842
J. Phys. Chem. 1992, 96, 842-845
by any mechanism, will always lead to an apparent threshold behavior of the net current for the fuel-forming reaction in the external circuit of the cell. Therefore, the only mechanism by which this threshold can be eliminated is to totally suppress all of the dark reactions, Le., to set k, = 0 in eq 14 and kh;l = 0 in eq 2. The kh;l = 0 assumption is not consistent with microscopic reversibility arguments for the interfacial hole-transfer process.z7 These assumptions of k, = 0 and kh;l = 0 are not justified for real semiconductorsystems, because there are always back-reaction pathways by which the majority carriers can transfer into the solution; additionally, recombination reactions of the majority carriers with photogenerated minority carriers inside the semiconductor cannot be neglected and set to In the dark, the presence of a potential difference between the semiconductor electrode and the metal counterelectrode will ensure that the carrier concentrations in the semiconductor deviate from their equilibrium values; any recombination mechanisms that tend to restore these concentrations to their equilibrium positions must be considered in determining the net current flowing through the external circuit. Even if no trapping sites are present, thermal generation/recombination processes such as radiative luminescence will act to restore the carrier concentrations toward their equilibrium values.z8 The second-order rate law for eqs 1 and 2 also need not apply to the semiconductor/liquid junction for a threshold to occur, because any interfacial charge-transfer process will possess a corresponding reverse rate that cannot be neglected. Thus, the above treatment always predicts a threshold in fuelforming ability, regardless of the actual values of the interfacial minority or majority carrier charge-transfer rate constants in the system. Some of the evidence supporting the stochastic model for interfacial charge transfer has been obtained from the observation of high quantum yields for photocurrent collection at the nTiOz/liquid interface and from experiments indicating the possibility of unthermalized, hot hole transfer at TiOz/H20junct i o n ~ . ~Within ~ the kinetic framework developed above, the (27) Note that khcl can be neglected in calculating the total 'back reaction" rate when kh;'[A-] is small compared to knn,[A]. However, when using detailed balance to obtain the forward vs backward rate constants for the valence band process (eq 3), the ratio k h t / k b [ l must be considered, and kb;' can therefore never be set to zero for a finite equilibrium constant for this charge-transfer process. The value of kbcl becomes important when considering the lower limit on the value of the total anodic back reaction rate when k,n,[A] is small compared to the valence band anodic hole injection flux. (28) Blakemore, J. S. Semiconductor Statistics; Dover: New York, 1987. (29) Kasinski, J. J.; Gomez-Jahn, L. A,; Faran, K. J.; Gracewski, S. M.; Miller, R. J. D. J . Chem. Phys. 1989, 90, 1253.
observation of a photoelectrolysisthreshold is predicted regardless of the value of kht; in fact, the assumption of eq 7 that ps a pso is more rigorous for larger values of kht. The hole continuity expression of eq 7 implies that as long as competing recombination pathways are sufficiently small, the condition khgs[A-] = 4ro will be met, and the quantum yield for collected minority carriers will approach unity regardless of the actual value of kht. This condition is obviously expected to hold at large positive values of Em, because the strong interfacial electric field will reduce n, and thereby suppress all bimolecular interfacial recombination rates at the semiconductor/liquid junction. If khtis very small, then ps must build up relative to pm in order for current continuity to be satisfied; this will simply act as an overpotential for hole transfer and will act to increase the light intensity threshold value at which the photovoltage developed by the semiconductor is sufficient to sustain the water electrolysis reaction. Thus,the observation of a threshold for photoelectrolysis does not directly address the value of khtor the issue of whether thermalized or hot hole transfer has occurred at the n-SrTi03/ NaOH(aq) interface. In conclusion, the present experiments indicate that charge transfer at the n-SrTi03/5.0 M NaOH(aq) interface can be described using a conventional kinetic treatment of the collective behavior of photoexcited semiconductor carriers. The free energy of the illuminated SrTi03/NaOH(aq) interface is a function of the light intensity incident on the semiconductor surface, and only when the photovoltage exceeds the electromotive force for production of 1 atm of Hz(g) and 1 atm of O,(g) from HzO can a sustained photoelectrolysisof water be effected. The experimental data for SrTi03-based potentiostatic and nonpotentiostaticcells are consistent with this conclusion. Straightforward expressions have been presented that offer an intuitive, chemical kinetic approach to this behavior, and this kinetic treatment appears to describe satisfactorily the behavior of typical regenerative photoelectrochemicalcells and photoelectrosyntheticcells explored to date. The observation of a photoelectrolysis threshold does not directly address the question of whether interfacial charge transfer occurs from a thermalized carrier distribution, and such conclusions require independent experiments that probe the carrier dynamics and thermalization times in the system of interest. Acknowledgment. We thank the National Science Foundation, Grant CHE-8814263, for support of this work. A.K. also is grateful to the US.Department of Education for a graduate fellowship. Registry No. H 2 0 , 7732-18-5; SrTi03, 12060-59-2; 02,7782-44-7; H2,1333-74-0; NaOH, 1310-73-2; Pt, 7440-06-4.
Theory of Phase Separation in Mixtures with Lower Critical Solution Temperature Ming Yu and Hidm Nishiumi* Chemical Engineering Laboratory, Hosei University, Koganei, Tokyo, Japan 184 (Received: May 31, 1991;
In Final Form: August 12, 1991) A thermodynamic theory of phase separation in mixtures possessing LCST is presented which incorporates (1) chemical equilibrium theory to account for cross association between solute and solvent, (2) Flory-Huggins theory for the difference of molecular volumes, and (3) NRTL equation for the interaction between molecules. The theoretical predictions of binodal curves over a wide temperature range are in good agreement with experimental findings in some binary aqueous solutions. The model can also reproduce the flat coexistence curves near the consolute points.
Introduction Some binary liquid mixtures which form a single, homogeneous phase at low temperatures possess a lower critical solution temperature (LCST) above which phase separation occurs. Lower critical solution points are not confined to mixtures of small molecules but, in fact, have been observed in some complex
mixtures, such as the aqueous solutions of surfactants and of water-soluble Because of its theoretical and practical (1) Schick, M. J. Nonionic Surfactants; Surfactant Science Serries, Vol. 1; Marcel Dekker: New York, 1967. (2) Warr, G. G.; Zemb, T. N.; Drifford, M. J . Phys. 1990, 94, 3086.
0022-3654/92/2096-842%03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 843
Phase Separation in Mixtures
by solving simultaneously eq 3 and following material balance equations
TABLE I: Values of Factor q, comwnent water
2-isobutox yethanol 1-ethylglycerol 3-isopropyl ether
1-propoxy-2-propanol 2,6-dimethylpyridine
41
0.9200 5.0462 6.6381 5.0462 4.4693
(4) ir
importance, the theoretical description of lower critical solution points in binary liquid mixtures has advanced signifmltly in recent years.+l2 There exist several competing descriptions of LCST phenomenon. In the one of these models, this phenomenon is explained as due to hydrogen bonds between unlike species. At low temperatures two liquids are completely miscible due to the strong hydrogen bonds, but as the temperature rises, molecular rotation increases, causing the hydrogen bonds to break. If dispersion-force interactions between unlike species are weaker than those between like species, the system splits into two phases. Although this explanation seems less general, the LCST phenomenon in many mixtures can be explained with this theory successfully. In the previous work,13 by combining the ideal of multiplestate interaction with NRTL equation, we could correlate LLE quantitatively along the coexistence curve close to or remote from both lower and upper critical conditions in binary liquid mixtures. The main ideal of the multiple-state interaction theory9 is that the local interaction can be divided into van der Waals-like and hydrogen bonds. In this work we use another physically more transparent approach to take into account the effect of hydrogen bonds on phase separation. Within this approach, the relevant thermodynamic parameters have more evident physical significance than in the previous one, and its application to polymer solutions and the generalization to treat multicomponent systems are straightforward.
outline of the Thermodynamic Model Consider a solution of Nw solvent molecules (water) and Ns solute molecules at temperature T and pressure P. We model the excess Gibbs free energy GE as consisting of two additive parts GEphys and GEchem, that is
GE = GEchem + GEphys
where asand aW are the "apparent" volume fraction of chemical species and r is the ratio of molecular volume of solvent to that of solute. The molecular volume of component is represented with the volume factor qi in UNIQUAC equation,I4which is obtained basically from crystallographic measurement. The values of qi for the substances concerned in this work are given in Table I. The temperature dependence of the association constant K is given by the Gibbs-Helmholtz equation PS
AH R P
if AH, the enthalpy of formation of complex SiWj, can be regarded as independent of the temperature, eq 6 may be integrated:
K = exp(-AH/(RT)
+ C)
(7)
Here C is a constant. From Flory-Huggins theory,Is it can be shown that if reaction 2 is included, the apparent activity coefficients of solute and solvent can be expressed as In ySchem = In
In yWchem = In
%w, i+jr
aw,
as,+ In xs + 1.0 - Os,- -- - (8) @SlWJ
r
awl+ In xw + 1.0 - a w l - - r@sl i +jr
(9)
B. Physical Contribution. The physical contribution may be represented by any physical theory model, for example, the Scatchard equation, the van Laar equation, or the Wilson equation. In this work, NRTL equation16is chosen to represent the physical contribution.
(1)
A. Chemical Contribution. It is assumed that there exists following cross association or solvation reaction an equilibrium constant due to the hydrogen bonds between solvent and solute i s I + j W , + SiWj
(2) where the NRTL parameter for ij pair rij is defined as rij =
(3)
where i and j are "average" association numbers. awl,and aSIw are the 'true" volume fraction of molecular species SI, WI, and kiwi respectively. The true volume fraction can be obtained (3) Walker, J. A.; Vause, C. A. Sei. Am. 1987, 253, 98. (4) Tager, A. Physical Chemistry of Polymers; Mir: Moscow, 1972. (5) Saeki, S.;Kuwahara, N.;Nakata, M.; Kaneko, M. Polymer 1976,17, 685. (6) Karlstrom, G. J. Phys. Chem. 1985, 89, 4962. (7) Prange, M. M.; Hooper, H. H.; Prausnitz, J. M. AIChE J . 1989, 34, 1595. (8) Goldstein, R. E.; Walker, J. S.J . Chem. Phys. 1983, 78, 1492. (9) Goldstein, R. E. J . Chem. Phys. 1984,80, 5340, 1985,83, 1246; 1986, 84, 3367. (10) Wheeler, J. C. J . Chem. Phys. 1975, 62, 433. Anderson, G. R.; Wheeler, J. C. Ibid. 1978, 69, 2082. Wheeler, J. C.; Anderson, G. R. Ibid. 1980, 73, 5778. ( 1 1 ) Kim, Y. C.; Kim, J. D. Fluid Phase Equilib. 1988, 44, 229. (12) Thurston, G. M.; Blankschtein, D. J.; Fisch, M. R.; Benedek, G. B. J. Chem. Phys. 1986,84,4558. (13) Yu, M.; Nishiumi, H. Manuscript in preparation.
(e, - ejj)/(RT)
and
Gji = exp(-ajirji) In this work, nonrandom parameters aji = aij = 0.2.
Results and Discussion In the present theory, for each binary pair there are six parameters: two NRTL parameters (el2- e22)and (e21- ell), two "average" association numbers i and j , enthalpy of formation AH, and the constant C in eq 7. Because i and j are average association numbers, it is possible for i and j not to be equal to integers. For (14) Sorensen, J. M.; Arlt, W. Liquid-Liquid Equilibrium Dam Collection, Binary Systems; Chemistry Data Series, Vol. V, Part 1; DECHEMA: Frankfurt/Main, FRG, 1979. (1 5) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (16) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd d.;Prentice-Hall: Englewood Cliffs, NJ, 1986.
Yu and Nishiumi
844 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 50
TABLE E Values of Binary Parameters l
(e12- e22)/
(e21 - ell)/
R i AHIR C 2731.4291 2.6205 6013.1353 -17.9452 2870.4307 2.1825 5711.4319 -16.6314 -192.7779 2597.4406 2.2937 5119.0628 -14.5812 -510.5773 2797.5390 2.4158 5366.3187 -14.6612 a 1. 2-Isobutoxyethanol and water. 2. 1-ethylglycerol 3-isopropyl ether and water. 3. 1-propoxy-2-propanol and water. 4. 2,6-Dimethylpyridine and water. mixture"
R -415.0799 66.0799
1 2 3 4
20
0.00
0.05
0.10 0.15 0.20 MOLE PER CENT OF I11
0.25
Figure 1. LLE for the system of 2-isobutoxyethanol(l) and water (2).
80
I?
I I
701
0 EXPERIMENTAL DATA OF DAVISON ET AL. 0 CINOLUTE POINT FAOM THE THEORY
60
Figure 3. LLE of 2-propanol (1) and water (2).
I
190,
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0 EXPERIMENTAL SATA
MOLE PER CENT I1 1 Figure 2. LLE for 1-ethylglycerol 3-isopropyl ether (1) and water (2).
the sake of simplicity, in this work i is arbitrarily set equal to 1.0. The other parameters can be determined from experimental LLE data. The coexistence curve is obtained by solving i = 1,2
8
0
140
/J/ "\
0 OF CQuSoLUTE F L A POINT M ET AL.
\ o
FROM THE THEORY
(14)
where single and double primes refer respectively to the bottom and top phase and ai is the activity of component i. The critical concentration, xc, and critical temperature, T,,can be found by solving simultaneously the following equations a2Ag/ax2 = o
(15)
Figure 4. LLE of 2,6,-dimethylpyridine(1) and water (2).
o
(16)
That means that the cross association can account for not only the existence and speclfic value of LCST but also the ''nonclassical* nature that occurs at the consolute points and makes the coexistence curves flat near the points. B. Phase Diagrams with Both LCST and UCST. For systems with a LCST, the mixtures are homogeneous below the LCST and form a two-phase systems if the temperature is higher than the LCST. At an even higher temperature, the ordinary mixing entropy will dominate, thus making the systems homogeneous again and closing the solubility gap. Figures 3 and 4 show the closed-loop coexistence curves and the consolute points of two binary mixtures calculated from the theory together with experimental points. The model parameters used in the calculation for each system are also given in Table 11. The model reproduces fairly well temperature-composition diagrams which are asymmetric and closed in a loop. The coexistence curves from the model near both LCS points and UCS points are also very flat. The agreement between theory and experiment in lower critical regions is better than in upper critical regions because more attention have been paid to LCS points than to UCS points in the calculation. All the experimental data used in Figures 1-4 are from DECHEMA Chemical Data Series.I4 The algorithm does exhibit a problem, however. In determining the coordinates of LCST and UCST, eqs 15 and 16, two highly nonlinear equations, must be solved simultaneously by numerical iteration, which cannot always guarantee convergence in the cases of UCST.
a3~g/ax3=
where Ag, the molar Gibbs energy of mixing, is given by the sum of the molar excess Gibbs energy of mixing and the ideal Gibbs energy of mixing. In the following, three types of examples of complex phase diagrams are calculated. First, systems exhibiting LCSTs are studied. There then follows a discussion of systems exhibiting closed-loop coexistence curves of liquid-liquid equilibrium. A. Phase Diagrams with LCST. Mixtures in which there exist significant hydrogen bonds between unlike species may exhibit partial miscibility with a LCST. Complete miscibility a t temperatures below the LCST is attributed to the cross association due to the hydrogen bonds. Because association reaction is exothermic, an increase in temperature decreases the equilibrium constant K. A decrease in K entails a smaller yield of SiWj or the break of some hydrogen bonds between solvent and solute, causing mutual solvability to decrease the phase separation to occur. Figures 1 and 2 show the coexistence curves and lower critical solution points for two binary liquid mixtures calculated from the theory together with the experimental mutual solubility data. The model parameters used in our calculation for each system are given in Table 11. It is important to note that the AH obtained from regressing experimental data have magnitudes and signs consistent with the physical premise of our model. It can be seem that the theoretical coexistence curves agree with experimental data quite well and the curves near the consolute points are very flat, also.
J. Phys. Chem. 1992.96, 845-854
Conclusions In & work we have pro* a theory Of phase in mixtures LCST* The Of LaT is due to strong, highly directional interaction forces, such as hydrogen bonds. The model incorporates (1) chemical association theory to account for the hydrogen bonds, (2) Flory-Huggins theory for the volume difference between components, and (3)
845
NRTL equation for the interaction between m o l d e s . The model gives a good representation of binodal curves over wide temperature range. The flat coexistence curves near consolute points because of the Ynonclassical*nature can be r e p r o d u d well by the theory* Acknowledgment. M.Y.is grateful to the HOSE1 University for a fellowship.
Selective Adsorption of Simple Mixtures in Slit Pores: A Model of Methane-Ethane Mixtures in Carbon Ziming Tant and Keith E. Gubbins* School of Chemical Engineering, Cornel1 University, Zthaca, New York 14853 (Received: June 7, 1991)
We report a study of Lennard-Jones mixtures in model carbon pores having parallel walls. The fluid potential parameters were chosen to model methane and ethane, and the 10-4-3 model was used for the solid-fluid potential. A density-functional theory with the nonlocal density approximation was used for the calculations. We focused on the selectivity of ethane relative to methane for a wide range of system parameters. Different types of selectivity isotherm were found, which can be explained microscopically in terms of intermolecular and surface forces. Macroscopically,each type of isotherm corresponds to a certain range of temperature in relation to the capillary critical temperatures. At very low temperatures where layering transitions occur, the selectivity isotherm is steplike. The density profiles show a strong surface segregation of ethane from methane.
1. Introduction When a mixture is codmed in a small pore, the range of surface phenomena that exist is considerably wider than for pure fluids. While some properties (capillary condensation, layering transitions, critical behavior, etc.) are natural extensions of those for a pure fluid, there are properties, such as selectivity and segregation, unique to mixtures. These properties are of particular interest to separation processes based on adsorption. Although problems involving mixtures are much more frequently encountered in industrial processes, the behavior of mixtures is relatively less well understood than that of pure fluids, because of the addition of composition as a variable. Recent developmentsin theories for nonuniform fluids make it possible to investigate the behavior of simple mixtures in mimpores. While simulation methods provide a reliable approach for such studies, bhheories are much less expensive and hence allow us to carry out extensive calculations in a short time. In this work, we report a study of a Lennard-Jones (LJ) mixture in slitlike pores. The system is modeled on methane and ethane in porous carbon with graphite pore walls. For this system, documented experimental data are limited.' Szepesy et al? have measured the mixture in activated carbon at 293.15 K and 100 kPa for a range of bulk composition. They found that the selectivity of ethane relative to methane (see section 3) increases as the bulk mole fraction of methane increases. For a different brand of activated carbon, Costa et aL3 reported results at the same temperature but wering a range of pressure up to 100 kea. In this pressure range, the total amount adsorbed was found to increase as the concentration of methane decreases. The work of Reich et a1.4 has w e r e d a range of higher pressures (140-2000 kPa) for several bulk mole fractions and temperatures (212.7-301.4 K). Their results also show that the selectivity decreases as the pressure increases. Their results indicate that the selectivity can decrease as the bulk methane concentration increases. Since these experimental data are within a rather narrow range of bulk conditions, it is difficult to draw a general conclusion regarding the variation of the selectivity against the 'Present address: Exxon Research and Engineering Co., P.O. Box 101, Florham Park, NJ 07932.
0022-36S4/92/2096-845$03.00/0
bulk pressure and bulk mole fraction. Moreover, it is difficult to see how selectivity changes with varying pore size from direct measurements, since the carbons are poorly characterized. Only a few theoretical studies of LJ mixtures in pores using molecular theories have been In one of our earlier ~ t u d i e swe , ~ reported gas-liquid phase diagrams for LJ Ar and Kr in cylindrical pores. Later? we investigated the selectivity of Kr relative to Ar in slitlike pores. However, the work was concentrated on a narrow range of subcritical conditions. The present study is the continuationof these two previous studies and covers a wide range of temperature, pressure, and bulk composition. Based on earlier simulation and theoretical studies of both pure and mixed LJ fluids in pores,528 several features are expected for
(1) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1990. ( 2 ) Szepesy, L.; Illes, V. Actu Chim. Hung. 1963,35, 245. (3) Costa, E.; Sotelo, J. L.; Calleja, G.; Marron, C. AIChE J . 1981, 27, 5. (4) Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 336. (5) Tan, Z.; van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 62, 1213. (6) Heffelfinger, G. S.; Tan, Z.; Marini Bettolo Marconi, U.; van Swol, F.; Gubbins, K. E. Mol. Simul. 1989, 2, 393. (7) Wendland, M.; Heinbuch, U.; Fischer, J. Fluid Phase Equilib. 1989, 48, 259. (8) Tan, Z.; Gubbins, K. E.; van Swol, F.; Marini Bettolo Marconi, U. Proe. Third Int. Con/. Fundam. Adsorption; Sonthofen, F R G Engineering Foundation: New York, 1991; p 919. (9) Sokolowski, S.; Fischer, J. Mol. Phys. 1990, 71, 393. (10) Nicholson, D. In Characterization of Porous Solids IC RodriguezReinoso et al., Eds.; Elsevier Science: Amsterdam, 1991; p 3. (1 1) Marini Bettolo Marconi, U.; van Swol, F. Mol. Phys. 1991,72,1081. (12) Finn, J. E.; Monson, P. A. Mol. Phys. 1991, 72, 661. (13) Magda. J. J.;TmU, M.; Davis, H. T. J. Chem. Phys. 198!4,83, 1888. (14) Piotrovskaya, E. M.; Smirnova, N . A. Preprint, 1987. (15) Schoen, M.; Diestler, D. J.; Cushman, J. H. J. Chem. Phys. 1987.87, 5461. (16) Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 573. (17) Heffelfinger, G. S.; Van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 61, 1381.
1992 American Chemical Society
