According to that method, the system of equations is solved in an iterative way, starting from a homogeneous concentration profile {f& = @A}. The iteration is terminated when the difference between the values obtained in the last two iteration steps is smaller than 10V5MIN of $A. As in our previous works on monolayer adsorption, we now face the problem of a shortage of appropriate experimental data for analysis. In spite of the extensive calorimetric investigations now being carried out in many laboratories, experimental excess isotherms and the related heats of immersion have been reported only in a few cases [ 23,44-461. There are no such data available
329 TABLE 1 The values of the parameters r, A$, A,, A., to. end c obtained by fitting best the theoretical expressions for n;,,Qw developed in the preeent work, to the Wolfram-Padi’s experimental data for (benzene [A] + methanol [B ] ) mixture adsorbed on Weanal” Adsorption model
Monolayer adsorption on a homogeneous solid surface Monolayer adsorption, random topography, CA approach Monolayer adsorption, patchwise topography. CA approach Monolayer adsorption, random topography, RJ approach Monolayer adsorption, patchwise topography, F&Japproach Double-layer adsorption, random topography, FLJapproach Double-layer adsorption, patchwise topography, RJ approach
r
NO (mm01 g-‘)
A, (kJ molP’)
A” (kJ molP’)
1.36
10.79
-4.96
-3.32
-7.18
1.51
9.73
-4.25
-3.94
- 10.4
2.48
2.29
6.47
-4.67
-4.95
-6.52
2.50
1.28
11.31
-0.65
-3.44
-4.63
2.64
2.15
8.17
-3.47
-2.48
-2.13
2.40
2.18
6.51
4.90
-5.43
4.27
2.3
9.5
4.90
-3.12
3.14
0.94 -0.06
e0 (kJ mol-‘)
c (kJ mol-‘)
0
in the case of badly mixing liquid mixtures (showing large positive deviations from Raoult’s law) adsorbed on strongly heterogeneous solid surfaces. The only experimental data which we had at disposal were our own data measured a few years ago in Professor Wolfram’s laboratory at the Lorand Eiitvijs University in Budapest [ 471. These were the excess isotherms and heats of immersion for the (benzene (A) + methanol (B) ) mixture adsorbed on a silica gel “Reanal”, (BET area 369 m2 g-l) at 298 K. The experimental excess isotherm represents the III-type according to the Shay-Nagy classification [ 191. The (benzene + methanol) mixture is relatively badly mixing, so according to Lane’s [ 21, and our own [ 141 investigations an effective multilayer formation should take place in that adsorption system. However, in order to prove the importance of various physical factors affecting the behaviour of excess isotherms and heats of immersion in that system, we have performed our numerical analysis by accepting various adsorption models. The results so obtained are presented in Table 1, and in Figs 1-5. Figure 1 shows the agreement between theory and experiment which is obtained by assuming a monolayer adsorption on a homogeneous solid surface. Figure 2 shows our experimental data plotted as xi (1-xi) *(n&) -’ versus x:. That representation should be linear in the case of molecules of equal sizes adsorbed on homogeneous solid surfaces in a monolayer fashion. The agreement obtained is evidently poor. Next, Fig. 3 shows the agreement which is obtained for the model of monolayer adsorption on a heterogeneous solid surface, using the CA approach. It can be seen from Fig. 3 that taking into account the effect of surface heterogeneity does not strongly affect the behaviour of the theoretical excess isotherm, compared to the improvement which is obtained in the case of the theoretical heat of immersion. Although the theoretical curve does not yet match
330
.
.
.
Fig. 1. Comparison between the experimental and theoretical excess isotherms and heats of immersion for the (benzene (A ) + methanol (B ) ) mixture adsorbed on the silica gel “Reanal” at 298 K. The circles on the left part of the figure denote the experimental data for n>,, and those on the right are the experimental data for heats of immersion Qti. The solid lines denote appropriate theoretical curves corresponding to the model of a homogeneous solid surface. (Calculated by using the parameters collected in Table 1. ) Fig. 2. The experimental tion of xi.
data for the excess isotherm plotted in the form: x!j & ( nit) -’ as a func-
the experimental data, there is an essential improvement in the form of that function. In the case of random surface topography, that curve is concave at high concentrations of benzene in the equilibrium bulk phase, thus imitating correctly the behaviour of the experimental data for the heat of immersion. Figure 4 shows the same calculation as previously, but this time the RJ
I
0.5
I
0.5
__b A
Fig. 3. Comparison between theory and experiment obtained by means of the CA approach: (- - - - ) ) monomonolayer adsorption on a heterogeneous surface with patchwise topography; (layer adsorption, random topography. Other notations as in Fig. 1.
331
Fig. 4. Comparison between theory and experiment obtained by means of the FU approach: (- - - - ) monolayer adsorption, patchwise topography; ( -) random topography.
approach was applied. Looking solely at the agreement between the theoretical and experimental excess isotherms, one might come to the surprising conclusion that the patchwise model should be extremely good to represent the nature of silica gel surfaces. However, the behaviour of the related theoretical curve for the heat of immersion strongly suggests an opposite and less surprising conclusion. Note that the RJ approach still further improves the character of the theoretical curve for the heat of immersion in the case of the random surface topography model. However, the discrepancy between the theoretical and the experimental points is still clearly visible. Finally, Fig. 5 shows the agreement which is obtained by means of the RJ approach for the model of a double-layer adsorption on a heterogeneous solid surface. Looking at the behaviour of the theoretical heats of immersion, one can see
Fig. 5. Comparison between theory and experiment obtained by means of the RJ approach for the double-layer model: (- - - - ) patchwise topography; ( -) random topography.
332
that the model of the patchwise topography is completely unrealistic, contrary to the random model which reproduces very well the behaviour of the experimental heats of immersion. Figure 5 also shows clearly that only this model of a double-layer adsorption makes a good fit to experimental heats of immersion. (Since none of the adsorption models applied here yields a good agreement between the theoretical excess isotherm n$, and the last experimental point at the highest concentration of benzene in the bulk phase, we believe that this is due to a large experimental error in that case. There is yet another reason, why we believe the model of a double-layer adsorption on a surface characterized by the random surface topography. The more exact RJ approach yields 6.51 mmol g-’ as the surface capacity for methanol (“monomer”) adsorption. The surface area occupied by one methanol molecule is, for that model, 2.18 times smaller than that occupied by one benzene molecule. According to Kagiya et al. [ 481 one benzene molecule adsorbed on a silica gel occupies 30 A”. Accepting that estimation we arrive at the conclusion that our silica gel has a surface area of 493 m2 g-l. That result is closest to the value of 369 m2 gg’ estimated independently by means of the BET method. Looking at Fig. 5, one can also see that an assumption of a certain kind of surface topography slightly affects the behaviour of the theoretical excess isotherm, compared with that effect in the case of the heat of immersion. This means that the best fit procedure applied to appropriate theoretical expressions for the surface excess, does not offer much hope for distinguishing between random andpatchwise surface topography. A similar situation has been reported by Rudzinski et al. [ 39,401 in the case of gas adsorption on heterogeneous solid surfaces. Thus, once again, one sees the importance of investigating the calorimetric effects of adsorption at solid/solution interfaces. Figures 6 and 7 demonstrate that the contributions to the overall surface excess n>, and to the overall heat of immersion Qwt from the second adsorbed layer are absolutely essential in the case of the (benzene + methanol) mixture adsorbed on silica gel. Table 1 suggests that there is a strong anisotropy of mixing near the silica gel surface, as indicated by the very different values of parameters A, and A,. This may be due to the different orientation of adsorbed molecules in the first adsorbed layer. Figure 8 shows the surface activity coefficients for the first, yf , and the second, r:, adsorbed layer evaluated according to the expressions [ 141, in Y; =~~[A,(&,)2
+A,(&t)2l
(41a) (4lb)
\,
m
.=cao . t
t
(Bl
.
.
-10
0.5
.
.
.
-
.
)(b A
Fig. 6. The contribution to the overall surface excess nit from the second adsorbed layer (part A) calculated for random surface topography, using the parameters collected in Table 1. The circles in part B denote the experimental surface excess, and the solid line denotes the evaluated contribution to n;, from the first adsorbed layer. Fig. 7. The contributions to the heats of immersion Q,, from the second (part A), and the first (part B) adsorbed layer. The black circles in part B denote experimental values for QU,t.
0.5 -(p'
At
0.5 -
($2 At
Fig. 8. The surface activity coefficients in the first, yi , and in the second, yf , adsorbed layer, calculated according to Eqns (41a,b) and (42a,b), using the parameters collected in Table 1.
334
ln ri
=~TLW?L(~~t -@id +A,(&d2
+MW21
(@a)
using the parameters collected in Table 1. (In the case of random surface topography). It is interesting to note in addition that the best-fit value of the parameter r obtained here is almost the same as the one which we assumed apriori in our previous work [ 291 in the case of (benzene + methanol) mixture adsorbed on the charcoal “Carbo Medicinalis”. So from the results obtained in the present work we may conclude that a simultaneous combined analysis of excess isotherms and heats of immersion is a very powerful tool for investigating the nature of the actual solid/solution interfaces. -
We very much regret that one of the authors - Professor Erwin Wolfram could not take part in preparing the final version of the present paper.
ACKNOWLEDGMENTS
Financial
support from CPBR 3.20.63.
LIST OF SYMBOLS
+,,~a, o!
numbers of the nearest-neighbours adsorption in the adjacent lattice plane, respectively “interchange energy” parameter
sites in the same and
A,, = az%RT A,= az~RT R gas constant temperature area1 fraction in the i-th lattice plane (i= 1, . . ., L) and volume frac;,@A tion in the bulk phase of k-species (k=A, B) overall area1 fraction of k-species for a heterogeneous surface ;;; the first correction term to the overall isotherm l&3 the CA adsorption isotherm [ Eqn ( 22) ] fi&=~~,@Lc=@f4f4~=~3 r ratio of surface areas occupied by one molecule A and B in the first lattice plane equilibrium constant for i-th adsorbed layer Ki energy of adsorption of single molecule k ck A parameter defined by Eqn (4)
335
molecular partition function of the k component in the i-th lattice plane and in the bulk phase adsorption energy from solution E the most probable value of adsorption energy CO lr function defined by Eqn (18) EC function defined by Eqn ( 28 ) EIP differential distribution of the number of lattice sites among various values of adsorption energy antiderivative of x( E) X(4 antiderivative of E*X(E) V(E) heterogeneity parameter number of lattice sites in each lattice plane &O overall surface excess nT4, overall heat of immersion Q wt heat of immersion in the pure component B QB interaction term describing the change in the pair interaction due to Qin formation of solid/solution interfaces the CA heat of immersion Q","t D function defined by Eqn (32b) Di(Zm) the 2m-th derivative of $4 with respect to t surface activity coefficients for the i-th lattice plane rk
qk,qk
X74
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