J. Phys. Chem. 1993,97, 7730-1135
7730
Thermodynamic Relations for the Noncalorimetrical Determination of the Heat of Adsorption in Multicomponent Systems C. Wittrock and H,-H. Kohler' Institute of Physical and Macromolecular Chemistry, University of Regensburg, 0-8400 Regensburg, Germany Received: December 7, 1992; In Final Form: March 16, 1993
The molar heat of adsorption, Ahad,of a single solute in ideal dilute solution can be determined indirectly from the temperature dependence of the adsorption isotherm by means of the so-called Clausius-Clapeyron equation. The present paper deals with the generalization of this equation to systems with several adsorbing components. The generalization must allow for the nonstoichiometry of the adsorption process; that is for the fact that, in general, the amounts of the adsorbing substances exchanged between the bulk phase and the surface phase are not coupled by constant ratios. Instead, these ratios change with the composition of the bulk solution. These changes can be represented by a path P in the space of the bulk solute concentrations. To make the value of the differential molar heat of adsorption unique, it must be related to this path P,which leads to the definition of a corresponding heat Ah$. With use of this definition the Clausius-Clapeyron equation is generalized on the level of the individual components as well as on the level of the adsorption process as a whole. The latter approach turns out to be much more fruitful for practical applications. Several examples are discussed in detail.
1. Introduction
2. Defdtion of the Differential Molar Adsorption Enthalpy
In the case of an ideal dilute solution, the molar heat of adsorption, Ahad,of a single substance 2 dissolved in a solvent 1 can be determined indirectly from the temperature dependence of the adsorption isotherm by use of the so-called ClausiusClapeyron equation,l-l7
2.1. Enthalpy Change of the System. In general, a surface phase may exchange substance with either of the adjacent bulk phases. For simplicity,we assume that this exchangeis restricted to a single bulk phase, more precisely, to an ideal dilute liquid solution consisting of n components. Component 1 is assumed to be the solvent. The other bulk phase, whether solid, liquid, or gaseous, is assumed to be inert. The amounts of substance of component i present in the active bulk phase and in the surface phase are denoted by n; and nl, respectively. nl is also called a surface excess. The precise values of n; and np depend on the location of the Gibbs dividing surface. To remove this ambiguity we use, instead of the nys, relative surface excesses with respect to solvent 1. Following common notation: we denote this relative surface excess of component i by nz1 and the according bulk quantity by n;'. This kind of indexing will be also used for other extensive quantities. Note that the relative surface excess nll becomes identical with the absolute surface excess nl if the position of the dividing surface is such that nf = 0. By virtue of definition we have, independent of the position of the dividing surface,
where c2 denotes the bulk concentration and I'z,l the relative surface excess concentration of the adsorbing component 2, with an additional subscript 1 indicating that the surface excess of 2 is taken relative to that of species 1. Since I'z,lis kept constant on the right-hand side of eq 1, the noncalorimetricaldetermination of the adsorption enthalpy given by this equation is also called isosterical. Fluctuations around a constant value of 1'2,1 resemble fluctuations in crystallization of a dissolved or gaseous substance. Therefore, eq 1 may be considered as an immediate consequence of the van't Hoff reaction isobar or, alternatively, of the original Clausius-Clapeyron equation for a gas-/solid-phase tran~ition.~.6 Considerable difficulties, however, are met if one tries to generalize eq 1 to systems with several adsorbing components. Difficulties are mainly due to the nonstoichiometry of the adsorption process,that is, to the fact that, in general, the amounts of the adsorbing substances exchanged between the bulk phase and the surface phase are not coupled by constant ratios. Instead, these ratios change with the composition of the bulk solution. In the past, literature has mainly dealt with single-solutecases.Thus the nonstoichiometry of the adsorption process has been widely ignored. This may be the reason for inconsistencies appearing between calorimetrically and isosterically determined values of the adsorption enthal~y.~-PTaking nonstoichiometry fully into account, we present in this paper a generalization of the ClausiusClapeyron equation to multicomponent adsorption. It is our intention to present the final results in a form easily applicable to cases of practical interest. The basic idea of our treatment has already been used, more intuitively, to solve a special problem in ref 10.
* To whom all correspondence should be addressed. 0022-3654/93/2091-1130$04.00/0
ny,l = 0
(2) Throughout this paper we consider a plane surface. The pressure p and the surface area A are assumed to be constant and therefore will not be handled as variables. Thus, the enthalpy H of the system can be written in the form H = H(T,n;l,$,) (3a) We assume a materially closed system without chemical reactions. Hence dn, = dn;,
+ dn;,
=0
(3b) At constant temperature, we obtain from eqs 3a,b with eq 2 n
d H = c ( h ; - h:) dn;, i=2
where hi"and hp are the partial molar enthalpies of component i in the surface phase and in the bulk phase. They are defined 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vo1. 97, No. 29, 1993 7731
Heat of Adsorption in Multicomponent Systems
4
4
b)
Along path P this becomes
r,,1= r,,l(T,c,,c~(c,),...,~(c2))
(8b)
or briefly
q l = rcl(T,c,)
Figure 1. Examples of a path P corresponding to different choices of c3 = 4(c2) for a three-component system (n = 3). The concentration space is given by the c ~c3-plane. , (a) c3 = c2, (b) c3 = constant, (c) c3 = co - c2, co = constant.
i = 2,3,
...,n
(8c) Atconstant T,eq8ccanbeinterpretedasasetof(n- 1) adsorption isotherms associated with path P and relating each I'r,lto CZ. On eliminating c2, we obtain from eq 8c, for a given path P and a given temperature,
As in this equation, partial derivatives taken along a path P will Although we have defined hl and h: with respect to the relative surface excess nl,, we write these quantities without the additional subscript 1. It will be shown under Summary and Discussion that this subscript is in fact superfluous. We now introduce the partial molar adsorption enthalpy, Ah;d, by
and the surface excess concentration,
I'j.1,
be indexed by P. As an example we consider the case of competitive Langmuir adsorption of two components 2 and 3 ( n = 3). The relations 8a now read CdK2
A
rz~ = lr11
+ cz/Kz + c3/K3
where K2 and K3 are the temperature-dependent dissociation constants and f is the maximum surface excess concentration assumed to be temperature-independent. Forming from eq 9a the differentials
by
Since A is assumed to be constant, we have dn;, = A dI'i,l
(44
With eqs 4b and 4d, eq 3c becomes
and using c3 = g(c,),we obtain
n
dH/A = z A h f d dr,,l I-,
Note that, according to eqs 3b,c, Ah;d can be also written as
2.2. Exchanged Amounts of Substance. In an ordinary chemical reaction the ratio of the differential changes dn, and dn, of any two components i and j is constant and given by -dn, = - vi d"i vi Here vi and vj are the (sign-dependent) stoichiometrical coefficients of componentsi and j . In general, there is no such coupling in adsorption. Instead, the ratio dn:,/dni, or, equivalently, dI',,l/dI',,I changes with changes in the bulk-phase composition. Since at constant temperature, the bulk composition of an n-component systemis uniquelydeterminedby (n- 1)-independent concentrations, such changes correspond to a path P in the according ( n - 1)-dimensional concentration space. For our purpose we describe the bulk composition (and thus assume the concentration space to be spanned) by the solute concentrations c2, ..., c,. A path P then can be defined by a set of (n - 2)independent conditions of the form (see Figure 1) c, = Cp(c2)
i = 3,4,
..., n
(7)
For convenience the name path and the symbol P will also be used for this set. Equality of the chemical potentials in the bulk and the surface phases implies
r,,l= l"i,l(T,cz,...,c,)
i = 2, 3, ..., n
(84
which illustrates that aI'3,,/dI'~,~l~,p depends on the state of the bulk phase ( T , C ~ , and C ~ )additionally on the direction of the path P,given by d 2. In the following section we will try to establish generalizations of eq 1 more appropriate for practical applications. 3.2. Generalizationon the Level of the AdsorptionProcess. On inserting eq 19 into eq 1la and changing the order of summation in the double sum, we obtain
T
d In ci
+
(14)
Hence, We now establish some relations which will help to simplify the double sum. Using again the theorem of Schwarz, we find from eq 16 On the other hand, the differential of the Gibbs energy of the surface phase is
dG" = -S" d T +
n
E&'dnl,
From eq 15 we obtain for changes taking place along path P at constant temperature
1-2
Application of the theorem of Schwarz gives
With eqs 13b, 15, and 17, we obtain from eq 13a (cf. ref 6)
In analogy to eq 4b, we define As:d = si"- s:.
Because of &' =
With eq 13b the condition of the sustained equilibrium, eq 13a, yields
If we insert eq 22 into eq 23a, the right-hand side of this equation becomes identical with the inner sum of the double sum of eq 21.
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7733
Heat of Adsorption in Multicomponent Systems We then use eq 23b and obtain from eq 21
rz,1
On comparison with eq 21 one notes that the partial derivatives of the pFs on the right-hand side have been replaced by partial derivatives of the bulk concentrations which, operationally, are much easier to handle. Equation 24 can be used to determine the value of Ah$ from the (2n - 1) differential changes d In ci, drj,1, and dT. These (2n - 1) changes are subject to the (n - 1) equilibrium conditions of eq 18 (which in turn result from eq 13a). With these conditionsbeing taken into account, the system still has n degrees of freedom. Since eq 24 should be applicable for any d T # 0, one degree of freedom must be left over. Thus a set Q of (n - 1) conditions still can be chosen freely in the (2n - 1)-dimensionalspace of the bulk and the surfaceconcentration, cfand I'j,l ( i , j = 2,3, ...n), and of temperature T. Together with the (n - 1) equilibrium conditions, this set Q defines a path (onedimensional manifold) in this space. We consider the points on this path as a function of the independent variable T. (Similarly, condition P given by eq 7 has generated a path in the (n - 1)dimensional space of the bulk concentrations ci, with c2 playing the role of the independent variable.) Therefore, for a given Q, the cI)s and I'j,l's can be related to d T by
4
'4
d In c
dT (25) d T dI'j,l = Q Q Using these relations, we obtain from eq 24, after changing subscript j into i, the following general equation for Ah$: d In ci=
Of course, for a given P, the value of Ah$ resulting from this relation does not depend on the choice of Q. Seen under experimental aspects, the choice of Q regulates the changes of bulk and surface composition accompanying changes of the temperature. One possible choice is dI',,l = 0 for i = 2,3, ,.., n. The right-hand side of eq 26a then reduces to the first sum, and we return to the unfavorable situation of eq 20b discussed earlier. So, to get more useful formulas, we have to look for more favorable choices of Q. Recall that the desired quantity, Ah$., is defined on the basis of a path P that regulates changes of the bulk-phase composition at a given temperature. The most natural way to obtain experimental data at other temperatures is to maintain these regulations. In other words this means that all changes in bulk and surface composition needed to determine A h $ , are obtained by measuring, at neighboring temperatures T and T dT, the adsorption isotherms defined by P and given by eq 8c. Thus, with the (n - 2) conditions of P taken over into Q, a single condition, say q, remains free. Q therefore can be written as the pair (P, q ) , and eq 26a becomes
+
-
A h $ = - R i ( ng dd l n c
$3'3)
P,qdr2.1
1 I
,
TtdT
I d In c2
+
Incz/co
Figure 2. Illustration of the mathematical quivalence of eqs 27c and 27d for a two-component system (n = 2). co is a reference concentration. At a given temperature change dT, the differential change d In c2 at constant rZ,l is related to the differential change dI'2.1 at constant cz by dl''2~!~~ = -tan a d In c~lr+,.On substituting tan a = (8I'2,1/8 In c2)zand dividing by dT, one obtains (8 In c2/8T)lrz, = -(a In C Z / ~ I ' ~ . I ) I ~ ~ I ' ~ , I / 8q,k,, which proves the equivalence of eqs 27c and 27d.
(cf. eq 7) c:,&..,$ = constant (274 that is under variation of c2 alone. For every i > 2 we have a In ci/aI'2,11Tp= 0 and a In ci/d71p,, = 0. Thus eq 26b reduces to
where the subscript cph2 abbreviates condition 27a. The unspecified condition q can be used to regulate the relation between I'zJ,c2, and T. We choose r2,1= constant. Then eq 27b becomes
In this special case nearly perfect formal agreement with eq 1 is obtained. Regardless of how many components are taking part in the adsorption process, only a single adsorption isotherm, namely, that relating r2.1 to c2, needs to be measured at neighboring temperatures T and T + dT. Notice the great reduction in experimental expenditure compared with the requirements of eq 20b, especially for large values of n. More generally, one concludes the following from eq 27a: If v bulk concentrations are kept constant in P, then the noncalorimetrical determination of Ah$ requires the surface excess concentrations of the remaining (n - 1 - v) substances only to be kept constant while temperature is varied. Thus the experimental expenditure of the noncalorimetrical determination of the differential heat of adsorption decreases with an increasing number of bulkconcentrations kept constant during adsorption. Another interestingchoiceofqiscz= constant. In thiscaseeq27b becomes
According to this result, Ah$ can be obtained from the inverse slopeof the adsorption isothermof component 2 (which determines d In c2/dI'z,l) and the temperature dependence of 1'2,1 obtained under the condition of constant bulk composition. Assume, for example, that the surface phase, too, is ideal. Then c2 = K 2 r ~ 1 , where K2 is the dissociation constant. In remarkable symmetry with eq 27c, eq 27d then becomes
T,P
P,q ar2,1
(26b)
T,P
Suppose, for example, we are interested in Ah$ for P given by
The mathematical equivalence of eqs 27c and 27d is illustrated in Figure 2. Further details of eqs 26b, 27b, and 27c will be
Wittrock and Kohler
7734 The Journal of Physical Chemistry, Vol. 97,No. 29, 1993
eq 26b gives
discussed in the next section. 4. Special Cases of Two-Component Adsorption 4.1. Competitive Langmuir Adsorption. We now complete the analytical treatment of the example of competitive Langmuir adsorption repeatedly considered above. In Langmuir adsorption the partial molar adsorptionenthalpy equals the standard value given by the van't Hoff equation:
Ah;d
-RF
+
5. Summary and Discussion
a In K,
aT
On the other hand, with eq 9a transformed into n
which differs by a factor of (1 + 8) from the ordinary ClausiusClapeyron equation given by eq 1. This factor allows for the increase in entropy produced by the dissociation of an electrolyte particle into (1 8) ions.
n
eq 28a is also easily obtained from the isosteric relation of eq 20a. The indirect determination of the differential molar heat of adsorption,Ah$,, now will be discussed for the two special cases: (a) c: = constant and (b) 4 = c2. (a) P Given by c: = Constant. This path P is illustrated in Figure lb. To be able to use eq 27b, we need cz as a function of rZ.1 and cj. Equation 9a yields
With eq 28a we now obtain from eq 27c
which agrees with eq 12b. (b) P Given by c: = c2 = c. This path P i s illustrated in Figure la. Using for q the condition rz,l= constant, we obtain from eq 26b, irrespective of whether adsorption is of the Langmuir type or not,
Formally, this equation is more complicated than eq 20b (which for n = 3 contains only two terms). But it has the remarkable advantage of asking only for partial derivatives taken under the condition P, i.e. at c: = c2 = c. In the special case of competitive Langmuir adsorption we find directly from eq 12a
Of course, the same result can be obtained from eq 30a, but calculation is much more complicated. This is not quite unexpected, since eq 30a has been derived as a special tool for the indirect empirical determination of Ah$ and not for purely analytical calculations. 4.2. Electrolyte Adsorption. As a last example, we consider adsorption of a completely dissociated nonsymmetrical electrolyte K'+Ai- of bulk concentration c. The system consists of the solvent, the cationic species (symbolized by index +) and the anionic species (symbolized by index -). These play the role of components 1, 2, and 3. Then with dr-,, = B dr+,, (electroneutrality), and q given by a constant value of r = r+,l=
As adsorption processes are generally nonstoichiometric in nature, the ratio of the adsorbed amounts dnyl, and dnul of two substances i and j is not constant, but depends on t i e actual changes in bulk composition. If the solution consists of n components, these changes correspond to a path P in the (n 1)-dimensional space of the independent bulk concentrations c2,c3,..,,cn.Such a path has (n - 2) degrees of freedom (in the case of charged components one degree is used up by electroneutrality, so, effectively, there are only (n - 3) degrees of freedom). It is easy to see that the differential molar adsorption enthalpy, too, depends on the course of the path P. In the past, the theoretical treatment of adsorption has focused on singlesolute (or singleelectrolyte) systems without any degree of freedom. Thus, the nonstoichiometry and the path dependence of the heat of adsorption were ignored. Taking this dependence fully into account, we present a definition of the differential molar adsorption enthalpy, Ah$,, for the multicomponent case. On this basis we treat the main topic of this paper: the noncalorimetrical determination of Ah;$ from the temperature dependence of the adsorption process or, in other words, the generalization of the Clausius-Clapeyron equation given by eq 1 to multicomponent adsorption. A first generalizationof the Clausius-Clapeyron equation can be obtained on the level of the individual components. This leads to an isosteric expression for the partial molar adsorption enthalpy Ah;d, eq 20a, and from there to the expression for Ah$,in eq 20b. In the case of multicomponent systems, the isosteric conditions of eq 20b imply that under variation of temperature all surface excess concentrationsare kept constant by appropriate variations ofthe bulkconcentrations. Theaeconditionsaredifficultto realize in practice. Therefore, an effort is made to generalize eq 1 on the level of the adsorption process as a whole and not simply on the level of individual components. Thus Ah$ is found to be related to changes of T,ct, and Ti,,by the general expression given in eq 24. Experimental practice suggests to subject these changes to the same restrictions that define path P (cf. eq 7). Taking the restrictions following from the condition of sustained equilibrium (eq 13a) into account, a single degree of freedom, q, is left over, which can be used to formulate an additional condition relating changes of the cis and rj,l's to changes of T. This leads to eq 26b, which is the central result of our treatment. Compared with eq 20b, this equation considerably reduces the experimental expenditure in many cases of practical interest. Formally, the final expressions obtained for Ah$ may be quite different from that of the simple Clausius-Clapeyron equation, eq 1. For example, eq 3 1 shows that adsorption of a symmetrical electrolyte produces an additional factor of 2. On theother hand, if only a single solute concentration, say CZ, is varied while all others are kept constant one concludes fromeq 27c that the original Clausius-Clapeyronequation, after minor modifications, applies to this case of multicomponent adsorption as well. As can be seen from the definition of Ah$, in eq 1l a (or, perhaps, more clearly, from the example of eq 12b), one should be careful, however, not to confuse Ah$ with the (partial) molar adsorption enthalpy of component 2, Ahid. In our treatment we have used relative surface excesses nyl, but have omitted the additional subscript 1 in the symbol Ah;d of the
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 1135
Heat of Adsorption in Multicomponent Systems partial molar adsorption enthalpy. We will show now that this omission is justified and, for this purpose, temporarily add subscript 1. Equation Sa then becomes
Ahrd = Ah$
i=z
(34)
(35a)
Acknowledgment. This work has been financially supported by the Deutsche Forschungsgemeinschaft (DFG). We thank Dr. J. Seidel, Bergakademie Freiberg,for stimulatingdiscussions.
Written with general surface excesses, ni', the corresponding relation reads (33a)
I- 1
where r r = n J / A . The partial molar adsorption enthalpy Ah:d now is given by
7
i = 1,2,
...,n
(33b)
Tq&,
which should be compared with eq Sb. Instead of eq 11b we now have, as a definition of the differential molar adsorptionenthalpy Ah;d, dH/A = Ah;d d r ,
ri and ri,,are related by4 ci
ri = rZl+ rl;
i = 1,-.,n
The second term corresponds to a displacement of the Gibbs dividing surface by a distance Ax into the bulk region relative to the position yielding rl = 0. Thus, eq 3Sa can be also written as
ri = q l+ ciAx
(33))
With this relation we obtain from eq 33a n
dH/A = x A h ; d dFi,l
+ x A h ; d c id(Ax)
1-2
(36)
i- 1
and, by subtracting eq 32, get n
x ( A h ; d - Ah;:) dI'i,l P Z
+ ( x A h ; d c i )d(Ax) = 0
c1
The first of these two equationsjustifies the omission of subscript 1 in the main text. The second equation states that a pure displacement of the Gibbs dividing surface does not have any enthalpy effect (thus confirming what one would expect intuitively). While, according to eqs 38a,b, the partial molar enthalpies are invariant against displacements of the Gibbs dividing surface. the differential enthalw Ahtd. according to eu 34, clearly varies with changes of dI'z produced by changesof Ax; cf. eq 35b. However, in ideal diluted solutions these changes in dI'z are mostly small compared with the values of r2,1usually encountered. In any case, experimentally determined values of rZare mostly based on the assumption of a constant bulk volume Vb. Since this condition nearly matches with eq 2, the experimental values of rz are approximately identical with r2,1. The present treatment deals with adsorptionfrom ideal diluted bulk solutions. With slight variations this treatment can be adapted to ideal concentrated solutions. The thermodynamic treatment of real solutions, however, is a bit more complicated and will be subject of a separate contribution.
P Z
dH/A = x A h ; d d r ,
(38a)
Ahrd = - CnA h $ - ci
dH/A = x A h $ dI'i,l
=
for i # 1
(37)
i= 1
SincedI'i,l and d(Ax) provide n-independent differential changes, all coefficients written in brackets are necessarily zero. Thus we conclude
References and Notes (1) Mehrian, T.; Keizcr, de A.; Lyklema, J. Lungmuir 1991, 7, 3094. (2) Everett, D. H. Trans. Faraday Soc. 1950,46,453. (3) Denoyel, R.; Rouquerol, F.; Rouquerol, J. J. Colloid Interface Sci. 1990, 136 (2), 375. (4) Defay, R.; Prigogine, J.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans, Green & Co. LTD: London, 1966; p 50. ( 5 ) Wedler,G. Adsorption; Verlag Chemie: Weinheim, Germany, 1970. (6) Hall, D. G. In Adsorptionfrom Solution; Ottewill, R. H., Rochester, C. H., Smith, A. L., Eds.;Academic Prcss: London, 1983; p 31. (7) Woodbury, G. W.; Noll, L. A. ColloidSurJ 1988, 33, 301. (8) Partyka, S.; Rudzinski, W.; Brun, B. Lungmuir 1989, 5, 297. (9) Killmann, E.; Eckart, R. Makromol. Chem. 1971,144,45. (10) Kohler, H.-H. In Coagulation and Floccularion; Dobias, B., Ed.; Dekker: New York, 1993; p 2. (11) Broier, P.; Kiselev, A. V.; Lesnik, E. A,; Lopatkin, A. A. Zh. Fiz. Khim. 1969,43 (6), 1519. (12) Roque-Malherbe, R. J. Therm. Anal. 1987, 32 ( S ) , 1361. (13) Shekhovtsova, L. G.; Fomkin, A. A.; Bakaev, V. A. Izv. Akad. Nauk SSSR, Ser. Khim. 1987,10,2347. (14) Jannakoudakis, D. A.; Nikitas, P. J.; Pappa-Louisi, A. K. Chem. Chron. 1981, 10 (4), 293. (15) Zabransky, M. Chem. Zvesti 1975, 29 (3), 350. (16) Cointot, A.; Cruchaudet, J.;Simonot-Grange,M. H. Bull.Soc. Chim. Fr. 1970, 2, 497. (17) Lason, M.; Terlecka, M. Zesz. Nauk. Akad. Gorn-Hutn. im. Stanislawa Staszica. Gorn. 1977, 85, 97.
