J. Phys. Chem. 1990, 94, 6489-6495
6489
The Compensating-Mixture Model for Multicomponent Systems and Its Application to Ion Exchange P.K. de Bokx* and H. M. J. Boots Philips Research Laboratories, P.O. Box 80000, 5600 J A Eindhouen, The Netherlands (Received: January 19, 1990)
The term "compensating mixture" is introduced to specify a mixture of components that belong to the same compensation class. A compensation class is defined as a group of similar experimental systems described by the same compensation temperature: T, AH/AS,where the enthalpy and entropy changes refer to a phase equilibrium. The sufficient conditions for systems to belong to the same compensation class, obtained from a statistical analysis, are stated. Various expressions for thermodynamic properties of compensating mixtures are derived. Experimental data pertain to the ion-exchangeequilibrium of alkali-metal ions between an aqueous solution and a solid exchanger. Experimental results for the isosteric enthalpy and entropy of exchange as a function of composition are compared with calculated values. A good agreement is observed. Other treatments of multicomponent mixtures (viz., ideal- and regular-solution models) are shown to yield results that do not agree with experiment. For a binary mixture, an equation of state is derived, illustrating the physical significanceof the Compensation temperature of a class.
Introduction In a previous paper we have demonstrated how the thermodynamic properties of systems that are members of the same compensation class can be calculated relative to a single reference.' Phenomenologically, a compensation class can be defined as a group of similar experimental systems described by the same compensation temperature T , AHIAS. Here AH and AS are the enthalpy and entropy changes in a given process. As a practical example, we have shown that pairs of alkali-metal ions belong to the same compensation class with regard to the ion-exchange equilibrium between a solid ion exchanger and an aqueous solution.* It was argued that in ref 1 that a sufficient condition for systems to belong to the same compensation class can be found in the similarity of the interaction potentials. This similarity can be made explicit. In the theoretical section it is explained that differences in potential energy between members of the same class should be attributable to a specific interaction potential. Moreoever, these differences in potential energy can be ascribed to differences in one structural parameter, i.e., a parameter that is independent of temperature and composition. In calculating thermodynamic properties of systems that belong to the same compensation class, no reference is made to a lattice model and the random-mixing assumption (or Bragg-Williams approximation) is not made. This implies that the interaction energy is allowed to vary with temperature. By contrast, in a regular-solution model the interaction energy is temperature-independent. In the following the abbreviation C M (compensating mixture) will be used to designate a mixture of compounds that belong to the same compensation class. As changes in interaction energy with temperature and composition are encompassed in the C M model, one is in a position to calculate equilibrium thermodynamic properties at any temperature and composition once the above-mentioned structural parameters are known. In other words, the C M model stipulates that the structural parameters are invariant with respect to the number of components in a mixture, the composition of the mixture, and the temperature; values for these parameters obtained under any given set of experimental conditions should apply to all temperatures and compositions. It is precisely this property of the C M model that is put to the experimental test in the present work. The C M model has been tested previously in a study of the ion-exchange equilibrium. The equilibrium in question is that between an ideal electrolyte solution and a surface phase consisting ( 1 ) Boots, H . M . J.; de Bokx, P. K. J . Phys. Chem. 1989, 93, 8240. (2) de Bokx, P. K.; Boots, H.M. J. J . Phys. Chem. 1989, 93. 8243.
of a compensating mixture of ions. Linear-elution chromatography was used as the experimental technique. By definition, the application of this technique is limited to conditions in which one of the ions of the exchanging pair is present in great excess. Structural parameters derived from these experiments were thus obtained at the extremes of the composition range. Using frontal chromatography as the experimental technique, we here present results of experiments over the entire mole fraction range. Isosteric (constant composition) enthalpies and entropies of exchange, derived from exchange isotherms at different temperatures, are compared with calculated values using structural parameters obtained from elution experiments. Differences between the CM model and other models of multicomponent systems, viz., idealand regular-solution models, are illustrated. As we believe the surface area to enter into the problem of dispersed-phase equilibria explicitly, we seek to compare the C M adsorption model on the one hand with the Langmuir (ideal)3 and Fowler-Guggenheim ( r e g ~ l a rtreatments )~ of surface layer equilibria on the other. For a binary mixture, an equation of state is derived, illustrating the physical significance of the compensation temperature of a class.
Theory In ref 1 it was shown that systems belong to the same compensation class, if the following general conditions are fulfilled: ( a ) the temperature should be high enough f o r MaxwellBoltzmann statistics to apply and ( b ) the internal energies should be unaffected by the environment. Conditions a and b refer to the assumptions generally made in the statistical treatment of dense phases. Also, the potential energy (v) of a system can always be written as a sum of terms involving N-body potentials
u = Cui(ii)+ Cuij(ii,ij) + ... i
i300 K), it was protected against dissolution by installing a guard column containing the underivatized silica. All salts used were nitrates of suprapur or of a comparable quality (Merck, Darmstadt, FRG). High-purity deionized water was used to make up to the solutions.
Results and Discussion Adsorbed amounts as a function of the concentration ratio in solution were measured by frontal chromatography. The results for Li-Na, Li-K, and Li-Rb exchanges are presented in Table I. The experimental values are to be compared with the calculated data given in the same table. Mole fractions in the surface phase were obtained by numerically solving for x i in eq 1 1. Values for the structural parameters were taken from elution experiments reported previously? This implies that these f values were obtained under conditions where the surface concentrations of one of the exchanging ions can be neglected. The set of structural parameters used is given in Table 11. From the good agreement between calculated and experimentally observed surface concentrations, it is concluded that thefvalues obtained at the extremes of the composition range are also applicable at all intermediate mole fractions. In Table I1 structural parameters are tabulated as 2aCf, fRbRb), where Rb is arbitrarily chosen as the reference. As only differences between structural parameters can be measured, the value of the factor a cannot be determined. This need not bother us too much. All experimental observables contain this same factor so that a consistent set of structural parameters suffices for their evaluation. In Figure 1 the exchange isosteres for K-Li exchange are plotted. The data were obtained by measuring exchange isotherms at five different temperatures. Data at constant surface composition were obtained by fitting the exchange isotherms to an equation of the same form as e q l 1. The concentration ratio in solution at a given surface composition was calculated by using
6492 The Journal of Physical Chemistry, Vol. 94, No. 16, 1990
1I In
de Bokx and Boots
, 3.0
X ' X '
2.0
0
1
1 .o
0
.- -./.-.9 3.0
0.6
.lo ____)
.-O,? 3.5
-.-.-.--
T-'
(K-'x i O 3 ) -20
-1.0
-/.-
-C
0.9
Figure 3. Isosteric en opies of K-Li exchange deriv from Figure 1. Error bars denote one standard deviation. The solid line is calculated by using eq 14. The dashed line indicates ideal and regular behavior.
-2.0
Figure 1. Adsorption isosteres for K-Li exchange at different fractional coverages ( x i ) .
-4
2
4
SdThm(kJmol)-
Figure 4. Plot of 0vs AGTh for the alkali-metal exchanges. Column: Chromsep Ionosper C. The data are evaluated at Thm= 319.1 K. The observed linear compensation corresponds to a compensation temperature of 540 f 30 K.
Figure 2. Isosteric enthalpies of K-Li exchange derived from Figure 1. Error bars denote one standard deviation. The solid line is calculated according to eq 12. The dashed line indicates ideal behavior.
the fitted parameters. Relative changes in surface concentrations with temperature become exceedingly small at high surface concentrations. Therefore, isosteres referring to fractional coverages of 0.7,0.8, and 0.9 were obtained from isotherms of Li-K rather than K-Li exchange. They refer to measured Li surface concentrations. Isosteres at intermediate coverages hence have the highest experimental uncertainty. The isosteric enthalpy of exchange corresponds to the slope of the exchange isosteres. Figure 2 shows the enthalpy of exchange as a function of surface composition. Error bars in the figure correspond to one standard deviation as obtained by regressing In xLi/xk onto 1/ T . The solid line indicates enthalpies of exchange calculated by using eq 12 and the data given in Table 11. The good agreement between calculated and experimentally observed
values is evident. Also indicated in the figure (dashed line) is the curve for an ideal solution, Le., no interaction between ions in the surface layer and hence a constant enthalpy of exchange. The height of the line is, of course, arbitrarily chosen. Specifying to a two-dimensional phase, it is seen that this approach, which corresponds to assuming Langmuirean adsorption behaviour, is inadequate. As both the CM model and the regular-solution model predict a linear dependence of the isosteric heat of exchange on surface composition, no distinction between the two can be made on the basis of the enthalpy of exchange. However, from the intercepts of the isosteres presented in Figure 1 the isosteric entropies of exchange can be evaluated. The results are presented in Figure 3. Again, error bars denote one standard deviation. The solid curve was calculated by using the CM model (eq 14). The dashed curve indicates the ideal configurational entropy. Note that nonideality in our case not only shifts the entropy of exchange along the ordinate axis but also induces deviations from symmetry about YK = 0.5. Here, differences between models that make the random-mixing approximation and those that do not become apparent. By inspection of eqs 11-14, it can also be appreciated that the C M model implicitly offers an explanation for the often made observation that the regular-
Compensating-Mixture Model for Multicomponent Systems solution model frequently predicts free energy changes correctly, but the description of enthalpy and entropy changes separately is rather p00r.l~ Figure 4 shows a plot of flvs AeTb (i.e., the Gibbs free energy evaluated at the harmonic mean of the experimental temperatures) for monovalent cation exchange on a silica-based column. The results were obtained by using elution chromatography. They hence pertain to the linear part of the exctange isotherm. The results are_ presected as a plot of AH vs ACThmrather than as a plot of AH vs AS to avoid correlation between abscis!a and ordinate values. It has been argued that regressing AH onto AS leads to results that are heavily confounded with statistical error.I4 The compensation temperature can be obtained from the slope (c) of the line in Figure 4 according to T, = Th/( 1 - (l/c)). The result 540 f 30 K is in good agreement with the value obtained for the resin material (480 f 20 K). It should be emphasized that silica-based ion-exchanger material is very different from the resinous exchanger. The exchange site density of the silica column is more than an order of magnitude lower than that of the resinous exchanger. Although the value for the latter appears to be unrealistically high, probably due to the use of Kr adsorption as a measure of surface area, the constancy of the compensation temperature seems to support the conclusion drawn earlier that T, is a property of a class of compounds, independent of the type and the site density of the exchanger material used. The invariance of the compensation temperature with the site density cannot be easily explained without a more detailed model of the ion-exchange process. For the time being we will treat it as a very intriguing experimental observation. The invariance of T, with the site density can be used to gain a better understanding of the physical significance of the compensation temperature of a class. To that end, we derive the equation of state for the two-dimensional surface phase: y(p,x,,T). Changing the surface tension means in our case that the twodimensional density p of adsorption sites is changed, while the number of sites remains the same and all sites remain occupied. Therefore, we express the surface tension in terms of the density p of (occupied) sites where N is the number of (occupied) sites and A is the free energy change,' given by A = A.
+ N k B T x x pIn x p + N f i E x d c 4 ( f ,
In this way one easily derives the entropy-related terms that occur in models with a variable number of empty sites.15 There is a similarity between the definitions of the Boyle temperature of a pure compound and the compensation temperature. At the Boyle point the second virial coefficient vanishes, so that the system looks ideal, while at the compensation point the difference in interactions in different systems of the same compensation class vanishes, so that the mixtures in the compensation class behave ideally. This can be understood by using two experimental results: the linearity of $ with T (eq 6 ) and the invariance of T, with the site density. From eq 19 it can be seen that the surface tension of any mixture is equal to that of the reference at T = T,. This does not necessarily imply ideal behavior of the surface phase at T = T,. All deviations from ideality of the reference are absorbed into y o . Throughout the literature on surface phases the distinction between mobile and localized layers is made. The C M model differs from the regular-solution description of a surface layer in the way interactions between sorbed particles are treated. In order to calculate the total interaction energy, a site model, specifying the number of nearest neighbors, is applied by Fowler and Guggenheim. They refer to their model as regular localized monolayers. As no reference to a lattice model is made in our derivation of the CM, it is of interest to pursue the question of whether a distinction between the two extremes in mobility can be made on the basis of our experiments. To do so, let us first compare expressions for the quantity we can measure: p A - pBfor a two-dimensional ideal gas on the one hand and an ideal lattice gas on the other. In the first case p A = R T In NA and pB = R T In NB Hence
where x is the mole fraction of component A. We should stress that only the quantity p A - pBhas physical significance in our case; adding a particle A without removing a particle B is prohibited by the restriction imposed by electrical neutrality. For an ideal lattice gas we have
- 1) (18)
P 4
P
The Journal of Physical Chemistry, Vol. 94, No. 16, I990 6493
Hence
Applying eq 17 to eq 18, the equation of state becomes
which may be shown to be consistent with the Gibbs adsorption equation: P-' d r = C x , dcL,
(20)
In the process of ion exchange, the change in surface tension contains only the interaction term, which is what one would expect. In the usual formulas for the surface tension one finds entropyrelated terms, but there empty adsorption sites are allowed. In varying the surface area, one varies the total number of sites (N), while keeping the area per site ( u ) constant. Then A contains the extra contribution k B T [ nIn n ( N - n) In ( N - n ) ]
+
where n is the number of occupied sites. Then the surface tension is proportional to the chemical potential of sites =
Y dN ",*
(13) See e&: Hildebrand, J. H.; Prausnitz, J . H.; Scott, R. L. Regular and Related Solurions; Van Norstrand: New York, 1970; Chapter 7. (14) Krug, R. R.; Hunter, W. G.; Grieger, R. A. J. Phys. Chem. 1976,80, 2335, 2341.
where Nto,is the total number of available sites. To highlight the difference in individual chemical potentials, allowance is made for empty sites in the derivation of eq 23. The equations will, of course, still hold if the restriction of full coverage is made. The important conclusion is that the only relevant chemical potential p A - wB is the same for localized and mobile surface layers in the ideal case. For nonideal mixtures there is always the ideal contribution. Deviations from ideality are caused by interactions between sorbed particles that do not have a direct bearing on the mobility of the sorbed particles. Following an early conclusion by Everett,I6 we state that discrimination between mobile and localized adsorption is not readily possible on the basis of equilibrium data. Although enthalpies of exchange are quite small (several kJ/mol), these values could still represent small differences between very large numbers. Obviously, a kinetic argument is needed to tackle the problem; Le., information in the height of the energy barriers between adjacent sites is required. Limiting ourselves to chromatographic techniques, kinetic information can be obtained from band broadening in the elution mode. Unfortunately, the second statistical moment of a concentration dis(15) Spamaay, M. J. The Electrical Double Luyer; Pergamon Press: Oxford, 1972; Chapter 2. (16) Everett, D. H. Trans. Faraday SOC.1950, 46, 942.
6494 The Journal of Physical Chemistry, Vol. 94, No. 16, 1990
Conclusions 1. By use of frontal chromatography as the experimental technique, the CM model has been corroborated with respect to the invariance of structural parameters with temperature and composition. Values for structural parameters obtained with any set of conditions (Le., at the extremes of the composition range using elution chromatography) can be used successfully to predict thermodynamic properties at all other temperatures and compositions. 2. The CM model for two-body interactions as defined in the theoretical section leads to temperature-dependent interaction energies. In cases where enthalpy and entropy changes can be considered independent of temperature, interaction energies are linearly dependent on temperature. Through this temperature dependency both the variation of the enthalpy of exchange and that of the entropy of exchange with coverage can be described well, in contrast with models in which random mixing is assumed.
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6 X'Rb
-
0.8
1
Outline of the properties of a compensating binary mixture. Isotherms were calculated for Rb-Li exchange according to eq I 1. From left to right the curves correspond to temperatures of 100, 200, 300, and
Figure 5.
480 K.
tribution in elution chromatography is determined by contributions from several rate processes from which the contribution due to rate of the actual ion-exchange process cannot be easily separated. To solve the problem of surface mobility, there must therefore be direct measurement of the surface diffusivity. Due to the interactions between sorbed particles, there may exist a critical temperature below which the monolayer becomes unstable and will split up into two separate surface phases. In regular solutions the critical temperature is solely determined by differences in interaction energies between like and unlike particles, in such a way that a critical temperature exists if the interaction energy between unlike particles is larger than that between like particles and that the critical temperature does not exist in the opposite case. This dependence of the critical temperature solely on interaction energies is due to the assumption of random mixing made in the regular-solution model: the differences in interaction energy are assumed to have no influence on the distribution of adsorbed particles, Le., on the surface entropy term. As this assumption is not made in the CM model, we will now show how the interpretation of the critical temperature is modified. For simplicity we restrict ourselves to two-component systems. The condition for a critical point is a horizontal point of inflection in the curve of x' vs xs, Le., 8x'/8xS = 0. Writing xB as 1 - xA and applying the differentiation to eq 1 1 , we find
where Tdt is the critical temperature, all other quantities having been defined in the theoretical section. It is seen that there exists a positive Tcri,not only for positive values of adAB (interactions between like particles larger than those between unlike particles) but also for negative values of adAe as long as 2 a d < ~ -4RTc ~
de Bokx and Boots
(25)
The consequence of not using the random-mixing assumption hence is that there is an additional tendency toward phase separation caused by the surface entropy term. In Figure 5 the properties of a binary compensating mixture are summarized by plotting calculated isotherms according to eq 11. For the calculated example (Rb-Li) a critical temperature does not exist because adAB< 0 and the condition of eq 25 is not met. At increasing temperatures the surface phase becomes progressively less enriched in Rb relative to the solution composition. At the compensation temperature the solution composition at equilibrium is equal to the surface composition over the entire mole fraction range. It would seem appropriate to use the term azeotropic line for the exchange isotherm at the compensation temperature.
Appendix BlanderIO takes second-order contributions into account in his derivation of compensation effects in ionic mixtures. The effect originates from the interaction of two positive (negative) ions with the same negative (positive) ion. In our case a similar effect may be present, in the sense that two adsorbed ions may interact indirectly via the wall potential. This point is addressed here. For conformal solutions the second-order theory has been worked out by Brown,* who also discusses the subtle point of the randommixing approximation in second-order contributions. According to ref 1 the free energy, relative to the free energy of a reference system, is given by A - A o = A , i x + ( U - U o ) - l / z @ [ ( ( U - U o ) 2) (U-Uo)2] (AI) where p = l/(kBT) and where the ensemble average of a quantity X in the reference system is given by 1 ( X )= df, ...d f , . X ( f l ,...,iN) exp(-pU0) (A2)
-J ZO
with Zo the configurational partition function in the reference system Zo = j d f ,...dfN exp(-pUo)
(A31
The average energy is expanded to second order in structural parameters f - fo
where we specialized to a system containing two types of exchanging ions. In the following we shall make use of the identity
+ XBVB -f0)12 = + XBVB-fo12 - X A X B V A -feI2 (A5) xAVA so that, up to second order [xAVA
(U -
-fo)
uop = p [ x A V A -f0I2
+ XBvB -fO)*
- xAxBCfA
-fBI21
(au/a!2
('46) For the evaluation of eq A1 we also need
J . Phys. Chem. 1990, 94, 6495-6503
where the term containing AI accounts for the i = j contribution in the sum
6495
between the chemical potential of the mixture and a standard chemical potential, which we choose to be the chemical potential for xA = 0. Then the free energy difference of interest is
The averages in eqs A7 and A8 are defined by where
Thus, the second-order contribution is partly symmetric and partly antisymmetric under the operation of interchanging A and B. This could provide an explanation of the lack of symmetry in the experiment, if it were not for the universality of the compensation temperature within a compensation class. As has been explained in ref 1, compensation only occurs if one term dominates or if the temperature dependence of all dominating terms is the same. We do not see a physical reason for the temperature dependence of Ai to be the same as the temperature dependence of or of A. Therefore, the asymmetry of the experimental result cannot be explained as a second-order effect of interactions with the resin.
(&/an
Photophysics of Poly(2,3,4,5,6-pent af luorostyrene) Film Donald B. O’Connor, Gary W. Scott,*,+ Department of Chemistry, University of California, Riverside, Riverside, California 92521
Daniel R. Coulter, Vincent M. Miskowski, and Andre Yavrouian Space Materials Science and Technology Section, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91 109 (Received: December 27, 1989)
The temperature-dependent steady-state emission, emission polarization anisotropy, and fluorescence kinetics of poly(2,3,4,5,6-pentafluorostyrene)film are reported. Two interconverting excited-state conformations of the chromophore have been identified. The fluorescence of the higher energy conformation results from excitation on the red edge of the polymer absorption band at temperatures below 180 K. The energy barrier for conversion of the higher energy conformer to the lower energy conformer is estimated to be E/hc = 27 f 7 cm-’. Electronic energy migration is not evident in this polymer.
1. Introduction There has been significant research activity in recent years aimed at understanding the photophysical and photochemical role of electronic energy transfer in long-chain, amorphous polymers containing aromatic pendant chromophores.Id Examples include polystyrene, poly(vinylnaphthalene), and poly(vinylcarbazo1e). One particular area of concern involves understanding the mechanisms of both photostabilization and photodegradation’ in such polymeric systems. Photostable polymeric materials are used widely in the solar energy industry especially as coatings for solar collectors, optical elements, and protective containers.* We have previously reported on the role of electronic energy transfer in the photophysics of polystyrene copolymerized with a photostabilizing c h r o m o p h ~ r e . ~ * ~ ~ The photophysics of these long-chain polymers containing aromatic chromophores are similar in many ways. The absorption spectra of these polymers closely resemble the absorption spectra of their respective chromophores. For example, polystyrene ab-
(1) OConnor, D. 9.;Scott, G.W. Laser Studies of Energy Transfer in Polymers Containing Aromatic Chromophores. In Applications of Lasers in Polymer Science and Technology; Fouassier, J. P., Rabek, J. Eds.; CRC Uniscience: Boca Raton, FL, in press. (2) Guillet, J. Polymer Phorophysics and Photochemistry: An Introduction to the Study of Photoprocesses in Macromolecules; Cambridge University Press: London, 1985. (3) Phillips, D. Ed. Polymer Photophysics: Luminescence, Energy Migration and Molecular Motion in Synthetic Polymers; Chapman and Hall: London, 1985. (4) Kloepffer, W. Aromatic Polymers in Dilute Solution. Elecironic Energy Transfer and Trapping ACS Symp. Ser. 358 (Photophys. Polym.); American Chemical Society: Washington, DC, 1987; pp 264-285. ( 5 ) MacCallum, J. R. Significance of Energy Migration in the Phorophysics of Polystyrene; ACS Symp. Ser. 358 (Photophys. Polym.); American Chemical Society: Washington, DC, 1987; pp 301-307. (6) Fredrickson, G. H.; Andersen, H. C.; Frank, C. W. Electronic Excited-State Transport and Trapping on Polymers Chains. Macromolecules 1984,
‘NASA-ASEE Summer Faculty Fellow, Jet Propulsion Laboratory, 1988 and 1989.
Photostabilization of Polymers: Principles and Applications; Wiley: New York, 1975.
sorption is much like a monoalkyl-substituted benzene absorption. This observation is indicative of weak interaction in the ground
17,5449. (7) Ranby, B.; Rabek, J. F. Photodegradation, Photooxidation, and
0022-365419Ol2094-6495%02.50/0 0 1990 American Chemical Society
